ABSTRACT Title of dissertation: GRAVITY GRADIOMETER AIDED INERTIAL NAVIGATION WITHIN NON-GNSS ENVIRONMENTS Justin A. Richeson Doctor of Philosophy, 2008 Dissertation directed by: Professor Darryll J. Pines Department of Aerospace Engineering Gravity gradiometer aiding of a strapdown inertial navigation system (INS) in the event of Global Navigation Satellite System (GNSS) signal loss, or as a com- plement to an INS/GNSS system, is proposed. Gravity gradiometry is ideal for covert military applications where a self contained, passive, spoof-free aid is de- sirable, and for space navigation near planetary bodies and moons where GNSS is unavailable. This dissertation provides the rst comprehensive discussion on gravity gradiometry fundamentals, map modeling, and regional and altitude e ects on the gravitational gradient signal for use as a navigation aid. A thorough methodology to implement strapdown and stabilized gravity gradiometer instruments (GGIs) into an autonomous extended Kalman lter is also presented in the open literature for the rst time. Lastly, a brief discussion on extraterrestrial navigation using gravity gradiometry is given. To quantify the potential performance for future gravity gradiometer instru- ments as an INS aid, extensive Monte Carlo simulations of a hypersonic scramjet cruise missile were performed. The results for the 1000 km range mission indicate that GGI updates signi cantly improve the navigation accuracy of the autonomous INS. The sensitivities of the system to variations in inertial measurement unit (IMU) quality, gravity eld variation, GGI noise, update rate, and type are also investi- gated along with a baseline INS/Global Positioning System (GPS). Given emerging technologies that have the potential to drastically decrease gradiometer noise levels, a hypothetical future grade gravity gradiometer aided INS is shown to bound root- mean-square (RMS) position errors at 0.336 m, velocity errors at 0.0069 m/s, and attitude errors at 0.00977 , which is comparable to the nominal INS/GPS system with 10 sec updates. The performance of two subsonic cases is also investigated and produced im- pressive passive navigation accuracy. A commercial aircraft simulation using a fu- ture grade GGI provided RMS errors of 0.288 m in position, 0.0050 m/s in velocity, and 0.0135 in attitude. A low altitude and velocity gravity gradiometer based sur- vey simulation similarly showed sub-meter RMS position errors of 0.539 m, velocity errors of 0.0094 m/s, and attitude errors of 0.0198 . GRAVITY GRADIOMETER AIDED INERTIAL NAVIGATION WITHIN NON-GNSS ENVIRONMENTS by Justin A. Richeson Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2008 Advisory Committee: Professor Darryll J. Pines, Chair & Adviser Professor Mark J. Lewis, Co-Adviser Associate Professor Kenneth Yu Assistant Professor J. Sean Humbert Professor Shapour Azarm, Dean?s Representative Adjunct Faculty Ryan P. Starkey, Currently at University of Colorado, Boulder c Copyright by Justin A. Richeson 2008 ACKNOWLEDGMENTS There are many people who have encouraged and aided me throughout my life, and ultimately in achieving my doctorate, and to those people I would like to say \Thank you." The list of people that follow is fairly extensive, but is by no means intended to be complete. For that reason, if I have forgotten to include someone, I apologize in advance. First, I would like to express my gratitude to my family for their support throughout my life?s ambitions. I?d particularly like to thank my brother, Chase, for sticking by my side through the good and bad times. I?d like to thank my mother, Jackie, for her constant support and encouragement (and also for the laptop that?s been the workhorse throughout my graduate years). And lastly, I?d like to thank my father, Glen(n), for constantly pushing me in life to be the best I could be and for instilling me with a hardy work ethic. Without your \perseverance," few of my academic accomplishments would have been possible. Along the same lines, I?d like to thank my grandparents Alice and Scottie Lacy (\Ma" and \Pa") and Margot \Grandma" Marmor for everything they have done for me over the past twenty-six years. All three of you have shown by example how to be strong independent people no matter the circumstances. Ma and Pa, you have taught me that an academic education can never be a substitute for common sense, and that even the most educated individual can be humbled by someone ii with less \book smarts." And to Grandma, thank you for the countless trips to the Smithsonian museums in D.C., and speci cally to the National Air and Space Museum which undoubtedly lay the seed for my current profession. And while not related biologically, I would like to thank Harry Walker who ran the Child?s Garden day care center where my brother and I spent countless hours before and after elementary school and during summer breaks. Chase and I joke that you were like a second father to us, but in all seriousness, the lessons you taught me in my formative years have de nitely shaped my entire life. And for that, I deeply thank you. On a less serious note, I?d like to thank my high school friends who made those years much more enjoyable. In particular, Don and the Kunkels, Matt and the Palmers, Vince and the Beachleys, Matt Heron, Dave Bilbrough, and Darrin Mal , thank you for all the good times. I?d also like to acknowledge my high school calculus and physics teachers, Mrs. Kubic and Mr. Hopkins, for encouraging me to pursue engineering in college and being the best teachers in high school. In my eight-and-a-half years at the University of Maryland (undergraduate and graduate), I have been in uenced by countless people who I would like to thank for helping me succeed and for making this time of my life so memorable. Starting with people from undergrad, I?d like to thank Jesse Colville, Matt Strube, Andy Zang, Ben Hein, Tim Beasman, Dan Shafer, Dave Giger, Suzanne Ruddy (MacKusick), Sadie Micheal, Kirstin Hollingsworth, Alexandra Langley, Will Becker, Jackie Rielly (Schmoll), and everyone else in the 2003 ENAE class who put in the long hours with me to try and excel in every team project and homework assignment thrown iii at us. I?d also like to thank Chad Tucker, Will Becker, Chris Wei, Pat Taylor, Joe Sivak, Mike \Jerome" McAndrew, Lenny Reisner, and Maurice for the \non- academic" college experiences. From my junior and senior years in the rotorcraft center, I?d like to thank Nick Rosenfeld, Julie Blondeau (Samuel), Ben Berry, Alice Ryan, and many others for their camaraderie, and to Jinsong Bao, Paul Samuel, Felipe Bohorquez, Matt Tarascio, and Ron Couch (among others) for showing me the required work ethic and leadership needed to be a successful graduate student. In graduate school, there have been countless o cemates and classmates who have helped me immensely with my coursework and research. In particular, Jesse Colville, Kerrie Smith, and Andy Zang, if it weren?t for you three working together with me on homework and projects, the rst two years of grad school would have been a nightmare. Also, to all of my o cemates from JMP 1116 (Rama Balar, Adam Beerman, Josh Clough, Qina Diao, Amardip Ghosh, Dan Hoult, Andrew Johnston, Inna Kurits, Jamie Meero , Dave Minton, Vijay Ramasubramanian, Marc Ruppel, Neal Smith, Greg Stamp, and the honorary o cemate, Falcon Rankins), thank you for the long random discussions at the conference table and for the input on the o ce caricature murals. Speci cally, Adam|thank you for keeping up on entertainment gossip so I had someone to talk to about the newest episode of The O ce or Heroes and for the memorable Friday night WiiWas; Neal|thank you for being up for any much needed distraction to go get co ee or talk outside, and for explaining numerous fundamental compressible ow concepts to me; and Vijay|thank you for being more than willing to proofread something I wrote or bounce ideas o of, and more importantly, there to vent to and o er advice on any number of topics you iv may (or may not) have been familiar with. While technically a fellow graduate student for several years, I?d like to thank Suneel Sheikh for his immeasurable aid and mentorship in completing this disserta- tion (and for the extra discretionary income through ASTER Labs). Thank you for the many hours helping me grasp Kalman ltering concepts, IMU error models, and your invaluable advice on living life in general. But even more so, I?d like to thank you for your uncompromising values and endless supply of encouragement, support, and all around positivity (including acknowledging my very modest contributions to the ag football team). I hope that in my lifetime I am able to learn even a small part of the interpersonal skills I?ve seen you exhibit. I would also like to thank Ryan Starkey for all the same reasons I mentioned above with Suneel. Even when you were staring down a deadline, you were always more than willing to let someone sit down with you and talk about their research or any non-technical issue at length. Your work ethic, down-to-Earth personality, and all-around generosity are truly amazing characteristics that have become a template in my mind of what a true scholar is. And to my dissertation committee (including Ryan), I am sincerely grateful for the praises, recommendations, and criticisms I have received from you all through- out this process. The feedback in my pre-defense and defense truly made this a sounder dissertation. On a more personal note, I would like to thank Dr. Yu for his friendliness and many long discussions on my long-term aspirations. To Dr. Lewis, thank you for allowing me to attend you group meetings even though I wasn?t technically your student, being happy to answer any question I might ask, and for v \re-vectoring" my research topic into the eld of navigation which has been a much more rewarding experience. And last, but de nitely not least, I would like to thank Dr. Pines for the countless hours spent discussing any and every topic imaginable, for being a good friend, and for tasking me with the novel gravity gradiometer aided INS topic. To the aerospace engineering sta , I would like to give a collective thank you. And speci cally, I?d like to thank Becky Sarni for helping me track down Dr. Pines, and to Pat Baker for solving every problem I had in record time (e.g. campaigning the graduate school for my MS/BS credits). On a non-technical note, there are many caricature artists I have had the fortune of working with and/or for over the past 8+ years. The summers, weekends, and late night gigs were always a welcome break from my engineering day job. Starting with my mentors at Kaman?s at Six Flags America, thank you Jim Toth and Ryan Holeman for teaching me the art and business of \big-heads-and-little- bodies." And to everyone from the 1999 season (Don Kunkel, Jerry Gaylord, Luci Pereira, Jason Vaughn, Richard Warr, Reggie Butler, Charles Matthews, Lindsey Bailey Harris, Mia, etc.), thank you for the memorable summer. Don and Jerry, thanks for being amazing friends, two of the nicest people I?ve ever known, and for serving as a constant reminder that I should leave the artwork to professionals. Luci, thank you for everything and for the best back-handed compliment of all time when you told me \your picture would be great, if you knew how to draw." Also, thank you to the other Kaman?s employees I worked with over the remaining years: Kevin and Katie Grey, Craig, Domo, et al. Lastly, I?d like to thank all the local Washington vi D.C. area caricature artists who?ve shared some laughs (and often good sketches) with me: Mike Hassan, Chris Metzger, Dan Smith, Peter Scott, Rick Wright, Jason Levinson, and everyone else I?ve worked with the past several years. The last person I must thank is my girlfriend of the past 4+ years, Lauren Belliveau. Thank you for taking a chance on me when I was just starting grad school, and for putting up with me the past 50 months (including my long hours, late nights, cranky sleep-deprived mornings, stressed out times, co ee overdoses, loud drumming, and countless other things I?ve annoyed you with). Thank you for also picking up the chores when I?ve been busy with work or just plain burnt out. Thank you for making me take vacations and see di erent parts of the world even when I initially resisted. Thank you for convincing me to get Houdini. Thank you for coming out to L.A. with me for an uncertain future. In general, thank you for everything. This research was supported by the Space Vehicle Technology Institute (SVTI), one of the NASA Constellation University Institute Projects (CUIP), under grant NCC3-989, with joint sponsorship from the U.S. Department of Defense. Appreci- ation is expressed to Dr. Darryll Pines, director of the SVTI at the University of Maryland; Claudia Meyer of the NASA Glenn Research Center, program manager of CUIP; and to Dr. John Schmisseur and Dr. Walter Jones of the U.S. Air Force O ce of Scienti c Research. vii TABLE OF CONTENTS List of Tables xii List of Figures xvi List of Symbols xxi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Gravity Gradiometer Instruments . . . . . . . . . . . . . . . . 4 1.2.1.1 First Generation Airborne GGIs . . . . . . . . . . . 5 1.2.1.2 Current Generation GGIs . . . . . . . . . . . . . . . 12 1.2.1.3 Gravity Gradiometer Instrument Speci cations . . . 19 1.2.2 Gravity Gradiometer Aided Inertial Navigation . . . . . . . . 19 1.2.2.1 Real-Time Determination of the Gravity Anomaly . 21 1.2.2.2 Gravity Gradiometer Surveying . . . . . . . . . . . . 24 1.2.2.3 Gravitational Gradient Map-Matching . . . . . . . . 28 1.2.3 Other Gravity Gradiometer Instrument Applications . . . . . 31 1.2.3.1 Close-Loop Satellite Attitude Re nement . . . . . . 31 1.2.3.2 Arms Treaty Veri cation . . . . . . . . . . . . . . . . 32 1.2.3.3 Underground Bunker or Void Detection . . . . . . . 33 1.2.3.4 All Accelerometer Inertial Navigation . . . . . . . . . 34 1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 Gravity Map Model 42 2.1 Gravity Gradiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Gravity Map Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2 Terrain Elevation Contributions . . . . . . . . . . . . . . . . . 48 2.2.3 Minimum Altitude to Neglect Terrain E ects . . . . . . . . . . 49 2.2.3.1 Normalization of Terrain DD Computation . . . . . 55 2.2.4 Gravitational Gradient Biases . . . . . . . . . . . . . . . . . . 58 2.3 Gravitational Gradient Characterization . . . . . . . . . . . . . . . . 60 2.4 Simulation Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4.1 Gravity Map Resolution . . . . . . . . . . . . . . . . . . . . . 73 2.4.1.1 Horizontal Resolution . . . . . . . . . . . . . . . . . 73 2.4.1.2 Vertical Resolution . . . . . . . . . . . . . . . . . . . 76 2.4.2 Simulated Gravity Field Maps . . . . . . . . . . . . . . . . . . 79 2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 viii 3 Hypersonic Vehicle Model 84 3.1 JHU/APL Axisymmetric Scramjet Model . . . . . . . . . . . . . . . . 85 3.1.1 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.1.1 O -Design Mass Capture . . . . . . . . . . . . . . . 90 3.1.1.2 Maximum Geometric Contraction Ratio . . . . . . . 91 3.1.1.3 Thrust Coe cient . . . . . . . . . . . . . . . . . . . 93 3.1.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.2.1 Pro le Drag Coe cient . . . . . . . . . . . . . . . . 95 3.1.2.2 Additive Drag Coe cient . . . . . . . . . . . . . . . 103 3.1.2.3 Normal Force Coe cient . . . . . . . . . . . . . . . . 105 3.1.3 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Mass Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.1 Internal Volume Calculation . . . . . . . . . . . . . . . . . . . 110 3.2.2 Determination of Mass Model Design Parameters . . . . . . . 118 3.2.3 Modeled Axisymmetric Hypersonic Missile Summary . . . . . 121 3.3 Trim State Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.3.1 Free Body Diagram & Cruise Dynamics . . . . . . . . . . . . 124 3.3.2 Simulated Pitch and Roll Rates . . . . . . . . . . . . . . . . . 127 3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4 Inertial Navigation System 134 4.1 Coordinate Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.1.1 Earth-Centered-Inertial Frame . . . . . . . . . . . . . . . . . . 134 4.1.2 Earth-Centered-Earth-Fixed Frame . . . . . . . . . . . . . . . 136 4.1.3 Navigation Frame . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.1.4 Body Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 139 4.2.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . 139 4.2.2 ECEF to ECI Transformation . . . . . . . . . . . . . . . . . . 142 4.2.3 Navigation to ECEF Transformation . . . . . . . . . . . . . . 143 4.2.4 Body to Navigation Transformation . . . . . . . . . . . . . . . 146 4.3 Inertial Navigation Equations . . . . . . . . . . . . . . . . . . . . . . 148 4.3.1 Arbitrary Frame Equations . . . . . . . . . . . . . . . . . . . 148 4.3.2 Navigation Frame Equations . . . . . . . . . . . . . . . . . . . 150 4.3.3 Navigation Mechanization . . . . . . . . . . . . . . . . . . . . 154 4.3.3.1 Body-to-Navigation Frame Quaternion . . . . . . . . 154 4.3.3.2 Fourth-Order Runge-Kutta Integration . . . . . . . . 156 4.4 Inertial Navigation Error Equations . . . . . . . . . . . . . . . . . . . 159 4.4.1 Position Error Equations . . . . . . . . . . . . . . . . . . . . . 159 4.4.2 Attitude Error Equations . . . . . . . . . . . . . . . . . . . . . 161 4.4.3 Velocity Error Equations . . . . . . . . . . . . . . . . . . . . . 165 4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.5 Inertial Measurement Unit Model . . . . . . . . . . . . . . . . . . . . 170 4.5.1 IMU Error Model . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.5.2 IMU Speci cations . . . . . . . . . . . . . . . . . . . . . . . . 177 ix 4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5 Gravity Gradiometer Instrument Model 184 5.1 Accelerometer-Based GGI Measurements . . . . . . . . . . . . . . . . 184 5.1.1 Gravitational Gradient Transformation Matrix . . . . . . . . . 187 5.1.2 Inertial-to-Accelerometer Frame Rotation Rate . . . . . . . . 191 5.1.3 Rotating, Stabilized GGI Measurements . . . . . . . . . . . . 193 5.2 Modeled Twelve-Accelerometer GGI . . . . . . . . . . . . . . . . . . 197 5.2.1 Strapdown Gravity Gradiometer Instrument . . . . . . . . . . 202 5.2.2 Stabilized Gravity Gradiometer Instrument . . . . . . . . . . . 208 5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6 Monte Carlo Simulation Results 213 6.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.2 Monte Carlo Set Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 6.3.1 Gravity Gradiometer Aided INS . . . . . . . . . . . . . . . . . 223 6.3.1.1 Monte Carlo Results . . . . . . . . . . . . . . . . . . 225 6.3.1.2 Sensitivity Results . . . . . . . . . . . . . . . . . . . 242 6.3.2 Global Positioning System Aided Navigation . . . . . . . . . . 258 6.3.2.1 Monte Carlo Results . . . . . . . . . . . . . . . . . . 260 6.3.2.2 Sensitivity Results . . . . . . . . . . . . . . . . . . . 267 6.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 7 Conclusions and Future Work 284 7.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 285 7.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 286 A Global Gravitational Maps 289 B Thrust Coe cient Curve Fits 300 C Extended Kalman Filter Model 309 C.1 Filtering Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 309 C.2 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 C.2.1 Random Constant . . . . . . . . . . . . . . . . . . . . . . . . 313 C.2.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 C.2.3 First Order Gauss-Markov Process . . . . . . . . . . . . . . . 314 C.3 Linear Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 315 C.4 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 C.5 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . 319 C.5.1 System Linearization . . . . . . . . . . . . . . . . . . . . . . . 320 C.5.2 Discrete Kalman Filter . . . . . . . . . . . . . . . . . . . . . . 323 C.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 x D Global Positioning System Model 330 D.1 GPS Satellite Constellation . . . . . . . . . . . . . . . . . . . . . . . 330 D.2 GPS Satellite Vehicle Position and Velocity . . . . . . . . . . . . . . . 335 D.3 GPS Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 D.3.1 Visibility Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 D.3.2 Pseudorange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 D.3.3 Pseudorange Rate . . . . . . . . . . . . . . . . . . . . . . . . . 344 D.3.4 Geometric Dilution of Precision . . . . . . . . . . . . . . . . . 348 D.4 GPS Receiver Error Model . . . . . . . . . . . . . . . . . . . . . . . . 351 D.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 E Additional Monte Carlo Results 356 E.1 Dead Reckoning Results . . . . . . . . . . . . . . . . . . . . . . . . . 357 E.2 Gravity Gradiometer Aided Navigation . . . . . . . . . . . . . . . . . 359 E.3 Global Positioning System Aided Navigation . . . . . . . . . . . . . . 379 Bibliography 392 xi LIST OF TABLES 1.1 Gravity Gradiometer Instruments . . . . . . . . . . . . . . . . . . . . 20 2.1 Simulated Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.2 Simulated Cruise Values . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.3 Gravitational Gradient Map Parameters . . . . . . . . . . . . . . . . 79 2.4 Gravitational Gradient Map Storage Requirements . . . . . . . . . . 80 3.1 Capture Area Curve-Fit Coe cients . . . . . . . . . . . . . . . . . . . 92 3.2 Maximum Contraction Ratio . . . . . . . . . . . . . . . . . . . . . . . 93 3.3 Shock Angles for 12.5 Cone . . . . . . . . . . . . . . . . . . . . . . . 101 3.4 Additive Drag Coe cient Data . . . . . . . . . . . . . . . . . . . . . 104 3.5 Scramjet Design Validation for M1 = 4.0, = 0.0 , ER = 1.0 . . . . 106 3.6 Scramjet Design Validation for M1 = 8.0, = 0.0 , ER = 1.0 . . . . 107 3.7 Modeled Scramjet Design Parameters . . . . . . . . . . . . . . . . . . 122 3.8 Scramjet Missile Design Summary . . . . . . . . . . . . . . . . . . . . 122 3.9 Scramjet Initial Trim Angles and Excess Fuel . . . . . . . . . . . . . 131 4.1 World Geodetic System 1984 Properties . . . . . . . . . . . . . . . . 145 4.2 Navigation and Tactical Grade Acceleromter Speci cations . . . . . . 179 4.3 Navigation and Tactical Grade Gyro Speci cations . . . . . . . . . . 180 4.4 Simulated Navigation and Tactical Grade IMU Speci cations . . . . . 181 6.1 Steady State Error Versus Monte Carlo Set Size . . . . . . . . . . . . 218 6.2 Dead Reckoning Navigation Accuracy after 1000 km Cruise, High n Variation Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.3 INS/GGI Monte Carlo Test Matrix . . . . . . . . . . . . . . . . . . . 225 xii 6.4 INS/GGI \Best" and O -Nominal Simulation Parameters . . . . . . . 225 6.5 \Best" Gradiometer-Aided INS Case . . . . . . . . . . . . . . . . . . 227 6.6 Tactical Grade IMU, Gradiometer-Aided INS Case . . . . . . . . . . 229 6.7 \Low" Gravity Gradient Variation Gradiometer-Aided INS Case . . . 230 6.8 Increased Noise Gradiometer-Aided INS Case . . . . . . . . . . . . . 233 6.9 Stabilized Gradiometer-Aided INS Case with 10 sec Updates . . . . . 234 6.10 Strapdown Gradiometer-Aided INS Case . . . . . . . . . . . . . . . . 236 6.11 Commercial Aircraft INS/GGI Case . . . . . . . . . . . . . . . . . . . 238 6.12 Commercial Aircraft INS/GGI Case w/ Increased Noise . . . . . . . . 240 6.13 GGI Survey INS/GGI Case . . . . . . . . . . . . . . . . . . . . . . . 241 6.14 GGI Survey INS/GGI Case w/ Increased Noise . . . . . . . . . . . . 243 6.15 INS/GPS Nominal and O -Nominal Simulation Parameters . . . . . 259 6.16 INS/GPS Monte Carlo Test Matrix . . . . . . . . . . . . . . . . . . . 259 6.17 Nominal Global Positioning System Case . . . . . . . . . . . . . . . . 262 6.18 Tactical Grade IMU Global Positioning System Case . . . . . . . . . 263 6.19 Pseudorange Only Global Positioning System Case . . . . . . . . . . 264 6.20 Commercial Aircraft INS/GPS Case . . . . . . . . . . . . . . . . . . 266 6.21 GGI Survey INS/GPS Case . . . . . . . . . . . . . . . . . . . . . . . 268 6.22 Hypersonic INS/GGI Postion MRSE (m) Sensitivity to GGI Noise . . 281 6.23 Hypersonic INS/GGI Sensitivities to n Variation and Mach Number 282 B.1 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 1 . . . . . . . 302 B.2 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 2 . . . . . . . 303 B.3 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 3 . . . . . . . 304 B.4 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 4 . . . . . . . 305 xiii B.5 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 6 . . . . . . . 306 B.6 Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 8 . . . . . . . 307 B.7 Thrust Coe cient Curve-Fit Corrections when (A4=A2) = 3 . . . . . 308 D.1 Simulated GPS Parameters . . . . . . . . . . . . . . . . . . . . . . . 332 D.2 GPS-24 Satellite Constellation, from Ref. [7] . . . . . . . . . . . . . . 334 D.3 User Position for GPS Satellite Visibility Analysis . . . . . . . . . . . 342 D.4 Precise Positioning System Error Model, P/Y Code, from Ref. [7] . . 344 D.5 Simulated GPS Receiver Clock Parameters . . . . . . . . . . . . . . . 352 E.1 Dead Reckoning: Navigation Grade IMUs, High Variation . . . . . 357 E.2 Dead Reckoning: Navigation Grade IMUs, Low Variation . . . . . . 358 E.3 Dead Reckoning: Tactical Grade IMUs, High Variation . . . . . . . 358 E.4 Dead Reckoning: Tactical Grade IMUs, Low Variation . . . . . . . 359 E.5 INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 6 . . 361 E.6 INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 7 . . 362 E.7 INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 8 . . 363 E.8 INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 6 . . 364 E.9 INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 7 . . 365 E.10 INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 8 . . 366 E.11 INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 6 . . 367 E.12 INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 7 . . 368 E.13 INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 8 . . 369 E.14 INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 6 . . 370 E.15 INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 7 . . 371 E.16 INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 8 . . 372 xiv E.17 INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 6 . 373 E.18 INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 7 . 374 E.19 INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 8 . 375 E.20 INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 6 . 376 E.21 INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 7 . 377 E.22 INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 8 . 378 E.23 INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 6 . . . . . . . . . 380 E.24 INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 7 . . . . . . . . . 381 E.25 INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 8 . . . . . . . . . 382 E.26 INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 6 . . . . . . . . . 383 E.27 INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 7 . . . . . . . . . 384 E.28 INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 8 . . . . . . . . . 385 E.29 INS/GPS: Updates, Nav. Grade IMUs, Mach 6 . . . . . . . . . . . 386 E.30 INS/GPS: Updates, Nav. Grade IMUs, Mach 7 . . . . . . . . . . . 387 E.31 INS/GPS: Updates, Nav. Grade IMUs, Mach 8 . . . . . . . . . . . 388 E.32 INS/GPS: Updates, Tac. Grade IMUs, Mach 6 . . . . . . . . . . . . 389 E.33 INS/GPS: Updates, Tac. Grade IMUs, Mach 7 . . . . . . . . . . . . 390 E.34 INS/GPS: Updates, Tac. Grade IMUs, Mach 8 . . . . . . . . . . . . 391 xv LIST OF FIGURES 1.1 Aided Inertial Navigation using an Extended Kalman Filter, from Ref. [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Recent Scramjet Research Programs . . . . . . . . . . . . . . . . . . 3 1.3 Draper Cylindrical and Spherical GGI Schematics, from Ref. [28] . . . 7 1.4 Hughes Rotating Torsional GGI, from Ref. [33] . . . . . . . . . . . . . 9 1.5 Accelerometer Null Bias vs. Rotation Rate (Cycles/Hour), from Ref. [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Bell / Textron Rotating Accelerometer GGI, (Schematic, GGI w/ Sta- bilized Platform, and Internal Umbrella Con guration) from Ref. [47] 12 1.7 University of Maryland Superconducting Gravity Gradiometer, from Ref. [63] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 UMD SAA (a) and Cryostat (b), from Ref. [66] . . . . . . . . . . . . 16 1.9 UWA OQR, from Ref. [19] . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 ESA GOCE EGG, from Ref. [75] . . . . . . . . . . . . . . . . . . . . 18 1.11 Block Diagram of INS/GGI for Real-Time Determination of Gravity Anomaly, from Ref. [88] . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.12 Gravitational Gradient Survey Integration Methodologies, from Ref. [96] 25 1.13 Horizontal (a) and Vertical (b) Position Error vs. Cruise Altitude, from Ref. [20]; originally from Ref. [36] . . . . . . . . . . . . . . . . . 29 1.14 Map-Matching GGI/INS using an Extended Kalman Filter . . . . . . 35 1.15 Simulated Gravity at Nominal Latitude and Altitude for Mach 7 Tra- jectories, \High" Variation (solid), \Low" Variation (dashed) . . . . . 36 2.1 Spherical Harmonic Classi cations (a) Zonal (b) Sectoral (c) Tesseral, from Ref. [109] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2 Schematic of Modeled Gaussian Mountain . . . . . . . . . . . . . . . 50 2.3 Terrain Contribution to DD . . . . . . . . . . . . . . . . . . . . . . . 51 xvi 2.4 Optimal T=h to Maximize Terrain DD . . . . . . . . . . . . . . . . 53 2.5 Terrain Contribution to Maximum DD for Various Peak Terrain and User Altitude Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 Point Mass Gravitational Gradient Contribution . . . . . . . . . . . . 59 2.7 Inline Gravitational Gradients at Surface (a) NN (b) EE (c) DD . . 61 2.8 O -Diagonal Gradients at Surface (a) NE (b) ND (c) ED . . . . . . 63 2.9 East-Down Gravitational Gradient at Three Altitudes . . . . . . . . . 64 2.10 DD Standard Deviation, log10(E o), at Surface . . . . . . . . . . . . . 65 2.11 Minimum, Mean, and Maximum Gravitational Gradient Standard Deviation vs. Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.12 Simulated Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.13 Simulated Gravity at Nominal Latitude and Altitude for Mach 7 Tra- jectories, High Variation (solid), Low Variation (dashed) . . . . . 70 2.14 Horizontal Spherical Harmonic Error Due to Linear Interpolation, with 500 m Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.15 Horizontal Spherical Harmonic Error Due to Linear Interpolation vs. Map Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.16 Vertical Spherical Harmonic Error Due to Linear Interpolation, with 160 m Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.17 Vertical Spherical Harmonic Error Due to Linear Interpolation vs. Map Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1 Scramjet Missile Geometry and Thermodynamic Stations . . . . . . . 88 3.2 Scramjet Missile Front View . . . . . . . . . . . . . . . . . . . . . . . 97 3.3 Detail of Scramjet Missile Cowl . . . . . . . . . . . . . . . . . . . . . 100 3.4 Scramjet Missile Volume De nitions . . . . . . . . . . . . . . . . . . . 109 3.5 Detail of Scramjet Missile Inlet . . . . . . . . . . . . . . . . . . . . . 111 3.6 Detail of Scramjet Missile Isolator . . . . . . . . . . . . . . . . . . . . 112 3.7 Scramjet Missile Fin Detail . . . . . . . . . . . . . . . . . . . . . . . 116 xvii 3.8 Scramjet Fuel Mass Fraction vs. (Vf=Vtotal) . . . . . . . . . . . . . . . 119 3.9 Scramjet Mass vs. (Vf=Vtotal), Mach 7 inlet . . . . . . . . . . . . . . . 120 3.10 Scramjet Missile Free Body Diagram . . . . . . . . . . . . . . . . . . 124 3.11 Trim Roll Rate, Pitch Rate, and Equivalence Ratio (a) Mach 6 (b) Mach 7 (c) Mach 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.12 Trim Speci c Forces, Mach 7 . . . . . . . . . . . . . . . . . . . . . . . 132 4.1 Earth-Centered-Inertial Coordinate System, from Ref. [115] . . . . . . 136 4.2 Navigation Frame (North-East-Down) Coordinate System, modi ed from Ref. [115] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.3 Body Frame Coordinate System and Euhler Angles, from Ref. [145] . 138 4.4 Coordinate System Transformation, from Ref. [1] . . . . . . . . . . . 145 4.5 Fourth-Order Runge-Kutta Schematic, from Ref. [147] . . . . . . . . . 158 5.1 Schematic of GGI with 2 Rotating Accelerometers . . . . . . . . . . . 195 5.2 Schematic of GGI with 4 Rotating Accelerometers . . . . . . . . . . . 196 5.3 Schematic of Modeled Twelve-Accelerometer GGI . . . . . . . . . . . 198 6.1 Normalized Steady State Filter Error vs. Monte Carlo Set Size . . . . 220 6.2 Normalized Steady State Error vs. Monte Carlo Set Size . . . . . . . 220 6.3 Normalized Steady State MRSE vs. Monte Carlo Set Size . . . . . . . 221 6.4 Sample \Best" Stabilized Gradiometer-Aided INS Simulation . . . . . 227 6.5 Sample Tactical Grade IMU Gradiometer-Aided INS Simulation . . . 229 6.6 Sample \Low" Gravity Gradient Trajectory Gradiometer-Aided INS Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.7 Sample Increased Noise Gradiometer-Aided INS Simulation . . . . . . 233 6.8 Sample Stabilized Gradiometer-Aided INS Simulation with 10 sec Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 xviii 6.9 Sample Strapdown Gradiometer-Aided INS Simulation . . . . . . . . 236 6.10 Sample Commercial Aircraft INS/GGI Simulation . . . . . . . . . . . 238 6.11 Sample Commercial Aircraft INS/GGI Simulation w/ Increased Noise 240 6.12 Sample GGI Survey INS/GGI Simulation . . . . . . . . . . . . . . . . 241 6.13 Sample GGI Survey INS/GGI Simulation w/ Increased Noise . . . . . 243 6.14 INS/GGI Steady State MRSEs for Mach 7, High n Variation, (a) Position (b) Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.15 INS/GGI Steady State Attitude MRSEs for Mach 7, High n Varia- tion, (a) Monte Carlo (b) Filter . . . . . . . . . . . . . . . . . . . . . 247 6.16 Normalized INS/GGI Steady State MRSE for Stabilized GGI w/ Nav. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . . . 249 6.17 Normalized INS/GGI Steady State Attitude MRSE for Stabilized GGI w/ Nav. Grade IMUs, (a) Monte Carlo (b) Filter . . . . . . . . . 251 6.18 Normalized INS/GGI Steady State MRSE for Stabilized GGI w/ Tac. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . . . . 253 6.19 Normalized INS/GGI Steady State Attitude MRSE for Stabilized GGI w/ Tac. Grade IMUs, (a) Monte Carlo (b) Filter . . . . . . . . . 255 6.20 Normalized INS/GGI Steady State MRSE for Strapdown GGI w/ Nav. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . . . 256 6.21 Normalized INS/GGI Steady State Attitude MRSE for Strapdown GGI w/ Nav. Grade IMUs, (a) Monte Carlo (b) Filter . . . . . . . . . 257 6.22 Sample Nominal Global Positioning System Simulation . . . . . . . . 262 6.23 Sample Tactical Grade IMU Global Positioning System Simulation . . 263 6.24 Sample Pseudorange Only Global Positioning System Simulation . . . 264 6.25 Sample Commercial Aircraft Global Positioning System Simulation . 266 6.26 Sample GGI Survey Global Positioning System Simulation . . . . . . 268 6.27 INS/GPS Steady State MRSE for Mach 7 Simulations, (a) Position (b) Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 xix 6.28 INS/GPS Steady State Attitude MRSE for Mach 7 Simulations, (a) Monte Carlo (d) Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 271 6.29 Normalized INS/GPS Steady State MRSE w/ & _ Measurements and Nav. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . 273 6.30 Normalized INS/GPS Steady State MRSE w/ & _ Measurements and Tac. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . 274 6.31 Normalized INS/GPS Steady State MRSE w/ Measurements and Nav. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . . . 276 6.32 Normalized INS/GPS Steady State MRSE w/ Measurements and Tac. Grade IMUs, (a) Position (b) Velocity . . . . . . . . . . . . . . . 278 A.1 Inline Gravitational Gradients at 10 km (a) NN (b) EE (c) DD . . . 290 A.2 O -Diagonal Gradients at 10 km (a) NE (b) ND (c) ED . . . . . . 291 A.3 Inline Gravitational Gradients at 100 km (a) NN (b) EE (c) DD . . 292 A.4 O -Diagonal Gradients at 100 km (a) NE (b) ND (c) ED . . . . . . 293 A.5 Inline n, log10(E o), at Surface (a) NN (b) EE (c) DD . . . . . . . 294 A.6 O -Diagonal n, log10(E o), at Surface (a) NE (b) ND (c) ED . . . 295 A.7 Inline n, log10(E o), at 10 km (a) NN (b) EE (c) DD . . . . . . . 296 A.8 O -Diagonal n, log10(E o), at 10 km (a) NE (b) ND (c) ED . . . 297 A.9 Inline n, log10(E o), at 100 km (a) NN (b) EE (c) DD . . . . . . . 298 A.10 O -Diagonal n, log10(E o), at 100 km (a) NE (b) ND (c) ED . . . 299 C.1 Extended Kalman Filter, from Ref. [9] . . . . . . . . . . . . . . . . . 310 D.1 Perifocal Coordinate System, from Ref. [115] . . . . . . . . . . . . . . 335 D.2 GPS Visibility Angles, From Ref. [7] . . . . . . . . . . . . . . . . . . 339 D.3 Simulated GPS Satellite Visibility . . . . . . . . . . . . . . . . . . . . 341 D.4 Simulated GPS Geometric Dilution of Precision . . . . . . . . . . . . 350 xx LIST OF SYMBOLS A = Area, m2 Abw = Wetted body area, m2 Acle = Cowl lip leading edge area, m2 Acsurf = Cowl surface area, m2 Acx = Axially projected cowl area, m2 (Ai=A1)max = Maximum geometric contraction ratio Aq = Quanternion 4 4 skew symmetric matrix of !bnb, rad/s Awall = Combustor wall area, m2 A0 = Capture area, m2 (A0=Ai) = Capture area ratio (A0=A1) = Inlet contraction ratio (A4=A2) = Combustor expansion ratio (A5=Ai) = Base-to-inlet area ratio (A5=A0) = Base-to-capture area ratio a = Speci c force vector, m/s2 ae = Semimajor axis of Earth, m aGPS = Semimajor axis of GPS orbit, m aT = Amplitude of Gaussian height distribution, m2 a1 = Freestream speed of sound, m/s2 a, b, c = Polynomial curve- t coe cients (Ch. 3) b = Bias _b = Drift be = Semiminor axis of Earth, m CD = Total drag coe cient, referenced to A5 xxi CDadd = Additive drag coe cient, referenced to A5 CDcw = Cowl wave drag coe cient, referenced to A5 CDf = Body skin friction drag coe cient, referenced to A5 CDle = Cowl leading edge drag coe cient, referenced to A5 CDtw = Tail wave drag coe cient, referenced to A5 CDtf = Tail skin friction drag coe cient, referenced to A5 CD;0 = Pro le drag coe cient, referenced to A5 CD = Induced drag coe cient, referenced to A5 CN = Normal force coe cient, referenced to A5 Cn;m, Sn;m = Fully normalized spherical harmonic coe cients CT = Thrust coe cient, referenced to A5 Cts = Coordinate transformation matrix from s-frame to t-frame c = Speed of light, m/s2 cij = ith row and jth column element of coordinate transformation cov() = Covariance operator D = Drag, N dle = Cowl lip leading edge diameter, m E[ ] = Expectation operator Emin = Minimum elevation angle for GPS satellite visibility, rad E o = E otv os unit, = 10 9 s 2 ER = Equivalence ratio e = Error quantity, estimated - truth e2 = First eccentricity of the Earth ellipsoid squared eGPS = First eccentricity of GPS orbit F = Externally applied force vector, N F = Linearized state error dynamics matrix xxii F1, F5 = Freestream and Nozzle exit stream thrust, respectively, N f = Nonlinear state dynamics vector f() = Arbitrary function = Probability density function (App. C) fe = Flatness of the Earth ellipsoid fstoich = Stoichiometric fuel-to-air ratio G = Universal gravitational constant, m3/(kg-s2) GM = Gravitational constant of Earth, m3/s2 g = Gravitational acceleration vector, m/s2 g = Gravity acceleration vector, m/s2 H = Linearized measurement error matrix (H=C) = Hydrogen-to-Carbon ratio h = Nonlinear measurement vector h = Altitude, m hT = Height of Gaussian terrain distribution, m h0;h 1;h 2 = Allan variance parameters I = Identity matrix iGPS = Inclination of GPS orbit, rad Jn = nth unnormalized zonal harmonic K = Kalman gain matrix k = Slope evaluation of nonlinear dynamics L = GGI measurement vector equivalent of L, E o L = Length, m (Ch. 3) = GGI measurement tensor, E o (Ch. 5) L0 = GGI tensor measurement with angular accelerations, E o Ltotal = Total scramjet missile length, m xxiii lb = GGI accelerometer baseline vector, m lb = Magnitude of GGI accelerometer baseline, m M = Mach number Mdes = Inlet design Mach number Me = Radius of curvature in the meridian, m m = Mass, kg = Order of spherical harmonic model (Ch. 2) _mf, _m0 = Fuel and Capture mass ow rate, respectively, kg/s N = Number of grid nodes (Ch. 2) = Normal force, N (Ch. 3) Ne = Radius of curvature in the prime vertical plane, m n = Gyro noise state, rad/s = Degree of spherical harmonic model (Ch. 2) nmax = Maximum degree of spherical harmonic model nl = Nonlinearity P = Error covariance matrix PGPS = Period of GPS orbit Pn;m = Fully normalized associated Legendre function P1 = Freestream static pressure, Pa Q = Discrete process noise covariance matrix q = Body-to-navigation frame quaternion qw = White noise power spectral density q1 = Freestream dynamic pressure, Pa R = Measurement noise covariance matrix Re = Mean curvature, m Ri = Rotation matrix about the ith axis xxiv R1 = Gas constant for air, J/kg-K r = Position state vector rj = Position of jth GPS satellite vehicle, m r = Radius, m riso;in, riso;out = Interior and exterior isolator boundary, respectively, m Sf = GPS receiver clock frequency PSD, s2/s3 S = GPS receiver clock phase PSD, s2/s SF = Scale factor s = Horizontal or axial distance, m T = Thrust, N Tan = 6 6 vector transition matrix from n- to a-frame T1 = Freestream temperature, K t = Time, s uGPS = Argument of latitude of GPS satellite vehicle, rad V = Volume, m3 (Vf=Vtotal) = Fuel volume fraction VL = GGI white noise matrix, E o Vtotal = Total scramjet missile volume, m3 vj = Velocity of jth GPS satellite vehicle, m/s v = Velocity, m/s2 w = White process noise wfin = Tail n width, m x = State vector x;y = Arbitrary state variables (App. C) y = Measurement vector xxv Greek = Angle of attack, rad GPS = Satellite half-angle for GPS SV visibility test, rad = First order Gauss-Markov process time constant, 1/s GPS = Earth half-angle for GPS SV visibility test, rad s = Cone shock angle, rad = Gravitational gradient vector equivalent of , E o = Gravitational gradient tensor, E o fpa = Flight path angle, rad 1 = Ratio of speci c heats in freestream = Discrete incremental quantity = Laplacian operator (Ch. 2) = Small-angle misalignment matrix, rad (Ch. 4) r = Gradient operator = Perturbation (small error) quantity b = Pitch angle, rad GPS;j = GPS satellite-to-user visibility angle, rad g = Accelerometer-to-gradiometer frame rotation angle, rad s = Scramjet forebody half-cone angle, = Longitude, rad = Mean = White measurement noise = Density, kg/m3 = GPS pseudorange measurement, m (Ch. 6 & App. D) _ = GPS pseudorange rate measurement, m/s = Standard deviation xxvi = Error state transition matrix = Latitude, rad b = Roll angle, rad g = Gravitational potential, m2/s2 n = Skew symmetric matrix equivalent of n (), rad nin = Skew symmetric matrix equivalent of nin (), rad b = Yaw angle, rad n = Small-angle body-to-navigation frame rotation error, rad nin = Small-angle inertial-to-navigation frame rotation error, rad GPS = Right ascension of the ascending node of GPS SV, rad tsr = Skew symmetric matrix equivalent of !tsr (), rad/s !e = Earth?s rotation rate, rad/s !tsr = Angular velocity of the r-frame with respect to the s-frame, with coordinates in the t-frame, rad/s Superscripts a = Accelerometer b = Body frame or Strapdown instrument e = Earth Centered, Earth Fixed frame g = Gyro or GGI frame (Ch. 5) i = Earth Centered Inertial frame or Stabilized instrument n = North-East-Down Navigation frame PQW = Perifocal frame T = Transpose = Trim value = A priori quantity xxvii + = A posteriori quantity b = Estimated quantity e = Measured quantity Subscripts comb = Scramjet combustor cone = Scramjet inlet cone cowl = Scramjet cowl cyl = Scramjet external, constant-radius portion of body D = Down = Drag (Ch. 3) E = East f = Final = Fuel (Ch. 3) = GPS clock drift (App. D) fin = Scramjet tail n fin;tip = Scramjet tail n tip gm = Gauss-Markov H = Homogeneous solution i = ith array element = Scramjet inlet (Ch. 3) iso = Scramjet isolator j = jth array element or jth satellite vehicle k = kth time epoch L = GGI N = North xxviii = Normal force (Ch. 3) noz = Scramjet nozzle P = Particular solution r = Position ref = Reference str = Structural surf = Scramjet surface (area) T = Terrain u = GPS user receiver clock v = Velocity w = White noise 1 = Freestream 0 = Initial 1 = Scramjet isolator entrance (Ch. 3) 2 = Scramjet isolator exit / combustor entrance (Ch. 3) 4 = Scramjet combustor exit / nozzle entrance (Ch. 3) 5 = Scramjet nozzle exit / base (reference) (Ch. 3) = GPS clock phase = Orientation error ! = Angular rate Acronyms AGARD = Advisory Group for Aeronautical Research and Development CHAMP = CHAllenging Minisatellite Payload DCR = Dual Combustor Ramjet ECI = Earth Centered Inertial xxix ECEF = Earth Centered, Earth Fixed EGM96 = Earth Geopotential Model, 1996 EKF = Extended Kalman Filter ESA = European Space Agency FFT = Fast Fourier Transform FTG = Full Tensor Gradiometer GDOP = Geometric Dilution of Precision GGI = Gravity Gradiometer Instrument GNSS = Global Navigation Satellite System GOCE = Gravity Field and Steady-State Ocean Circulation Explorer GPS = Global Positioning System GRACE = Gravity Recovery and Climate Experiment IMU = Inertial Measurement Unit INS = Inertial Navigation System JHU/APL = Johns Hopkins University / Applied Physics Laboratory LEO = Low Earth Orbit MRSE = Mean Radial Spherical Error NASA = National Aeronautics and Space Administration NED = North, East, Down NGS = National Geodetic Survey NOAA = National Oceanic and Atmospheric Administration RMS = Root Mean Square SQUID = Superconducting QUantum Interference Device SV = Satellite Vehicle UERE = User Equivalent Range Error WGS84 = 1984 World Geodetic System xxx Chapter 1 Introduction 1.1 Motivation High accuracy position, velocity, and attitude estimation is required for many aerospace vehicle mission objectives. Such accuracy requirements are typically achieved by an integrated inertial navigation system (INS) consisting of inertial measurement units (IMUs) and an external aid to prevent error accumulation due to uncompensated instrument errors and geodetic/geophysical uncertainties,1{6 See Fig. 1.1 from Ritland.6 Of the common available aids, all-weather Global Navigation Satellite Systems (GNSS), speci cally the Global Positioning System (GPS), have become the most popular means to limit INS errors.1,2,4,6{9 GNSS aiding does have the noted disadvantages that it relies on expensive segments (space, control, and user) that require constant maintenance and monitoring, data rates are relatively low, the greatest accuracy requires constant satellite tracking, satellite geometry can yield poor performance (especially in altitude), orientation information requires multiple antennas, and the weak signal can be easily jammed or spoofed. Further- more, GNSS aiding is ine ective for exploration missions far from Earth, such as the Moon or Mars. 1 Figure 1.1: Aided Inertial Navigation using an Extended Kalman Filter, from Ref. [6] In the event of non-GNSS environments, the INS must integrate the inertial measurements with any uncompensated errors which can produce unacceptable state estimation, divergence, and/or loss of vehicle. During these periods, several other aids may be used for robustness.6 Vision based systems, which are becoming in- creasingly prevalent with unmanned reconnaissance vehicles, require optical access that may be infeasible aboard some systems such as hypersonic cruise vehicles, and are susceptible to weather variations. Vision aids may also have high computational requirements that are prohibitive with current technology. Terrain aids typically emit a radar or laser that can be sensed by other users which is undesirable for covert military missions. And like optical systems, terrain aiding often relies on matching the sensor measurement to a pre-surveyed map that may be sensitive to temporal variations and anomalies, such as uctuations in water or sand. This dis- sertation proposes the use of a completely self contained, passive system that relies on gravity gradiometry for INS aiding that exhibits none of the issues above. 2 Figure 1.2: Recent Scramjet Research Programs An airbreathing hypersonic cruise vehicle was chosen as the primary example mission for such a system due to the interest following the successful X-43A,10{12 HyShot,13{15 and HyFly/FASST tests,16 and its relevance to the X-51 Scramjet En- gine Demonstrator { WaveRider (SED-WR) program.17,18 Furthermore, gravity gradiometer instrument (GGI) aiding for scramjet applications has two distinct ad- vantages over low-speed/low-altitude ight. First, because the vehicle is traveling at high speeds (Mach 6{8) and the GGI produces only a moderate update rate ( 1 Hz), the gravitational gradients are sampled at position intervals >1500 m that al- low for greater gravity variations than those that would be observed for low velocity vehicles. Second, the high altitudes required for scramjet cruise ( 22{26 km) at- tenuates high frequency gravity anomalies found at lower altitude so the system is less susceptible to terrain anomalies.19,20 Unfortunately, current gradiometer systems are too massive and sizeable for many airborne applications, so continued improvements must be made to make 3 GGIs a viable airborne INS aid. For these reasons, and to show applicably to a wider range of missions, two subsonic cases are also simulated with less rigor. The rst represents a commercial aircraft at cruise conditions, and the second is a low speed and altitude gravity survey mission. 1.2 Previous Work 1.2.1 Gravity Gradiometer Instruments The rst gravity gradiometer instrument (GGI) was invented by Hungarian physicist Lor and (Roland von) E otv os in the late 1880s using a specialized tor- sion balance to investigate gravitational phenomenon.21 His extensive and ground- breaking research in gravitational gradiometry led to the naming of the E otv os as the fundamental unit of the gravitational gradient. (1E o 10 9s 2, which is physi- cally equivalent to measuring the gradient of 10 grains of sand 1 cm away, assuming 1 grain of sand 1 milligram.)22 Over the past century, gravity gradiometer in- struments have evolved from torsion balances to precisely machined sensors that are typically based on nite di erencing linear accelerometers or torsion beams. This subsection summarizes the history and research in developing airborne and space-borne gravity gradiometer instruments for navigation aiding and survey- ing. The original 1960s airborne GGIs are described rst, followed by the current generation airborne GGIs. Superconducting and cold atom interferometer GGIs are also discussed as they are believed to be the enabling technologies to improve future 4 generation gradiometer sensitivities. 1.2.1.1 First Generation Airborne GGIs The 1960s brought about many innovative technological breakthroughs in the eld of inertial navigation systems. As the error sources of inertial measurement units became better understood, and more precise and accurate accelerometers and gyroscopes became available, \the Air Force Geophysics Laboratory (then called Air Force Research Laboratory) sought to develop e cient approaches for mapping the short-wavelength features of the Earth?s gravity eld over large geographic ar- eas" to improve the navigation performance of autonomous inertial systems.23 This issue arises due to Einstein?s equivalence principle which states that inertial ac- celerations are indistinguishable from accelerations caused by a gravitational eld. Therefore, accelerometers are unable to measure acceleration, and instead measure speci c forces which is the acceleration of the system in an inertial frame minus the gravitational acceleration in the inertial frame: ai = _vi gi: (1.1) Because of this issue, an INS using accelerometers must include an estimate of the true gravitational eld in which the vehicle operates in order to calculate the true vehicle accelerations. Until this point in time, the uncertainty attributed to the accelerometer errors, a, was su ciently large that the gravitational modeling errors, even for simple models, were safely negligible. Mathematically, _vi = ai + gi ai: (1.2) 5 With improvements in accelerometer design and fabrication, INS analysts began to speculate that gravitational errors would need to be modeled and compensated for.5 One approach was to use a gravity gradiometer instrument to update the INS?s gravity model, usually by spatial integration of the gravitational gradients in real time (See Sec. 1.2.2.1). However, with the publication of Kalman?s seminal papers,24,25 the growth of other INS aids, and the di culty in design and manufacture of a robust, sensitive, small GGI, gravity gradiometer aided inertial navigation became largely forgotten. Fortuitously, with the rise of oil prices and the potential pro t of mineral exploration (speci cally diamond mines), GGI development is being actively pursued by several commercial enterprises to provide a fast, low cost surveying and prospecting service to these industries. (The current exploration instruments are discussed in detail in the next section.) Here, the rst generation airborne gravity gradiometer instruments developed simultaneously in the 1960s and 1970s by the Charles S. Draper Laboratories, Hughes Research Laboratories, and Bell Aerospace / Textron are reviewed. All three GGIs were pursued under Department of Defense (DoD) funding as a poten- tial airborne INS aid with the goal of producing an airborne GGI with a noise level of 10 E o moving-window averaged at a data rate of 10 sec.26 6 Figure 1.3: Draper Cylindrical and Spherical GGI Schematics, from Ref. [28] Charles S. Draper Laboratory Floated GGI The Draper Lab?s oated gravity gradiometer instrument was initially de- veloped as a \feasibility" cylindrical model27 and later into a spherical model. Trageser28 describes the feasibility model in detail along with test results which proved its surprising sensitivity to puddles on the roof of the laboratory.Because the instrument was stationary (unlike the other two 1960s GGIs), its noise level was limited primarily by the thermal noise oor. For benchtop tests using a human st the cylindrical Draper GGI produced a 0.99 E o mean root-mean-square (RMS) error. When using a 100 kg lead ball the RMS was 1.15 E o with a 10 sec integration and 0.50 E o for a 120 sec integration. Draper?s spherical GGI built on lessons learned from their cylindrical model and focused on \attaining [a] high degree of mass balance, temperature control and material stabilities" to meet the DoD speci cations.29 The spherical design was also relatively immune to platform jitter compared to the feasibility model. The 10 cm, 0.7 kg spherical GGI used two silver- lled tungsten proof masses attached to either 7 end of a sphere oated in Freon 113 to produce a pair of torques that could be used to measure two o -diagonal elements of the gravitational gradient tensor, see right portion of Fig. 1.3.30 A cluster of three GGIs in an umbrella con guration could then be used to measure the full tensor.26,28 The spherical GGI development began in 1974 and was rst tested in Septemeber 1976. In laboratory experiments with a 100 kg lead mass, this second Draper GGI produced near-thermal noise noise limits (0.085 E o experimental vs. 0.045 E o theory) with a bias stability less than 1 E o over several days.30 Grubin later proposed adding a second pair of proof masses to the Draper GGI in order to measure three components of the gravitational gradient tensor.31 This would also allow the use of only two oated GGIs to measure the full gradient tensor, and the orientation of the instruments could be optimized for robustness. While current airborne GGIs are all based o of the Bell/Textron instrument, as described shortly, the Draper Lab notion of producing an extremely sensitive non- rotating gradiometer is the basis for research of superconducting and cold atom interferometer GGIs. Hughes Research Laboratory Rotating Torsional GGI Hughes Research Laboratory took a drastically di erent approach to devel- oping its gradiometer sensor. Hughes quickly rotated a precisely manufactured cruciform shape of four proof masses and thin arms to measure the gravitational gradients using torque di erences. The premise was to measure the gravitational 8 Figure 1.4: Hughes Rotating Torsional GGI, from Ref. [33] gradients at a su ciently high frequency that the linear and angular motions had negligible error contributions.32 This resulted in Hughes having to overcome var- ious precision manufacturing issues, and to contend with bearing noise, a highly isoelastic structure, and material stabilities.29 A summary of the instrument and its idealized performance (perfectly matched masses and lengths) is presented by Bell et al.32 They show that the GGI mea- surement dynamics are uniquely driven by the gravitational gradients at twice the instrument rotation rate. Berman continued this analysis in two papers where he investigated errors that occur at multiples of the rotation rate since they would be seen as gravitational gradients.33,34 The four rotating proof mass modes (gravita- tional gradient, torsional, and two orthogonal translations) are also modeled along with the error due to a center of gravity o set and general asymmetry. A summary of the Hughes rotating torsional GGI is as follows. The overall instrument has an approximately 4.5" (12 cm) diameter, the proof masses are 0.75" 9 (2 cm) cubed, and the arms connecting the masses are 1.5" (4 cm) long 0.75" (2 cm) wide 0.050" (.13 cm) thick. The cruciform sensor has a resonance frequency of 200 cycle per second and is rotated at 6000 revolutions per minute.32 Bell Aerospace / Textron Rotating Accelerometer GGI The rotating nite di erenced accelerometer gravity gradiometer instrument developed and tested by the Bell Aerospace Division of Textron, Inc. is easily the most published rst generation airborne GGI, and has continued to live on to the present day.20,26,29,35{41 The Bell / Textron GGI is based on summing two nite di erenced accelerometer pairs tangentially mounted on a (relatively slow) rotat- ing disc. The instrument was originally used aboard Trident eet submarines for improved navigation accuracy during periods of prolonged submersion,42 and later tested for surveying feasibility on land, rail, and in the air.38 Most recently, it has been developed into an airborne surveying tool for prospecting and mineral explo- ration as detailed in the next subsection. Metzger29 presented a thorough review of research and development practices for the Bell / Textron GGI. He explained that this gradiometer had a 15 cm baseline between accelerometers and was rotated at a 1/4 Hz frequency to decrease the power of the Bell Model IV accelerometer?s turn-on bias as seen in Fig. 1.5. The scale factor error then became the predominant source of GGI error and was corrected with two feedback loops: one that balances the accelerometers in each pair, and one that balances the two pairs. This e ectively matched the scale factor error of 10 Figure 1.5: Accelerometer Null Bias vs. Rotation Rate (Cycles/Hour), from Ref. [29] three accelerometers to that of the fourth at every revolution. Misalignments were also corrected at 1/4 Hz, and further error compensation was achieved by shaking and dithering the instrument at speci c frequencies.20 Laboratory tests conducted in the mid 1970s showed that this instrument was able to produce noise levels of 2 E o with a 10 sec moving window average.29 The Bell / Textron GGI, like the Hughes instrument, measured the di erence of two on-diagonal gravitational gradients and another o -diagonal gradient as a result of the instrument?s rotation, see Fig. 1.6 and Sec. 5.1.3 for details. In order to measure the full gravitational gradiometer tensor, three instruments were \sym- metrically positioned about the vertical axis, with each gradiometer inclined at the same ?umbrella? angle; i.e., the spin axis of each instrument is oriented at the same angle away from the vertical, analogous to the spindles of an umbrellas,"23 as shown 11 Figure 1.6: Bell / Textron Rotating Accelerometer GGI, (Schematic, GGI w/ Sta- bilized Platform, and Internal Umbrella Con guration) from Ref. [47] in the bottom right of Fig. 1.6. The entire umbrella con guration was also rotated at a much slower 500 /hr.40 Richman43 proposed a nite di erenced accelerometer GGI that was based on using two separated IMUs on an airplane. The issues of aircraft exure, among others, likely prohibited this concept from being developed further. 1.2.1.2 Current Generation GGIs Bell / Textron Derived GGIs The Bell / Textron instrument technology was eventually acquired by Lock- heed Martin which has since produced several current-generation rotating GGIs including eight-accelerometer partial tensor GGIs (BHP Billiton?s Falcon Airborne Gravity Gradiometer (AGG)44,45 and the Arms Control Veri cation Gravity Gra- 12 diometer (ACVGG)39,46), and full tensor GGIs similar to the original Bell / Textron GGI (Bell Geospace?s 3-D Full Tensor Gradiometer (3D-FTG)47,48 and ARKeX?s FTGeX45). The BHP Billiton FalconTM AGG44,49{51 was the rst commercial system to use a gradiometer for airborne surveying and mineral deposit exploration. The instrument development began in 1993 with initial studies. Although the results weren?t very promising, the decision to move forward with construction of the Falcon AGG was made in March 1994. The rst mission was own in late 1997, and the rst system, Einstein, was operational in 1999 using a Cessna Grand Caravan airplane. Since then three other systems have been delivered, including one aboard a helicopter for improved spatial resolution because of the reduced ight speed. One major drawback of this airborne GGI is that it only includes one rotating disc, so that only a partial-tensor measurement is made. The Arms Control Veri cation Gravity Gradiometer is similar to the Falcon AGG in that it is comprised of a single rotating disc with eight-accelerometers that produces a partial-tensor measurement. The ACVGG was developed under funding by the Defense Threat Reduction Agency46 and the Defense Nuclear Agency20 for surface gravity surveys. While little has been published on this instrument, it is reported to have a 30 cm baseline between accelerometer pairs (twice that of the Bell / Testron GGI) and it?s rotation rate is \dramatically increased" compared to the original 1/4 Hz frequency.20 From these improvements, the projected stationary survey noise is said to be 1 E o at a 1 Hz update rate.46 A similar instrument was patented in 1994 by Ho meyer and A eck that provides some further detail into 13 eight rotating accelerometer GGIs.39 Bell Geospace, who had acquired exclusive rights to the Bell / Textron tech- nology and performed several marine surveys, took notice at the potential pro t of BHP Billiton?s airborne survey system and began work on their own.52 The primary advantage of Bell Geospace?s Airborne 3D Full Tensor Gradiometer (Air-FTGR ) to the Falcon AGG is the ability to measure the full gravitational gradient tensor.53 Like the original Bell / Textron GGI, the 3D FTG sensor uses three GGIs each with four rotating accelerometers to provide a full gravitational tensor observation aboard either airborne or marine missions (Marine-FTGR ). Also, like the Falcon AGG, the Air-FTG system performs airborne surveys on a Cessna Grand Caravan equipped with various technologies to accurately map terrain variations and eliminate their gravitational contributions to the GGI signal during post-processing. The result is a detailed map of gravitational anomalies below the survey area that warrant further investigation if they exhibit mineral or oil deposit characteristics.48,54 Unfortunately, current airborne GGI exploration systems are able to discriminate only major dia- mond formations because smaller formations are masked by the instrument?s noise oor.54 ARKeX is another commercial organization that has developed an airborne full-tensor gradiometer for mineral exploration (FTGeX) with the aid of Lockheed Martin. This Oxford Instruments Superconducting Ltd. spin-o reports that their Cessna-based survey system has been used extensively since its deployment in Spring 2005.55{57 And currently, ARKeX is pursuing the development of a superconducting GGI that will decrease the instrument?s noise oor at least a factor of ten so that 14 smaller mineral formations may be discovered. Lastly, a room temperature, stationary GGI was brie y mentioned by Glea- son20 to have been investigated by researchers at the Johns Hopkins University / Applied Physics Laboratory around 1995. The goal noise level was reported to be 0.1 E o up to a 100 Hz data rate. Unfortunately, to the present author?s knowledge, this instrument has never been discussed again in the open literature. Superconducting Airborne GGIs Gravity gradiometer instruments saw over an order-of-magnitude improvement in noise sensitivity with the use of superconducting technologies in the early 1980s. With the incorporation of superconducting quantum interference devices (SQUIDs) and wire electric discharge machining (EDM), incredibly precise measurements of accelerometer proof masses were possible. A revolutionary single axis supercon- ducting gravity gradiometer (SGG) at the University of Maryland produced a noise oor of 1 E o/pHz in 1987.58,59 Further development provided by NASA funding led to a three-axis SGG with an improved noise of 0.02 E o/pHz, Fig. 1.7.60{63 This NASA SGG was meant for a global survey mission aboard a satellite, and was able to reduce its noise three orders of magnitude below the room-temperature Bell / Textron derived GGIs by keeping the instrument at a cryogenic temperature of 4 K which enhances the mechanical linearity of the proof mass de ections. The University of Maryland?s Superconducting Angular Accelerometer (UMD SAA, Fig. 1.8 (a)) and the University of Western Australia?s Orthogonal Quadrupole 15 Figure 1.7: University of Maryland Superconducting Gravity Gradiometer, from Ref. [63] (a) (b) Figure 1.8: UMD SAA (a) and Cryostat (b), from Ref. [66] 16 Responder (UWA OQR, Fig. 1.9) are two superconducting GGIs designed for air- borne surveying.19,64{66 Both instruments use angular, instead of linear, accelerom- eters as their basic means of measuring the gravitational gradients because the angular accelerometers are more robust to demanding aircraft dynamics. The UMD SAA has a predicted airborne surveying performance of 0.34 Eo at a 1 sec update rate.64 The main issue with both of these systems is the need to enclose the GGI in a closed-loop refrigerant system, or cryostat, to maintain the 4 K operating tem- perature, as shown in Fig. 1.8 (b). Also, a sophisticated stabilized platform must be incorporated to isolate the GGI from the vehicle?s dynamics. Both the UMD SAA and UWA OQR are being pursued by ARKeX and Gedex to design an airborne survey system with a 1 E o/pHz design goal. The ARKeX Exploration Gravity Gradiometer (EGG) is reported to only measure the vertical gravitational gradient,45,55,56,67 while the Gedex High-De nition Airborne Gravity Gradiometer (HD-AGGTM) is likely a full-tensor GGI based primarily on the UMD SAA that includes funding support from De Beers.68{73 The European Space Agency?s Gravity Field and Steady-State Ocean Cir- culation Explorer (ESA GOCE) satellite will incorporate an Electrostatic Gravity Gradiometer (EGG) as its primary payload to map Earth?s gravitational eld spa- tially and temporally.74,75 The di erence between the GOCE EGG and many of the other superconducting GGIs is that the EGG uses capacitance (i.e. voltage) to measure the accelerometer?s proof mass displacements whereas the UMD and UWA GGIs rely on inductance (i.e. current).74 The noise speci cation for this 137 kg, 1.32 m 0.9 m 0.9 m state-of-the-art instrument is an impressive 3mE o/pHz 17 Figure 1.9: UWA OQR, from Ref. [19] Figure 1.10: ESA GOCE EGG, from Ref. [75] with a data rate of up to 10 sec.74 The University of Texas?s operational Gravity Recovery and Climate Exper- iment (GRACE) mission uses twin satellites as the proof masses of an e ective gravity gradiometer instrument with a 220 km baseline.76 While not a supercon- ducting GGI per se, this novel system shows the current state of the art in satellite gravity gradiometry. The last superconducting GGI to mention is a novel sensor from Gravitec that is based on measuring the behavior of a superconducting string.45,77{80 The size (400 30 30 mm) and mass (0.5 kg) of this gradiometer is tremendously smaller than the other surveyed instruments and its goal noise level of 5 E o/pHz at 5{10 Hz measuring all o -diagonal graviational gradients is also encouraging. Unfortunately, the open literature does not provide much detailed information on how exactly this 18 GGI is intended to perform, nor how it will overcome the challenges present in the others sensors. But, if this GGI comes to fruition, it could revolutionize current airborne GGI survey systems. Cold Atom Interferometer GGIs Innovative research in cold atom interferometry (based on work that has al- ready garnered two Nobel Prizes in Physics)81,82 may lead to dramatically reduced GGI noise levels.83{85 The premise of a cold atom accelerometer is to cool 109 cesium atoms to 2 K so that their wave-like properties can be exploited when the atoms are launched vertically. Then using interferometric methods, the gravita- tional acceleration on the drag-free atoms can be measured with laser light pulses. A vertical GGI that incorporates two such cold atom accelerometers using the same cesium atoms produced a noise of 30 E o/pHz with its 1.4 m baseline. If the baseline were increased to 10 m, the noise oor is predicted to decrease to 4 mE o/pHz.85 1.2.1.3 Gravity Gradiometer Instrument Speci cations Table 1.1 summarizes the instruments above and their speci cations. 1.2.2 Gravity Gradiometer Aided Inertial Navigation The use of gravity gradiometer instruments as an INS aid has been identi ed and investigated since the 1960s. Most of the early research focused on real-time determination of the gravity anomaly to provide improved un-aided inertial naviga- 19 Table 1.1: Gravity Gradiometer Instruments Gradiometer Dev elop er Noise, 1- E o Data Rate, sec Rotating Accel. GGI Bell Aerospace /T extron 2(Lab.), 10 (Air) 10 Rotating Torqu eGGI Hughes Researc hLab oratory 0.5 (Goal) 10 Floated GGI Drap er Lab oratory 1(Lab.) 10 Falcon AG G Lo ckheed Martin /BHP Billiton 3 Post Surv ey ACV GG Lo ckheed Martin 1 1 3D FTG Lo ckheed Martin /Bell Geospace 5 Post Surv ey FTGeX Lo ckhe ed Martin /A RKeX 10 (Goal) 1 UMD SGG (Space) Univ. of Maryland 0.02 (Lab.) 1 UMD SAA (Air) Univ. of Maryland 0.3 (Lab.) 1 UW A OQR Univ. of W estern Australia 1(Lab.) 1 Exploration GGI ARKeX 1(Goal) 1 HD-A GG Gedex /UMD /UW A 1(Goal) 1 Electrostatic GGI Europ ean Space Agency 0.001 (Goal) 10 Cold Atom Interferometer Stanford Univ. /JPL 30 (Lab.) 1 tion (i.e. dead-reckoning) accuracy. Eventually, GGI-based gravity mapping mis- sions were proposed and researched so that these high accuracy and resolution maps could be stored onboard non-GGI aided INSs. Then, once these gravity eld maps 20 were available, it was identi ed that they could be used for matching onboard GGI measurements to the stored map for position updating. While the work in this dis- sertation is focused on simulating map-matched GGI/INS systems, a high accuracy gravity eld model is a necessity. Therefore, a review of the literature pertaining to the determination of gravity anomalies using a GGI/INS (which is a precursor to mapping missions), GGI/INS mapping and surveying, and map-matched GGI/INS technology is presented in the subsequent subsections. A brief summary of some other novel applications of gravity gradiometer instruments is also given. 1.2.2.1 Real-Time Determination of the Gravity Anomaly The 1975 AIAA Guidance and Control conference in Boston, MA held a spe- cial session on gravity gradiometry technology that produced many seminal papers in this eld.28,35,86{88 Gerber86 investigated the e ect of GGI errors (white noise, time-correlated noise, random constant bias, and random constant drift) on a one- dimensional integrated INS. He found that transient errors were dominated by GGI biases and long-term errors were governed by noise near the Schuler frequency. Gru- bin,87 the next paper in the session, presented a similar paper with 1-D simulations that looked at the e ect of gravity anomalies on an INS?s position accuracy with and without an onboard GGI. Grubin examined the e ects of GGI biases, scale factors, and misalignments along with gravity anomaly bias and Schuler resonance, and he concluded that the GGI bias produced the largest INS position error. One primary di erence between these two papers is in their modeling of the gravity eld. Gerber 21 Figure 1.11: Block Diagram of INS/GGI for Real-Time Determination of Gravity Anomaly, from Ref. [88] chose to take a stochastic approach while Grubin used a simple deterministic model with several randomly placed masses. The third paper in the session, by Heller and Jordan,88 proved that GGI aiding of a reference gravity eld was optimal (Fig. 1.11), as had been assumed by the previous two papers. The authors compared the simulated INS performance of a reference ellipsoidal gravity model aided with GGI updates and the performance of direct integration of the GGI signal with the reference gravity model acting as an aid with a nite update rate. They concluded that the main issue with the latter system is that the error growth exhibits a diverging random walk trend because of the integration of the GGI noise. The other two papers in this 1975 session are reviewed elsewhere in this chapter. Trageser?s paper28 on the Draper GGI was discussed on pg. 7, and Metzger and Jircitano?s paper35 on a map-matching GGI/INS will be discussed on pg. 28. Over the years, the interest in a GGI-aided INS for real-time gravity anomaly 22 determination has decreased as the di culty in the instrument design was identi- ed. Occasionally, people have revisited the subject and added some modest gains to the state of the art. Zondek89 is notable for extending this use of a GGI/INS to an Earth-orbiting satellite as a means to improve its ephemerides. Speci cally, the paper focuses on measuring and correcting for high-frequency orbit errors due to unmodeled gravitational phenomenon. Wells and Breakwell90 derived several one- dimensional lters (one Kalman and two Weiner-Hoph) to blend GGI and Doppler velocity measurements into an INS. The Weiner-Hoph lters were necessary to im- plement a higher-order nonlinear gravity model that would be impossible to include in a Kalman lter due to its inherent assumption of linearized error dynamics. Hopkins91 investigated the e ect of a GGI-aided INS to reduce anomalous gravitational errors during GPS outages. Unlike the work in this dissertation, there is no comparison of the GGI measurement to a stored gravity map. Instead, Hopkins only looks at whether or not to model the gravitational errors in a GPS/INS by using a gradiometer as an additional sensor. He shows that the gravity error modeling and the added GGI provide further re nement of the navigation solution during GPS blackouts. Shingu92,93 simulated a GGI-aided INS to estimate unmodeled gravity errors for a more robust inertial navigation. His chosen application was a long-range au- tonomous rocket trajectory where a portion of the Earth?s mass was concentrated in a single location unbeknownst to the rocket?s INS. Most recently, Jekeli94 simulated a GGI-aided INS consisting of future-grade IMUs for accurate long term dead-reckoning as part by a Defense Advanced Research 23 Projects Agency (DARPA) concept. He showed that the unmodeled gravitational disturbances from a simulated mountainous region would produce 5 km of horizon- tal position error after one hour dead-reckoning with future-grade IMUs. However, with an integrated, onboard GGI with 0.1 E o noise updating at 1 Hz, the position er- rors decreased to only 5 m|a three order of magnitude improvement. This reference is particularly notable for its clear derivation of a strapdown gradiometer?s linearized measurement errors and the integration of a GGI into a 6 degree of freedom INS. This dissertation extends Jekeli?s work by including map-matching of the GGI mea- surement to a stored gravity map, adding orientation e ects to the derivation of the GGI measurement errors, and derivation of a stabilized GGI with its linearized error measurement. Furthermore, the system error state transition matrix is calculated more computationally e cient in this work as compared to Jekeli. Kwon and Jekeli95 also investigated the DARPA problem of accurate long-term dead reckoning from the viewpoint of using a high-resolution onboard gravity map instead of an onboard GGI. Using the future-grade IMUs, they showed by simulation that ground data needed to be gridded at a 2 arcmin resolution with accuracy of 5 mgal (5 10 5 m/s2) or better in order to provide the goal of 5 m position error after one hour of free-inertial dead reckoning. 1.2.2.2 Gravity Gradiometer Surveying Gravity gradiometry technology has been proposed as an alternative or re- placement to traditional gravimeter surveying. The major advantages to a GGI- 24 Figure 1.12: Gravitational Gradient Survey Integration Methodologies, from Ref. [96] based survey system are: Improved high frequency observability due to the gravitational gradients being the derivative of the gravitational acceleration. More information available since n is a 3 3 tensor measurement while the gravimeter?s acceleration is at most a 3-element array. GGI, unlike gravimeter, measurements are decoupled from linear accelerations so that accurate estimation of these accelerations is unnecessary. Jordan96 was the rst to research using an airborne gradiometer for fast, large- scale survey missions. He compared three methods of using the GGI measurements, see Fig. 1.12: a simple integration of only the gradient of the vertical gravitational acceleration with respect to the velocity vector ( xD) to yield scalar gravity anoma- lies; integration of the gradients of the gravitational acceleration with respect to the velocity vector ( xN; xE; xD) to yield a vector of the gravity anomaly; and optimal integration of the full tensor. By simulating a mission over a salt dome eld, Jordan 25 showed that at least three components of the gradient tensor should be measured, but that the accuracy improvement from using three to nine elements was rather small. He also concluded that a GGI-based survey can save time and money because a 18 km-spaced GGI survey was comparable to a 8 km- spaced airborne gravimeter survey. Sensitivities to track spacing, GGI noise, vehicle speed, and survey altitude were also presented. A decade later, Brzezowski and Heller23 discussed the error sources of a GGI survey mission in detail. The error contributions are, in summary: Gradiometer System Errors: GGI noise; environmentally induced errors; navigation, attitude, and attitude-rate uncertainties; and gimbal, vehicle, and limited nearby-object compensation. Discrete Sampling E ects: aliasing of frequencies higher than half the sampling frequency, which determines the spatial resolution of the survey when multiplied by the aircraft speed. Limited Data Extent: the limitation in determining the low frequency signal content because the far- eld gravity is not measured. Then, using three gravity eld characteristics (low, medium, and high variations), Brzezowski and Heller showed that it is easier to compute the de ections of the vertical of the gravitational vector than to compute the magnitude of the gravita- tional disturbance. Furthermore, \rougher," or highly variant, gravity elds caused larger survey errors. And surprisingly, limited data extent produced about half the modeled survey error. 26 Jekeli38 summarized the Gravity Gradiometer Survey System (GGSS) experi- mental test program of 1983{1989. The GGSS used a full tensor Bell / Textron GGI mounted in a large conversion van to investigate the usefulness of the system for air- borne, road, and rail surveying. The airborne GGI tests were the rst of their kind and were accomplished by loading the van aboard an aircraft and then ying tracks over a 315 km 315 km area at a 700 m altitude. The test was plagued by poor GPS coverage and of the 128 tracks own, only 19 were chosen for analysis. Rather surprisingly, this rst ight test of an airborne GGI produced average noise levels of only 10 E o with a 10 sec average, while 3{6 E o was expected. Moreover, the results of the GGSS program likely motivated early development of the commercial airborne GGI survey systems. Jekeli97,98 later compared airborne gravimetry and gradiometry survey errors in the frequency domain. He reiterated the potential bene ts of an airborne GGI system but noted that the current limitation is the self-generating noise of the instrument, which is over an order of magnitude of other GGI system errors. He concluded that the future of gravity surveys would preferably lie with GGIs because future gradiometer-based surveys only require improvements in IMU quality, while gravimeter-based surveys also require improvements in real time kinematic (RTK) GPS to compensate for the vehicle?s accelerations. (A full technical report on this work is also available from Ohio State University.)99 27 1.2.2.3 Gravitational Gradient Map-Matching Although not much has been published in the open literature, map-matched gravity gradiometer aiding dates back to the 1975 AIAA Guidance and Control conference. A one-dimensional covariance analysis by Metzger and Jircitano35,100 estimated INS position accuracy based on gravity and gradient map-matching for mobile systems (4{240 m/s). The premise was to simulate an initial mapping mission and then compare the steady state lag of a second GGI/INS mission following the same trajectory. They show that the gravitational gradients are preferable for map- matching since their shorter correlation distances (4,600 m vs. 37,000 m) produced higher frequency signals and thus ner spatial resolutions. Increases in instrument noise and vehicle velocity, or decreases in map record length, are shown to degrade performance. The velocity e ects are due to the assumption that the GGI produces measurements at every 10 seconds, so the initial mapping mission produces coarser maps when simulated at higher velocities. For the work presented in this disserta- tion, the gravity map is a xed resolution regardless of vehicle velocity so position error is less sensitive to cruise speed. In 1990, A eck and Jircitano36 presented three-dimensional results of an in- tegrated INS/GGI simulation for low speed airborne and submarine systems. Un- fortunately, the depth of the presented analysis was minimal, most likely due to the proprietary nature of the original work and the classi cation of some of the tech- nology. Indeed, the vast majority of their simulations are shown as simple block diagrams. Nevertheless, the results are quite promising as shown in Fig. 1.13. This 28 (a) (b) Figure 1.13: Horizontal (a) and Vertical (b) Position Error vs. Cruise Altitude, from Ref. [20]; originally from Ref. [36] 29 dissertation?s research is essentially a continuation of A eck and Jircitano?s work but with the noted di erence that the current work presents the detailed analyses that are omitted in the reference. Furthermore, this present work extends the anal- ysis to the hypersonic regime and includes simulation of future-grade GGIs that the reference did not consider. The following year Jircitano and Dosch proposed and patented a Gravity Aided INS (GAINS) using a GGI and a vertical gravimeter for covert submarine naviga- tion.37,101 This concept built on the prior two references and includes speci cs in the modeling of the gravity eld, but their lter implementation is again shown as only a block diagram. From the schematic, the lter states included GGI, gravimeter, depth sensor, and IMU instrument errors along with the standard INS position, ve- locity, and attitude error dynamics. The lter also includes states for the estimated gravitational eld potential, acceleration, gradient and third order derivative. With their covert navigation system, the authors showed that modern submarines could produce position errors as low as 30 m. More recently, Zhang et al.102 simulated a map-matched GGI/INS for an au- tonomous underwater vehicle (AUV), similar to the prior papers. Unfortunately, like the previously cited papers, the presented analysis consists of only ow charts and results. Their results show an estimated position accuracy of tens of meters for their AUV concept. A di erent GGI/INS map-matching approach was taken by Archibald for his doctoral work.103 He implemented neural networks to match large-area noisy and truth gravitational gradient and magnetic eld maps. While INS simulations were 30 shown to motivate his research, the neural network was not integrated with the INS. Regardless, the novel concept of matching large-scale geophysical quantities could be used as an initial estimate of a user?s position state. Lastly, Gleason20 discussed many of the practical issues of a GGI/INS for navigation and terrain avoidance at length. The paper focuses on extrapolation of gridded gravity data at a given altitude using Fast Fourier Transforms (FFTs), terrain elevation, and density assumptions. He also showed that the at-altitude gravitational eld can be optimally estimated when ground gravity data is available. Some of the other issues discussed were the e ects of vehicle velocity and altitude, gradiometer noise level and data rate, and the design of a low-pass lter to reduce high frequency instrument errors. 1.2.3 Other Gravity Gradiometer Instrument Applications Some other novel applications for gravity gradiometer instruments are sum- marized below to show the versatility of this relatively unknown sensor. 1.2.3.1 Close-Loop Satellite Attitude Re nement Roberson104 proposed using a hypothetical di erenced accelerometer GGI to compute the radial (i.e. vertical) gravitational gradient of the Earth so that ne tilt orientation could be achieved aboard an orbiting satellite. He motivated his research by the discovery that the primary radial gradient produced a torque on satellites that caused their long axis to be aligned with the gradient. Thus, because 31 the orientation of this gradient is well known, it could be exploited for high accuracy attitude determination. He also derived error equations for the GGI assuming scale factor, bias, and misalignments in accelerometer pairs that pointed radially away from the Earth?s center. While his derivations are informative to show the e ects of the accelerometer errors on the overall GGI error, their derivation is rather confusing and limited to a partial-tensor measurement ( ND; ED; DD). Diesel105 showed that a single rotating accelerometer could theoretically pro- vide a measurement of the gravitational gradient as the IMU sensor records mea- surements at various locations. The periodic measurement could then be used to nely estimate the spacecraft?s tilt errors because the tangential gradient signal is zero in the vertical and horizontal directions. Diesel also comments on a controller to stabilize the system, the lter power spectral density, sensor dynamics and an error analysis. 1.2.3.2 Arms Treaty Veri cation The use of a GGI to estimate the mass properties of arms treaty-limited sys- tems was rst proposed by Parmentola.106 He investigated using a GGI to take an e ective gravitational X-ray of two Tomahawk-scale cruise missiles; one with a simulated conventional warhead, and one with a nuclear warhead. By unobtrusively scanning the gravitational gradients 0.5 m away from the missile, the warhead type could be unambiguously determined. Gray, Parmentola, and LeSchack22 expanded this work by using a least squares 32 approach to invert noisy GGI measurements to estimate the mass properties of an object. The paper discusses the numerical issues of calculating the measurement matrix pseudo-inverse and provides background on the multipole expansion for the mass properties. Then by simulating objects moving on an assembly line near a GGI, it was shown that this system could be a viable tool in monitoring arms treaty-limited objects. Moreover, it was concluded that the GGI estimated non- uniform and less spherical objects less accurately than a uniform sphere; however, this result may be heavily dependent on their choice of a multipole expansion in estimating the object?s mass distribution. Determination of asteroid and comet mass distributions using a GGI has also been proposed.107 Although this is not a treaty-limited object, the problem formu- lation is quite similar. 1.2.3.3 Underground Bunker or Void Detection The notion of using a GGI system for underground bunker or void detection is essentially the same as using it for surveying or exploration. The premise is to take a gravitational survey of an area along with its terrain elevation and back out any anomalous features. The di erence is now that instead of an increased gravitational potential for the exploration missions, there is a decrease due to the void. Romaides et al.46 undertook an experimental ground-based validation study using the ACVGG (see pg. 13) and these principles. Preliminary tests were per- formed over a subway car storage facility in Cambridge, MA that showed the clear 33 presence of the underground tunnel. A full-scale survey was then performed at Vandenberg Air Force Base over the Missile Alert Facility (MAF) bunkers using the ACVGG and a state of the art gravimeter. Although the bunkers were heav- ily reinforced with concrete that helped compensate for the absence of mass, the GGI-based survey unambiguously resolved the MAF location, unlike the gravimeter survey. The results from this study are particularly promising since they show a viable tool for determining the location of underground bunkers in the global war on terror. 1.2.3.4 All Accelerometer Inertial Navigation Gravity gradiometer instruments have also been proposed as an extension to gyro-less all-accelerometer inertial navigation systems.108 Zorn presented two pa- per on this topic that envisioned a 12-accelerometer GGI to measure speci c force, angular accelerations, and the full-tensor gravitational gradient tensor (Section 5.2 discusses a similar GGI, and pg. 186 explains how the angular acceleration is observ- able). Zorn?s rst paper40 summarizes the concept and derives the applicable mea- surements for the system. The second paper41 continues the work by simulating two INS/GPS systems to investigate the sensor requirements for the all-accelerometer INS to be comparable to a tactical-grade INS with gyros. For a constant altitude, speed, and turning radius simulation, it was shown that the all-accelerometer INS (which is essentially a GGI) needed a 10 6 improvement in the accelerometer bias stability and white noise level to yield a comparable tactical-grade INS/GPS navi- 34 Figure 1.14: Map-Matching GGI/INS using an Extended Kalman Filter gation accuracy. This vast reduction in accelerometer error is directly attributed to the need to accurately measure the angular acceleration of the vehicle so that the attitude could be determined after integrating twice. 1.3 Objective The objective of this dissertation is to show the potential bene t of a novel gravity gradiometer aided inertial navigation system. The premise is to compare GGI measurements with an onboard gravity eld map to produce delta-position cor- rections through an extended Kalman lter implementation, as shown schematically in Fig. 1.14. Conceptually, this system functions in much the same manner as a terrain-based map-matching INS aid. The main di erence is that the emitting radar or laser sensor of a terrain-based system is replaced with the self-contained, passive gravity gradiometer instrument. Furthermore, instead of making a single range measurement (or possibly a Doppler as well), a GGI can make up to six non-symmetric measurements of the gravitational gradient tensor. In order to quantify the performance of the INS/GGI system a characterization 35 0 2004006008001000 ?0.034 ?0.033 ?0.032 ?0.031 g N Gravity Vector, m/s 2 0 2004006008001000 ?5 0 5 x 10 ?4 g E 0 2004006008001000 9.725 9.73 9.735 g D Downrange, km 0 2004006008001000 ?1535 ?1530 ?1525 ?1520 ?1515 ?1510 ? NN Inline Gradients, Eo 0 2004006008001000 ?1535 ?1530 ?1525 ?1520 ?1515 ?1510 ? EE 0 2004006008001000 3040 3045 3050 3055 3060 3065 ? DD Downrange, km 0 2004006008001000 ?15 ?10 ?5 0 5 10 ? NE Cross Gradients, Eo 0 2004006008001000 ?25 ?20 ?15 ?10 ?5 0 ? ND 0 2004006008001000 ?15 ?10 ?5 0 5 10 ? ED Downrange, km Figure 1.15: Simulated Gravity at Nominal Latitude and Altitude for Mach 7 Tra- jectories, \High" Variation (solid), \Low" Variation (dashed) 36 of the gravitational gradients is rst undertaken. Of utmost importance are the gradient variations since the slope of the gravitational gradients are used for the position updates. In other words, if the gradients were constant, there would be no discernable features in the signal to derive position knowledge, see dashed gradients in Fig. 1.15 compared to the solid curves. Conversely, if the gradient variation is su ciently large compared to the GGI noise level, these changes may be exploited to update the INS position estimate. The primary questions to be answered are: How much do the gravitational gradients vary? Which gravitational gradient varies the most? Where do the gravitational gradients most? How does altitude e ect the gradient variations? As a corollary to the gravitational gradient characterization study, and the lack of available terrain elevation data for this work, a rst-order analysis of when one may neglect local terrain e ects is carried out. In order to broaden the applicability of this study, a parametric \mountain" is simulated and its vertical gravitational gradient is computed for a variety of dimensions and user altitudes. Then, to reduce one of the independent variables, the \mountain" width is optimized to provide the maximum gravitational gradient so that the terrain contributions may be estimated as a function of terrain peak and user altitude. After these fundamental gravitational gradient questions are answered, the map-matching GGI-aided INS simulations are performed to quantify the potential performance and sensitivities of future INS/GGI systems. The rst objective to this goal is the modeling of the inertial measurement unit signals. A hypersonic 37 scramjet is chosen for the majority of the simulations. The o -design aerodynam- ics and propulsion characteristics are used to calculate the trim conditions over a 1000 km range cruise. Then, the trim angles are nite di erenced to calculate the body-to-navigation frame portion of the gyro signal. The assumed trajectory and cruise velocity and altitude are used to calculate the accelerometer signals for the simulation. The details of integrating a gravity gradiometer instrument into an inertial navigation system using an extended Kalman lter implementation are also pre- sented thoroughly for the rst time. Extensions and modi cations to traditional INS simulations are documented to enable stable lter performance. In the event of lter divergence, the reason (numerical truncation error) is identi ed and solutions are proposed. Furthermore, the simulated gravity gradiometer instrument measurement and their linearized errors are derived thoroughly for the rst time in the open litera- ture. Speci cally, the inclusion of tilt errors and the conversion from the GGI tensor measurements to vector measurements to allow for the lter implementation is per- formed for the rst time. Completely new derivations for a stabilized GGI are also performed for this dissertation work. The hypersonic INS/GGI navigation system sensitivities are identi ed through the use of numerous Monte Carlo simulation con gurations. The varied design parameters are: instrument noise, update rate, and type (strapdown or stabilized); gravitational gradient variation; IMU quality (navigation or tactical grade); and Mach number. Also, to compare with current technologies, an INS/GPS system 38 is simulated. The e ects of GPS measurements (pseudorange with or with out pseudorange rate), update rate, and IMU quality are also quanti ed. Two subsonic INS/GGI cases are also simulated to show the potential per- formance on current platforms. A commercial aircraft mission and a GGI-survey mission are both simulated using the \best" INS/GGI design parameters from the hypersonic simulations. These cases are also important because current GGIs are too large and heavy for missile-class vehicles, but can be used aboard these two current subsonic vehicles. 1.4 Dissertation Outline This dissertation consists of seven chapters and ve appendices, organized as follows. Chapter 2 discusses some fundamental aspects of gravity gradiometry includ- ing the gravitational potential and acceleration, and the centripetal components of gravity. A brief review of how the gravity eld is typically modeled and how it was modeled for this work is presented. The parametric terrain study to estimate when local terrain contributions may be negelected from the computed gravity map is also undertaken in this chapter. Then the gravitational gradient sensitivities to altitude and location on the Earth are shown. Lastly, the simulated trajectories used in the Monte Carlo simulations are detailed along with a study to estimate the stored map?s linear interpolation error as a function of grid resolution. The third chapter presents the hypersonic vehicle model used for the ma- 39 jority of the simulations. The aerodynamic and propulsion characteristics from a JHU/APL reference are described and the implemented curve- t calculations are given. A simple mass model is then determined so that the trim conditions can be computed over the assumed trajectories. The trim pitch and roll angles are lastly nite di erenced to produce part of the simulated gyro signals for the INS simulations. Chapter 4 details the intertial navigation system model. Standard coordinate frames and transformations are reviewed, then the navigation equations are derived from rst principles. The INS linearized error dynamics are also derived thoroughly. And lastly, the simulated acclerometer and gyro measurement errors are presented along with a survey of current tactical and navigation grade IMU speci cations. The modi cations to traditional INS simulations in order to integrate a GGI aid is also identi ed in this chapter. The fth chapter presents a thorough methodology for modeling many grav- ity gradiometer instruments. Then the rotating, stabilized GGI measurements are derived clearly for reference purposes since many references provide at most a con- fusing derivation of this type of GGI measurement. Next, the assumed twelve- accelerometer GGI is described. Strapdown and stabilized GGI measurements and their linearized error equations are also derived comprehensively for the rst time. Chapter 6 explains the Monte Carlo simulation set up and assumptions. A parametric analysis on the e ect of Monte Carlo set size is undertaken, and then the INS/GGI and baseline INS/GPS results are presented for numerous system con gurations. 40 Chapter 7 summarizes the contributions of this work to the state-of-the-art and recommends a set of work for future study. Several appendices are also included in this dissertation to supplement the content of the main body. Appendix A consists of additional global gravitational gradient maps at various altitudes, which are used in the analyses of Ch. 2. Appendix B supplements Ch. 3 with additional polynomial curve ts for the thrust coe cient propulsion calculations. Appendix C presents an overview of the extended Kalman lter model implemented for this work. A review of the Kalman lter assumptions, stochastic processes, and linear system dynamics are also shown. And a new method to calculate the gyro noise portion of the error state transition matrix is discussed as well. Appendix D details the modeled nominal 24-satellite GPS constellation and the assumed measurements for the baseline INS/GPS analyses. Lastly, App. E lists the extensive mean-radial-spherical-error results from the Monte Carlo simulations. 41 Chapter 2 Gravity Map Model This chapter presents the model used to simulate the gravity eld for this work. Section 2.1 rst presents a brief review of the gravity potential and its derivatives. Section 2.2.1 and 2.2.2 then describe how the gravity eld is typically modeled using spherical harmonic and local terrain models. Section 2.2.3 next estimates when the local terrain e ects may be omitted from the gravitational eld model, and Sec. 2.2.4 discusses and estimates other vehicle self-generated bias sources. A global-scale characterization of the gravitational gradients using a spherical harmonic model is performed in Sec. 2.3 to identify trends in the gradients for use as a navigation map- matching aid. Then the two chosen simulation trajectories are detailed in Sec. 2.4 along with studies to determine the stored gravity eld grid spacing. Lastly, Sec. 2.5 summarizes the contributions and results from this chapter. 2.1 Gravity Gradiometry The gravitational potential is de ned as109,110 g ZZZ V (r0) jr r0jdV; (2.1) 42 where is the density of the attracting mass at r0 and r is the vehicle (or \user") location. The rst derivative of the the potential yields the gravitational vector: gn r g = 0 BB BB BB @ gN gE gD 1 CC CC CC A ; (2.2) where it has been assumed that the coordinates are in the local North-East-Down navigation frame, see Sec. 4.1.3. The second derivative produces the gravitational gradient tensor: n rrT g = 0 BB BB BB @ NN NE ND NE EE ED ND ED DD : 1 CC CC CC A (2.3) The trace of n, or equivalently the Laplacian of the potential, is equal to Fourier?s equation:109 4 g = NN + EE + DD = 4 G (r); (2.4) where G is Newton?s gravitational constant and here is the density at the user. Because the density of the Earth (mean crust 2,670 kg/m3)110 is much greater than the atmosphere (at sea level 1.2 kg/m3), it is common to assume a \free-air" gravitational potential so that the gravitational attraction of the air is neglected. Thus, Fourier?s equation is now 4 g = NN + EE + DD = 0; (2.5) which is Laplace?s equation. The general solution to Laplace?s equation is an in nite harmonic summation, to be discussed in the next section. 43 Before continuing, a word on nomenclature is in order. Throughout much of the literature the term \gravity gradient" has been used. In the strict sense, the quantity that is being referred to is the gravitational gradient|not the gravity gradient. Gravity is de ned as the sum of the gravitational and centrifugal potentials (or accelerations, or gradients). It will be shown later that the measurement made by a gravity gradiometer is a combination of the gravitational gradient and noise in the form of angular rates and accelerations, see Sec. 5.1. The terms \gravity gradient" and \gravity gradiometer" in some sense are misnomers, as the quantities that are being used are more accurately gravitational gradients. For completeness, the centripetal potential is the analytic function:111 c = ! 2 e 2 (rcos ) 2; (2.6) where !e is Earth?s rotation rate, is the latitude of the user, and r is the radius from the user to the center of the Earth. The rst derivative of Eq. (2.6) is then the centripetal acceleration due to the Earth?s rotation. 2.2 Gravity Map Modeling Gravitational eld maps are typically computed by the summation of a spher- ical harmonic model to capture low frequency, long range gravitational e ects and integration of local terrain elevation to capture the high frequency, short range ef- fects.95 The centripetal portion is then added to compute the gravity acceleration vector. And since the centripetal e ects are simple analytical functions, this section will focus solely on the more complicated modeling of the gravitational potential 44 and its derivatives. 2.2.1 Spherical Harmonics The general solution to Laplace?s equation, Eq. (2.5), is an in nite harmonic summation. In spherical coordinates:109,112 g(r; ; ) = GMr h 1 + 1X n=2 ae r n nX m=0 C nm cos(m ) + Snm sin(m ) P nm( ) i ; (2.7) where is the colatitude (= =2 latitude), is the longitude, n and m are the degree and order of the fully normalized coe cients ( Cnm, Snm), and Pnm( ) is the fully normalized associated Legendre function. The equation is referenced to a given gravitational parameter GM which is the universal gravitational constant times the total mass of the attracting body, and ae is a reference radius (for Earth ae = 6,378,137 m).113,114 In practice, the series is truncated at a maximum degree, nmax, based on the available coe cient set. There are three designations for spherical harmonics.109,115 The rst designa- tion is a zonal harmonic when m = 0, Fig. 2.1 (a). These harmonics are indepen- dent of and related to the well known unnormalized J harmonics by (Vallado,115 pg. 518): Jn = Cn;0p2n+ 1: (2.8) The J2 term is by far the largest harmonic term (zonal or otherwise) as it accounts for the bulk ellipsoidal shape of the Earth. The second special designation is a sectoral harmonic which occurs when m = n, Fig. 2.1 (b). These harmonics are 45 (a) (b) (c) Figure 2.1: Spherical Harmonic Classi cations (a) Zonal (b) Sectoral (c) Tesseral, from Ref. [109] independent of latitude and account for gravitational variations in slices parallel to the meridian. Lastly, when 0 < m < n the harmonics are designated as tesseral and essentially account for potential variations in a checkerboard-like pattern of alternating mass distributions, Fig. 2.1 (c). While spherical harmonic models allow for calculation of global gravitational potentials, their usefulness is limited primarily by the nite spatial resolution of the model. And although there is no universal de nition for the spherical harmonic model?s resolution, a convenient and common de nition is the half wavelength of the maximum zonal harmonic with respect to the equatorial radius:116 S:H:M:res = aen max 20 10 6 m nmax : (2.9) Many spherical harmonic gravity models have been published for Earth with the most extensive being the EGM96 set that includes coe cients up to degree and order 360.114 Recent work such as the GFZ Potsdam CHAMP (Challenging Minisatellite Payload) and the University of Texas GRACE (Gravity Recovery and 46 Climate Experiment) satellite missions are producing higher accuracy coe cients; however, their maximum degree and order is currently only 140 (CHAMP)117 or 200 (GRACE).118 And since the resolution is proportional to 1=nmax, EGM96 is about twice as ne as GRACE?s GGM02C.118 Therefore, to produce the highest resolution gravitational maps, the EGM96 coe cient set was chosen. The calculation of the spherical harmonic potential and its derivatives (vec- tor and gradient tensor) is rather straight forward except for the calculation of the associated Legendre functions.111,119 For the present work, the National Oceanic and Atmospheric Administration (NOAA) / National Geodetic Survey (NGS) pro- gram geopot97.v0.4e.f was modi ed to produce gridded gravity accelerations and gravitational gradients at user de ned latitude, longitude, and altitude ranges.120,121 The code calculates the Legendre functions and derivatives using an e cient itera- tive Clenshaw summation.119,122 However, this version of the program may not work or be as e cient for future higher degree coe cient sets because of numerical preci- sion limitations. Instead, the new geopot07.f123 may be used for nmax 2190, or the Fortran 95 package SHTOOLS by Wieczorek124 may be used up to nmax 2800. Another approach would be to implement one of the algorithms presented by Holmes and Featherstone.125 As mentioned, the spherical harmonic models alias higher frequency (and ner resolution) terrain contributions due to its nite summation of Eq. (2.7). A discus- sion on methods to account for the terrain e ects is presented in the next section, and a fundamental investigation as to when the terrain e ects may be neglected is performed in Sec. 2.2.3. Section 2.2.4 then discusses other gravitational gradient 47 biases. 2.2.2 Terrain Elevation Contributions Referring back to the spherical harmonic gravitational potential, Eq. (2.7), and the gradient tensor de nition, Eq. (2.3), one can show that to rst order the gravitational gradients are inversely proportional to the cube of the distance between the vehicle and attracting mass because the gradients are the second derivative of the potential, which is itself inversely proportional to the distance. Mathematically, g GMr ! @ 2 g @r2 = DD 2GM r3 : (2.10) Therefore, because the gradients attenuate proportional to distance cubed, local terrain elevation variations will be negligible at su ciently high altitudes compared to a given gradiometer instrument?s noise level. Conversely, for low altitude appli- cations, such as surveying missions, the signal-to-noise ratio due to the local terrain variation is large enough that it should be included in the computed gravity map. In these latter cases digital elevation maps and an assumption of the ter- rain?s density can be used to account for local terrain contributions. Jekeli and Zhu surveyed several algorithms including prism, fast Fourier transform (FFT), and ordinary numerical integration methods for two terrain data sets and found that FFTs can produce accurate models at a constant gridded altitude with low compu- tation time.110 Gleason also presented an FFT method that included several discrete density layers and showed that one can optimally estimate gravity at altitude with knowledge of surface gravity data and terrain elevation.20 Thus, when terrain ef- 48 fects are not negligible (i.e. low altitudes and/or low GGI noise), an FFT model such as those described in the above references should be included in gravity eld mapping. A limitation of the current work is the omission of such terrain e ects in the simulated gravity eld which causes a reduction in the available gradient signal frequencies. This is especially true for the subsonic GGI survey simulation since its assumed altitude is only 100 m. 2.2.3 Minimum Altitude to Neglect Terrain E ects This section presents a parametric study of when terrain e ects may be ne- glected for a given GGI noise level and user altitude. To bound the analysis, a single hypothetical mountain directly below the user is simulated and its vertical gravitational gradient is calculated. The vertical component was chosen as it is the largest component of the gradient tensor as shown later in Fig. 2.11. The mountain is assumed to have a zero-mean, normal, Gaussian height distribution, see Eq. (C.8) on pg. 313: hT(s) = aT T p2 exp s 2 2 2T ; (2.11) where hT is the height of the mountain, s is the horizontal distance from the ori- gin of the mountain, aT and T are the amplitude and standard deviation of the distribution, respectively. Figure 2.2 shows a schematic of the mountain for several design parameters. A Gaussian distribution was chosen because it qualitatively es- timates the shape of a mountain and its distribution is uniquely de ned by only two parameters, thereby facilitating the parametric nature of this study. 49 ?6 ?4 ?2 0 2 4 6 0 0.5 1 1.5 2 s / ? T h T h T (0) = 1.0, ? T = s h T (0) = 1.0, ? T = 2s h T (0) = 0.5, ? T = s User = (0, h) Figure 2.2: Schematic of Modeled Gaussian Mountain The vertical gravitational gradient is calculated according to:110 DD = G T ZZ A T33dA; (2.12) where the integration in the vertical direction has been performed so that T33 = 8 >>> < >>> : 1 (h hT)2 1 h2; s = 0 1 s2 " h hT rh h h T rh 3 hr 0 + h r0 3# ; s6= 0 (2.13) where h is the user altitude, and r2h s2 + (h hT)2; (2.14) r20 s2 +h2: (2.15) For a given peak altitude, hT(0), and standard deviation, T, the amplitude is computed by aT =p2 hT(0) T: (2.16) 50 log 10 (h T (0) / h) log 10 ( ? T / h) 100 Eo10 Eo 1 Eo 0.1 Eo 0.01 Eo 1 mEo ?4 ?3.5 ?3 ?2.5 ?2 ?1.5 ?1 ?0.5 ?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 2 Figure 2.3: Terrain Contribution to DD Then DD is calculated by numerically integrating Eq. (2.12) and (2.13) by: DD = G T 6 TX si=0 (T33;i+1 +T33;i)si s+ (2T33;i+1 +T33;i) s 2 3 ; (2.17) where s is a 2,001-element equispaced array of the horizontal distance from the origin to six times the standard deviation parameter and T33;i is the value from Eq. (2.13) at si. Also, it is assumed that G = 6:6742 10 11 m3/(kg s2) and T = 2,670 kg/m3. To investigate a wide range of terrain possibilities, the simulated mountain?s standard deviation and peak altitude were normalized by the user altitude and varied logarithmically. (Section 2.2.3.1 proves that the gradient calculation, as formulated, is uniquely de ned by T and hT(0)=h.) The vertical gravitational gradient due to the simulated mountains are shown in Fig. 2.3. This plot can be used to estimate 51 when terrain e ects may be neglected by: 1. Identifying the largest terrain object(s) from an elevation map and estimating the peak, hT(0), and standard deviation, T, of the object. 2. Normalizing hT(0) and T by the predicted user?s ight altitude, h. 3. Locating the corresponding vertical gravitational gradient in Fig. 2.3 for the estimated hT(0)=h and T=h. 4. Then, the terrain e ects may be neglected for modeling purposes if the esti- mated DD is su ciently less than the GGI noise. It should be noted, that many mountains actually sit on large plateaus, which act as an altitude bias that has no e ect on the terrain contribution of the gradient signal. Indeed, it can be shown that an in nite uniform sheet of mass has zero vertical gravitational gradient. In this regard, when estimating a mountain?s characteristics, the peak should be referenced to the map?s minimum elevation; not necessarily mean sea level. Figure 2.3 also presents the interesting trend that the largest gravitational gradient occurs when T=h 1:29 for values of hT(0)=h < 10 2 and decreases to unity as hT(0) approaches h, as shown in Fig. 2.4. Therefore, low altitude surveys are more susceptible to more compact terrain variations, while higher applications are more sensitive to \wider mountains." The optimal T=h ratios which maximize DD are a function of hT(0)=h only; however, at high altitudes T grows large enough that these features would be accounted for in the spherical harmonic model. Thus, a constraint of 3 T ( a)=nmax 55 km for nmax =360, from the spherical harmonic resolution de nition of Eq. (2.9), can be set to limit the maximum standard deviation 52 Figure 2.4: Optimal T=h to Maximize Terrain DD parameter. Figure 2.5 uses the optimal T=h ratios and the T 18.6 km constraint to calculate the maximum DD for a variety of peak terrain and user altitude con g- urations. This gure allows one to not have to estimate T=h for a given mission and terrain elevation map. Therefore, using Fig. 2.5 one can estimate the terrain contribution to the vertical gravitational gradient for various missions given only the peak terrain and user altitude. Then, if the user?s GGI noise level is su ciently above the estimate in the gure, the terrain e ects may be neglected. For example, a commercial aircraft cruising at a 10 km altitude using a current- grade airborne GGI with an 5 E o noise level would be a ected by mountains approximately 100 m and taller. And a satellite in a 300 km altitude orbit with a space-grade 0.01 E o GGI noise level would only be a ected by terrain e ects 53 Terrain Peak, m User Altitude, m 0.1 mEo 1 mEo 0.01 Eo 0.1 Eo 1 Eo 10 Eo 100 Eo 1 1e1 1e2 1e3 1e4 1e2 1e3 1e4 1e5 1e6 h T (0) > h ? T = 18.6 km Figure 2.5: Terrain Contribution to Maximum DD for Various Peak Terrain and User Altitude Conditions 54 greater than about 500 m tall. The hypersonic cases simulated in this work are at an 24 km altitude and have GGI noise levels 0.1 E o. Therefore, according to Fig. 2.5, terrain e ects of 10 m and less should produce gradients above the noise GGI noise oor and should thus be included in future work. For reference purposes, a 100 m altitude corresponds to an 30 story building, the largest mountain in the contiguous United States is Mount Whitney at 4.4 km, and the largest in the world is Mount Everest at 8.8 km from sea level. 2.2.3.1 Normalization of Terrain DD Computation The proof of non-dimensionalizing the Gaussian distribution parameters by the user altitude will be shown in parts. The dependence of hT(0)=h and T=h on the horizontal distance (s), terrain elevation (hT), rh, r0, and T33;i components will be derived in succession. Then these components will be substituted into Eq. (2.17) to show that, as posed, DD is indeed a function of only the two normalized Gaussian distribution design parameters. The horizontal distance array is assumed to be equally-spaced from 0 to six times the standard deviation parameter: s = 0 : 6 T2000 : 6 T : (2.18) This can be rewritten in terms of T=h by factoring out the user altitude: s = h 0 : 62000 T h : 6 T h hfs T h ; (2.19) where the second equality denotes that s is equal to a function of only T=h mul- tiplied by the user altitude, h. Similarly, the increment of the horizontal distance 55 array is: s = 6 T2000 = h 62000 T h hf s T h : (2.20) The Gaussian terrain elevation is, from Eq. (2.11): hT(s) = hT(0) exp " 12 s T 2# ; (2.21) where the amplitude aT has been replaced by Eq. (2.16). Then, substituting the horizontal distance, Eq. (2.18), into the terrain elevation equation results in hT(s) = hT(0) exp 12 0 : 62000 : 6 2! : (2.22) The exponential portion is now shown to be independent of the standard deviation parameter because of the choice of the assumed horizontal distance array. And the terrain elevation can be rewritten as: hT(s) = h h T(0) h exp 12 0 : 62000 : 6 2! hfhT h T(0) h : (2.23) The distances r0 and rh can then be derived as functions of only hT(0)=h and T=h multiplied by the user altitude. Starting with Eq. (2.15), and substituting in Eq. (2.19), r0 =ps2 +h2 = h p f2s ( T=h) + 1 hfr0 ( T=h) (2.24) And substituting Eq. (2.19) and (2.23) into (2.14) yields rh = p s2 + (h hT)2 = q h2f2s( T=h) + [h hfhT(hT(0)=h)]2 = h q f2s( T=h) + [1 fhT(hT(0)=h)]2 hfrh h T(0) h ; T h : (2.25) 56 Then these terms can be substituted into the calculations of T33;i. For the singularity case of s = 0, T33;0 = 1(h h T) 2 1 h2 = 1 h2 1 [1 fhT (hT(0)=h)]2 1 1h2fT33;0 h T(0) h : (2.26) When s6= 0, there are two main components to T33;i: (h hT)=rh and h=r0. The rst component can be rearranged as h hT rh = h(1 hT=h) rh = h(1 fhT (hT(0)=h)) hfrh (hT(0)=h; T=h) fT33;i1 h T(0) h ; T h : (2.27) The second component is h r0 = h hfr0 ( T=h) fT33;i2 T h : (2.28) Then, from Eq. (2.13), T33 for s6= 0 can be rewritten as T33;i = 1h2f2 s( T=h) fT33;i1 f3T33;i 1 fT33;i2 +f3T33;i 2 1h2fT33;i h T(0) h ; T h : (2.29) Finally, the vertical gradient can be found from Eq. (2.17) using the change of integration variable from s to s=h and substitution of the relations above: DD = G T 6 T=hX (s=h)i=0 (T33;i+1 +T33;i) s h i s h h2 + (2T33;i+1 +T33;i) 13 s h 2 h2 = G T 6 T=hX (s=h)i=0 h2 1 h2fT33;i+1 + 1 h2fT33;i fsf s +h2 2 1h2fT33;i+1 + 1h2fT33;i f2 s 3 G T 6 T=hX (s=h)i=0 f DD h T(0) h ; T h : (2.30) 57 The altitude dependence in the T33 calculations are thus e ectively canceled by the choice of de ning s as a function of T, which caused DD to be independent of h. 2.2.4 Gravitational Gradient Biases Nearby masses, such as the vehicle?s structure, fuel, and payload, must also be accounted for in the INS/GGI lter since they can produce non-negligible gravita- tional gradients. Many of these vehicle masses produce essentially constant gravita- tional gradient biases since they consist of constant masses that are positioned at a constant distance from the instrument. However, other self-generated gradients are time-varying such as fuel consumption and slosh, control surface ( n) de ections, and passenger movement in the case of a commercial aircraft system. To use the proposed INS/GGI navigation system e ectively, the onboard lter must estimate and compensate each one of these additional biases accurately so that the external gravitational eld may be used for position updates. In order to estimate the gravitational gradient contribution from a variety of bias sources, the generating bodies are modeled as simple point masses. Then, using Eq. (2.10) on pg. 48: @2 g @r2 = 2GM r3 ; the gravitational gradient contribution from a point mass is a function of only the bias?s mass, M, and its distance from the user, r. Figure 2.6 plots the magnitude of the gradient for masses from 0.1 kg to 1 metric ton (1,000 kg) at a distance of 10 cm to 100 m. Each mass has a slope of 3 on the log-log axes because of the 58 Figure 2.6: Point Mass Gravitational Gradient Contribution inverse-cube relation of the point mass gradient to the displacement distance. For example, a 10 kg mass produces a 1 E o gravitational gradient at 1 m and a 0.001 E o gradient at 10 m. Furthermore, a 0.1 kg mass 1 m from a GGI is equivalent to a 100 kg mass at a 10 m distance. Also, as the displacement is increased 2.154 (i.e. 3p10 ), the gravitational gradient bias decreases an order of magnitude. And lastly, for a given distance, the gravitational gradient is directly proportional to the bias source?s mass, so that a larger mass produces a linearly proportionate larger gradient bias. As another example, if a 100 kg ( 220 lb) person is 2.5 m from a GGI, they would produce 1 E o gravitational gradient. Then, as the person walks away from the instrument, their gravitational gradient measurement would be 0.1 E o at about 5.4 m, 0.01 E o at 12 m, and 0.001 E o at 25 m. Then, in terms of GGI noise levels, a 59 1 E o sensor would have to account for all 100 kg and larger masses 2.5 m and closer, and a 0.001 E o sensor would have to compensate for all masses 100 kg within 25 m of the GGI. The same 0.001 E o GGI would also have to account for all 10 kg masses within 12 m, and 1 kg masses within 5.4 m of the instrument. Therefore, for a given GGI noise level, all self-generated vehicle biases should be taken into account, and ideally the largest time-varying biases should be placed farthest away from the GGI so that their errors are inherently reduced. 2.3 Gravitational Gradient Characterization In this section the regional and altitude e ects of the gravitational gradients are investigated on a global scale. The following gures plot components of the gradient tensor using the full 360 degree and order EGM96 spherical harmonic model and the modi ed geopot97 code at given altitudes. As discussed, the EGM96?s nite resolution aliases high frequency terrain e ects so that the true gravitational gradients at low altitudes are most likely larger than those presented. Nevertheless, the spherical harmonic model allows for identi cation of global areas of interest and trends. Referring to Fig. 2.7 (c), the gravitational gradients are intuitively highest and vary most rapidly in mountainous ranges, as seen over the Rockies, coast of South America, and Himalayas. More surprisingly, areas in the Paci c Ocean also produce noticeable gradients, particularly around Indonesia and west of Japan. Therefore, it can be seen that unlike other map matching aids (e.g. terrain or vision), gravity 60 (a) (b) (c) Figure 2.7: Inline Gravitational Gradients at Surface (a) NN (b) EE (c) DD 61 gradient aiding can be applicable over bodies of water. Figures 2.7 and 2.8 plot all six gravitational gradients at the Earth?s surface. The rst set of gures correspond to three inline gradients, and the second set are the three o -diagonal gradients. The inline gradients are plotted with a colorbar varying from 20 E o from the mean, and the o -diagonal gradients are plotted with a 10 E o variation to compare how much the gradient signal varies. Comparing the inline gradients, it is apparent that the vertical gravitational gradient varies more noticeably than either NN or EE. This result is due to the coupling of the three inline gradients by Laplace?s equation, Eq. (2.5) on pg. 43, so that DD = ( NN + EE): (2.31) The regions where the gradients are the largest and vary the most is, however, relatively independent of the gradient. This is also true for the three o -diagonal gradients. Comparing the o -diagonal components in Fig. 2.8, it is apparent that the North-East gravitational gradient varies the least while the North-Down and East- Down components have about the same variation. The reason for this is that the ND and ED gradients are the spatial derivatives of the vertical gravitational accel- eration (gD). The NE component, however, is the horizontal derivative of the other horizontal gravitational acceleration, which is typically orders of magnitude smaller than gD. The North-Down component also has a prominent low-frequency variation in the latitudinal (North/South) direction due to the bulk oblateness of the Earth. 62 (a) (b) (c) Figure 2.8: O -Diagonal Gradients at Surface (a) NE (b) ND (c) ED 63 Figure 2.9: East-Down Gravitational Gradient at Three Altitudes To qualitatively show the e ects of altitude attenuation on the high frequency gravitational gradient signals, Fig. 2.9 plots the East-Down gradient at three al- titudes focusing on North and South America. Comparing the surface and 10 km plots, it is apparent that many of the high frequency components of the gravitational signal have been removed as the altitude was increased. This is particularly notice- able in moderate areas like Canada and Brazil. As the altitude is increased from 10 km to 100 km, almost all distinguishable gravitational gradient variations are removed. At this altitude, only the largest mountain ranges produce subtle changes in the gravitational gradient. Thus, increasing one?s altitude acts to smooth the gravitational gradient signal and thereby reduce its usefulness for map-matching. This trend is common for all six gradients. (Appendix A includes additional plots of the global gravitational gradients at various altitudes.) 64 Figure 2.10: DD Standard Deviation, log10(E o), at Surface To further investigate the regional and altitude e ects on the gradiometer aided INS system, standard deviations were computed over a horizontal 220 km 220 km moving window (corresponding to 5 5 n grid points). This analysis aids in quantifying the expected gravitational gradient signal variations over a region of interest and helps to quantify the altitude trends. Figure 2.10 logarithmically plots the standard deviation of the vertical gradient at the Earth?s surface, which was calculated from Fig. 2.7 (c). Similar standard deviation plots were computed for all six gradients at several altitudes and are shown in Appendix A. The geographical regions of high and low gradient variation are essentially the same for each n com- ponent, but the magnitude of the standard deviations are functions of the gradient components and altitude. Figure 2.11 summarizes the global minimum, mean, and maximum gravitational gradient standard deviations for the computed components as a function of altitude. The EE and NN standard deviations are approximately equal, so the East-East component is omitted in the gure. 65 0 1e1 1e2 1e3 1e4 1e5 5e5 10 ?5 10 ?4 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 Altitude, m Min., Mean, & Max. ?? ? DD ? ND ? ED ? NN ? NE ? EE ~ ? NN LEO Satellites Commercial Aircraft UAVs Airborne Gravity Surveys Figure 2.11: Minimum, Mean, and Maximum Gravitational Gradient Standard De- viation vs. Altitude 66 Referring to Fig. 2.11, the vertical gradient ( DD) has the largest variation as expected because it is the largest component of the gradient tensor. The spa- tial derivatives of the vertical gravitational acceleration (i.e. @gD=@N = ND and @gD=@E = ED) have the next largest variation due to gD being the largest com- ponent of the gravitational vector. The other inline gradients, NN and EE, have variations that are about half those of DD due to the coupling of these gradients due to Laplace?s constraint, as shown in Eq. (2.5) and (2.31). Lastly, the NE com- ponent has the least variation because it is the longitudinal derivative of gN, which is itself quite small. The change in the ND trend at the highest altitudes is caused by Earth?s oblateness which yields comparably large ND variations at the equator and poles, See Fig. A.10 (b) on pg. 299. It should be noted that the gradient variations at lower altitudes in Fig. 2.11 are rather conservative because of the aliasing of terrain e ects. However, even with this omission, the gradient signal variations are on the same order of magnitude as the airborne GGI noises in Table 1.1. The gradient signal variation is an important part of the navigation performance because it is what allows the Kalman lter to make delta corrections to the observable system states. As an analogy to terrain based systems, if the user is standing in a at region, it is nearly impossible to gain any orientation or position information. Conversely, if the user is in a hilly terrain, one can estimate where they are in relation to their local elevation map. The same general concept holds for gradient navigation|that performance is proportional to signal variation. However, the notion of preferring high gravitational variations is in direct contrast with typical inertial navigation because gravity variations are usually 67 seen as sources of error to the INS. Therefore, if the signal-to-noise ratio were to increase by decreasing the GGI noise, it appears that one could theoretically achieve an improvement in navigation performance using a GGI-aided INS for airborne applications, especially over regions with a strong gravitational gradient (high variations). The same can be said for Low-Earth-Orbit (LEO) satellites. The current primary issues limiting an INS/GGI system are: 1. The lack of high resolution, accurate gravitational maps over many regions. 2. The prohibitive size, weight, and noise oor of current GGIs. 3. The absence of open-literature algorithms to optimally blend GGI measure- ments into an inertial navigation system. This dissertation makes the assumptions that the rst two issues are solved at some point in the future and focuses on the algorithm development and quanti cation of the potential performance of future INS/GGI systems. 2.4 Simulation Trajectories In order to investigate the sensitivity of the gravitational gradient signal vari- ation on the INS/GGI navigation performance, two 1000 km constant latitude tra- jectories in the contiguous United States were chosen. The \High" gravity gradient variation trajectory follows the Northern border of Wyoming, and the \Low" gradi- ent trajectory follows near the Northern border of Kansas into Missouri and Illinois. Figure 2.12 plots these two trajectories over the logarithmic standard deviation plot 68 Table 2.1: Simulated Trajectories Latitude Longitude High Gradient Variation 45.0 [ 113:0 : 100:3 ] Low Gradient Variation 38.0 [ 100:0 : 88:6 ] Figure 2.12: Simulated Trajectories of the vertical gradient at a 10 km altitude. Figure 2.13 plots the gravity accelera- tion and the gravitational gradients for the two trajectories at the Mach 7 altitude. The constant latitude and the longitude ranges are summarized in Table 2.1. The 1000 km range was chosen as it is approximately the maximum range that a cruise missile can travel without violating arms treaties. The choice of a constant latitude, Eastern cruise was so that the longitude rate is constant for a given velocity, see Eq. (4.51) on pg. 152. The two subsonic cases were simulated to 69 0 2004006008001000 ?0.034 ?0.033 ?0.032 ?0.031 g N Gravity Vector, m/s 2 0 2004006008001000 ?5 0 5 x 10 ?4 g E 0 2004006008001000 9.725 9.73 9.735 g D Downrange, km 0 2004006008001000 ?1535 ?1530 ?1525 ?1520 ?1515 ?1510 ? NN Inline Gradients, Eo 0 2004006008001000 ?1535 ?1530 ?1525 ?1520 ?1515 ?1510 ? EE 0 2004006008001000 3040 3045 3050 3055 3060 3065 ? DD Downrange, km 0 2004006008001000 ?15 ?10 ?5 0 5 10 ? NE Cross Gradients, Eo 0 2004006008001000 ?25 ?20 ?15 ?10 ?5 0 ? ND 0 2004006008001000 ?15 ?10 ?5 0 5 10 ? ED Downrange, km Figure 2.13: Simulated Gravity at Nominal Latitude and Altitude for Mach 7 Tra- jectories, High Variation (solid), Low Variation (dashed) 70 ranges such that the lter achieved steady state operation at about halfway into the simulation. The commercial aircraft range was set to 200 km, and the GGI-survey case range was 50 km. For the hypersonic scramjet simulations, the cruise altitude was calculated so that the vehicle ew at a constant one-atmosphere dynamic pressure according to the 1976 standard atmosphere model.126 The velocities and altitudes were calculated as follows. Given a freestream Mach number, the cruise velocity is v1 = M1a1 = M1p 1R1T1; (2.32) where a1 is the freestream speed of sound at altitude, 1 is the ratio of speci c heats and is assumed to be 1.4, R1 = 287 J/kg-K is the gas constant for air, and the atmospheric temperature causes the altitude dependence. Then, using the de nition of dynamic pressure, the velocity can also be found as q1 12 1(h)v21 ! v1 = s 2q1 1(h): (2.33) These two equations can be set equal to each other and squared to yield 2q1 = M21 1( 1R1T1) = M21 1P1(h); (2.34) where the ideal gas equation has been used for the second equality. This relation can now be solved for pressure as a function of the prescribed dynamic pressure and Mach number: P1(h) = 2q1 1M21 : (2.35) A bisection method is used to numerically calculate the altitude so that the pres- sure as de ned by the 1976 standard atmosphere model is equal to the given Mach 71 Table 2.2: Simulated Cruise Values Mach 6 Mach 7 Mach 8 747-100 GGI Survey Altitude, m 22,043.8 24,040.2 25,785.3 10,000.0 100.0 Velocity, m/s 1,778.43 2,084.22 2,391.29 250.0 40.0 Range, km 1,000.0 1,000.0 1,000.0 500.0 300.0 Final Time, sec 565.20 482.45 420.60 802.60 1,252.15 number and the assumed dynamic pressure constraint. Equation (2.32) is then used to calculate the East velocity with the at-altitude freestream temperature from the standard atmosphere calculation. Table 2.2 summarizes the cruise altitudes, East- ern velocities, range, and total simulation time for the three cruise Mach numbers simulated and the two subsonic cases. For the two subsonic cases, the velocities and altitudes were chosen to ap- proximate a commercial aircraft and a GGI-based survey/exploration mission. The commercial aircraft mission is assumed to be a Boeing 747-100 class vehicle which cruises at approximately 250 m/s (M1 0.84) at a 10,000 m ( 33,000 ft) alti- tude.127 The GGI survey mission assumes a 100 m altitude based on the Falcon AGG speci cation for their average xed wing system.y The velocity is estimated to be 40 m/s, which is slightly higher than the stall speed of a Cessna Grand Caravan (61 knots 31.4 m/s)128 since this is the platform for most airborne GGI exploration systems, and reduced speed improves the spatial resolution of the survey. The details of the gravity eld maps that were computed and stored for the INS simulations are summarized in Sec. 2.4.2. But rst, the methodology to determine yhttp://falcon.bhpbilliton.com/falcon/speci cations.asp, cited 6 Nov. 2007 72 the map resolution is presented in the following section. 2.4.1 Gravity Map Resolution 2.4.1.1 Horizontal Resolution For a given altitude, the gravitational potential and horizontal (latitudal and longitudal) derivatives are linear combinations of sine and cosine waves, see Eq. (2.7). To determine the chosen resolution on the gravity map in the horizontal directions, and to estimate the error due to the linear interpolation between grid points, a MAT- LAB script was written that calculated the residual error between a ne, \truth," cosine wave and the linearly interpolated, \estimated," cosine wave for various res- olutions: ehoriz =jcos(sT) \cos(sT)j; (2.36) where the truth resolution is sT = 0 : 100N horiz : : (2.37) The number of node points for the coarse cosine wave was de ned by Nhoriz = ae =nmaxhoriz map res ; (2.38) where the numerator is the spherical harmonic resolution and the denominator is the variable gravity map resolution. The bounds of [0 : ] for the independent cosine wave variable, s, were chosen because it is half a period and thus corresponds to the highest half-wavelength of the spherical harmonic resolution as de ned in Eq. (2.9) 73 Figure 2.14: Horizontal Spherical Harmonic Error Due to Linear Interpolation, with 500 m Resolution on pg. 46. The coarse cosine wave was created with a resolution of Nhoriz, or s = 0 : N horiz : : (2.39) The estimated cosine wave was then linearly interpolated to each of the truth nodes by \cos(sT) = cos(s)i + cos(s) i+1 cos(s)i si+1 si (sT si); (2.40) and the error residual between the coarse and ne cosine waves was computed by Eq. (2.36). Figure 2.14 plots the horizontal residuals for the case of horizmap res = 500 m, other resolutions exhibit similar trends but varying magnitudes and periods. For all cases, the maximum error occurs near the 0 and locations of cos(s) because the slopes are most nonlinear at these points. 74 Figure 2.15: Horizontal Spherical Harmonic Error Due to Linear Interpolation vs. Map Resolution The resolution of the estimated cosine curve, horizmap res, was varied from 10 m to 1 km and the maximum residual for each case was stored and plotted as Fig. 2.15. As seen is this gure, as the map spacing decreases, the error due to linear interpolation decreases as one might expect. Also, it is apparent that as the map resolution continues to become more and more ne, there is diminishing returns in terms of the error residual. Therefore a compromise between acceptable error and gravity map storage size must be made (as the horizontal resolution is decreased by half, the storage requirements increase by a factor of four because of the two horizontal dimensions). For this work an error limit of 0.01% was chosen, which leads to a horizontal spacing of 500 m and an linear interpolation error of 0.00983%. 75 2.4.1.2 Vertical Resolution The vertical resolution was determined by investigating the nominal magni- tude of the vertical gravitational acceleration, gradient, and third order derivative as functions of altitude. Each of these functions were also normalized by the nom- inal altitude for the Mach 6 cruise trajectory (hnom = 22,043.8 m) because the gravitational quantities are largest at lowest altitudes. (The subsonic cases are dis- cussed at the end of this section.) The assumed simplistic normalized gravitational acceleration is gD(h) gD(hnom) " g0 a e ae +h 2# = " g0 a e ae +hnom 2# = a e +hnom ae +h 2 ; (2.41) where ae = 6;378;137 m for Earth?s semimajor axis.113,114 The normalized vertical gradient is DD(h) DD(hnom) 2g 0a2e (ae +h)3 = 2g 0a2e (ae +hnom)3 = a e +hnom ae +h 3 : (2.42) And the normalized gradient derivative is @ DD @h 6g 0a2e (ae +h)4 = 6g 0a2e (ae +hnom)4 = a e +hnom ae +h 4 : (2.43) The altitudes investigated were 1200 m below the nominal altitude to 1200 m above, which corresponds to approximately the nal 1- lter altitude error for the Mach 6 cruise simulation with a tactical grade IMU and no external measurements. Again, coarse and ne normalized gravitational parameters were computed using the relations above and the linear interpolation residuals were calculated. Figure 2.16 plots the residuals for the three gravitational quantities with a vertical 76 Figure 2.16: Vertical Spherical Harmonic Error Due to Linear Interpolation, with 160 m Resolution resolution of 160 m (Nvert = 16). The third derivative of the gravitational potential, @ DD=@h, produces the largest linear interpolation errors because it has the greatest exponential compared to DD and gD. Instead of varying the vertical resolution, as done above with the horizontal resolution, the number of node points was varied. Nvert was investigated from 11 to 20, corresponding to a vertical resolution of 218 m to 120 m. The maximum residuals are summarized in Fig. 2.17 along with the 0.01% error constraint. From Fig. 2.17, the vertical resolution is set by the third derivative of the gravitational potential at a value of 160 m, or Nvert = 16, and the maximum linear interpolation error is 0.00991%. For the commercial aircraft case with the assumed 10,000 m altitude, a 160 m grid resolution causes a linear interpolation error of 0.00992%. For this case, only the best INS/GGI system was simulated, which has position errors on the order of 77 Figure 2.17: Vertical Spherical Harmonic Error Due to Linear Interpolation vs. Map Resolution a meter. Therefore, only 3 altitude grid points were used for the map: one at the nominal altitude, and one each plus/minus the 160 m vertical grid resolution. The gravity gradiometer instrument survey mission altitude is 100 m, so this was set as the grid resolution for this gravity eld map since negative altitudes would not be physical. With the survey altitude of 100 m and the 100 m resolution, the linear interpolation error was calculated to be 0.00621%. Again, only 3 grid points were used for this case since only the best INS/GGI system was simulated and the position errors (< 1 m) are much less than the grid resolution. 78 Table 2.3: Gravitational Gradient Map Parameters High Gravitational Gradient Variation Trajectory M1 Latitude, N Longitude, N Altitude, m Nh 6 44.982 { 45.018 9 -113.0 { -100.2705 2005 20843.8 { 23243.8 16 7 44.9865 { 45.0135 7 -113.0 { -100.2768 2004 23080.2 { 25000.2 13 8 44.991 { 45.009 5 -113.0 { -100.2832 2003 25065.3 { 26505.3 10 Low Gravitational Gradient Variation Trajectory M1 Latitude, N Longitude, N Altitude, m Nh 6 37.982 { 38.018 9 -100.0 { -88.5774 2005 20843.8 { 23243.8 16 7 37.9865 { 38.0135 7 -100.0 { -88.5831 2004 23080.2 { 25000.2 13 8 37.991 { 38.009 5 -100.0 { -88.5888 2003 25065.3 { 26505.3 10 2.4.2 Simulated Gravity Field Maps Table 2.3 summarizes the inputs to the modi ed geopot97 code. The varia- tion in the nominal latitude, nal latitude, and nominal altitude parameters were computed from the nal 1- lter position errors of these states using a tactical grade INS with no external updates. Because faster simulations traveled the 1,000 km range in a shorter period of time, their dead-reckoning position error has less time to grow and therefore the position errors are less. This also results in smaller storage requirements for the faster simulations because the nite spatial resolutions were held constant. The latitude resolutions were converted from position to degrees by [ ] = [m]a e 180 [rad] : (2.44) 79 Table 2.4: Gravitational Gradient Map Storage Requirements M1 6 7 8 Ngrid 288,720 182,364 100,150 File Size, KB 70,207 44,345 24,354 The longitude resolutions were converted by [ ] = [m]a e cos nom 180 [rad] : (2.45) The storage requirements for the tabulated ; ;h; g;gn; and n components using eight signi cant gures is summarized in Table 2.4, where Ngrid = N + N + Nh. The les were saved as ASCII text, and the storage requirements could have been reduced if it were saved as a binary le. The subsonic cases were simulated over only the high n variation trajectory. Their horizontal grid parameters were kept the same as the Mach 8 case and the altitude nodes were de ned as described at the end of the last section. 2.5 Chapter Summary This chapter presents a review of gravity gradiometry, spherical harmonics, and local terrain integration methods for the purpose of gravity eld modeling in Sec. 2.1{2.2.2. Section 2.2.3 then estimates when the high frequency local terrain e ects may be neglected compared to a given GGI noise level. Using a parametric model for the analysis, it is shown that the gravitational gradient contribution of the mountain can 80 be computed by only two parameters (width and height of the terrain feature) since the user altitude can be used to normalize these values, as shown in Sec. 2.2.3.1. And it is found that wider mountains have a larger e ect on higher altitude missions, whereas narrower features with the same peak height produce larger gradients for lower altitude missions. Also, the width parameter is then optimized to maximize the gravitational gradient so that the vertical gradient contribution from a terrain feature can be estimated by only the user altitude and the peak height of the feature, as shown in Fig. 2.5. If the estimated gradient is su ciently less than the GGI noise level, then the terrain e ects may be neglected from the computed gravitational map. Section 2.2.4 next estimates the gravitational gradients due to a variety of point masses at various distances from a GGI. It is shown in Fig. 2.6 that when the GGI noise level is reduced an order of magnitude, the GGI is able to measure a mass 1/10 the original mass at the original distance or a mass the same size as the original mass but at a distance 2.154 times the original distance. Therefore, for the GGIs simulated later in Ch. 6 ( L = 0.1{0.001 E o), almost all vehicle mass distributions would need to be accounted for in a real GGI navigation system. However, for the simulations preformed in this work, these additional vehicle-generated gradient contributions are ignored because of the low delity mass model in Sec. 3.2. Also, it is noted that stationary mass distributions produce a constant gravitational gradient bias that can be calibrated by the INS/GGI, and only moving masses need to be estimated and corrected for by the onboard lter. For this latter case, it is suggested to maximize the separation distance between the GGI and the moving mass so that 81 its measured gradient is reduced. The gravitational gradient characterization studies in Sec. 2.3 serve to answer the four questions posed on pg. 37: First, it is shown that the gravitational gradient signal variation is on the same order of magnitude as current airborne and space-borne GGI noise lev- els, i.e. 1{10 and 0.001-0.01 E o, respectively. The signal-to-noise ratio for lower altitude airborne applications is quite conservative, however, because the high frequency uctuations from local terrain e ects are not modeled. Therefore, there is enough information that an airborne gravity gradiometer map-matching aid should be able to provide position information to an INS. The order in which the gravitational gradient tensor components vary, from greatest to least, is DD, ND and ED, NN and EE, and NE; as shown in Fig. 2.11 on pg. 66. In the global gravitational gradient plots it is shown that n varies most in mountainous regions, as one might expect. However, it is also shown that there are noticeable areas of high gradient variation over certain bodies of water. Lastly, Fig. 2.11 shows that altitude does not attenuate the magnitude of the gradient variation below about 1,000 m. But, the low altitude trends shown in the gure would likely change if local terrain e ects were included in the analysis. Section 2.4 lastly discusses the two chosen trajectories over the USA (\high" and \low" gradient variations) for the simulations performed in Ch. 6. The calcula- tion of the three hypersonic cruise vehicles? altitudes and velocities are also shown 82 in this section, as are the assumptions for the two subsonic simulations (see Table 2.2). Section 2.4.1 then details the gravity map resolution studies that determined the 500 m horizontal and 160 m vertical grid spacing so that the linear interpolation error is less than 0.01%. Lastly, Sec. 2.4.2 summarizes the stored gravity eld maps that are needed for the computation of the hypersonic missile trim states in the following chapter, the gravity vector and the gravitational gradients for the INS in Ch. 4, and the gradient portion of the simulated GGI measurements in Ch. 5. 83 Chapter 3 Hypersonic Vehicle Model The hypersonic vehicle modeled in this work is based primarily on a 1982 Advisory Group for Aeronautical Research and Development (AGARD) report by the Johns Hopkins University / Applied Physics Lab (JHU/APL) which presents a surprisingly candid, in-depth design methodology to model the propulsion system and aerodynamics of parametric scramjet missiles.129 This dissertation digitized and curve- t many of the results of this reference so that they may be implemented as a parametric rst-order hypersonic missile design tool. Several extensions to the original report are also derived and implemented to produce realistic truth inertial measurement unit signals. First, volume calculations and a parametric mass model are added. Then, trim state relations are newly derived for numerical computation of the missile?s trim pitch, roll, and equivalence ratio at a point in time. The trim state calculations are used to numerically integrate the missile?s mass properties over a 1,000 km (540 nautical mile) range simulation as fuel is being burned and the vehicle accounts for variations in the acceleration due to gravity. The pitch and roll trim states are lastly nite di erenced so that they may be used as part of the gyro signals in the inertial navigation system simulation. The following section describes the methodology to calculate the propulsion 84 and aerodynamic characteristics of the axisymmetric scramjet missile as set forth in the cited reference.129 Appendix B supplements this section with numerous thrust coe cient curve ts. The last part of Sec. 3.1 then validates the implemented Fortran code with the reference?s example calculations. Section 3.2 explains the parametric mass model used for the analysis and the calculation of the internal volume of the missile. This section also details some of the additional assumptions pertaining to the dimensions of the vehicle. Section 3.3 next derives the trim state relations and presents how these results are used to simulate the INS truth gyro and accelerometer signals, and the last section summarizes the chapter results. 3.1 JHU/APL Axisymmetric Scramjet Model The JHU/APL AGARD report129 presents a design study for a hypersonic scramjet missile that is boosted to Mach 4 on a rst stage rocket and then accelerates to a Mach 8 cruise. The missile is assumed to be launched from a volume constrained box, so that the total length is a xed 4.0 m and the diameter is a xed 0.50 m. The inlet may be either a full axisymmetric chin inlet (as assumed in this work) an aft inlet, or a sector of one of these two inlets with an on-design Mach number, Mdes, of 6, 7, or 8. In this work, the scramjet is assumed to cruise at its on-design Mach number at one of the three values in the report, i.e. M1 = Mdes = 6, 7, or 8. From analyzing the many tradeo s in the inlet designs, the reference129 con- cludes that the full axisymmetric chin inlet allows for maximum capture area and engine thrust. Therefore, this type of vehicle inlet is chosen for the scramjet models 85 in this dissertation. The e ect of on-design inlet Mach number was not thoroughly investigated here because of the assumption that the cruise M1 = Mdes. Also, for the simulations in this dissertation, the missiles are assumed to y at their constant on-design Mach number so that the o -design M1 e ects could have been neglected. However, for design purposes, the report?s assumed operating range of Mach 4{8 and an angle of attack from 0{10 are maintained for calculating the maximum geometric contraction ratio in Sec. 3.1.1.2. Other design choices in the implemented scramjet model are detailed in their respective sections. The results in the JHU/APL AGARD reference129 were rst scanned from a micro che-sourced hard copy of the report and stored as image les. Engauge Dig- itizer 2.14,y a free software package, was then used to automatically and manually identify feature points on the graphical results of the image les. The software al- lows the user to de ne arbitrary axes on the image le so that rotation and scaling of the feature points are computed internally in the program. The data sets were then imported into MATLAB and curve t using the polyfit.m function and an assumption of either quadratic polynomials or linear segments. The coe cients pro- duced from the MATLAB curve ts are listed in the appropriate discussions below and in App. B for the thrust coe cient calculations. The interpolation between various curve ts is also discussed in the following subsections. The rst subsection describes the calculation of the propulsion system and the assumptions used in the reference report?s analyses. The next subsection details the aerodynamic model and the multitude of drag terms included. The last subsection yhttp://digitizer.sourceforge.net/ 86 validates the implementation of these two subsections as a computer code with example calculations in the appendix of the reference report. 3.1.1 Propulsion According to the 1982 JHU/APL report,129 \the results presented and assump- tions used . . . are based on experimental data and analytical techniques developed from testing and analyzing scramjet engines over the past 20 years." And while this reference is somewhat dated when taking into account the developments in scram- jet technologies over the past 25 years,130{132 it is arguably the most in-depth open literature reference regarding the full design of a hypersonic scramjet. And the in- clusion of the o -design propulsion and aerodynamics (which is absent from most references) allows for calculation of the trim states. The reference uses the integral form of the mass, momentum, energy, and species conservation equations at several discrete thermodynamic stations to com- pute the thrust coe cient of the hypersonic scramjet.129 The rst station is the freestream, \1," ow station that is una ected by the hypersonic vehicle. The properties at this station are uniquely de ned by the velocity of the vehicle and an assumption of the atmosphere at altitude. The reference states that all calculations were performed at a 15,240 m (50,000 ft) altitude for simplicity, and that variations in the altitude would have had only a minor e ect on the overall thrust coe cient. For this work, the 1976 standard atmosphere is modeled to calculate the freestream pressure, density, and temperature for a given altitude.126 The next station, \0," 87 Figure 3.1: Scramjet Missile Geometry and Thermodynamic Stations consists of the properties just behind the initial cone forebody?s oblique shock. The area of this station, A0, is the capture area that is ingested into the engine and is at most equal to the geometric inlet area, Ai. The \1" station is at the entrance of the constant area isolator (or di user) and accounts for the total pressure losses from the inlet shocks. The \2" station is at the exit of the constant-area isolator (or equivalently, the combustor entrance) and compensates for the shock train in the isolator. The combustor is modeled between stations \2" and \4" as a constant expansion area section. And lastly, the nozzle is modeled from station \4" to \5," with its exit area, A5, serving as the reference area for all the propulsion and aero- dynamic force coe cients. Figure 3.1 illustrates the thermodynamic stations on a Mach 7-designed scramjet. The freestream and exit properties are then used to calculate the thrust coef- cient referenced to the capture area, A0, at a given freestream Mach number, angle of attack, and equivalence ratio by (CT)ref = 0:98F5 F1 P1(A5 A0)q 1A0 (3.1) where F5 is the nozzle stream thrust with an 98% e ciency, F1 is the freestream 88 stream thrust using the calculated capture area, P1 is the freestream static pressure, and q1 is the freestream dynamic pressure. The reference plots the thrust coe cient as a function of the inlet contraction ratio, (A0=A1), for the following discrete design parameter con gurations: Base-to-capture area ratio: (A5=A0) = 1, 2, 3, 4, 6, and 8. Combustor expansion ratio: (A4=A2) = 2, 3, and 4. Freestream Mach number: M1 = 3 to 8 in intervals of 1. Equivalence ratio: ER = 0.25, 0.50, 0.75, and 1.00 or the maximum ER to cause thermal choking. Mathematically, the thrust coe cient as shown in the reference appendices is T q1A0 ref = f A 0 A1; A5 A0; A4 A2;M1;ER : (3.2) There is also an implicit dependence on angle of attack in the thrust coe cient calculations because the inlet contraction ratio is calculated by A 0 A1 = A 0 Ai A i A1 max ; (3.3) where the mass capture area ratio, (A0=Ai), is a function of Mdes, M1, and as detailed in the following subsection. The maximum geometric contraction ratio, (Ai=A1)max is calculated from the assumed range of operating conditions of M1 = [4:8] and = [0:10 ] and is explained further in Sec. 3.1.1.2. The base-to-capture area ratio, (A5=A0), is also dependent on Mdes, M1, and because it is calculated by A 5 A0 = A 5 Ai = A 0 Ai ; (3.4) 89 and (A0=Ai) is dependent on these parameters. (A5=Ai) is a constant, user-de ned design parameter, which in this work is set to 1.10 to maximize the capture area and thus engine thrust. Therefore, for a given on-design inlet Mach number, Mdes, and a current M1 and , (A0=A1) and (A5=A0) are calculated by Eq. (3.3) and (3.4). Then, with the user-de ned combustor expansion ratio, (A4=A2), and a given equivalence ratio, the thrust coe cient may be computed. The details of how these calculations are implemented is explained in Sec. 3.1.1.3. Lastly, for the sake of completeness, the primary assumptions made in the reference?s thrust coe cient computations are:129 1. A constant area isolator, so that A2 = A1. 2. RJ-5 fuel with a 100% combustion e ciency. 3. A combustor wall-to-entrance area ratio, (Awall=A4), of 40. 4. Inlet and combustor ows are in thermochemical equilibrium. 5. Combustor wall heat transfer is neglected. 6. Nozzle exit thrust e ciency of 98% for an expansion one-third between the frozen and equilibrium chemistry solutions. 3.1.1.1 O -Design Mass Capture The reference plots the o -design mass capture as a function of angle of attack from 10 to 10 for the three on-design Mach numbers (Mdes = 6, 7, 8) and six freestream Mach numbers (M1 = 3{8 in intervals of one). Because the scramjets in this work are assumed to have a full axisymmetric chin inlet, the o -design capture 90 area is a function of only the magnitude of the angle of attack. So, when the digitized graph was curve t, the capture areas for the negative angles of attack were treated as if they were positive. The result is a quadratic function of angle of attack at a given Mdes and M1: A 0 Ai ref = 8 >>> >< >>>> : 1:0 if M1 Mdes & < min;Mdes;M1 aMdes;M1 2 +bMdes;M1 +cMdes;M1 else (3.5) where the coe cients for the quadratic are given in Table 3.1 along with the min values at the givenMdes andM1 nodes. The rst part of the if-statement in Eq. (3.5) is a constraint to ensure that the capture area ratio is never greater than 1, which may be calculated when the inlet is oversped (M1>Mdes) at low angles of attack. After the capture area has been calculated at the desired angle of attack, cubic interpolation is used to compute (A0=Ai) at the desired freestream Mach number using the four closest M1 nodes so that (A0=Ai) = f(M1; ). 3.1.1.2 Maximum Geometric Contraction Ratio The maximum geometric contraction ratio is computed by taking the low- est geometric contraction ratio over the assumed operating range so that the inlet operating conditions are not violated for any point in the design space. The max- imum contraction area ratio, (A0=A1)max, is computed in the reference using an empirical relation for the inlet kinetic energy e ciency and an approximation of the total pressure recovery of the cone inlet.129 The data points from the graphical representation of the maximum capture area are listed in Table 3.2. The geomet- 91 Table 3.1: Capture Area Curve-Fit Coe cients Mach 6 Inlet M1 a6;M1 b6;M1 c6;M1 min;6;M1 3 -0.50381096e-3 0.0 0.62345823 | 4 -0.92407015e-3 0.0 0.74196463 | 5 -0.13984166e-2 0.0 0.87788013 | 6 -0.10985493e-2 -0.10907126e-1 1.0184941 1.4761375 7 -0.35091049e-2 0.27913499e-1 0.94327056 3.9772961 8 -0.30688169e-2 0.24114687e-1 0.96142167 5.6218745 Mach 7 Inlet M1 a7;M1 b7;M1 c7;M1 min;7;M1 3 -0.33911231e-3 0.0 0.55158020 | 4 -0.81493479e-3 0.0 0.66104957 | 5 -0.11643156e-2 0.0 0.77025633 | 6 -0.16964511e-2 0.0 0.89003444 | 7 -0.14945940e-2 -0.99412493e-2 1.0146784 1.24389711 8 -0.41068464e-2 0.29071650e-1 0.94410395 3.5394128 Mach 8 Inlet M1 a8;M1 b8;M1 c8;M1 min;8;M1 3 -0.19959882e-3 0.d0 0.49430093 | 4 -0.67340679e-3 0.d0 0.58781782 | 5 -0.11137407e-2 0.d0 0.69188719 | 6 -0.14964941e-2 0.d0 0.79606842 | 7 -0.22808079e-2 0.d0 0.89916475 | 8 -0.19133374e-2 -0.60703864e-2 1.0059433 0.78489235 92 Table 3.2: Maximum Contraction Ratio M1 3 4 5 6 7 8 (A0=A1)max 3.15567 5.20698 7.15743 8.36812 8.50000 7.98145 ric contraction ratio, (Ai=A1), is then calculated over the operating design space of M1 = [4:8] and = [0:10 ] using cubic interpolation of the nodes of Table 3.2 for (A0=A1)max and the analysis in the previous subsection for (A0=Ai). The minimum value of the geometric contraction ratio over the entire operating regime is then used as the maximum allowable geometric contraction ratio. Mathematically,129 A i A1 max = min A 0 A1 max = A 0 Ai : (3.6) 3.1.1.3 Thrust Coe cient The thrust coe cient is then calculated after the contraction ratio, (A0=A1), and base area-to-capture area, (A5=A0), values are computed at the free stream Mach number and angle of attack of interest. The contraction ratio, (A0=A1), is calculated by multiplying the capture area ratio in Sec. 3.1.1.1 at a given M1 and with the maximum geometric contraction ratio in the previous subsection, as shown in Eq. (3.3) on pg. 89. And the base area-to-capture area ratio is computed by Eq. (3.4) again using the (A0=Ai) at the desired Mach number and angle of attack. The thrust coe cient (referenced to the capture area) is calculated as a func- tion of (A0=A1) by the polynomial curve ts in App. B. The reference plots results for combustor expansion ratios, (A4=A2), of 2, 3, or 4. However, only the combustor 93 expansion ratios of 3 and 4 were curve t in this work, since they are shown to prevent thermal choking over a much larger operating range than the (A4=A2) = 2 results. Also, this work assumes a value of (A4=A2) = 3 to allow for increased inter- nal vehicle volume over (A4=A2) = 4, and because the higher combustor expansion ratio has little e ect on the thrust levels. The thrust coe cient is calculated at each of the four closest base area-to- capture ratio (A5=A0), freestream Mach number, and equivalence ratio nodes for the given combustor expansion ratio, (A4=A2) = 3. After the thrust coe cients are calculated for the 64 nodes (4 (A5=A0), 4 M1, and 4 ER), these values are re-referenced to the base area by T q1A5 = T q1A0 ref = A 5 A0 node (3.7) and interpolated cubicly to the computed (A5=A0) at a given freestream Mach num- ber and angle of attack, i.e. (A5=A0) = f(M1; ). This process results in 16 thrust coe cients at 4 M1 and 4 ER nodes, which are then cubicly interpolated to the desired free stream Mach number and lastly the equivalence ratio. Thus, the thrust coe cient is CT T q1A5 = f (M1; ;ER): (3.8) 3.1.2 Aerodynamics The aerodynamics forces modeled in the reference129 are comprised of three major sources: the pro le drag coe cient, the inlet additive drag coe cient, and the normal force coe cient. Each component, and the pro le drag subcomponents, 94 will be presented in detail in the following subsections. Many of the missile dimen- sions will also be presented in the pro le drag subsection, and used again in the computation of the missile volume in Sec. 3.2.1. 3.1.2.1 Pro le Drag Coe cient The pro le drag is the drag present at all angles of attack and is sometimes referred to as the axial force. For this model, the pro le drag is the sum of the wave drag caused by the 6 cowl angle, the leading edge drag due to the 0.254 cm (0.1") bluntness of the cowl, the exterior body (skin) friction, and the wave and friction drag from the four tail surfaces. Each of the pro le drag components are detailed below along with any assumptions made by the reference source.129 Cowl Wave Drag The cowl wave drag is caused by the oblique shock from the 6 cowl angle assumed by the reference.129 The cowl wave drag was calculated using \a nite di erence solution of the hyperbolic equations of motion for a steady inviscid ow," (Waltrup et al.,129 pg. 8-5) which is the same technique used for the o -design capture area and inlet additive drag calculations. The cowl wave drag is normalized by the axially projected cowl area, Acx, and plotted in the report as a function of freestream Mach number for design Mach numbers from 4 to 8 in intervals of 1. The cowl wave drag exhibits two di erent trends, one when the inlet is undersped (M1 Mdes) which is a function of Mdes, 95 and one when the inlet is oversped (M1>Mdes) which is independent of Mdes. The reason for this behavior is that when the inlet is ying slower than the shock-on- cowl-lip inlet-design Mach number, the initial shock o the 12.5 conical forebody is outside the cowl and the cowl sees a ow eld dependent on this initial shock. When the inlet is oversped, the forebody shock is inside the cowl and the cowl wave drag is just a function of the freestream ow. For a Mach 6 designed inlet ying at a freestream Mach number lower than Mach 6, the curve- t cowl wave drag is C Dcw Acx 10 4 1 in2 ref = 0:020828418M21 0:28005453M1 + 2:3813260: (3.9) For an undersped Mach 7 inlet, the curve- t cowl wave drag is C Dcw Acx 10 4 1 in2 ref = 0:012006681M21 0:19139457M1 + 2:0817794: (3.10) And for the Mach 8 inlet ying below Mach 8, the curve- t cowl wave drag is C Dcw Acx 10 4 1 in2 ref = 0:011497814M21 0:18189467M1 + 1:9660494: (3.11) Lastly, for an oversped inlet with any Mdes, the curve- t cowl wave drag is C Dcw Acx 10 4 1 in2 ref = 0:024652242M21 0:44480915M1 + 3:2354515: (3.12) The reference?s presented cowl wave drag coe cient is then re-referenced to the base area in terms of metric units by: CDcw D cw q1A5 = C Dcw Acx 10 4 1 in2 ref (Acx) 10 4 in m 2 ; (3.13) where 1 inch equals 0.0254 meters, and Acx is the cowl area projected axially. 96 Figure 3.2: Scramjet Missile Front View The axially projected cowl area is simply the di erence between the base area and the inlet area, as shown in Fig. 3.2 of the front view of the modeled vehicle. From the given reference base area, A5 = r25 = 0:785m2; (3.14) (using the reference base radius r5 = 0.5 m) and the user de ned geometric design parameter, (A5=Ai), the axial cowl area is Acx = A5 Ai = A5 1 (A5=Ai) 1 : (3.15) For this work, (A5=Ai) is assumed to be 1.1 so that the inlet is able to capture nearly all of the incoming ow. Therefore, Acx = 0.0714 m2 for the scramjet designs in this dissertation. 97 Cowl Leading Edge Drag The cowl leading edge drag is caused by the bow shock that is formed from the reference?s assumed cowl lip diameter of 0.254 cm (0.1").129 The way in which the reference calculates this quantity is not reported, but it is graphically represented as a function of M1 for the three inlet design Mach numbers and referenced to the cowl leading edge area. The relationships follow linear trends that include a sharp drop in the drag coe cient when the inlet is near the on-design Mach number because the initial cone shock is on the cowl at this condition. The drag is then essentially constant when the inlet is oversped because the cone shock no longer a ects the cowl?s leading edge drag. For Mach 6 on-design inlets, the curve ts for the cowl leading edge drag are C Dle Acle 10 3 1 in2 ref = 8 >>> >>>< >>> >>>: 0:98467225M1 + 2:5726761; M1 5:89020 14:454016M1 + 93:270209; else 0:0033134638M1 + 3:8911951; M1 6:17372 (3.16) Similarly, the Mach 7 designed inlet curve ts are C Dle Acle 10 3 1 in2 ref = 8 >>> >>> < >>> >>> : 0:97465857M1 + 2:8212132; M1 6:8018133 15:048172M1 + 111:33128; else 0:0042186311M1 + 3:8984785; M1 7:1323400 (3.17) 98 And the Mach 8 inlet polynomial- t calculations are C Dle Acle 10 3 1 in2 ref = 8 >>>> >>< >>>> >>: 0:97924800M1 + 2:9478391; M1 7:7620733 15:065858M1 + 127:50517; else 0:00093863892M1 + 3:8535373; M1 8:1997933 (3.18) The cowl leading edge drag is then referenced to only the base area by CDle D le q1A5 = C Dle Acle 10 3 1 in2 ref (Acle) 10 3 in m 2 ; (3.19) The cowl lip leading edge area, Acle, is calculated by multiplying the circum- ference of the inlet and the diameter of the cowl lip leading edge: Acle = (2 ri)dle; (3.20) where dle = 0.00254 m (0.1") and the inlet radius is computed from the inlet area, ri = p Ai= = s A5 (A5=Ai); (3.21) which is 0.477 m following the (A5=Ai) = 1.1 assumption in this work. Body Friction Drag The reference computes the exterior surface?s body (skin) friction \assuming a smooth adiabatic wall with a fully developed turbulent boundary layer" (Waltrup et al.,129 pg. 8-20). The reference?s body friction is normalized by the wetted body area and plotted as a function of freestream Mach number. The quadratic polynomial curve t for the skin friction is C Df Abw 10 6 1 in2 ref = 0:028936761M21 0:83731029M1 + 6:3921607: (3.22) 99 Figure 3.3: Detail of Scramjet Missile Cowl And in metric units, referenced to the base area: CDf D f q1A5 = C Df Abw 10 6 1 in2 ref (Abw) 10 6 in m 2 (3.23) where Abw is the wetted body area that consists of the sum of the cowl surface area and the constant radius, \cylinder," surface area. The cowl surface area is found by using the reference?s 6 cowl angle129 and trigonometry to be Acsurf = (ri +r5) q L2cowl + (r5 ri)2; (3.24) where the cowl length is Lcowl = r5 ritan 6 ; (3.25) and the square root term in Acsurf is the hypotenuse of the cowl. The \cylinder" surface area is the length of the missile excluding the cone 100 Table 3.3: Shock Angles for 12.5 Cone Mach Number 6 7 8 Conical Shock Angle, 16.63569672 15.90885885 15.41968796 forebody and cowl lengths multiplied by the circumference of the base: Acylsurf = (2 r5)Lcyl; (3.26) where Lcyl = Ltotal Li Lcowl; (3.27) Ltotal = 4.0 m is assumed in the reference,129 Lcowl is calculated from Eq. (3.25), and Li is calculated from the on-design shock on cowl lip condition (i.e. whenM1 = Mdes and = 0). From geometry, see Fig. 3.5 on pg. 111, the inlet length is Li = ritan s ; (3.28) where s is the oblique conical shock angle from the 12.5 cone forebody. The shock angle for the shock on cowl lip condition is a function of the on-design Mach number and the cone half-angle (12.5 ). s can be found by numerically solving the Taylor- Maccoll equations from an initial shock angle to the cone surface and then iterating the shock angle until the computed cone surface angle converges to the desired cone angle (See Anderson,133 Ch. 10 for details). For this work, an online java program developed by Chris Hood134 at the University of Colorado, Boulder was used for these computations and the results are summarized in Table 3.3. 101 Now, using Eq. (3.24){(3.28) and Table 3.3, the body wetted area is Abw = Acsurf +Acylsurf: (3.29) Tail Drag The last pro le drag contributions come from the wave and skin friction from the four tail ns. The reference computes the tail wave drag based on two- dimensional ow over a 15 wedge with a 55 sweep angle using the freestream Mach number.129 The tail?s 0.2581 m2 (400 in2) surface area skin friction is computed with the same assumptions as the body friction above. From these assumptions, many higher-order e ects are omitted including the ow distortion from the forebody, cowl and missile body, and the tail?s contribution to the lift force. With these omissions, the tail wave drag, Dtw, and friction drag, Dtf, are only a function of the freestream Mach number: D tw q1A5 10 3 1 in2 ref = 0:010669708M21 0:21896876M1 + 9:8720353; (3.30) D tf q1A5 10 3 1 in2 ref = 0:026468258M21 0:75490733M1 + 5:6466228; (3.31) The actual tail drag coe cients referenced to the base area (CDtw and CDtf) are found by multiplying the curve t calculations by 10 3 and (in/m)2. 102 Pro le Drag Summary To recapitulate, the total pro le drag coe cient for a given inlet design Mach number and current freestream Mach number is CD;0 D 0 q1A5 = CDcw +CDle +CDf +CDtw +CDtf; (3.32) where Eq. (3.13), (3.19), (3.23), (3.30), and (3.31) are used for the components of the pro le drag. 3.1.2.2 Additive Drag Coe cient The inlet additive drag is the wave drag caused by the conical shock o of the inlet forebody. This quantity is calculated by integrating the pressure along the streamtube behind the forebody shock using the same numerical procedure as the capture area and cowl wave drag computations.129 The reference129 plots the additive drag coe cient normalized by the inlet area for various design Mach number (Mdes = 4, 6, 7, 8), freestream Mach number (M1 = 3{8 in intervals of 1), and angle of attack ( = 0 , 5 , 10 ) con gurations as a function of inlet \smile" angle. The report129 states that full axisymmetric chin inlets ( smile = 360 ) are the most e cient con guration since they maximize mass capture to the inlet, so only these inlets were curve t and modeled. For code implementation, the additive drag is calculated by quadratical inter- polation from the design Mach number?s node values, Table 3.4, to the desired angle of attack at each of the M1 nodes. These values are then cubicly interpolated to the desired freestream Mach number. A constraint is also added to ensure the 103 Table 3.4: Additive Drag Coe cient Data Mach 6 Inlet M1 = 0 = 5 = 10 3 0.0397868 0.0398905 0.0398908 4 0.0246806 0.0252112 0.0265942 5 0.0117020 0.0122356 0.0141486 6 0.0 0.00372576 0.00787309 7 0.0 0.0 0.0 8 0.0 0.0 0.0 Mach 7 Inlet M1 = 0 = 5 = 10 3 0.0491538 0.0491329 0.0496144 4 0.0342637 0.0349434 0.0372098 5 0.0215513 0.0225873 0.0250972 6 0.0102914 0.0112934 0.0148257 7 0.0 0.00395732 0.00901193 8 0.0 0.0 0.00610441 Mach 8 Inlet M1 = 0 = 5 = 10 3 0.0565045 0.0561253 0.0548474 4 0.0417301 0.0425830 0.0454651 5 0.0298079 0.0307331 0.0341246 6 0.0186616 0.0201850 0.0241256 7 0.00881249 0.0109387 0.0154648 8 0.0 0.00130182 0.0108252 104 additive drag coe cient is at least zero and never negative, as might be calculated from the interpolation procedure. After the interpolations, the additive drag is re-normalized from the inlet area to the base area by CDadd D add q1A5 = D add q1Ai ref = A 5 Ai ; (3.33) where the base area to inlet area ratio, (A5=Ai), is again one of the design parameters. 3.1.2.3 Normal Force Coe cient The normal force coe cient is arguably the most de cient aspect of the aero- dynamic model presented in the reference.129 The simple analytic relation for the normal force coe cient is given as: CN N q1A5 = ( 0:3M1 + 5:4) ; (3.34) where is in radians. This relation is said to be \representative of those computed for other hypersonic missile designs," Waltrup et al.,129 pg. 8-20. Improvements to this normal force coe cient could include the e ects from the geometric design parameters and tail n de ections. However, for the rst order system analysis in this work, this relation was assumed to reasonably accurate. 3.1.3 Code Validation This subsection validates the implemented Fortran 90/95 program using the previous two subsections? curve- t aerodynamics and propulsion with several ex- ample calculations in the appendix of the JHU / APL reference.129 The reference 105 Table 3.5: Scramjet Design Validation for M1 = 4.0, = 0.0 , ER = 1.0 Parameter Code Reference Error, % CDcw 0.028220 0.0281 0.4292 CDle 0.026047 0.0261 0.2027 CDf 0.028290 0.0285 0.7351 CDtw 0.009166 0.0092 0.3600 CDtf 0.003050 0.0030 1.6828 CDadd 0.010327 0.0104 0.7057 CD 0.105102 0.1053 0.1879 CT 0.490926 0.4872 0.7647 calculations use a Mach 6 inlet design (Mdes = 6) and a base-to-inlet area ratio, (A5=Ai), of 2.39 for the scramjet design parameters and investigates its Mach 4 and 8 performance. Speci cally, the reference computes the drag and thrust coe cient values for Mach 4, 0 ; Mach 8, 0 ; and Mach 8, 5 conditions with an equiv- alence ratio of 1.0. The trim equivalence ratio for the Mach 8, 5 angle of attack cruise is also estimated in the reference. For all cases, the assumed operating de- sign range was M1 = [4:8] and = [0:10 ], which dictates the maximum geometric contraction ratio as discussed in Sec. 3.1.1.2. The results of the example calculation for the Mdes = 6, M1 = 4.0, 0.0 angle of attack, 1.0 ER condition are listed in Table 3.5. The drag coe cient components, total drag coe cient, and thrust coe cient are listed for the Fortran code?s calculated values, the reference?s values,129 and the percent error between the two.y As shown in Table 3.5, the computed drag coe cient components are less than yThere are two typographical errors in the reference?s calculation of the drag coe cient.129 106 Table 3.6: Scramjet Design Validation for M1 = 8.0, = 0.0 , ER = 1.0 Parameter Code Reference Error, % CDcw 0.022209 0.0221 0.4927 CDle 0.015460 0.0155 0.2599 CDf 0.012472 0.0123 1.4006 CDtw 0.008803 0.0088 0.0358 CDtf 0.001301 0.0013 0.1025 CDadd 0.0 0.0 | CD 0.060245 0.0600 0.4089 CT 0.198546 0.2043 2.8166 1% in error except for the tail wave drag, which has a rather small contribution to the overall drag coe cient. Thus, the overallCD error for this operating condition is only 0:188% compared to the reference?s calculations. The thrust coe cient|which requires more extensive curve- tting and interpolation in the Fortran program| produces a larger error of 0.765%. The drag and thrust coe cient results for the Mach 8, 0 , ER = 1.0 oper- ating condition for the Mach 6 inlet design is listed in Table 3.6. The components of the drag coe cient are typically under 1% in error, and the overall CD is over- predicted by only 0.409% from the reference.129 The thrust coe cient, however, is under-predicted 2.82% compared to the reference?s calculations. Considering the The rst typo is in the additive drag coe cient calculation. The reference reports that \CDadd = 0:248=2:39 = 0:0104," which is mathematically incorrect. The rst value should be 0.0248. The second typo is with the total drag coe cient, CD. The reference reports it as 0.1503 instead of 0.1053, which is the value that is computed if the individual drag elements are summed. 107 error in computing the numerous curve- ts and interpolating between the 64 thrust coe cient nodes (see Sec. 3.1.1.3), the approximately 3% error is quite good. Fur- thermore, this under-prediction allows for a slightly conservative estimate of the scramjet?s thrust level. The third example calculation presented in the reference129 is for a Mach 8 cruise at a 5 angle of attack and ER = 1.0. The only di erence in the total drag coe cient between this case and the M1 = 8, = 0 case in Table 3.6 is the induced drag caused by the normal force coe cient. This drag term is CD = CN sin( ); (3.35) which equals 0.0228 for M1 = 8 and = 5 , where CN is calculated by Eq. (3.34) on pg. 105. Therefore, the total drag coe cient as computed by the implemented code is 0.830627, the reference?s CD = 0.828, and the error is 0.3172%. The thrust coe cient for the M1 = 8, = 5 con guration is the same as for the Mach 8, = 0 case, so the error is again 2.82%. Lastly, the reference129 estimates the trim equivalence ratio, ER , for the M1 = 8, = 5 condition. The trim ER as computed by the reference is the value when CD = CT at the given M1 and . As de ned in this manner, the Fortran program?s ER = 0.448 and the reference?s trim equivalence ratio is 0.437 thus resulting in a 2.52% error between the two results. This error is primarily attributed to this dissertaion?s code under-predicting the thrust coe cient as shown in Table 3.6. Overall, the comparison with the Fortran program?s calculation of the aerody- 108 Figure 3.4: Scramjet Missile Volume De nitions namic and propulsion forces are in excellent agreement with the example calculations in the source reference.129 The total drag coe cients are predicted within 1% and the thrust coe cients are at worse in error of less than 3%. 3.2 Mass Model The mass model implemented in this work is based o the parametric model used by Starkey in his doctoral work.135 The model assumes that the initial total vehicle mass, m(t0), is comprised of only fuel and a constant density structure that ideally compensates for all subsystem masses that would be included on the vehicle. The division of the two mass components is de ned by the fuel volume fraction, (Vf=Vtotal), and an assumption for the fuel density, f, and structural density, str: m(t0) = (1 (Vf=Vtotal))Vtotal str + (Vf=Vtotal)Vtotal f: (3.36) The total volume, Vtotal, is calculated from summing the internal volume of the scramjet missile. A summary of the di erent scramjet volumes is illustrated in Fig. 3.4. The cone volume consists of the 12.5 forebody, Vcone;1, and the internal 109 portion of the isolator, Vcone;2. The isolator volume is a sum of the 6 cowl volume, Vcowl = Viso;1, the constant area cowl volume, Viso2, and the exterior portion of the isolator, Viso;3. The combustor volume, Vcomb, nozzle volume, Vnoz, and the volume of the four tail ns, Vfin, are the other components of the total vehicle volume, Vtotal. The following subsection details how each of these components are evaluated. Section 3.2.2 then presents a discussion on the fuel choice used in this work and how the structural density, str, and fuel volume fraction, (Vf=Vtotal), were chosen to meet the design goals of a approximately 1,100 kg initial mass and 25{30% initial fuel mass fraction. 3.2.1 Internal Volume Calculation This subsection details the component volume calculations for the modeled scramjet. The dimensions and volumes are described in the following order: cone inlet, isolator and cone rear, combustor, nozzle, and tail ns. A summary of these volumes concludes this subsection. Inlet The inlet is a 12.5 cone whose maximum radius, rcone, is calculated by rst computing the isolator area: A1 = A i A1 1 max Ai = (r2i r2cone); (3.37) where (Ai=A1)max is the maximum geometric contraction ratio as de ned in Eq. (3.6) on pg. 93, and the second equality is a result of the geometry, as shown in Fig. 3.5. 110 Figure 3.5: Detail of Scramjet Missile Inlet Solving the second equality for the maximum cone radius yields rcone = q r2i A1= : (3.38) The length of the inlet cone from the tip to where the ow begins to be turned downwards is also found from geometry: Lcone = rconetan(12:5 ): (3.39) The volume of the 12.5 cone prior to turning the ow downward is then analytically Vcone;1 = 3r2coneLcone: (3.40) Isolator / Rear of Cone The volume of the 6 cowl from the inlet radius, ri, to the base radius, r5, is Vcowl = Viso;1 = 3Lcowl(r2i +rir5 +r25) r2iLcowl; (3.41) where Lcowl is calculated in Eq. (3.25) on pg. 100. The volume of the constant cross-sectional area portion of the cowl/isolator between Li + Lcowl and Lcone, see Fig. 3.5 and 3.6, is Viso;2 = r25 r2i (Lcone Li Lcowl): (3.42) 111 Figure 3.6: Detail of Scramjet Missile Isolator The constant-area isolator dimensions are calculated with the assumption that the rear of the cone forebody (i.e. the inner portion of the isolator) is a quadratic polynomial: riso;in(s) = aiso s Liso 2 +biso s Liso +ciso; (3.43) where s is the axial length of the isolator beginning where the cone directs the ow down (s = [0:Liso]), see Fig. 3.6. The coe cients for the isolator?s inner geometry are a result of known constraints and several assumptions to make the isolator appear similar to other JHU/APL designs.136,137 The rst constraint is that the initial value of the isolator equals the maximum cone radius: riso;in(0) = rcone = ciso: (3.44) Then substituting Eq. (3.44) into (3.43) and adding the constraint that the cone radius at the end of the isolator is zero: riso;in(Liso) = aiso +biso +rcone = 0 ! aiso = biso rcone: (3.45) Now it is assumed that biso = 0, thus aiso = rcone, to make the isolator look similar to the schematic presented in a 2001 JHU/APL design137 which uses the 112 same methodologies as the original 1982 reference.129 Substituting these coe cients into the isolator quadratic equation, riso;in(s) = rcone s Liso 2 +rcone: (3.46) Furthermore, the isolator length is assumed to be Liso = 2Li Lcone; (3.47) again so that the overall missile dimensions approximately correspond to the 2001 JHU/APL paper by Waltrup.137 Now that the interior curve of the isolator is de ned, the exterior curve is calculated using the isolator?s constant area assumption. Therefore, A1 = const = r2iso;out r2iso;in ! riso;out(s) = q A1= +riso;in(s)2; (3.48) where riso;in(s) is given by Eq. (3.46). A close up of the isolator geometry for the Mach 7 inlet design is shown in Fig. 3.6 on pg. 112. The volume of the rear of the cone is numerically integrated by Vcone;2 = Z Liso 0 r2iso;in(s)ds = LisoX s=0 " r iso;in(si) +riso;in(si 1) 2 2 (si si 1) # ; (3.49) which is essentially summing small segments of a cylinder, and si is the ith element of the 100-element equispaced s array from zero to Liso. The isolator?s outer volume is similarly numerically integrated by Viso;3 = r25Liso Z Liso 0 r2iso;out(s)ds = LisoX s=0 ( " r25 r iso;out(si) +riso;out(si 1) 2 2# (si si 1) ) : (3.50) 113 Combustor The combustor is modeled as a linear expansion speci ed by (A4=A2) from the isolator exit to the nozzle entrance. The length of the combustor is calculated using the assumption that the combustor wall area is 40 times that of the combustor entrance area, i.e. (Awall=A2) = 40, as given in the primary reference.129 The combustor wall?s surface area is then Awall = (r2 +r4) q L2comb + (r4 r2)2; (3.51) where r2 = riso;out(Liso) and the combustor exit radius, r4, is A 4 A2 = r 2 4 r22 ! r4 = r2 p (A4=A2) (3.52) Solving Eq. (3.51) for the combustor length gives Lcomb = s Awall (r2 +r4) 2 (r4 r2)2; (3.53) and the combustor volume is then Vcomb = r25Lcomb 3Lcomb r22 +r2r4 +r24 : (3.54) The gures in this chapter show the modeled combuster with an expansion ratio, (A4=A2), of three with a Mach 7 designed inlet. Nozzle The nozzle is modeled as a quadratic polynomial in a manner similar to the internal isolator geometry with two known boundary conditions, rnoz(0) = r4 and 114 rnoz(Lnoz) = r5, and a known length: Lnoz = Ltotal (Lcone +Liso +Lcomb): (3.55) Using the quadratic polynomial equation, rnoz(snoz) = anoz s noz Lnoz 2 +bnoz s noz Lnoz +cnoz; (3.56) where snoz = [0 : Lnoz], substituting in the boundary conditions, and solving for the rst and third coe cients yields anoz = r5 bnoz r4 (3.57) cnoz = r4: (3.58) The midpoint value is then rnoz(Lnoz=2) = 14anoz + 12bnoz +cnoz = 14 (r5 +bnoz + 3r4) = 14 (3r5 +r4); (3.59) where the last equality is an assumed constraint so that the nozzle geometry is qualitatively similar to typical nozzles. Solving Eq. (3.59) for the second coe cient yields: bnoz = 2 (r5 r4): (3.60) All three coe cients are then substituted into Eq. (3.56) to produce the geometry of the nozzle boundary: rnoz(snoz) = (r4 r5) s noz Lnoz 2 + 2 (r5 r4) s noz Lnoz +r4: (3.61) The volume of this portion of the missile is the volume of an external cylinder 115 Figure 3.7: Scramjet Missile Fin Detail of radius r5 and length Lnoz minus the internal volume of the nozzle: Vnoz = r25Lnoz Z Lnoz 0 r2noz(snoz)dsnoz (3.62) = r25Lnoz LnozX snoz=0 r noz(snoz;i) +rnoz(snoz;i 1) 2 2 (snoz;i snoz;i 1); where snoz;i is the ith element in the 100-element equally spaced array from zero to Lnoz, and Lnoz as de ned in Eq. (3.55). Tail Fins The volume of the four tail ns is the nal component of the total volume of the modeled scramjet missile. Following the reference?s assumptions of a 15 wedge angle,129 the length of the n at the tip is Lfin;tip = tfintan(15 ); (3.63) where tfin is the thickness of the n and Fig. 3.7 shows the n dimensions. The reference129 also assumes a 5% thickness for the tail. For this work it is assumed 116 that this constraint is in relation to the length of the n at its base, so tfin = 0:05Lfin; (3.64) where Lfin is the tail n?s length where it is mounted to the missile body. Substi- tuting the n thickness into the tail?s length at the tip, Eq. (3.63), yields Lfin;tip = 0:05tan(15 )Lfin: (3.65) The width of the tail n, wfin, is computed from the reference?s assumption that the ns have a 55 sweep angle.129 Using the tangent of the sweep angle and the di erence in the n length at the base and tip, one has tan(55 ) = Lfin Lfin;tipw fin ! wfin = 1 (0:05=tan(15 )) tan(55 ) Lfin; (3.66) where Eq. (3.65) has been used for Lfin;tip. From the reference,129 the total planform area of the four ns is 400 in2, so each n?s planform area is 100 in2 or 0.06452 m2. Using the dimensions calculated above, the planform area is Afin = 0:5 (Lfin +Lfin;tip)wfin = Lfinwfin2 1 + 0:05tan(15 ) = L 2 fin 2 tan(55 ) " 1 0:05 tan(15 ) 2# : (3.67) Rearranging, the n base length can be solved for as: Lfin = s 2Afin tan(55 ) 1 (0:05=tan(15 ))2: (3.68) With Eq. (3.64), (3.66), and (3.68), the volume of a single tail n (which is the same for all missile designs) is Vfin = 0:5Lfin;tipwfintfin + 0:5 (Lfin Lfin;tip)wfintfin = 0:5Lfinwfintfin = 0:025L2finwfin = 0:00119 m3; (3.69) 117 where Lfin = 0.437 m from Eq. (3.68) and wfin = 0.249 m from Eq. (3.66). Internal Volume Summary The total scramjet volume is calculated by Vtotal = 2X i=1 Vcone;i + 3X i=1 Viso;i +Vcomb +Vnoz + 4Vfin; (3.70) where Eq. (3.40){(3.42), (3.49), (3.50), (3.54), (3.63), and (3.69) are used for the various component volumes. 3.2.2 Determination of Mass Model Design Parameters Now that the total missile volume can be calculated for a given Mdes, (A5=Ai), and (A4=A2), there are three quantities that need to be determined to calculate the initial mass of the modeled scramjet using Eq. (3.36) on pg. 109: the fuel density, f, average structural density, str, and fuel volume fraction, (Vf=Vtotal). JP-10 is assumed as the fuel for the scramjet missile. This fuel type was chosen based on Edwards? 2003 survey paper on aerospace propellants138 which states that JP-10 is \the only airbreathing-missile fuel in operational use by the United States at the present time." Furthermore, the fuel assumed in the JHU/APL report (RJ-5),129 had \cost and freeze-point limitations [that] prevented eld use." In hindsight, JP-7, which was used by the Mach 3+ SR-71138 and proposed for the Mach 5 X-51,18 may have proven to be a better fuel choice because of its improved regenerative cooling properties over JP-10. Regardless, JP-10?s density of 940 kg/m3 is used for the scramjet mass model in this dissertation (Horning,139 pg. 131). JP-10?s hydrogen- 118 Figure 3.8: Scramjet Fuel Mass Fraction vs. (Vf=Vtotal) to-carbon ratio, (H=C), is 1.61 which results in a stoichiometric fuel-to-air ratio of 0.07066 using (Heiser and Pratt,140 pg. 112): fstoich = 36 + 3(H=C)103[4 + (H=C)]; (3.71) which assumes air is comprised of 79% nitrogen and 21% oxygen. Four average structural densities are considered for the mass model: aluminum (2,700 kg/m3), titanium (4,507 kg/m3), steel (7,850 kg/m3), and tungsten (19,255 kg/m3). Figure 3.8 plots the fuel mass fraction as a function of the fuel volume fraction from 20{90% for the four structural densities using the relation m(t0) mf(t0) = mstr +mf(t0) mf(t0) = (1 (Vf=Vtotal)) (Vf=Vtotal) str f + 1; (3.72) and the JP-10 fuel density. Billig states, \The mass fraction of propellant for a rocket for a tactical missile is typically 50-70% of the initial weight as compared to 25-30% for the ramjet," (Jensen and Netzer,141 pg. 12) so the modeled scramjet?s 119 Figure 3.9: Scramjet Mass vs. (Vf=Vtotal), Mach 7 inlet fuel mass fraction would ideally fall in this range, which is shown as dashed lines in the gure. The second design constraint considered in the determination of the mass prop- erties of the scramjet model is its initial mass, m(t0). The initial mass for the Mach 7 inlet design is plotted as Fig. 3.9 for fuel volume fractions between 20 and 90% us- ing Eq. (3.36) on pg. 109. The estimated initial masses of two JHU/APL hypersonic missile designs (excluding their rocket booster masses) are plotted as dashed lines in the gure. The Mach 7.5 SCRAM missile initial mass is approximated as 850 kg and the Mach 4{6 Dual Combustor Ramjet (DCR) initial mass as 1,240 kg from Waltrup et al.?s survey of U.S. Navy high speed air-breathing propulsion systems.142 Therefore, the second design constraint imposed is an initial wet mass of 1,100 kg which is the average of these two designs. From Fig. 3.9, it is apparent that the initial mass of the scramjet is quite 120 sensitive to (Vf=Vtotal) near the m(t0) = 1,100 kg constraint. For an aluminum structure, the fuel volume fraction would be less than 20% which causes a fuel mass fraction of less than 10% as shown in Fig. 3.8. For a titanium structure, the fuel volume is about 50% for an initial mass of 1,100 kg and the resultant fuel mass fraction is 17.3%. A steel structure requires a 75% (Vf=Vtotal) to produce a 1,100 kg initial mass and thus a 22% fuel mass fraction. Lastly, the tungsten structure needs a fuel volume fraction over 90% to yield the desired initial mass, which is deemed unreasonably high. Although none of the modeled densities meet both design constraints, the titanium structure with a 50% fuel volume fraction was chosen as a compromise between the two constraints. 3.2.3 Modeled Axisymmetric Hypersonic Missile Summary The primary design parameters used to model the hypersonic, axisymmetric scramjet missiles in this work are listed in Table 3.7. The top portion of the table summarizes the main geometric design parameters for the vehicles. The middle portion summarizes the fuel and mass properties of the vehicle. And the bottom portion summarizes the cruise dynamic pressure and the operating ranges used to calculate the maximum contraction ratio, (Ai=A1)max, and the gridded aerodynamic and propulsion data for the trim calculations in the next section. Table 3.8 lists the resultant maximum contraction ratios and the operating point that sets this value, the total volume of the three missile designs, the initial fuel mass, and the initial vehicle mass. Again, the fuel mass fraction is 17.3%. 121 Table 3.7: Modeled Scramjet Design Parameters Design Parameter Symbol Value Units Total Length Ltotal 4.0 m Base (Reference) Radius r5 0.25 m Base-to-Inlet Area Ratio (A5=Ai) 1.1 | Combustor Expansion Area Ratio (A4=A2) 3.0 | Combustor Wall-to-Entrance Area Ratio (Awall=A2) 40.0 | Fuel Density (JP-10) f 940.0 kg/m3 Fuel Hydrogen-to-Carbon Ratio (JP-10) (H=C) 1.61 | Stoichiometric Fuel-to-Air Ratio (JP-10) fstoich 0.07066 | Structural Density (Titanium) str 4507 kg/m3 Fuel-to-Total Volume Ratio (Vf=Vtotal) 0.5 | Cruise Dynamic Pressure q1 101325 Pa Freestream Mach Number Range M1 4{8 | Angle of Attack Range 0{10 Equivalence Ratio Range ER 0.25{1 | Table 3.8: Scramjet Missile Design Summary Mdes (Ai=A1)max Total Volume Fuel Mass Total Mass 6 7.0178 at M1 = 4.0, = 0 0.3887 m3 182.7 kg 1058.5 kg 7 7.8768 at M1 = 4.0, = 0 0.4034 m3 189.6 kg 1098.7 kg 8 7.9814 at M1 = 8.0, = 0 0.4102 m3 192.8 kg 1117.1 kg 122 3.3 Trim State Calculation Detailed trim conditions are used to compute the trim pitch, roll, and equiv- alence ratio pro les along the 1,000 km cruise trajectory as fuel is burned and the vehicle compensates for changes in the gravity vector. The trim Euler angles are then nite di erenced to produce the body-to-navigation frame angular rates that are a portion of the overall simulated truth gyro signal. The accelerometer spe- ci c force measurements are more easily simulated for cruise since they are analytic functions of the trajectory. The assumed cruise trajectories in this work are due East at a constant velocity and altitude, see Sec. 2.4. Therefore, the yaw angle of the vehicle is held at 90 throughout the duration of the simulation. Also, because of the constant altitude assumption, the ight path angle, fpa, of the missile is always zero so that the pitch angle, b, is equivalent to the angle of attack, i.e. b + fpa = . The pitch angle, roll angle, and equivalence ratio are then the three variables used to trim the scramjet?s accelerations at each simulation epoch. Because of the low delity aerodynamics of the tail ns, and the absence of a high- delity mass model, the moment dynamics are not trimmed in this work. The next subsection derives the trim state calculations used in this work start- ing with the free-body diagram of the missile and the vehicle dynamics. The fol- lowing subsection presents the results of nite di erencing the trim pitch and roll angles along with the simulated navigation-frame speci c forces. 123 Figure 3.10: Scramjet Missile Free Body Diagram 3.3.1 Free Body Diagram & Cruise Dynamics The dynamics of the missile are computed in the North-East-Down navigation frame (see Sec. 4.1.3 for details on the NED coordinate frame). The forces acting on the vehicle are the thrust, drag, and normal force as shown in Fig. 3.10. The gravitational, centripetal, and Coriolis accelerations are accounted for separately in the speci c force term of the vehicle dynamics, see Eq. (3.84) on pg. 128. The scramjet thrust is assumed to be aligned with the body?s line of symmetry which is inclined from the Easterly cruise by the pitch angle (or equivently ). The vehicle drag is de ned as the sum of the pro le drag and additive drag and is opposite of the velocity vector, i.e. in the West direction. The normal force is nominally in the body frame?s \up" direction, but may also be rolled North or South to compen- sate for latitudinal gravitational variations during cruise. For all aerodynamic and propulsion force calculations, it is assumed that the vehicle is ying at a constant 1-atmosphere dyanmic pressure as discussed in Sec. 2.4. The sum of these forces in the navigation frame are calculated as follows. The thrust is aligned predominately in the East direction, but also has a small component 124 in the negative Down (\up") direction, so that Tn = T(M1; ;ER) (0; cos ; sin )T ; (3.73) where the functional dependence of Mach number, angle of attack, and equivalence ratio on thrust is explicitly shown, and the angle of attack is used in place of the pitch angle to coincide with the standard aerodynamic/propulsion nomenclature in the majority of this chapter. The drag is simply in the West, and is a function of the freestream Mach number and angle of attack (due to the additive drag): Dn = (0; D(M1; ); 0)T : (3.74) Lastly, the normal force has a small component in the West direction due to the vehicle?s angle of attack and its \up" component has a small portion of lift in the North/South axes: Nn = N(M1; ) ( cos sin b; sin ; cos cos b)T : (3.75) Then setting the sum of these forces equal to the mass times the speci c forces in the n-frame: maN = N( ) cos sin b (3.76a) maE = T( ;ER) cos D( ) N( ) sin (3.76b) maD = T( ;ER) sin N( ) cos cos b; (3.76c) which is a set of three nonlinear equations with three unknowns ( , b, and ER). The freestream Mach number dependence is dropped above because it will be as- sumed that the vehicle cruises at the constant, on-design Mach number throughout 125 the simulation. Also, the vehicle mass and speci c forces on the left hand side of the equations are known quantities for a given position and velocity by Eq. (3.84) discussed in the following subsection. Therefore, the trim states ( ; b;ER ) are computed by solving Eq. (3.76a){(3.76c) at each epoch along the trajectory given the calculated man. The trim roll angle, b, can be calculated as a function of the trim angle of attack, , using Eq. (3.76a): b = sin 1 ma N N( ) cos : (3.77) The trim angle of attack (or trim pitch angle) is found by solving Eq. (3.76b) and (3.76c) for the thrust term and then dividing the Down equation from the East equation to remove the dependence on equivalence ratio: tan = N( ) cos cos b maDN( ) sin +D( ) +ma E : (3.78) Rearranging so that the roll angle term is isolated on the left hand side results in N( ) cos cos b = (N( ) sin +D( ) +maE) tan +maD: (3.79) Squaring this equation, and then using the fact that sin2 b + cos2 b = 1 with Eq. (3.77) for the sine term, the roll dependence can be removed so that the entire expression is only a function of the trim angle of attack: N2( ) cos2 cos2 b = N2( ) cos2 " 1 ma N N( ) cos 2# N2( ) cos2 (maN)2 = [(N( ) sin +D( ) +maE) tan +maD]2:(3.80) Then rearranging slightly, N2( ) cos2 = [(N( ) sin +D( ) +maE) tan +maD]2 + (maN)2; (3.81) 126 which is essentially a relationship for matching the lift and weight of the missile at the current epoch. This equation is only a function of the angle of attack and is numerically solved for the trim value, , using a bisection method to bound the trim from gridded aerodynamics tables of N( )M1=Mdes and D( )M1=Mdes and a false position method with cubic interpolation to re ne the value. The equivalence ratio is computing using Eq. (3.76b) after being rearranged slightly to T( ;ER) cos = N( ) sin +D( ) +maE: (3.82) This equation is e ectively a higher-order T = D constraint. The thrust?s trim equivalence ratio is numerically calculated using a bisection and false position meth- od with cubic interpolation and from Eq. (3.81). 3.3.2 Simulated Pitch and Roll Rates In order to calculate the trim states, the speci c forces, an, must be computed at each epoch along the trajectory. These quantities can be found from rearranging Eq. (4.47) on pg. 151 in terms of the speci c force vector: an = _vn + ( nin + nie) vn gn: (3.83) For the constant altitude, constant Eastern velocity trajectory assumed, _vn = 0, vn = (0; vE; 0)T, and _ = _h = 0. Then substituting these conditions into the expanded velocity dynamics of Eq. (4.49) on pg. 152 yields the trajectory?s speci c 127 force equation in the n-frame: an = 0 BB BB BB @ aN aE aD 1 CC CC CC A = 0 BB BB BB @ _ + 2!e vE sin gN gE _ + 2!e vE cos gD 1 CC CC CC A ; (3.84) where _ = vE=((Ne +h) cos ). The details of these derivations and how the ele- ments are calculated are explained in Ch. 4. The gravity acceleration vector, gn, is calculated using the 360 degree and order EGM96 spherical harmonic model and the modi ed geopot97 code as explained in Sec. 2.4. Table 2.2 on pg. 72 summarizes the cruise altitudes and velocities for the 1-atmosphere dynamic pressure, Mach 6, 7, and 8 designs. Then, starting with the initial scramjet masses (Table 3.8 on pg. 122), the rst trim state is computed from Eq. (3.81), (3.77), and (3.82). Using the trim equivalence ratio, ER , and the capture area at the design Mach number (which also equals the freestream Mach number) and trim angle of attack: A 0(Mdes;M1; ) = A 0 Ai Mdes;M1; A i A5 A5; (3.85) where (A5=Ai) = 1.1 in this work and A5 = 0.785 m2 using r5 = 0.25 m from the JHU/APL reference,129 the fuel mass ow rate is _mf = _m f _m0 _m0 = (ER fstoich) ( 1vEA 0); (3.86) where the stoichiometric fuel-to-air ratio is calculated by Eq. (3.71) on pg. 119 as 0.07066 and the freestream density at altitude is computed using the 1976 standard atmosphere model.126 The mass of the scramjet is next reduced by Euler integration 128 at the 20 Hz simulation time step: m(t+ t) = m(t) _mf t: (3.87) The trim states are then recalculated and this process is repeated for the entire 1,000 km cruise trajectory. The trim state and mass pro les are stored in memory and the pitch (angle of attack) and roll rates are computed by rst-order forward nite di erences of their respective trim values. The trim states, angular rates, and the mass pro les are written out as a text le for the 1,000 km downrange at the 20 Hz simulation time step. This le is then read in by the INS simulation and used as part of the simulated gyro signals. The total simulated truth gyro signals are a sum of the navigation-to-inertial and the body-to-navigation frame rotations: !bib = !bin +!bnb = Cbn!nin +!bnb: (3.88) where Cbn is computed using the INS truth quaternion as discussed in Sec. 4.3.3.1, !nin is calculated by Eq. (4.23) on pg. 144, and !bnb is found by Eq. (4.33) on pg. 147 using the roll and pitch rates ( _ b and _ b) calculated in this section. As a reminder, the yaw angle, b, is a constant 90 to ensure the Eastern ight, so its rate is always zero, i.e. _ b(t) = 0. Figure 3.11 plots the Mach 6, 7, and 8 trim roll and pitch rate pro les along with the trim equivalence ratio pro le over the 1,000 km simulation. The high gravitational gradient variation trajectory results are shown in red solid lines and the low n variation trajectory is shown in blue dashed lines. (Section 2.4 details the 129 (a) (b) (c) Figure 3.11: Trim Roll Rate, Pitch Rate, and Equivalence Ratio (a) Mach 6 (b) Mach 7 (c) Mach 8 130 Table 3.9: Scramjet Initial Trim Angles and Excess Fuel Mdes n Var. (t0) b(t0) mf(tf) 6 High 7.4721 -4.6525 2.70 kg 7 High 8.2970 -6.2255 54.38 kg 8 High 8.9666 -8.1503 6.51 kg 6 Low 7.4417 -3.7964 2.86 kg 7 Low 8.2524 -5.0580 54.43 kg 8 Low 8.9021 -6.6034 7.33 kg two trajectories). The trim roll rates are quite insensitive to the simulation Mach number because they compensate for the (approximately same) small North/South de ections of the gravity vector. The trim pitch rate is negative throughout the simulation since the vehicle pitches down as fuel mass is expended and the lift required to match the weight of the vehicle is decreased. The equivalence ratio also decreases along the trajectory since the thrust level increases and the drag decreases as decreases and also because the reduced vehicle mass requires less thrust compensation, as shown in Eq. (3.82). Table 3.9 lists the initial trim pitch, (t0), and roll, b(t0), angles along with the excess fuel at the end of the simulation. The initial trim angles increase with higher Mach number primarily due to the increased initial mass of the higher speed vehicles which requires more lift and thus a higher . The trim roll angle is coupled to the trim by Eq. (3.77) on pg. 126 so that as the trim pitch angle increases the vehicle must roll more to compensate for the gravity vector de ection towards the South. The excess fuel mass is listed to show that the 17.3% fuel mass fraction is 131 Figure 3.12: Trim Speci c Forces, Mach 7 su cient for each of the cruise simulations. Figure 3.12 plots the simulated speci c forces along the 1,000 km trajectory for the Mach 7 design. The Mach 6 and 8 systems are identical to these plots with the only di erence being the bias of the North and Down speci c forces caused by the ( _ + 2!e)vE term in Eq. (3.84) on pg. 128. The variation of the signals in Fig. 3.12 are attributed solely to the gravitational eld since the other terms in the speci c force calculations are constant along the trajectory and the centripetal acceleration is constant for a constant latitude and altitude cruise. 3.4 Chapter Summary This chapter details the hypersonic missile model used for the majority of the navigation simulations in Ch. 6. The rst section consists of extensive curve ts from a JHU/APL reference129 and an interpolation methodology to calculate the on- and o -design thrust, drag, and normal forces for several axisymmetric missile designs. (Appendix B includes additional thrust coe cient curve ts for the propulsion analysis.) Section 3.1.3 validates the implemented Fortran code to 132 the reference?s129 sample calculations and shows that the model used in this work is accurate to within 1% of the drag coe cient and 3% of the thrust coe cient computations. Section 3.2 extends the original JHU/APL report to include a simple para- metric mass model. Section 3.2.1 details the assumptions and calculations used to compute the total missile volume. Section 3.2.2 then explains the design constraints and assumptions used to determine the missile model?s mass properties of 50% JP- 10 fuel and 50% titanium structure by volume. The three scramjet designs (M1 = 6, 7, and 8) are nally summarized in Sec. 3.2.3. Section 3.3 incorporates the o -design angle of attack and equivalence ratio aerodynamic and propulsion performance developed in Sec. 3.1 and the mass model of Sec. 3.2 to compute the trim states along the 1,000 km cruise. Section 3.3.1 derives new trim relations to balance the forces of the vehicle using the angle of attack (pitch angle), roll angle, and equivalence ratio as the trim variables. Then Sec. 3.3.2 explains how the vehicle is numerically integrated along the 1,000 km range trajectory as mass is being expended and the vehicle is pitching, rolling, and throttling to maintain constant altitude and velocity cruise conditions. This section also details how the pitch (angle of attack) and roll angles are nite di erenced to produce the navigation-to-body rotation rates needed for a portion of the simulated gyro signals in the INS dynamics, which are explained in the following chapter. Furthermore, the trim accelerometer speci c forces are shown in this section, which are also derived thoroughly in the next INS chapter. 133 Chapter 4 Inertial Navigation System In this chapter, the inertial navigation system (INS) model is derived. The chapter begins with a summary of the relevant coordinate frames needed and the transformations between these frames. Then the navigation equations used to nu- merically integrate the inertial measurements into velocity, position, and orientation information are derived. Lastly, the inertial measurement unit (IMU) characteristics are given along with the equations used to model their error behavior. 4.1 Coordinate Frames 4.1.1 Earth-Centered-Inertial Frame The Earth-centered inertial, or \i-", frame is de ned in such a way that New- ton?s laws of motion hold. According to his First Law, a body at rest (or constant velocity) will remain at rest (or constant velocity) in the absence of applied forces. And, according to his Second Law, the time rate of change of momentum is equal to the sum of forces applied: d dt m i_ri = Fi; (4.1) 134 where mi is the inertial mass of the object, _ri is its linear velocity in the inertial frame, and Fi is the sum of the applied forces in the inertial frame. If the mass is constant, the more common mi ri = Fi (4.2) expression is found, where r is now the linear acceleration of the object. In order to use classical navigation theory, Newton?s Second Law must be modi ed with a gravitational term. This modi cation is necessitated by the fact that the gravitational eld is a kinematic force that induces accelerations that are independent of mass, which causes di erent behavior than the externally applied force, Fi, that Newton described (Jekeli,1 pg. 4). Thus, Eq. (4.2) becomes mi ri = Fi +mggi (4.3) where the second term is the force due to the gravitational eld from Earth?s mass attracting the user. Speci cally, mg is the gravitational mass and g is the gravi- tational vector. Now, invoking the Weak Equivalence Principle, which essentially states that an object will accelerate at the same rate regardless of its mass, one can equate the inertial and gravitational masses mi = mg = m. This gives the relation: ri = ai + gi; (4.4) where ai = F=m is the speci c force or acceleration due to applied forces in the in- ertial frame. Furthermore, a is the sensed quantity measured by the accelerometers because the proof mass in the accelerometer behaves identically to linear and gravi- tational accelerations (except in opposite signs). For a more thorough discussion of these concepts, the reader may consult Jekeli1 pg. 3{6. 135 Figure 4.1: Earth-Centered-Inertial Coordinate System, from Ref. [115] Now, the inertial frame is de ned as follows for terrestrial navigation. The Earth-Centered-Inertial (ECI) frame is a non-rotating rectilinear coordinate system with its origin located at the Earth?s center (See Fig. 4.1, from Vallado,115 pg. 157). The 1-axis (bI in the gure) points to the mean vernal equinox, the 3-axis (bK) is aligned with the Earth?s spin axis through the North pole, and the 2-axis (bJ) completes the right hand orthogonal coordinate system. Technically, this frame is not truly inertial. Although the coordinate system does not rotate with respect to the stars, the frame?s center is accelerating due to the Earth?s rotation around the Sun. 4.1.2 Earth-Centered-Earth-Fixed Frame The Earth-Centered-Earth-Fixed (ECEF) or \e-" frame is similar to the iner- tial frame except it rotates with the Earth. The coordinate system origin is still at 136 the Earth?s center and the 3-axis is aligned with the spin axis through the North pole; however, the 1-axis now points through the mean Greenwich meridian at the equator. The 2-axis completes a right hand orthogonal frame. This frame is partic- ularly notable for its use within the Global Positioning System?s satellite broadcast (App. D). 4.1.3 Navigation Frame The Navigation or \n-" frame is a local geodetic system that will serve as the primary frame of interest (See Fig. 4.2). The frame has its origin at the vehicle?s center of mass and it?s 1-axis points North, 3-axis points down (perpendicular to the local Earth ellipsoid?s surface), and the 2-axis completes the right hand convention such that it points East. Other n-frame variations exist such as East-North-Up and South-East-Up; however, North-East-Down is the most common orientation and will be used throughout this work. 4.1.4 Body Frame The Body or \b-" frame is also a local system whose origin is located at the vehicle?s center of mass. The 1-axis is aligned to point through the front, the 2-axis through the right, and the 3-axis down through the oor, as shown in Fig. 4.3. The inertial measurement units (IMUs) used in this study are all assumed to be strap- down systems where the accelerometers and gyros are aligned the b-frame axes. As discussed later in Sec. 4.5.1, the truth accelerometer and gyro coordinate systems 137 Figure 4.2: Navigation Frame (North-East-Down) Coordinate System, modi ed from Ref. [115] Figure 4.3: Body Frame Coordinate System and Euhler Angles, from Ref. [145] 138 were originally perturbed slightly from the true body frame to add uncompensated errors to the system. Unfortunately, this addition caused may of the INS/GGI sim- ulations to diverge, so these errors were decided to be removed for the simulations. 4.2 Coordinate Transformations 4.2.1 Fundamental Concepts This subsection explains the concepts used in the subsequent sections which describe speci c coordinate transformations. First, transformation (direction co- sine) matrices and their properties will be discussed. Then Euler angles and their small angle approximations will be presented. This section will conclude with the de nition of the time rate of change of a coordinate transformation matrix. Assume a single point de ned in two frames so that its three-element position vector is de ned as rs in the rst arbitrary frame and rt in the second arbitrary frame. The transformation matrix from the rst frame to the second, Cts, is de ned to satisfy rt = Ctsrs: (4.5) And the elements of the transformation matrix are ci;j = esi etj = cos( ); (4.6) where esi is the unit vector along the ith axis of the s-frame, etj is similarly de ned, and is the angle between esi and etj. Therefore, the elements of the transformation matrix are equivalent to the cosine of the angles between the frame axes, and thus 139 commonly referred to as a direction cosine matrix. The transformation matrix is also an orthogonal matrix and has the prop- erty that its inverse is equivalent to its transpose, which is also equivalent to the transformation matrix from the t-frame to the s-frame. Mathematically, Ct s 1 = Ct s T = Cs t: (4.7) The transformation matrix could also be formulated as a series of speci c rotations about three axes. Assuming the rst rotation is about the 1-axis of the s-frame, the second rotation is about the newly de ned \2-axis," and the last rotation is about the newest \3-axis," Cts = R3( )R2( )R1( ) (4.8) = 0 BB BB BB @ cos sin 0 sin cos 0 0 0 1 1 CC CC CC A 0 BB BB BB @ cos 0 sin 0 1 0 sin 0 cos 1 CC CC CC A 0 BB BB BB @ 1 0 0 0 cos sin 0 sin cos 1 CC CC CC A = 0 BB BB BB @ c( )c( ) c( )s( )s( ) +s( )c( ) c( )s( )c( ) +s( )s( ) s( )c( ) s( )s( )s( ) +c( )c( ) s( )s( )c( ) +c( )s( ) s( ) c( )s( ) c( )c( ) 1 CC CC CC A ; where c() denotes cosine and s() denotes sine in the last equality. The rotation angles can also be calculated from a given transformation matrix using its elements: = tan 1 c 3;2 c3;3 ; = sin 1 (c3;1); = tan 1 c 2;1 c1;1 : (4.9) If , , and are assumed to be small angles, then cos 1 and sin 140 (where is an arbitrary small angle). Neglecting second-order small angles, Cts 0 BB BB BB @ 1 1 1 1 CC CC CC A = 0 BB BB BB @ 1 0 0 0 1 0 0 0 1 1 CC CC CC A 0 BB BB BB @ 0 0 0 1 CC CC CC A I ; (4.10) where I is a 3 3 identity matrix and is the skew symmetric matrix of = ( ; ; )T which is equivalent to (), see Eq. (4.18). Using Eq. (4.7), Cst = Cts T I + (4.11) for small rotation angles. The transformation of a second order tensor from one frame to another requires two direction cosine matrix multiplications. This can be proven as follows. Starting with an arbitrary linear equation in the arbitrary s-frame, ys = Asxs (4.12) and the transformations for each vector, xs = Cstxt; ys = Cstyt; (4.13) where the t-frame is also arbitrary, one can substitute these relations into Eq. (4.12) above to get Cstyt = AsCstxt: (4.14) Then using the property that the inverse of the transformation matrix is equal to its transpose, Eq. (4.7), yt can be solved as yt = CtsAsCstxt = Atxt: (4.15) 141 Therefore, pre- and post-multiplication of a coordinate transformation matrix is needed to rotate a tensor from one frame to another, i.e., At = CtsAsCst: (4.16) The last concept to discuss is the time rate of change of Cts. It can be shown that (Jekeli,1 pg. 21) _Cts = Cts sts; (4.17) where sts is the skew symmetric matrix for the transformation matrix?s rotation rate !sts. If !sts = (!1;!2;!3)T, then sts is equivalent to !sts () and de ned as sts = 0 BB BB BB @ 0 !3 !2 !3 0 !1 !2 !1 0 1 CC CC CC A : (4.18) The terminology of the angular rate subscripts and superscripts should be clari ed at this point. An arbitrary rotation rate, !tsr, is the angular velocity of the arbitrary r-frame with respect to the s-frame, but with coordinates in the t-frame. The terminology for an arbitrary skew symmetric matrix, tsr, is the same. 4.2.2 ECEF to ECI Transformation Neglecting Earth?s polar axis motion and any nutation of its spin axis, the rota- tion between the Earth-Centered-Earth-Fixed (ECEF) and Earth-Centered-Inertial (ECI) frames can be estimated as a single rotation about the Earth?s spin axis, i.e., the 3-axis of both frames (Jekeli,1 pg. 22). The rotation rate of the Earth can also be assumed a constant, !e, and the total rotation between the e-frame and i-frame 142 is then !et. The rotation rate vector from ECEF to ECI, with coordinates in the e-frame is !eie = (0; 0; !e)T ; (4.19) and the transformation matrix from the i-frame to the e-frame is Cei = R3(!et) = 0 BB BB BB @ cos!et sin!et 0 sin!et cos!et 0 0 0 1 1 CC CC CC A : (4.20) 4.2.3 Navigation to ECEF Transformation The navigation frame can be transformed to the e-frame by two rotations. The rst rotation is about the n-frame?s 2-axis (East) to align the n-frame?s 3-axis with the Earth?s spin axis (ECEF 3-axis). A rotation about the spin axis is then performed to align the 1- and 2- axes. Denoting the geodetic latitude as and geodetic longitude as , the n-frame to e-frame transformation matrix is Cen = R3( )R2( 2 + ) = 0 BB BB BB @ sin cos sin cos cos sin sin cos cos sin cos 0 sin 1 CC CC CC A : (4.21) Solving Eq. (4.17) for the rotation rate vector gives !nen = _ cos ; _ ; _ sin T ; (4.22) where _ is the angular rate of latitude and _ is the longitude rate. The rotation rate of the navigation frame with respect to the inertial frame will also be used. This rotation rate is calculated by noting !nin = !nie +!nen and 143 !nie = Cne!eie. Therefore, using Eq. (4.19), (4.21), and (4.22), !nin = ( _ +!e) cos ; _ ; ( _ +!e) sin T : (4.23) To conclude this subsection, the relationship between the navigation frame position coordinates ( , , h) and the ECEF position coordinates will be given. This transformation will be useful in calculating the GPS measurements and the geometric dilution of precision (GDOP). This relationship is (Torge,143 pg. 99{100 or Jekeli,1 pg. 23) 0 BB BB BB @ re1 re2 re3 1 CC CC CC A = 0 BB BB BB @ (Ne +h) cos cos (Ne +h) cos sin (Ne(1 e2) +h) sin 1 CC CC CC A ; (4.24) where Ne is the radius of curvature in the prime vertical plane and e2 is the rst eccentricity of the Earth ellipsoid squared. For completeness, Me is the radius of curvature in the meridian, and all three properties are de ned as Ne = aap1 e 2 sin2 ; (4.25) e2 = pa2 e b2e a2e = 2fe f 2 e; (4.26) where ae is Earth?s semimajor axis, be is the semimajor axis, fe is the atness of Earth?s ellipsoid, and Me = ae(1 e 2) (1 e2 sin2 )3=2: (4.27) Figure 4.4 illustrates several of these parameters with reference to a simpli ed Earth ellipsoid, from Jekeli,1 pg. 23. 144 Figure 4.4: Coordinate System Transformation, from Ref. [1] Table 4.1: World Geodetic System 1984 Properties Parameter Name Symbol WGS84 Value Semi-Major Axis ae 6,378,137.0 m Semi-Minor Axis be 6,356,752.3142 m First Eccentricity Squared e2 6.694 379 990 14 10 3 Reciprocal of Flattening 1=fe 298.257 223 563 Earth?s Gravitational Constant GM 3.986 004 418 1014 m3/s2 Earth?s Rotation Rate !e 7.292 115 0 10 5 rad/s 145 The values for the Earth ellipsoid, de ned by the Department of Defense?s World Geodetic System 1984 (and used in the EGM96 spherical harmonic model) are summarized in Table 4.1.113 4.2.4 Body to Navigation Transformation The body to navigation frame transformation is comprised of three rotations. The rst rotation is a negative roll ( b) about the b-frame?s 1-axis, followed by a negative pitch ( b) rotation about the new 2-axis, and concluded with a negative yaw ( b) rotation about the newest 3-axis. Since matrix multiplication is not communicative (i.e., AB6= BA), the Euler angles of this transformation (roll, pitch, and yaw) are de ned by the order of rotations above, as shown in Fig. 4.3 on pg. 138. Mathematically, the body to navigation rotation matrix is Cnb = R3( b)R2( b)R1( b); (4.28) or when expanded: Cnb = 0 BB BB BB @ c( b)c( b) c( b)s( b)s( b) s( b)c( b) c( b)s( b)c( b) +s( b)s( b) s( b)c( b) s( b)s( b)s( b) +c( b)c( b) s( b)s( b)c( b) c( b)s( b) s( b) c( b)s( b) c( b)c( b) 1 CC CC CC A ; (4.29) where c() denotes cosine and s() denotes sine. The Euler angles can also be found using the Cnb coe cients, similar to Eq. (4.9), by b = tan 1 c 3;2 c3;3 ; b = sin 1 (c3;1); b = tan 1 c 2;1 c1;1 : (4.30) 146 The relationship between the Euler angle rates and the rotation matrix rates are also based on the speci c rotation order. If b1 is de ned as the frame that b rotates about and b2 as the frame that b rotates about, the transformation matrix can be rewritten as Cnb = Cnb2Cb2b1Cb1b : (4.31) The rotation rate of the matrix can also be decomposed into the three rotations due to each of the Euler angle rates ( _ b, _ b, _ b): !bnb = !bnb2 +!bb2b1 +!bb1b = Cbb1Cb1b2!b2nb2 +Cbb1!b1b2b1 +!bb1b = Cbb1Cb1b2 0 BB BB BB @ 0 0 _ b 1 CC CC CC A +Cbb1 0 BB BB BB @ 0 _ b 0 1 CC CC CC A + 0 BB BB BB @ _ b 0 0 1 CC CC CC A : (4.32) Using Eq. (4.28) and (4.31), it can be shown that Cb1b2 = R2( b) and Cbb1 = R1( b). Then, substituting back in and arranging, the rotation rate of the body-to-navigation frame transformation matrix is !bnb = 0 BB BB BB @ 1 0 sin b 0 cos b cos b sin b 0 sin b cos b cos b 1 CC CC CC A 0 BB BB BB @ _ b _ b _ b 1 CC CC CC A ; (4.33) where the Euler angle rates are given by the time rate of change of the trim pro les as discussed in Sec. 3.3.2. Sec. 4.3.3.1 dicusses the quaternion equivalent to this section, which is used for more stable numerical integration. 147 4.3 Inertial Navigation Equations In this section, the inertial navigation equations are derived and their nu- merical mechanization is presented. The navigation equations illustrate how the measured speci c forces and angular rates are integrated into meaningful position, velocity, and attitude information. The derivation of the equations are rst pre- sented for an arbitrary frame and then speci ed to the n-frame. Quaternions are also discussed as a means of integrating the body to navigation rotation matrix. Lastly, the classical fourth-order Runge-Kutta algorithm is described as it is used to integrate the inertial navigation system (INS) states and the rotation quaternion. 4.3.1 Arbitrary Frame Equations Before focusing on the rotating n-frame navigation equations, the dynamic equations in an arbitrary frame will be investigated. This subsection will thus illuminate the acceleration sources present for any possible coordinate system. The next subsection will then use these results in the derivation of the n-frame navigation equations that were implemented in this work. The derivation starts with a point in an arbitrary frame, ra, and the transfor- mation from an initial arbitrary frame to an inertial frame, Cia, so that ri = Ciara: (4.34) Then, using the chain rule to di erentiate Eq. (4.34) with respect to time gives _ri = Cia aiara +Cia_ra; (4.35) 148 where Eq. (4.17) on pg. 142 is used for the rotation matrix derivative term. Taking the derivative with respect to time again and arranging gives ri = Cia ra + 2Cia aia_ra +Cia aia aia + _ aia ra = ai + gi; (4.36) where Eq. (4.4) on pg. 135 is used for the second equality. Solving for the arbitrary- frame acceleration now gives ra = 2 aia_ra aia aia + _ aia ra + aa + ga; (4.37) where aa Cai ai and ga Cai gi. The rst term in Eq. (4.37) is the Coriolis acceleration due to an object having a velocity in a rotating frame. The rst part of the parenthetic term is the centrifugal acceleration felt by the object as the frame rotates, and the second term in the parenthesis is due to the frame?s angular acceleration. Equation (4.37) is a second order di erential equation describing the acceler- ation of a system that has the current velocity, _ra, and position states, ra, being forced be an aa and ga. Most second order systems are solved by splitting the system into two rst order systems as such: d dt_r a = 2 a ia_r a aia aia + _ aia _ra + aa + ga; d dtr a = _ra: (4.38) The system states (position and velocity, along with the transformation matrix) can now be solved using a numerical integration algorithm for rst order systems (Sec. 4.3.3). For strapdown accelerometers, the speci c force in the a-frame is calculated 149 by transforming the accelerometer readings in the b-frame to the arbitrary frame with the current estimated Cab rotation matrix: aa = Cabab: (4.39) The rotation matrix, Cab , is integrated according to Eq. (4.17) on pg. 142, and the rotation rate vector is given by !bab = !bib Cba!aia; (4.40) where !bib is the gyro measurement and !aia is the calculated rotation rate of the arbitrary frame to the inertial with coordinates in the a-frame using the current INS states. 4.3.2 Navigation Frame Equations This section uses the results from the previous section to derive the North- East-Down (n-frame) navigation equations. The navigation frame mechanization is noticeably di erent than other mechanizations1,2 (namely the i-frame or e-frame) because the velocity variables, vn = (vN;vE;vD)T, are not the time rate of change of the position variables, rn = ( ; ;h)T. Instead, the n-frame velocities are de ned as vn = Cne _re; (4.41) where _re is the ECEF velocity vector. The n-frame position is similarly de ned. Taking the derivative of Eq. (4.41) and using Eq. (4.17) for the rotation matrix derivative, _vn = Cne ene_re +Cne re: (4.42) 150 Substituting Eq. (4.37) on pg. 149 with the arbitrary frame now being the ECEF frame, noting that the Earth rotates at a constant rate ( _ eie = 0) and, by de nition, _re = Cenvn, we now have _vn = Cne eneCenvn 2Cne eieCenvn Cne eie eiere + an + gn; (4.43) where an = Cne ae and gn = Cne ge. Next, it can be shown that Cne eneCen = nne and similarly, Cne eieCen = nie using Eq. (4.16) on pg. 142. By inspection, one also has nne = nen. So, _vn = ( nen + 2 nie) vn + an + gn; (4.44) where gn gn Cne eie eiere; (4.45) which is the sum of the gravitational acceleration due to mass attraction and the centrifugal acceleration, more commonly referred to as gravity. Lastly, we can rear- range the angular rate matrices because nie = nin + nne, so nen + 2 nie = nen + nie + ( nin + nne) = nen + nie + ( nin nen) = nie + nin: (4.46) Substituting back into Eq. (4.44), the velocity navigation equations are nally _vn = ( nin + nie) vn + an + gn; (4.47) where the gravity term is de ned in Eq. (4.45). The skew symmetric matrices that multiply the velocity vector are found as fol- lows. The navigation frame to inertial frame rotation rate,!nin, is given by Eq. (4.23) 151 on pg. 144, and the ECEF to ECI rotation, !nie, is calculated by multiplying the transpose of Eq. (4.21) and!eie from Eq. (4.19) on pg. 143. With the skew symmetric de nition, Eq. (4.18), the matrices are ( nin + nie) = 0 BB BB BB @ 0 ( _ + 2!e) sin _ ( _ + 2!e) sin 0 ( _ + 2!e) cos _ ( _ + 2!e) cos 0 1 CC CC CC A : (4.48) The velocity navigation equations are then _vn = 0 BB BB BB @ _vN _vE _vD 1 CC CC CC A = 0 BB BB BB @ ( _ + 2!e)vE sin +vD _ +aN +gN ( _ + 2!e)(vN sin +vD cos ) +aE +gE vN _ ( _ + 2!e)vE cos +aD +gD 1 CC CC CC A : (4.49) The navigation equations for the position states are much simpler to derive. Starting with the navigation to ECEF position coordinate transformation, Eq. (4.24) on pg. 144, one substitutes the de nitions of Ne, Eq. (4.25), and Me, Eq. (4.27). Then di erentiating with respect to time and pre-multiplying by Cne , the transpose of Eq. (4.21), one has vn = 0 BB BB BB @ vN vE vD 1 CC CC CC A = 0 BB BB BB @ _ (Me +h) _ (Ne +h) cos _h 1 CC CC CC A : (4.50) Solving for the time rate of change of latitude, longitude, and altitude, the position navigation equations are: _rn = 0 BB BB BB @ _ _ _h 1 CC CC CC A = 0 BB BB BB @ vN Me +h vE (Ne +h) cos vD 1 CC CC CC A : (4.51) 152 With the velocity navigation equations, Eq. (4.49), there are now six nonlinear di erential equations that will be used to integrate the accelerometer readings into the navigation frame velocity and position components. However, the strapdown IMUs are rigidly xed to the body frame, so the rota- tion matrix Cnb must also be simultaneously computed to transform the accelerom- eter measurements into the n-frame (by an = Cnb ab). The di erential equation for this rotation matrix is given by Eq. (4.17) on pg. 142, which speci ed to the body-to-navigation frame transformation is _Cnb = Cnb bnb: (4.52) The angular rates for the rotation rate are found by subtracting the calculated navigation-to-inertial rate, !nin (Eq. (4.23) on pg. 144), from the gyro measurement, !bib: !bnb = !bib Cbn!nin: (4.53) The speci cs of the rotation matrix integration will be addressed in the mechaniza- tion section (Sec. 4.3.3). The gravity acceleration vector, Eq. 4.45 on pg. 151, is linearly interpolated from the gridded, precomputed gravity map using the full 360 degree and order EGM95 spherical harmonic model for the gravitational acceleration. The geopot97 program used for the spherical harmonic calculations adds the centrifugal potential, acceleration, and gradients by default to give the full gravity quantities. For the gravitational gradient portion of the precomputed maps, the centrifugal gradient contribution was commented out from the source code. This was also done for the 153 global gravitational gradient maps shown in this dissertation. 4.3.3 Navigation Mechanization This subsection addresses the numerical integration of the body-to-navigation frame transformation matrix and the six nonlinear navigation equations. The rota- tion matrix integration is performed by transforming the matrix into a quaternion, and a summary of its de nition and use is presented rst. The classical fourth-order Runge-Kutta method used to simultaneously integrate the quaternion, velocity, and position information is then described. 4.3.3.1 Body-to-Navigation Frame Quaternion Quaternions are another way to present orientation information in lieu of a transformation matrix. A quaternion is a four-element vector similar to a complex number with three imaginary components. The three imaginary parts essentially make up a single axis of rotation and the lone real part de nes the magnitude of the rotation about that axis.144,145 This is in contrast to the transformation matrix that can be decomposed into three rotations about three orthogonal axes, see Eq. (4.8) on pg. 140. (For a thorough survey of attitude represenations and their history, one may consult Phillips et al.145,146) The transformation between the rotation matrix and a quaternion is straight- forward and will be given without derivation (See Jekeli,1 pg. 13{18 for a more detailed discussion). If the ith row and jth column component of Cnb is ci;j, then the 154 quaternion equivalent isy q = 0 BB BB BB BB BB @ 1 2 p1 +c 1;1 +c2;2 +c3;3 (c2;3 c3;2) = (4q1) (c3;1 c1;3) = (4q1) (c1;2 c2;1) = (4q1) 1 CC CC CC CC CC A ; (4.54) where q1 is the rst component of the quaternion and represents the magnitude of the rotation. The remaining quaternion components represent the elements of the single rotation axis. Given a quaternion, the equivalent transformation matrix is can be found to be Cnb = 0 BB BB BB @ q21 +q22 q23 q24 2(q2q3 +q1q4) 2(q2q4 q1q3) 2(q2q3 q1q4) q21 q22 +q23 q24 2(q3q4 +q1q2) 2(q2q4 +q1q3) 2(q3q4 q1q2) q21 q22 q23 +q24 1 CC CC CC A ; (4.55) where qi is the ith component of the quaternion. The quaternion equivalent of the rotation matrix time derivative is _q = 12Aqq; (4.56) where Aq is the 4 4 skew symmetric matrix for the angular rate !bnb, Eq. (4.53) on pg. 153: Aq = 0 BB BB BB BB BB @ 0 !1 !2 !3 !1 0 !3 !2 !2 !3 0 !1 !3 !2 !1 0 1 CC CC CC CC CC A ; (4.57) yThis quaternion de nition is based on Jekeli,1 and it should be noted that other references may de ne the real and imaginary components of the quaternion in a slightly di erent manner. 155 and where !i is the ith component of !bnb. The body-to-navigation frame rotation matrix is integrated by rst converting the direction cosine matrix into a quaternion using Eq. (4.54). Equations (4.56), (4.57), and (4.53) on pg. 153 for the calculated rotation rate are used to integrate the quaternion elements simultaneously with the position and velocity states using the Runge-Kutta algorithm in the next section. After the integration is completed, the new quaternion is transformed back into a direction cosine matrix using Eq. (4.55). 4.3.3.2 Fourth-Order Runge-Kutta Integration This subsection describes the algorithm used to integrate the velocity, position, and attitude quaternion states. Numerical integration of a di erential equation is typically conducted by estimating the slope of the dynamical equation over a small interval and then using an Eulerian update. The choice of how one estimates the slope over the interval can be arbitrary, but the desire is to minimize the error between the true and estimated slopes. The error is often found by taking the Taylor series expansion of the function, and then a nite number of terms are used to minimize the resultant Taylor series error. The most common numerical integration algorithms are of the Runge-Kutta family. All Runge-Kutta methods are of the form (Chapra and Canale,147 pg. 695): xk+1 = xk + f(xk;tk; tk) tk; (4.58) where xk is the system state vector to be numerically integrated, t is the inte- gration time step, and the subscripts denote the discrete time epoch. The term f 156 is equivalent to the continuous dynamics of the system (i.e., _x) and is referred to as the increment function, which may be an explicit function of the current states, time, and time step. The increment function can also be envisioned as the estimated slope between the state vector at two epochs. Assuming there is no explicit depen- dence on time, as is the case for the dynamical systems in this work, and the time step is constant, the Runge-Kutta increment functions have the form (Chapra and Canale,147 pg. 695): f(xk; t) = a1k1 +a2k2 + +ankn; (4.59) where the coe cients ai and the vectors ki (the slope evaluations at di erent con- ditions) are chosen to minimize the Taylor series error. The most popular Runge-Kutta method is the classical Fourth-Order method (Chapra and Canale,147 pg. 701): xi+1 = xi + 16 (k1 + 2k2 + 2k3 + k4) t; (4.60) and the slope evaluations are performed according to k1 = f(xk) k2 = f(xk + 12k1 t) k3 = f(xk + 12k2 t) k4 = f(xk + k3 t): (4.61) Thus, there are four slope evaluations for each integration interval. The rst eval- uation is done at the original point, the second and third slopes are calculated at 157 Figure 4.5: Fourth-Order Runge-Kutta Schematic, from Ref. [147] the estimated midpoint, and the last slope is found from the estimated nal point (as represented in Fig. 4.5, from Chapra and Canale,147 pg. 701). These estimated slopes are then linearly combined using Eq. (4.60) to nd the state vector at the new time epoch, xk+1, from Eq. (4.58) with fourth-order accuracy in time. The state vector in this work, x, consists of the navigation position and veloc- ity states ( ; ;h;vN;vE;vD)T, the body-to-navigation frame quaternion, q, and the IMU and GPS states, which will be described in Sec. 4.5.1 and D.4, respectively. The slope evaluation vector, f, corresponds to the navigation equations given by Eq. (4.49) on pg. 152 and (4.51) on pg. 152, the quaternion time derivative given by Eq. (4.56) on pg. 155, and the IMU state rates in Sec. 4.5.2. The GPS states are integrated slightly di erently using Eq. (D.41) and (D.42) because of the process noise added to these states. The gravity vector is also calculated at each slope eval- uation for the given position using the stored gravity eld map. And the simulated IMU speci c forces, Eq. (3.84) on pg. 128, and angular rates, Eq. (3.88), pg. 129, 158 are assumed to be constant for each integration interval. 4.4 Inertial Navigation Error Equations To derive the necessary linear error dynamics for the extended Kalman lter, the navigation error equations will be di erentially perturbed. This perturbation technique follows the error de nition in Eq. (C.38) on pg. 323, which can be rewritten as bx(t) = x(t) + x(t): (4.62) This equation de nes the estimated states, bx, as the sum of the truth states, x, and a linear perturbation, or error, from the truth, x. It should be noted that the operator is equivalent to the perturbation or linear error about the true non- linear navigation equations. Also, the operator is communicative with the time di erential operator, d=dt (Jekeli,1 pg. 141). 4.4.1 Position Error Equations The position error equations are derived by a linear perturbation of the position navigation equations. To better explain the perturbation procedure, rst take a Taylor series expansion of the position dynamics, Eq. (4.51), about the true position and velocity states and de ne this quantity as the estimated position dynamics: _brn = _rn + @_rn @rn x=bx (brn rn) + 12 @2 _rn (@rn)2 x=bx (brn rn)2 + + @_rn @vn x=bx (bvn vn) + 12 @2 _rn (@vn)2 x=bx (bvn vn)2 + : (4.63) 159 Then, taking only the rst order terms and rewriting in terms of errors, the per- turbed position dynamics are _brn _rn = _rn = + @_rn @rn x=bx rn + @_rn @vn x=bx vn: (4.64) Introducing the shorthand notation of F_ab for the partial derivative matrix of _a with respect to b, the error dynamics can be written as _rn = F_rr rn +F_rv vn: (4.65) And the coe cient matrices for the partial derivatives of the position dynamics with respect to position and velocity are F_rr @_r n @rn = 0 BB BB BB @ @ _ @ @ _ @ @ _ @h @_ @ @_ @ @_ @h @_h@ @_h@ @_h@h 1 CC CC CC A ; F_rv @_r n @vn = 0 BB BB BB @ @ _ @vN @ _ @vE @ _ @vD @_ @v N @_ @v E @_ @v D @_h@v N @_h@v E @_h@v D 1 CC CC CC A : (4.66) For trajectories along an approximately constant latitude, the radii of curvature, Me and Ne, are constant, and the coe cient matrices for the position error can be derived as F_rr = 0 BB BB BB @ 0 0 vN (Me +h)2 vE sin (Ne +h) cos2 0 vE (Ne +h)2 cos 0 0 0 1 CC CC CC A (4.67) F_rv = 0 BB BB BB @ 1M e +h 0 0 0 1(N e +h) cos 0 0 0 1 1 CC CC CC A : (4.68) For the attitude and velocity error dynamics that follow, the linear perturba- tions will be calculated directly without an explicit Taylor series expansion. 160 4.4.2 Attitude Error Equations The errors in the attitude (i.e., the rotation matrix) equations will be ad- dressed next because these results will be used in the velocity error derivations of the following section. Three approaches are dominant in the literature to account for at- titude errors: the -angle,148,149 -angle,149,150 and quaternion formulations.151,152 The -angle approach uses the attitude errors between the estimated and true body frame while the -angle approach uses the angular errors between the navigation frame and the estimated platform frame as its error states, which is more typi- cally used for stabilized IMU suites. It has been shown that these two methods are equivalent by simulation, and are analytically related by:149 = + ; (4.69) where is the angular error between the estimated n-frame and true b-frame. Between these formulations, the -angle approach was chosen because it produces simpler, and thus computationally faster, error dynamics.149,153 The quaternion formulation has also been shown to be equivalent to the - angle error with an additional scale factor proportional to acceleration, which can be removed by normalizing the quaternion.151,152 The quaternion formulation was not chosen because its error dynamics are much more complicated than the other two approaches and it requires four, instead of three, error states with the added unit normalization constraint. As mentioned above, is the angular error between the estimated and true body frames, therefore its derivation comes from a perturbation about the true 161 body frame.1,148,153 Perturbing the body-to-navigation rotation matrix di erential, Eq. (4.52) on pg. 153, gives _Cnb = Cnb bnb +Cnb bnb: (4.70) Also, if the estimated rotation matrix is a small rotation from the true Cnb , then according to Eq. (4.10) on pg. 141, bCnb = (I n)Cnb = Cnb nCnb; (4.71) where n is the skew symmetric matrix representation of (), see Eq. (4.18) on pg. 142. Using the de nition that the rotation error is the estimated rotation minus the truth, the body-to-navigation rotation error matrix is Cnb bCnb Cnb = nCnb: (4.72) Now, di erentiating this error equation with respect to time gives an alternative relation for the rotation matrix error dynamics: _Cnb = _ nCnb n _Cnb = _ nCnb nCnb bnb; (4.73) where Eq. (4.52) on pg. 153 has been used for _Cnb in the last equality. Equating Eq. (4.70) and Eq. (4.73): Cnb bnb +Cnb bnb = _ nCnb nCnb bnb (4.74) Cnb bnb = _ nCnb; (4.75) where Eq. (4.72) was used to cancel the two terms between lines. Solving for _ n, _ n = Cnb bnbCbn = nnb; (4.76) 162 where Eq. (4.16) on pg. 142 is used for the second equality. Therefore, the right hand side is a skew-symmetric matrix, see Eq. (4.18), with elements equal to the vector !nnb = Cnb !bnb.154 The vector equivalent is then _ n = Cnb !bnb: (4.77) To get the body-to-navigation frame angular rate errors, we perturb Eq. (4.53) on pg. 153 to get !bnb = !bib Cbn!nin Cbn !nin: (4.78) The rotation error matrix Cnb is equivalent to Cbn = ( Cnb )T = ( nCnb )T = Cbn ( n)T = Cbn n; (4.79) where the property that a symmetric matrix is equal to the negative of its transpose has been used. Substituting Eq. (4.79) and (4.78) into (4.77), one has _ n = Cnb !bib + n!nin + !nin: (4.80) It can readily be shown that n !nin is equivalent to !nin n, see Eq. (4.97). Therefore, after some slight rearranging, the small error angles between the esti- mated body frame and the true body frame are governed by the dynamics: _ n = !nin !nin n Cnb !bib: (4.81) The rst term on the right hand side is the error due to the incorrect rotation of the navigation frame with respect to the inertial frame in n-frame coordinates, and is a function of the position and velocity states, see Eq. (4.23). The second term 163 is implemented as a skew symmetric matrix of !nin that couples the estimated n-to-i frame rotation rate with the angular errors, and the third term is the gyro sensor errors rotated into the n-frame. Section 4.5.1 details the simulated gyro error models. The !nin error is calculated by perturbing Eq. (4.23) on pg. 144, repeated here with the position rates substituted: !nin = 0 BB BB BB @ vE Ne +h +!e cos vNM e +h vE tan N e +h !e sin 1 CC CC CC A : (4.82) Because the angular rate is a function of the position and velocity states, its per- turbed (i.e., linearized error) value has the from !nin = F_ r rn +F_ v vn; (4.83) and the partial derivatives for the coe cient matrices are F_ r @! n in @rn = 0 BB BB BB @ !e sin 0 _ cos N e +h 0 0 _ M e +h _ cos 0 _ sin N e +h 1 CC CC CC A (4.84) F_ v @! n in @vn = 0 BB BB BB @ 0 1N e +h 0 1M e +h 0 0 0 tan N e +h 0 1 CC CC CC A ; (4.85) where the position dependence on Ne and Me have been neglected (Jekeli,1 pg. 154), and the position rates have been used for brevity, see Eq. (4.51) on pg. 152. 164 Using Eq. (4.81) to estimate the rotation errors, one can attempt to correct the estimated body-to-navigation rotation matrix, Cnb (Shin,155 pg. 46). With the small rotation assumption, Eq. (4.71) on pg. 162 can be solved for the true rotation matrix as Cnb = (I n) 1 bCnb: (4.86) To rst order, the true (or more accurately, the updated estimate of the) rotation matrix is bCn+b = Cnb (I + n) bCn b ; (4.87) where Eq. (4.11) on pg. 141 has been used. This relation is used within the Extended Kalman Filter to update the estimated rotation matrix after each measurement. Also, the n-angle errors are reset to zero after the update because of the assumption that there are no known systematic errors. 4.4.3 Velocity Error Equations The velocity error equations are also derived by a linear perturbation analysis. Beginning with the velocity dynamics, Eq. (4.47) on pg. 151, the velocity error dynamics have the form _vn = [( nin + nie) vn] + an + gn: (4.88) We will rst focus on the Coriolis error, which is a function of the position and velocity states only. Using Eq. (4.49) on pg. 152, neglecting the accelerometer read- ings and gravity acceleration, and substituting in the position rates from Eq. (4.51) 165 on pg. 152: ( nin + nie) vn = 0 BB BB BB @ v 2 E tan N e +h 2!evE sin + vNvDM e +h vNvE tan +vEvD Ne +h + 2!evN sin + 2!evD cos v 2 NM e +h v 2 EN e +h 2!evE cos 1 CC CC CC A : (4.89) Because of the dependence on position and velocity, the Coriolis error has the form [( nin + nie) vn] = F_vr rn +F_vr vn (4.90) = @[( n in + n ie)v n] @rn rn @[( n in + n ie)v n] @vn vn: Taking the partial derivates and assuming N and M are again constant, @[( n in + n ie)v n] @rn = (4.91)0 BB BB BB B@ v 2 E (Ne +h) cos2 2!evE cos 0 v2E tan (Ne +h)2 vNvD (Me +h)2 vNvE (Ne +h) cos2 + 2!e(vN cos vD sin ) 0 (vN tan vD)vE (Ne +h)2 2!evE sin 0 v 2 N (Me +h)2 + v2E (Ne +h)2 1 CC CC CC CA ; @[( n in + n ie)v n] @vn = (4.92)0 BB BB BB @ vD Me +h 2vE tan Ne +h 2!e sin vN Me +h vE tan Ne +h + 2!e sin vN tan +vD Ne +h vE Ne +h + 2!e cos 2vNM e +h 2vEN e +h 2!e cos 0 1 CC CC CC A : Using the position rates for _ and _ , and de ning the mean curvature Re pMeNe, 166 the partial derivatives can be simpli ed to F_vr = @[( n in + n ie)v n] @rn (4.93) = 0 BB BB BB @ _ cos + 2!e cos vE 0 vE _ sin vD _ R e +h vN _ cos + 2!e (vN cos vD sin ) 0 _ (vN sin +vD cos ) Re +h 2!evE sin 0 _ 2 + _ 2 cos2 1 CC CC CC A ; F_vv @[( n in + n ie)v n] @vn = 0 BB BB BB @ vD Re +h 2( _ +!e) sin _ ( _ + 2!e) sin vN tan +vDR e +h ( _ + 2!e) cos 2 _ 2( _ +!e) cos 0 1 CC CC CC A : (4.94) The velocity error dynamics due to the n-frame speci c force errors are at- tributed to two parts: the body-to-navigation frame rotation and the accelerometer sensors. Symbolically: an = Cnb ab +Cnb ab: (4.95) The rotation matrix error is given by Eq. (4.72) on pg. 162, so Cnb ab = nCnb ab = nan: (4.96) It can easily be veri ed that nan = 0 BB BB BB @ an2 n3 an3 n2 an1 n3 + an3 n1 an1 n2 an2 n1 1 CC CC CC A = 0 BB BB BB @ 0 an3 an2 an3 0 an1 an2 an1 0 1 CC CC CC A = an n: (4.97) The error from the speci c force term is now an = an n +Cnb ab; (4.98) 167 where ab are the errors caused by the accelerometer sensor triad, which will be derived in Sec. 4.5.1. The gravity error term is modeled by taking into account only the error in the gravitational acceleration. Because Earth?s rotation rate is known with high precision, and the position error is typically small, the gravity error due to the centripetal acceleration is neglected. This term could be added in future work. Using the full gravitational gradient tensor, the gravity error is modeled as: gn gn = n 0 BB BB BB @ Re 0 0 0 Re 0 0 0 1 1 CC CC CC A rn Fgr rn; (4.99) where the middle matrix approximates the Jacobian between the North-East-Down positions and the navigation position states, and the mean curvature of Earth at the current estimated position is de ned as Re pMeNe. The values of the grav- itational gradients are calculated by linearly interpolation from the stored, gridded gravity map to the current estimated position. 4.4.4 Summary The linearized error dynamics are used to update the lter error covariance matrix. The error dynamics are not actually integrated numerically, instead they are only used to compute the error state transition matrix as discussed at the end of Sec. C.5.1. The position, velocity, and attitude errors constitute nine of the twenty- six states that the lter estimates in its covariance matrix. The linear errors of these 168 states can be summarized as d dt 0 BB BB BB @ rn vn n 1 CC CC CC A = _xINS = 0 BB BB BB @ F_rr F_rv 0 F_vr +Fgr F_vv [ban ] F_ r F_ v [b!nin ] 1 CC CC CC A xINS + 0 BB BB BB @ 0 0 bCnb 0 0 bCnb 1 CC CC CC A 0 BB @ ab !bib 1 CC A; (4.100) where Eq. (4.67) and (4.68) de ne the position error matrices, F_rr and F_rv; Eq. (4.93), (4.99), and (4.94) de ne the velocity errors, F_vr, Fgr, and F_vv; and Eq. (4.84) and (4.85) de ne the attitude error matrices, F_ r and F_ v. The term [ban ] is a skew symmetric matrix of the lter-corrected speci c force measurements rotated from the body frame to the navigation frame using the current estimate of Cnb , i.e. ban = bCnbbab: (4.101) The [b!nin ] term is another skew symmetric matrix whose components are calculated from Eq. (4.23) using the current estimates of the position and velocity states. The last two terms on the right of Eq. (4.100) are due to the inertial mea- surement unit errors which are rotated into the navigation frame by bCnb . The lter is augmented with the IMU error states as described in the following section. This allows the lter to reduce some of the IMU errors in- ight with the information from the external INS aid. Also, the strapdown gravity gradiometer instrument requires an estimate of the angular velocity of the body frame which is calculated from the estimated gyro readings as described later in Sec. 5.2.1. 169 4.5 Inertial Measurement Unit Model This section presents the details of how the measured and lter-corrected IMU readings are simulated. The suite of inertial measurement units in this work con- sists of a triad of accelerometers and a triad of gyros. IMUs are able to produce uninterrupted speci c force and angular rate measurements in highly dynamic envi- ronments. Unfortunately, direct integration of these measurements (dead-reckoning navigation) results in navigation state error growth that may reach unacceptably high levels and even loss of vehicle. To ensure safe and reliable performance, an external aid with a nite accuracy is often blended into the IMU-only navigation solution through a Kalman lter approach. However, to produce an optimal Kalman gain, the IMU error sources must be modeled accurately. This next subsection describes the various IMU error sources and how they are modeled for this work. The following subsection surveys the current state-of-the-art in navigation and tactical grade inertial measurement unit speci cations and details how these values are used for the simulated IMU readings. 4.5.1 IMU Error Model The sources of error from an inertial measurement unit include uncompensated scale factors, biases, thermal e ects, nonlinearities, misalignments, non-orthogonal- ities, and electronic measurement noise.1,2 Each of these error sources are described as follows. Scale Factor errors cause measurement errors that are proportional to the 170 true speci c force of angular rate measurement. Theses errors are usually time invariant and are modeled as random constants. Bias error, also referred to as turn-on bias, bias repeatability, bias stability, drift bias (for gyros), or o set (for accelerometers), is the initially o set constant sensor reading that changes each time the instrument is turned on. Because of this, it is often modeled as a simple random constant. Thermal E ects in the mechanical properties of the IMUs can result in the scale factor and/or bias error to vary over time. In these cases, a correlated (Gauss- Markov) random walk process may be added to the random constant assumption of the scale factor and/or bias error. Nonlinearities in the IMUs can cause the linear input/output relationship of the sensor to vary o nominal, particularly during periods of high dynamics. These errors are often modeled by a sum of linear and/or quadratic accelerations with random constant coe cients (Farrell and Barth,2 pg. 216 & 218). For example, the accelerometer nonlinearity term may be modeled as nlaj = kaj;1(aa1)2 +kaj;2(aa2)2 +kaj;3(aa3)2 +kaj;4(aa1aa2) +kaj;5(aa1aa3) +kaj;6(aa2aa3): (4.102) where the a terms are the components of the truth speci c forces and the k terms are random constants. Misalignments and Non-orthogonalities result when the IMU package is not mounted perfectly along the body axes or when the IMU sensors themselves are not aligned su ciently with respect to each other. Since strapdown IMUs and their package are assumed to by rigidly attached to the vehicle, these errors are 171 modeled as a small error rotation matrix with random constants. For example, if the accelerometers were misaligned and non-orthogonal to each other in an a-frame, the body-frame speci c forces would be estimated as eab = (I a)eaa; (4.103) where a = 0 BB BB BB @ 0 d1 d2 d3 0 d4 d5 d6 0 1 CC CC CC A ; (4.104) and the d terms are random constants. Because of the non-orthogonalities, the a matrix is not necessarily skew symmetric. Instrument Noise is usually due to high frequency electronic noise that varies from each measurement reading. Since these variations happen at a frequency faster than the update rate of the sensor, they are often modeled as zero-mean white noises. For the gyro noise in this work a rst order Gauss-Markov is tuned with a small enough time constant that it approximates white noise. The accelerometer noise is modeled as white. Including all the errors above, the measured speci c force sensed by an ac- celerometer in its local a-frame is2 eaaj = 1 SFaj aaj baj nlaj waj ; (4.105) where aaj is the truth speci c force measurement, SFaj is the truth scale factor, baj is the turn-on bias, nlaj is the measurement nonlinearity as de ned above, and waj is the white noise of the sensor. The triad of accelerometer speci c forces would then be 172 estimated in the body frame by Eq. (4.103) using an estimate of the misalignments and non-orthogonalities. The gyro measurements would be implemented similarly. The issue with including all the error terms is that the Kalman lter must account for each of these errors to operate optimally. This entails augmenting the lter state vector with each coe cient in each error term, which would drastically increase the size of the lter and the computational costs. As a compromise, it is common to include only the largest sources of error such as IMU biases and noise, and to a lesser extent scale factors. For this work, IMU scale factor and bias states are augmented to the lter state vector along with a noise state for each of the gyros. Originally, misalignments and accelerometer nonlinearities were implemented into the simulation to provide uncompensated errors to the system that would test the lter robustness. Unfortunately, some of the INS/GGI simulations reduced the tilt errors to the point that the uncompensated misalignments caused divergence to occur. Therefore these errors were turned o for all Monte Carlo simulations presented. The non-orthogonalities and gyro nonlinearities are not widely reported in the surveyed IMU speci cations and were thus never pursued for implementation into the simulation. Following the assumption of no misalignments or non-orthogonalities, the IMUs are all aligned with the body frame axes. It is also implicitly assumed that the IMUs either reside at the center of mass of the vehicle or have already had compensation for any lever arm e ects on the readings. Therefore, along with the 173 neglection of the nonlinearities, the accelerometer speci c force measurement is eabj = 1 SFaj abj baj waj ; (4.106) and the gyro measurement is similarly, e!bib;j = 1 SFgj !bib;j bgj ngj ; (4.107) where ngj is the noise state that approximates a white noise process. The lter estimates the scale factors and biases of the IMUs (and the gyro noise) so that it assumes the IMU readings are eabj = 1 cSFaj babj bbaj ; (4.108) where the noise is compensated for in the error process noise matrix, Qk, and e!bib;j = 1 cSFgj b!bib;j bbgj bngj : (4.109) These relationships can then be solved for the estimated speci c forces and angular rates: babj = ea b j 1 cSFaj +bbaj; (4.110) b!bib;j = e! b ib;j 1 cSFgj +bbgj +bngj: (4.111) These estimated measurements are then processed by the INS when integrating the navigation states and computing the linearized navigation errors. In order for the lter to update its estimate of the IMU errors (which are ini- tialized to zero in the estimated state vector) their linearized errors must be derived 174 and the IMU states need to be added to the total lter state vector. The accelerom- eter speci c force error can be derived by subtracting the truth measurement (from rearranging Eq. (4.106)) from the lter-estimated measurement: abj = babj abj = eabj 1 cSFaj +bbaj ! eabj 1 SFaj +b a j +w a j ! = (1 SF a j ) (1 cSF a j) (1 cSFaj)(1 SFaj ) eabj + baj waj: (4.112) Noting that scale factors are on the order of 1 10 4 for the surveyed IMUs, the accelerometer errors can be approximated as abj ( SFaj )eabj + baj waj: (4.113) The gyro error can be similarly found, but with the slight di erence of the noise state: !bib;j ( SFgj )e!bib;j + bgj + naj: (4.114) The IMU error states added to the Kalman lter state vector are then: xa = SFax; SFay; SFaz; bax; bay; baz T (4.115) xg = SFgx; SFgy; SFgz; bgx; bgy; bgz; ngx; ngy; ngz T : (4.116) The accelerometer errors can also be re-written in vector form as ab = diag(eab); I T xa wa; (4.117) where diag(eab) is a diagonal matrix whose elements are the components of the un- corrected accelerometer measurements and I is a 3 3 identity matrix. The gyro errors in vector notation are !bib = diag(e!bib); I; I T xg: (4.118) 175 Equation (4.117) and (4.118) are then substituted back into Eq. (4.100) to produce the total linearized INS error dynamics. The dynamics for the scale factors and biases are all zero since they are modeled as random constants. The gyro noise dynamics follow the description in Sec. C.2.3 for the Gauss-Markov process estimating a white noise. In summary, d dt xg = 0; (4.119) and d dt xg = d dt 0 BB BB BB @ SFg bg ng 1 CC CC CC A = 0 BB BB BB @ 0 0 0 0 0 0 0 0 I 1 CC CC CC A xg + 0 BB BB BB @ 0 0 wg(t) 1 CC CC CC A : (4.120) The discrete implementation of the gyro errors is, see Sec. C.2.3, xg;k+1 = 0 BB BB BB @ 1 0 0 0 1 0 0 0 exp( t) 1 CC CC CC A xg;k + 0 BB BB BB @ 0 0 wgk 1 CC CC CC A ; (4.121) where the variance of the discrete driving noise is 2wg = (qwg= t) [1 exp( 2 t)]; (4.122) where qwq is the power spectral density of the gyro white noise and = 2:146=(2 t). The gyro noise states? portion of the lter discrete process noise covariance matrix is then Qk;ng = E h (wgk) (wgk)T i = 2wgI; (4.123) where I is a 3 3 identity matrix. 176 The accelerometer noise portion of the lter?s discrete process noise covariance matrix is a little more complicated to calculate. Using the velocity error dynamics, Eq. (4.100) on pg. 169, and the accelerometer errors, Eq. (4.117) on pg. 175, the acceleration error due to the accelerometer noise is _vnwa = Cnb wa: (4.124) The resultant velocity error can then be approximated by a simple Euler integration, see Eq. (C.29) on pg. 317, so that vnwa _vnwa t = Cnb wa t: (4.125) The discrete process noise covariance matrix for the velocity errors can now be calculated by Qk; vn = E h ( vnwa) ( vnwa)T i E h ( Cnb wa t) ( Cnb wa t)T i = CnbE h (wa) (wa)T i Cbn t2 = CnbCbn 2wa t2 = 2wa t2I; (4.126) where E h (wa) (wa)T i = 2waI is used in the second line. The rest of the discrete process noise covariance matrix is zero for the INS and IMU states. The GPS receiver clock states? process noise covariance is derived in Sec. D.4. 4.5.2 IMU Speci cations A survey of current IMU sensor speci cations was conducted to provide real- istic values for the simulated IMU instrument errors. Sensors manufactured by 177 Honeywell,156,157 Northrop Grumman,158 Astronautics Corporation of America?s Kearfott,159 BEI Technologies? Systron Donner,160 and the German iMAR (Iner- tiale Mess-, Automatisierungs- und Regelsysteme)161 are summarized in Tables 4.2 and 4.3, where the navigation grade IMUs are in the top portion of each table and the tactical grade sensors are in the bottom. This survey is not meant to be com- plete, but rather it is used to show the current (as of 2006) state-of-the-art in IMU sensor manufacturing. Using these IMU data sheets as reference, the navigation and tactical grade IMU speci cations used in this study are listed in Table 4.4. The modeled speci cations were chosen by taking the more aggressive performance spec- i cation in each IMU class. The units reported in the IMU speci cations can be a source of confusion, so the values used in the Monte Carlo simulations are detailed below. The truth uncompensated scale factor standard deviation is calculated by multiplying the value in Table 4.4 by 1 10 6. This term is dimensionless and needs no unit conversion. The standard deviation of the random constant accelerometer bias is converted to standard metric units by baj m=s2 = baj [ g] 1 10 6 1 9:81m=s2 1g (4.127) The 1- value for the navigation grade accelerometer turn-on bias is then 0.00147 m/s2 and the tactical grade value is 0.0491 10 3 m/s2. The gyro bias standard deviation is similarly converted to radians per second by bgj [rad=s] = bgj Hr rad 180 1Hr 3600sec : (4.128) 178 Table 4.2: Navigation and Tactical Grade Acceleromter Speci cations Man ufacturer Mo del Data Rate Bias Random W alk Scale Factor Non-Linearit y Alignmen t (H z) ( g ) ( g =p H z) (ppm ) ( g =g 2) (mr ad ) Honeyw ell HG9848 600 50 { 150 20 0.1{2 Honeyw ell HG9900 300 25 { 100 { 0.1{2 iMAR iNA V-FJI 1000 5{60 8 60{100 { { iMAR iNA V-R QH 2000 15{100 8 60{100 { { Kearfott KI-4901 { 400 85 500 15 0.1 Kearfott KI-4921 { 200 68 350 15 0.1 BEI DQI 600 200{1500 200 350 { 0.5 Honeyw ell HG1700 600 1000{2000 { 300 500 (ppm) { Honeyw ell HG1900 3600 1000 { 300 500 (ppm) { Honeyw ell HG1930 2400 4000 { 700 30 (ppm) { Honeyw ell HG9868 600 1000 { 300 20 0.4{12 iMAR iNA V-FMS 400 1000 { 300 { { Northrop LN-200 360 300{3000 { 100{5000 { { Northrop LN-600 700 200 { { { 0.1 Kearfott KI-4801 { 400 85 500 50 0.3 Kearfott KI-4920 { 400 85 500 15 0.3 179 Table 4.3: Navigation and Tactical Grade Gyro Speci cations Man ufacturer Mo del Data Rate Bias Ran dom W alk Scale Factor Non-Linearit y Alignmen t (H z) ( =H r) ( = pH r) (ppm ) (ppm ) (mr ad ) Honeyw ell HG9848 600 0.005 0.005 10 { { Honeyw ell HG9900 300 0.003 0.002 5 { { iMAR iNA V-FJI 1000 0.003{0.05 0.001{0.0025 30 5{10 { iMAR iNA V-R QH 2000 0.003{0.05 0.0013{0.005 10 { { Kearfott KI-4901 { 0.005 0.003 50 { 0.07 Kearfott KI-4921 { 0.04 0.01 75 { 0.1 BEI DQI 600 3{10 0.035 350 { 0.5 BEI DIGI-Q (est) 600 3{10 0.05 100 500 35 Honeyw ell HG1900 3600 0.3{30 0.1 150 { { Honeyw ell HG1930 2400 20 0.15 300 { { Honeyw ell HG1700 600 1{10 0.125{0.5 150 { { Honeyw ell HG9868 600 1 0.125 150 { { iMAR iNA V-FMS 400 0.75{3 0.15 300 { { Kearfott KI-4801 { 0.1{0.7 0.05 350 { 0.3 Kearfott KI-4920 { 0.1{0.7 0.05 120 { 0.3 Norh trup FOG-200 500 1 0.02 500 { { Northrop FOG-600 600 0.1{0.2 0.005 100 100{1000 1 Northrop FOG-1000 600 0.1 0.0035 100 50{1000 1 Northrop LN-200 360 0.1{1 0.02{0.15 100{500 { { Northrop LN-600 700 0.1 { { { 0.1 180 Table 4.4: Simulated Navigation and Tactical Grade IMU Speci cations Navigation Grade Tactical Grade Accel. Gyro Accel. Gyro Scale Factor 60 ppm 10ppm 300 ppm 200 ppm Turn-On Bias 15 g 0.003 =Hr 500 g 1.0 =Hr Sensor Noise 8 g=pHz 0.001 =pHr 100 g=pHz 0.02 =pHr Misalignment 0.1 mrad 0.1 mrad 0.3 mrad 0.5 mrad Non-Linearity 15 g=g2 | 20 g=g2 | and the navigation grade 1- bias is 1.45 10 8 rad/s (0.833 10 6 =s). The tactical gyro bias is 4.85 10 6 rad/s (0.278 10 3 =s). The instrument noise is sometimes referred to as the random walk parameter of the sensor because the result of integrating the noise is a random walk behavior of the sensor measurement. The actual value cited in most IMU data sheets is the square root of the white noise power spectral density (PSD). As shown in Eq. C.9 on pg. 313, the variance of a white process is equivalent to the PSD divided by the sampling rate. Therefore, the noise variance for each sensor is calculated as 2wa h m=s2 2 i = p qwa h g=pHz i 9:81 10 6m=s 1 g 2 = t; (4.129) qwg t (rad=s)2 = "pq wg[ = pHr] rad 180 r 1Hr 3600sec #2 = t; (4.130) where pqw is the parameter speci ed in Table 4.4, and the gyro noise value above is implemented as described at the end of the previous subsection. As shown in the two equations above, the value of the simulated noise variance is proportional to the frequency of the updates. IMUs are able to produce data rates up to several 181 thousand Hz, but for this work the simulation is updated at a frequency of only 20 Hz to keep the computational costs fairly low. This results in an accelerometer 1- noise level of 0.000351 m/s2 for nav.-grade IMUs and 0.00439 m/s2 for tac.-grade IMUs. The gyro noise as calculated from (4.122) and (4.130) is 0.130 10 5 rad/s (0.745 10 4 =s) for the navigation grade IMUs and 0.260 10 4 rad/s (0.149 10 2 =s) for the tactical grade gyros. (David Gaylor?s research report \Simulation of an Unaided INS in Orbit" was used as a reference for this discussion.)162 4.6 Chapter Summary This chapter reviews inertial navigation system fundamentals in Sec. 4.1 and 4.2. The INS state dynamics are then derived in Sec. 4.3, followed by their linearized errors in Sec. 4.4. These four sections follow standard INS formulations with only one modi cation for the use with a GGI aid|the velocity error dynamics due to gravity acceleration registration errors. This error source must include the full gravitational gradient tensor and a Jacobian between the coordinate frame of the gravity map and the n-frame?s latitude, longitude, and altitude states, as shown in Eq. (4.99) on pg. 168. This term di ers from typical INSs which only account for the vertical gravitational gradient of a point mass Earth. Section 4.5 next discusses the accelerometer and gyro error models simulated in this work. Section 4.5.1 reviews typical IMU error sources and details the scale factor, bias, and noise models used to corrupt the truth IMU measurements. For this work, the scale factor and bias of each accelerometer and gyro is augmented to the 182 lter state vector to allow for in- ight calibration of the IMU errors and for optimal calculation of the Kalman gain. The second modi cation to a traditional INS for GGI aiding is the need to model the gyro noise as a Gauss-Markov process that estimates white noise, and to augment the lter state with the gyro noises so that strapdown GGIs are able to estimate all gyro angular velocity errors. This results in the calculation of the gyro noise portion of the error state transition matrix having to include a full exponential term instead of just a rst order Taylor series expansion of the matrix exponential (as shown in Eq. (4.121) on pg. 176 and discussed further at the end of Sec. C.5.1). This estimation of the error state transition matrix is an improvement on Jekeli?s method94 which used a more computationally expensive procedure that truncated the matrix exponential expansion at 30 terms. Section 4.5.2 then surveys current navigation and tactical grade IMU error speci cations. The IMU errors simulated for the navigation analyses in Ch. 6 are then summarized in Table 4.4. The INS dynamics and error models presented in this chapter are used as the basis for the Monte Carlo simulations in Ch. 6 for both the INS/GGI and INS/GPS systems. The noisy GGI or GPS measurements are blended into the INS-estimated states through an extended Kalman lter as explained in App. C. The measurements are detailed in the next chapter for the gravity gradiometer instrument updates, and in App. D for the baseline GPS updates. 183 Chapter 5 Gravity Gradiometer Instrument Model As surveyed in Sec. 1.2.1, gravity gradiometer instruments (GGIs) have been manufactured in a multitude of con gurations. This chapter presents and derives the measurement observables and linearized error models for several nite-di erenced accelerometer-based GGIs. The rst section derives a general accelerometer-based GGI measurement formulation. These results are then used to derive the stabilized, rotating disc GGI measurements that many references state without derivation. Next, the envisioned non-rotating, 12-accelerometer GGI is discussed and its mea- surements are derived assuming the instrument is either strapped down to the body or stabilized with respect to inertial. The linearized strapdown and stabilized error equations are lastly comprehensively derived for the rst time so that they may be used in the extended Kalman lter simulations. 5.1 Accelerometer-Based GGI Measurements In an arbitrary, rotating \a-" frame, the sensed speci c force measured by a 3-axis accelerometer triad is, see Eq. (4.37) on pg. 149, aa = ra + 2 aia_ra + aia aia + _ aia ra ga; (5.1) 184 where ra is the accelerometer triad?s position, ga is the gravitational acceleration vector, aia is the skew symmetric matrix (see Eq. (4.18) on pg. 142) of the angular velocity from the a-frame to the i-frame, and _ aia is the corresponding angular accel- eration, all with coordinates in the a-frame. Now, assuming that two accelerometer triads are rigidly xed at a speci ed baseline (lb ra2 ra1) so that _lb = lb = 0, and the angular rates and accelerations are equivalent, these triads may be di erenced to yield a1 a2 = lb 2 aia_lb aia aia + _ aia lb g1 + g2 = a aia aia _ aia (ra2 ra1) L0a (ra2 ra1): (5.2) The second equality uses the assumption that the gravitational acceleration has a linear variation between the two accelerometer triads so that ga2 ga1 = a(ra2 ra1). This linear assumption is quite valid since typical GGI baselines are less than one meter (See Sec. 1.2.1), whereas gradient correlation distances are on the order of kilometers.87,88 With knowledge of the accelerometer positions, the gradiometer measurement can now be made: (a1 a2)=(r2 r1) = a aia aia _ aia L0a: (5.3) It is important to note that the gravitational gradient tensor cannot be directly measured; instead, the gradients are masked by centripetal and angular accelera- tions. To exploit the gravitational gradients for position aiding, these rotational e ects must be estimated or removed. 185 The angular acceleration may be easily removed, at least in theory, by aver- aging the GGI measurement with its transpose: 1 2 L0a + (L0a)T = a a ia a ia L a: (5.4) Or equivalently, the six non-symmetric tensor component measurements can be represented in vector notation as La 0 BB BB BB BB BB BB BB BB BB @ La11 La12 La13 La22 La23 La33 1 CC CC CC CC CC CC CC CC CC A = 0 BB BB BB BB BB BB BB BB BB @ a11 +!2y +!2z a12 !x!y a13 !x!z a22 +!2x +!2z a23 !y!z a33 +!2x +!2y 1 CC CC CC CC CC CC CC CC CC A ; (5.5) where !aia = (!x;!y;!z)T is the angular velocity. The angular accelerations may be removed in this manner because they are the only asymmetric term in the raw GGI measurement, whereas the gravitational gradient and the centripetal acceleration matrices are both symmetric. Another interesting corollary is that the angular accelerations can be observed directly by averaging the raw GGI measurement with the negative of its transpose: 1 2 L0a (L0a)T = _ a ia: (5.6) The observability of the angular accelerations, which may be integrated twice to produce orientation information, is the basis for using a GGI for all-accelerometer inertial navigation. For more on this topic see Sec. 1.2.3 on pg. 31 and the papers by Zorn.40,41 186 A short word on nomenclature is necessary at this point. The GGI mea- surement observable, a aia aia _ aia, is not typically given a dedicated symbol. Therefore, this work essentially follows the terminology used by Jekeli94 because his derivation was used as a starting point for many of the derivation in this chapter. The primary di erence between the nomenclature in this work and Jekeli?s is that he uses L to denote this GGI measurement. However, Jekeli notes that the gravita- tional gradient tensor is more easily observed when the angular acceleration term, _ aia, is not estimated. Thus, this work exclusively uses La a aia aia as the pri- mary GGI measurement and L0a as the raw, uncorrected measurement observable that includes the angular accelerations. To complete the generic GGI measurement formulation, two items need to be addressed: The calculation of the gravitational gradients from the stored navigation frame to the measured accelerometer frame, and the contributions to !aia. The gradient transformation will be discussed rst, and the rotation rates will follow. 5.1.1 Gravitational Gradient Transformation Matrix The pre-computed gravitational gradient tensor map is rotated transformed into the measurement frame by pre- and post-multiplication of a navigation-to- accelerometer frame direction cosine matrix, see Eq. (4.16) on pg. 142, a = Can nCan: (5.7) The coordinate transformation matrices may also be expanded to include an in- termediate gravity gradiometer instrument (\g-") frame so that rotating disc GGIs 187 may be investigated. The expanded transformation is then: a = Can nCna = CagCgn nCngCga: (5.8) This new g-frame can be specialized to the body frame for a strapdown GGI: a = CabCbn nCnbCba: (5.9) or to the inertial frame for a stabilized instrument: a = CaiCin nCni Cia: (5.10) The body-to-navigation frame rotation matrix is tracked by the INS. The Cin trans- formation is uniquely a function of position and time: Cin = CieCen = (Cei )TCen = R3( !et)R2( =2 + ) (5.11) = 0 BB BB BB @ sin cos( +!et) sin( +!et) cos cos( +!et) sin sin( +!et) cos( +!et) cos sin( +!et) cos 0 sin 1 CC CC CC A ; where Eq. (4.20) and (4.21) on pg. 143 have been used. The Cab and Cai rotations are based on the orientation of the accelerometer pairs. These gradient transformations will be further speci ed in the following sections. The coordinate transformation a = Can nCna can be alternatively written in vector notation as a = Tan n; (5.12) where a ( a11; a12; a13; a22; a23; a33)T (5.13) 188 and n ( n11; n12; n13; n22; n23; n33)T (5.14) are vectors of the diagonal and upper-diagonal elements of the gravitational gradient tensor. This new 6 6 transformation matrix, Tan, is comprised of the components of Can and can be derived in two ways as follows. The rst way to derive Tan is to symbolically compute a = Can nCna and then rearrange the components of a into a and factor out n into n. This results in Tan = 0 BB BB BB BB BB BB BB BB BB @ c211 2c11c12 2c11c13 c212 2c12c13 c213 c11c21 c11c22 +c12c21 c11c23 +c13c21 c12c22 c12c23 +c13c22 c13c23 c11c31 c11c32 +c12c31 c11c33 +c13c31 c12c32 c12c33 +c13c32 c13c33 c221 2c21c22 2c21c23 c222 2c22c23 c223 c21c31 c21c32 +c22c31 c21c33 +c23c31 c22c32 c22c33 +c23c32 c23c33 c231 2c31c32 2c31c33 c232 2c32c33 c233 1 CC CC CC CC CC CC CC CC CC A ; (5.15) where cij is the ith row and jth column of Can.y Looking at Eq. (5.15), it appears that there are some patterns imbedded in the matrix. To better understand where these patterns arise from, and to develop a more robust shorthand notation for populating Tan, start with a single component yThe transformation matrix elements here are in the opposite order as those in the author?s conference papers.163,164 This is because the elements were based on Cnb which is tracked by the INS quaternion. Here, it is based on Can which is essentially the transpose of Cnb if the GGI is a strapdown sensor, thus causing the cij in the conference papers to become cji here. Also, it was discovered after the conference proceedings that there is a small typo in these two references. The element in the rst row, second column of Tbn should be 2c11c21 in these papers. 189 of a, say aij. One can then say that this component is equal to the summation of n and some coe cients that may be written as partial derivatives: aij = 3X k=1 3X l=1 @xa i @xnk @xa j @xnl nkl; (5.16) where the partial derivative coe cients will be explained shortly. Expanding out this summation and combining coe cients for the symmetric elements of n, i.e. nkl = nlk, gives aij = @xa i @xn1 @xa j @xn1 n11 + @xa i @xn1 @xa j @xn2 + @xa i @xn2 @xa j @xn1 n12 + @xa i @xn1 @xa j @xn3 + @xa i @xn3 @xa j @xn1 n13 + @xa i @xn2 @xa j @xn2 n22 + @xa i @xn2 @xa j @xn3 + @xa i @xn3 @xa j @xn2 n23 + @xa i @xn3 @xa j @xn3 n33:(5.17) Now, the partial derivative coe cients may be thought of as the components of Can because xa = Canxn so each element of xa is xai = ci1xn1 +ci2xn2 +ci3xn3 = @xa i @xn1 xn1 + @xa i @xn2 xn2 + @xa i @xn3 xn3; (5.18) where cij is again the ith row and jth column of Can, and the second equality is using the notion that the transformation matrix is like a partial derivative of one element in one frame to another element in another state. Finally, substituting this notion that @xai=@xnj = cij into Eq. (5.17), one has aij = (ci1cj1) n11 + (ci1cj2 +ci2cj1) n12 + (ci1cj3 +ci3cj1) n13 + (ci2cj2) n22 + (ci2cj3 +ci3cj2) n23 + (ci3cj3) n33; (5.19) which is essentially a row of Tan multiplied by n. This equation is useful because it is a more convenient, and less error prone, way to implement Tan as compared to 190 Eq. (5.15) where all the rows have been written out. (Deriving this version of the Tan transformation matrix for the dissertation is how the errors in the conference paper Tbn transformations were discovered.) This Tan transformation matrix is a generic coordinate transformation for any n-frame symmetric matrix whose components are ordered as (11;12;13;22;23;33)T to the corresponding a-frame vector. The initial and nal coordinate frames are arbitrary as long as the cij coe cients used to populate the transformation matrix are consistent. In other words, if two arbitrary frames are used, say the \s" and \t" frames, then Tts can be computed using either formulation above by using the cij components of the known or calculated 3 3 Cts direction cosine matrix. Fur- thermore, this transformation matrix is computationally e cient because it exploits the tensor symmetry and is, most importantly, necessary for the linearized error formulations that follows. 5.1.2 Inertial-to-Accelerometer Frame Rotation Rate The inertial-to-accelerometer angular velocity, !aia, can be decomposed into three components: 1. The inertial-to-body frame rotation rate, !bib, which is measured by the on- board strapdown gyros. 2. The body-to-accelerometer frame rotation rate, !gbg, which accounts for rota- tion of the overall GGI with respect to the body axes. 3. The gradiometer-to-accelerometer frame rotation rate, !aga, which accounts for rotating disc GGIs, like the Bell/Textron models. 191 Mathematically, and including the appropriate coordinate frame transformations, !aia = CagCgb!bib +Cag!gbg +!aga: (5.20) This angular velocity can now be speci ed to strapdown and stabilized GGIs with either rotating or stationary accelerometers. For the strapdown GGI case, the gradiometer frame is assumed to be aligned with the body axes so that the \g" subscripts and superscripts may be replaced with \b." This results in !aia = CabCbb!bib +Cab!bbb +!aba = Cab!bib +!aba; (5.21) since Cbb = I and !bbb = 0. Therefore, a strapdown GGI must estimate the gyro measurements correctly and account for any accelerometer-to-body frame rotations and their rates to observe the gravitational gradient tensor. The stabilized GGI instead assumes that the g-frame is aligned with the inertial i-frame so that the \g-"scripts becomes \i"s: !aia = CaiCib!bib +Cai!ibi +!aga = Cai !iib +!ibi +!aga = !aga: (5.22) This equation essentially states that the rotation rate of the accelerometers with respect to the gradiometer frame is the only rate that needs to be estimated and removed from the GGI measurement in order to exploit the gradient tensor for position updates. 192 The last portion of the rotation rate that needs to be addressed is the accel- erometer-to-gradiometer frame rotation. As explained in the rst chapter, the Bell/Textron based GGIs use rotating accelerometers to modulate the gradient sig- nal to a higher frequency that exhibits lower system error. These instruments typi- cally rotate the accelerometers at a nominally constant angular velocity, !aga, in the direction out of the plane made by the four accelerometers. This results in !aga = (0; 0; !rot)T ; (5.23) where !rot is the nominal rotation rate of the disc. For non-rotating GGIs, as is the focus of this work, !aga = 0 nominally. 5.1.3 Rotating, Stabilized GGI Measurements This subsection uses the previous section?s results to derive the GGI mea- surement made by the Bell/Textron based instruments. (The Hughes Research Laboratory?s GGI also has the same resultant measurement; however, it is based on torque di erences.) The purpose of this subsection is, to attempt, to provide a straightforward derivation of this instrument?s measurement observable because most papers which reference this gradiometer either present a confusing derivation or none at all. It is the hope to also show the exibility of the previous section in deriving current and future gradiometer measurements. Beginning with the di erenced accelerometer equation, Eq. (5.2), in the accel- erometer-frame: (a1 a2) = a aia aia _ aia (r2 r1) = L0a (r2 r1): 193 Then assuming the accelerometer triads are single accelerometers whose sensitive axes are in opposite directions and are displaced from the gradiometer disc?s origin by one-half the instrument baseline in either direction (See Fig. 5.1), the accelerometer measurements and positions are aa1 = 0 BB BB BB @ 0 a1 0 1 CC CC CC A ; ra1 = 0 BB BB BB @ lb=2 0 0 1 CC CC CC A ; aa2 = 0 BB BB BB @ 0 a2 0 1 CC CC CC A ; ra2 = 0 BB BB BB @ lb=2 0 0 1 CC CC CC A : (5.24) Substituting these into the nite di erenced accelerometer equation above, 0 BB BB BB @ 0 aa1 +aa2 0 1 CC CC CC A = L0a 0 BB BB BB @ lb 0 0 1 CC CC CC A = lb 0 BB BB BB @ L0a11 L0a21 L0a31 1 CC CC CC A : (5.25) Since the accelerometers are only measuring the second entry of the this array, the GGI measurement can by found by taking that component and dividing it by the instrument baseline: (aa1 +aa2)=lb = L0a21 = a21 !ax!ay _!az ; (5.26) where the de nition of L0a has been used, and (!ax;!ay;!az)T = !aia. The rotating gravitational gradient tensor in the accelerometer frame is calcu- lated from the gradiometer frame by a = Cag gCga; (5.27) 194 Figure 5.1: Schematic of GGI with 2 Rotating Accelerometers where the gradiometer-to-accelerometer frame rotation matrix is, from Fig. 5.1, Cag = R3( g) = 0 BB BB BB @ cos g sin g 0 sin g cos g 0 0 0 0 1 CC CC CC A : (5.28) Carrying out the multiplication and only keeping the a21 element, a21 = g11 sin g cos g g12 sin2 g + g12 cos2 g + g22 sin g cos g = (1=2) ( g22 g11) sin 2 g + g12 cos 2 g; (5.29) and g = !rott, where !rot is the nominally constant rotation rate of the GGI disc. The Bell/Textron instrument is built and used on a stabilized platform, so its accelerometer-to-inertial angular velocity is just the rotation rate of the accelerom- eters with respect to the gradiometer frame. In other words, !aia = !aga = (0; 0; !rot)T: (5.30) 195 Figure 5.2: Schematic of GGI with 4 Rotating Accelerometers Thus, there is essentially no angular motion in the x-y plane of the a- or g-frames, and the only angular acceleration in the a-frame is caused by deviations from the nominal disc rotation rate. Mathematically, !x = !y = 0; (5.31) _!z = _!rot: (5.32) Therefore, after substituting Eq. (5.29) and (5.32) into Eq. (5.26), the stabi- lized, rotating disc gradiometer measurement is (aa1 +aa2)=lb = L0a21 = (1=2) ( g11 g22) sin 2 g g12 cos 2 g + _!rot: Furthermore, if a second pair of accelerometers are mounted 90 from the rst set 196 in the xa-ya frame (Fig. 5.2), they would measure (aa3 +aa4)=lb = (1=2) ( g11 g22) sin(2 g + 180 ) g12 cos(2 g + 180 ) + _!rot = (1=2) ( g11 g22) sin 2 g + g12 cos 2 g + _!rot: (5.33) Subtracting the second pair of accelerometers from the rst pair produces a gra- diometer measurement with twice the magnitude of a single accelerometer pair and with no angular acceleration errors (at least theoretically): [(aa1 +aa2) (aa3 +aa4)]=lb = ( g11 g22) sin 2 g 2 g12 cos 2 g: (5.34) The above equation is the measurement presented in many Bell/Textron GGI ref- erences with obtuse or absent derivations. However, because current laboratory gradiometers use non-rotating inertial measurement units, this type of sensor?s lin- earized error equation was not pursued. But for completeness sake, it was presented so that is could be used for reference purposes. 5.2 Modeled Twelve-Accelerometer GGI This section describes the envisioned gravity gradiometer instrument that was used in this research. The GGI is a set of three orthogonal accelerometer triads equally displaced from a central accelerometer in each of the gradiometer frame?s cardinal directions. Figure 5.3 illustrates this notional gradiometer where each ar- row represents a single accelerometer. The location and normalized speci c force measurement for each of the twelve accelerometers are: 197 Figure 5.3: Schematic of Modeled Twelve-Accelerometer GGI ra1 = ra2 = ra3 = (0; 0; 0)T; (5.35a) ra4 = ra5 = ra6 = (lb; 0; 0)T; (5.35b) ra7 = ra8 = ra9 = (0; lb; 0)T; (5.35c) ra10 = ra11 = ra12 = (0; 0; lb)T; (5.35d) 198 and aa1=a1 = aa4=a4 = aa7=a7 = aa10=a10 = (1; 0; 0)T; (5.36a) aa2=a2 = aa5=a5 = aa8=a8 = aa11=a11 = (0; 1; 0)T; (5.36b) aa3=a3 = aa6=a6 = aa9=a9 = aa12=a12 = (0; 0; 1)T: (5.36c) By di erencing pairs of accelerometers that are located at di erent locations and dividing by the instrument?s baseline distance, the full L0a tensor can be com- puted. For example, the L0a11 component of the uncorrected GGI measurement ma- trix is found by di erencing the two xa accelerometers that are separated in the xa direction, i.e. a4 and a1. Mathematically, a4 a1 = L0a (ra4 ra1);0 BB BB BB @ a4 a1 0 0 1 CC CC CC A = 0 BB BB BB @ L0a11 L0a12 L0a13 L0a21 L0a22 L0a23 L0a31 L0a32 L0a33 1 CC CC CC A 0 BB BB BB @ lb 0 0 0 1 CC CC CC A = lb 0 BB BB BB @ L0a11 L0a21 L0a31 1 CC CC CC A ; (5.37) where Eq. (5.2) on pg. 185 has been used. Because only the rst element of the array is measured by the accelerometers, one has a4 a1 = lbL0a11 ! L0a11 = (a4 a1)=lb: (5.38) Similar pairs of accelerometers may be di erenced to calculate all the elements in L0a: L0a = 1l b 0 BB BB BB @ (a4 a1) (a7 a1) (a10 a1) (a5 a2) (a8 a2) (a11 a2) (a6 a3) (a9 a3) (a12 a3) 1 CC CC CC A = a aia aia _ aia: (5.39) 199 As shown by Eq. (5.4) on pg. 186, the angular accelerations can be elimi- nated by averaging the L0a measurement with its transpose, which is equivalent to averaging o -diagonal elements. For example, (1=2)(L0a12 +L0a21) = [(a7 a1) + (a5 a2)]=(2lb) = (1=2) [( a12 !x!y + _!z) + ( a21 !x!y _!z)] La12 = [(a7 a1) + (a5 a2)]=(2lb) = a12 !x!y (5.40) The full La tensor is similarly found by La = 1l b 0 BB BB BB @ (a4 a1) 12[(a7 a1) + (a5 a2)] 12[(a10 a1) + (a6 a3)] (a8 a2) 12[(a11 a2) + (a9 a3)] sym (a12 a3) 1 CC CC CC A : (5.41) Therefore, it is apparent that the on-diagonal elements of La require only two ac- celerometers to measure, whereas the o -diagonal elements require four. The individual accelerometers? speci c force measurements of the GGI are not actually simulated. Instead, the overall measurement is computed for a given position (and orientation and rotation rate for a strapdown GGI). This modeling choice was made because the gradiometer?s manufacturer typically employs special feedback loops to correct the GGI?s internal accelerometer errors and only the overall instrument noise is speci ed. Also, because the accelerometers are stationary with respect to the gradiometer frame, it is assumed that Cag = Cga = I, as shown in Fig. 5.3. Thus, the noisy gradiometer measurement is simulated as eLa = eLg = Cgn nCng aia aia +VL; (5.42) 200 where n = n(rn) is linearly interpolated from the gridded, stored gravitational gradient map to the user?s true position, !aia is calculated according to the discussion in Sec. 5.1.2, and VL is a matrix of uncorellated, white Gaussian measurement noise of speci ed variance. It is implicitly assumed that the GGI is at the vehicle?s center of mass so that lever arm e ects are neglected. Also, higher order interpolation methods such as least-squares collocation, which optimally accounts for the error in the estimated gravity eld, could be implemented in place of the simplistic linear interpolation.95,165 The gradiometer matrix measurement, Eq. (5.42), can be alternatively written in vector form as eLa = 0 BB BB BB BB BB BB BB BB BB @ La11 La12 La13 La22 La23 La33 1 CC CC CC CC CC CC CC CC CC A = Tan 0 BB BB BB BB BB BB BB BB BB @ NN NE ND EE ED DD 1 CC CC CC CC CC CC CC CC CC A + 0 BB BB BB BB BB BB BB BB BB @ !2y +!2z !x!y !x!z !2x +!2z !y!z !2x +!2y 1 CC CC CC CC CC CC CC CC CC A + L; (5.43) where !j is the jth component of !aia, Tan is calculated by Eq. (5.15) on pg. 189 using Can = Cgn, and L is the vector of measurement noise. The estimated GGI measurement is computed similarly, but using the INS?s estimated position (and for the case of the strapdown GGI, orientation and rotation rate) and without an estimate of the noise vector. The overall twelve-accelerometer instrument may be physically strapped down to the body or stabilized on an inertial platform. Each con guration has its own 201 advantages and disadvantages, much like the tradeo s between strapdown of stabi- lized IMUs. The bene t of a strapdown GGI is that a massive, complex stabilized platform is unnecessary for the instrument-vehicle integration. The drawbacks are that the sensor must now be able to estimate the angular errors of the vehicle, which may cause numerical issues as will be shown in the Results chapter, and that the instrument must be robust enough to perform in a dynamic environment. When con- sidering the precision of current GGI?s proof mass displacement measurement and the dynamics of the airborne environment, the second issue (increased robustness) is probably the most di cult to overcome. This leads to why almost all gradiome- ters are integrated with a stabilized platform that isolates the sensor from the body dynamics. These platforms, currently, are on the order of a washing machine in size, and their isolation characteristics, along with the GGI fragility, are the limiting fac- tors on noise level reduction for airborne gravity gradiometry (H. J. Paik, Personal Communication, University of Maryland, College Park, May 14, 2007). The future potential performance of both classes of sensors are investigated and reported in the following chapters. But rst, their measurements and linearized error dynamics are derived for use by the extended Kalman lter in the next subsections. 5.2.1 Strapdown Gravity Gradiometer Instrument As mentioned above, the strapdown GGI is assumed to make measurements in the body frame with stationary accelerometers aligned to the body frame. Symboli- cally, the strapped down gradiometer measurement is found by substituting Eq. (5.9) 202 and (5.21) into Eq. (5.4) with Cag = I and !aba = 0. The result is Lb = Cbn nCnb bib bib: (5.44) The body-to-navigation frame rotation matrix, Cnb , is calculated by the truth inertial navigation system quaternion, and bib is the calculated by the truth simulated gyros. The modeled GGI measurement is corrupted by white, Gaussian noise and uses the truth INS and gyro states to calculate eLb at a given data rate. The strapdown GGI?s estimated measurement is modeled as bLb = bCbnb nbCnb b bibb bib: (5.45) Where bCbn is the INS estimate, b n = n(rn), and b bib is calculated from the gyro measurement after it has been corrected for by scale factor, bias, and noise errors (see Eq. (4.111) on pg. 174). The strapdown GGI residual is then Lb = bLb eLb; (5.46) after being reorganized into vector notation. The simulated strapdown gradiometer measurement and estimate could also by calculated using Eq. (5.43) with the appropriate transformation matrix (Cbn or bCbn), position (rn or brn) for the gradient interpolation, and gyro signal (!bib or e!bib). Of course, the noise vector would only by included in the simulated measurement, not the estimate. The linearized error will now be derived for the Kalman lter update equations. Typically, the gravity gradient errors consist of registration errors due to in- correct position knowledge, stored map errors, and instrument errors such as scale 203 factors, biases, nonlinearities, noise, etc.20,36 The simplifying assumption will be made that there are no map errors which implies that the spherical harmonic model is su ciently accurate at altitude. The assumption that the GGI errors are only a product of white noise is also made. (Red noise is sometimes used for low frequency deviations,99 but is neglected here because of the relatively fast update rates.) Since a GGI is an extremely sensitive instrument, the manufacturer would employ its own means to internally monitor and correct most other error sources. Therefore, only the registration and white noise instrument error are considered from this set. Other error sources occur due to incorrect rotation knowledge from the naviga- tion frame to the gradiometer frame and centripetal errors due to imprecise rotation rate knowledge. Jekeli94 included the centripetal terms, but the rotation matrix error contribution has not been thoroughly derived or investigated to the authors? knowledge. The following formulations will include both of these error sources along with the registration and white noise errors for use with a strapdown GGI. The case of a simulated stabilized GGI will then be derived from this expression. To derive the strapdown GGI Kalman lter measurement form, we will begin by linearly perturbing the gradiometer measurement, Eq. (5.44), to get Lb = Cbn ( n n + n n n)Cnb bib bib bib bib +VL = Cbn n +Ln Cnb Lb! +VL; (5.47) where Ln n n n n; (5.48) Lb! bib bib + bib bib (5.49) 204 are both symmetric matrices, n is the skew-symmetric matrix of rotation errors n = ( N; E; D)T, and VL is a matrix of instrument errors that includes the accelerometer di erencing errors. Following the assumptions above, the n-frame gravity gradients are only in error due to incorrect position knowledge, or registration errors. Mathematically, n = @ n @ + @ n @ + @ n @h h @ n @rn rn: (5.50) Since six observations are made for each full tensor GGI measurement, [@ n=@rn] is a 6 3 matrix that represents the third-order tensor of the gravity potential. Due to symmetry and Laplace?s constraint, this matrix could be computationally reduced to only include its seven independent components.36 However, for simple implementation purposes, the partial derivatives of n with respect to latitude, longitude and altitude are computed by second-order central nite di erences and linear interpolation to the estimated position. As mentioned before, higher order methods such as least-squares collocation would be preferable for a real system to account for errors in the estimated gravity eld.95,165 The error in the navigation-to-body frame rotation can be found by multiply- 205 ing out Ln : Ln n n n n = 0 BB BB BB BB BB B@ 2( ND E NE D) ND N + ED E +( NN EE) D NE N + ( DD NN) E ED D 2( ED N + NE D) ( EE DD) N NE E + ND D sym 2( ED N ND E): 1 CC CC CC CC CC CA (5.51) Then, rearranging the diagonal and upper-diagonal elements into the equivalent vector notations and factoring out n: 0 BB BB BB BB BB BB BB BB BB @ Ln ;11 Ln ;12 Ln ;13 Ln ;22 Ln ;23 Ln ;33 1 CC CC CC CC CC CC CC CC CC A = 0 BB BB BB BB BB BB BB BB BB @ 0 2 ND 2 NE ND ED NN EE NE DD NN ED 2 ED 0 2 NE EE DD NE ND 2 ED 2 ND 0 1 CC CC CC CC CC CC CC CC CC A 0 BB BB BB @ N E D 1 CC CC CC A ; (5.52) or more compactly Ln = @Ln @ n n: (5.53) Following a similar procedure, the rotation rate error contribution to the grav- 206 ity measurement is calculated using Eq. (5.49), which when expanded out is Lb! bib bib + bib bib = 0 BB BB BB @ 2(!y !y +!z !z) !y !x +!x !y !z !x +!x !z 2(!x !x +!z !z) !z !y +!y !z sym 2(!x !x +!y !y) 1 CC CC CC A ; (5.54) where (!x;!y;!z)T = !bib here. Then the vector form is Lb! 0 BB BB BB BB BB BB BB BB BB @ Lb!;11 Lb!;12 Lb!;13 Lb!;22 Lb!;23 Lb!;33 1 CC CC CC CC CC CC CC CC CC A = 0 BB BB BB BB BB BB BB BB BB @ 0 2!y 2!z !y !x 0 !z 0 !x 2!x 0 2!z 0 !z !y 2!x 2!y 0 1 CC CC CC CC CC CC CC CC CC A 0 BB BB BB @ !x !y !z 1 CC CC CC A @Lb ! @!bib !bib: (5.55) The linearized measurement errors for a strapdown gravity gradiometer can now be found. Substituting Eqs. (5.50), (5.52), & (5.55) into Eq. (5.47) with the transformation matrix Tbn (Eq. (5.15) using Cbn), and the gyro error states, Eq. (4.114) on pg. 175, gives Lb = Tbn @ n @rn rn +Tbn @Ln @ n n (5.56) @Lb ! @!bib h e!bib i SFg @Lb ! @!bib bg @Lb ! @!bib ng + L; where [e!bib ] is a diagonal matrix whose components are the measured rotation rate, and L is the vector of uncorrelated white measurement noise. The Kalman lter update matrix is implemented by using the coe cient matrices of Eq. (5.56) above 207 in the appropriate columns of the HbL matrix, and the remaining columns are lled with zeros. 5.2.2 Stabilized Gravity Gradiometer Instrument The stabilized gravity gradiometer instrument assumes that the stabilized plat- form continually aligns the gradiometer?s axes to the Earth-Centered-Inertial frame. Using Eq. (5.10) and (5.22) with the assumption that the accelerometers are sta- tionary with respect to the g-frame so that Cag = I and !aga = 0, the stabilized GGI truth measurement is Li = Cin nCni ; (5.57) because there are no GGI angular rates with respect to the inertial frame. The navigation-to-inertial frame direction cosine matrix is calculated by Eq. (5.11) on pg. 188 using the truth position. The stabilized GGI residual is calculated by subtracting the noisy measure- ment from the inertial navigation system?s estimated gradiometer reading: Li = bLi Li +VL ; (5.58) where Li +VL eLi is the simulated measurement with error, and VL is a matrix of uncorrelated measurement noise. The linearized error between the GGI measurement and INS estimate is found by linearly perturbing Eq. (5.57): Li = Cin ( nin n + n n nin)Cni +VL (5.59) = Cin n +Li Cni +VL; 208 where Li nin n n nin; (5.60) and nin is the skew symmetric rotation error matrix from the navigation-to-inertial frame with coordinates in the n-frame. This matrix is di erent than the traditional matrix n which is equivalent to nbn, or the n-to-b-frame rotation error. The navigation-to-inertial frame error can be calculated as follows. First, using the de nition of a small-error rotation, Eq. (4.72) on pg. 162, and taking its transpose, one has Cin = Cin nin: (5.61) Using Eq. (5.11) and linearly perturbing each element results in Cin = 0 BB BB BB BB BB B@ cos cos + sin sin ( +!e t) cos ( +!e t) sin cos + cos sin ( +!e t) cos sin sin cos ( +!e t) sin ( +!e t) sin sin cos cos ( +!e t) sin 0 cos 1 CC CC CC CC CC CA ; (5.62) which can be factored into Cin and the resultant small error rotation matrix: nin = 0 BB BB BB @ 0 sin ( +!e t) sin ( +!e t) 0 cos ( +!e t) cos ( +!e t) 0 1 CC CC CC A (5.63) which is the skew symmetric matrix of the error rotation vector nin = (cos ( +!e t); ; sin ( +!e t))T ; (5.64) 209 This error rotation vector is essentially the same as !nin rotation rate (Eq. (4.23) on pg. 144), but where the time di erential operator, d()=dt, has been replaced with the linear perturbation operator, (). nin can also be decomposed into INS error states: nin = 0 BB BB BB @ 0 cos 0 1 0 0 0 sin 0 1 CC CC CC A 0 BB BB BB @ h 1 CC CC CC A + 0 BB BB BB @ (!e=c) cos 0 (!e=c) sin 1 CC CC CC A (c bu) (5.65) or, more compactly as nin = @ n in @rn rn + @ n in @cbu c bu; (5.66) where !e is Earth?s rotation rate, c is the speed of light, and c bu is the user?s clock error, which for this work has the same dynamics as the GPS clock from Sec. D.4. The linearized stabilized gradiometer measurement error can now be found using a formulation similar to Eq. (5.56) but using the new rotation error equation above and omitting the gyro errors: Li = Tin @ n @rn + @Ln @ n @ n in @rn rn +Tin @Ln @ n @ n in @cbu c bu + L; (5.67) where Tin is calculated using Eq. (5.15) on pg. 189 with Cin as calculated in Eq. (5.11) on pg. 188. 5.3 Chapter Summary This chapter presents a methodology to derive the measurements for a vast array of GGI con gurations in Sec. 5.1. The methodology includes a new trans- formation matrix for converting on- and o -diagonal symmetric tensor components 210 from the North-East-Down navigation frame to an arbitrary frame in Sec. 5.1.1. And Sec. 5.1.2 includes the second part of the GGI measurement methodology which ac- counts for angular velocity e ects which essentially mask the gravitational gradients in the GGI measurements. Section 5.1.3 then uses the presented methodology to derive the rotating, stabilized GGI measurement produced by the Bell/Textron- derived GGIs as an example of its applicability. Section 5.2 details an envisioned 12-accelerometer, full-tensor GGI and derives the most comprehensive open-literature GGI linearized error models to date, includ- ing a new formulation for stabilized GGIs. As shown in Eq. (5.41) on pg. 200, for the 12-accelerometer GGI only two accelerometers are required to measure the inline (on-diagonal) gravitational gradients, but four accelerometers are necessary to mea- sure the o -diagonal gradients and remove the angular accelerations from the GGI observable. Section 5.2.1 then derives the measurement of a strapdown GGI and its linearized error model. The error derivation is based on Jekeli,94 however Eq. (5.56) on pg. 207 extends his derivation to include both the e ect of the navigation-to-body frame transformation, Tbn, and the error associated with this rotation, n. Section 5.2.2 lastly derives a new stabilized GGI measurement formulation and its linearized errors. This error model, Eq. (5.67), again includes rotation e ects and errors (Tin and nin), however this rotation error is comprised of position and time errors instead of orientation errors. Therefore, the error derivations of the strapdown and stabi- lized GGIs show that the strapdown sensor has direct observability of orientation and gyro errors, whereas the stabilized sensor has better observability of registration (position) errors through its rotation error term. These results produce a tradeo 211 between the two sensor types: A strapdown GGI-aided INS produces lower orientation and gyro errors over a stabilized sensor, but increased position errors because of the need to con- tinually estimate the orientation and gyro errors. A stabilized GGI-aided INS produces lower position errors than a strapdown sensor, but reduced orientation performance and gyro calibration because of the lack of observability of these states. The next chapter thoroughly quanti es the tradeo s between these two GGI types through extensive Monte Carlo simulations. 212 Chapter 6 Monte Carlo Simulation Results This chapter presents and discusses the Monte Carlo simulation and results to quantify the performance of the gradiometer aided inertial navigation system and the baseline INS/GPS system. The rst section gives an overview of how the simu- lations were run and how the data was reduced. The second section compares three Monte Carlo simulation set sizes and their e ect on how closely they model the Gaussian errors of the navigation system. The next section presents the INS/GGI and INS/GPS results. The rst half of each of the navigation aid?s results show a representative time history of a simulation and a detailed analysis of the navi- gation state errors. These results are shown as a preface to the second half of the results where the sensitivities of various system parameters are quanti ed in terms of their steady state mean-radial-spherical-errors (MRSEs). The conclusions from these results are then summarized in Sec. 6.4. 6.1 Monte Carlo Simulation For each Monte Carlo set and prior to any simulation run in the set, the stored gravity eld map and the body rate les (for the hypersonic cases) are opened and 213 read into memory.This allows each simulation of the Monte Carlo set to run faster because the le input is only required once. The only minor issue is that there is now an initial time lag on the order of several seconds for the code to read in these full les. For the subsonic cases, the body rates are set to zero along with the initial pitch and roll angles and the time lag is shortened slightly. The yaw angle for all cases is a constant 90 to ensure the Eastern cruise. After reading the le inputs, the inertial measurement unit speci cations were set to either use the navigation or tactical grade values from Table 4.4 on pg. 181. Next, the initial lter covariance matrix, P(0) was chosen to be a diagonal matrix with the following values for the diagonal elements. The lter position state variance was set to (10.0 m)2, and the latitude and longitude states were converted to radians by dividing by a, Earth?s equatorial radius. The velocity variances were set to (1.0 m/s)2, and the attitude states to (0.05 )2. The lter variances for the IMU scale factors and biases, and gyro noises, were set to their simulated IMU speci cations. The GPS receiver clock bias and drift variances were set to (15 m)2 and (0.5 m/s)2, respectively. This initial lter covariance was constant for all Monte Carlo sets and was tuned so that the lter would reach steady state operation as soon as possible for a wide variety of INS/GGI and INS/GPS con gurations. As discussed later in the results sections, this was not always the case. Following the set up of the initial covariance matrix, 1,000 Monte Carlo sim- ulations were run for a given set of system design parameters. (The sensitivity of increasing the set size to 10,000 simulations or decreasing to 100 simulations is discussed in the following section.) Each simulation in the set has its initial truth 214 position, velocity, and attitude states set so that they correspond to the correct Mach number, initial latitude and longitude for a given trajectory, and initial trim angles. The lter?s estimated position, velocity, and attitude states are also set to these values so that there are no initial errors for these nine states. The truth IMU states are then randomly initialized according to the IMU speci cations in Table 4.4. Misalignments and nonlinearities were originally implemented into the simulated IMU measurements to add uncompensated errors to the lter, but it was quickly discovered that the INS/GGI simulations were too sensitive to prevent diver- gence with these additional errors. Therefore, the misalignments and nonlinearities errors were not simulated in any of the following simulations. The GPS receiver?s truth bias and drift are randomly initialized with a 15 m and 0.5 m/s standard deviation, respectively. The last term randomly initialized for each simulation is a constant time o set between the simulation time and the GPS constellation time to allow for a variety of GPS geometries. The truth and estimated state vectors and the lter covariance matrix are numerically integrated at 20 Hz with the truth and estimated IMU measurements according to Ch. 4, and App. C and App. D. Then the simulated GGI or GPS measurements are made at a given update rate that is constant for a given Monte Carlo set. The noisy truth measurements are calculated using the current truth states and white noise is added. The lter also estimates a noise-free measurement at the same time using its current state estimate. The residual between these two measurements and the linearized measurement errors are then used to compute the Kalman gain to correct the estimated state vector and covariance matrix. The 215 process of propagating the truth and estimated states and updating the estimated states at a nite rate is continued until the truth longitude passes the given range requirement. In order to quantify the lter performance, the error in the 26 truth and lter- estimated states are computed at each epoch. The maximum error, sum of the error, and sum squared of the error of each state at each epoch are tallied for each of the simulations in the Monte Carlo set. Once all 1000 simulations are completed for a given set the mean, standard deviation, and maximum error of each state is computed at each epoch and written to a le. The same process is done with the lter?s 1- estimate of each state?s error using the diagonal elements of the lter covariance matrix, see Eq. (C.7) on pg. 312: b xi = p P(i;i): (6.1) The data in the Monte Carlo error le is reduced further by calculating the root-mean-square (RMS) error of each state at each epoch using: RMSi = q x2i + 2xi; (6.2) where xi is the mean error of the ith state at a given epoch, and xi is the standard deviation of the error at the same epoch. The RMS of the lter estimates are computed as well to identify if the lter is performing correctly. The mean RMS for the position, velocity, and attitude states are then cal- culated for several lter settling times and the presented results are tabulated as follows. The top half of each table calculates the mean RMS from 1/10 of the nal simulation time until the end of the simulation as a way of quantifying the 216 performance of the system while neglecting the initial lter settling transient. The bottom half of each table quanti es the steady-state navigation performance by calculating the mean RMS over only the last half of the simulation. The lter?s es- timated 1- standard deviations are denoted as \Cov." and are included to identify the e ectiveness of the lter and illuminate the presence of any divergence issues. The mean-radial-spherical-error (MRSE) is also calculated to quantify the overall position, velocity, and attitude error. The velocity MRSE is computed by: vMRSE = q v2N +v2E +v2D; (6.3) and the position and attitude MRSEs are found similarly. The latitude error is converted to crossrange error by multiplication of a, and the downrange error is found by multiplying the longitude error by acos( nom), where nom is the constant truth latitude. (Appendix E tabulates the position, velocity, and attitude MRSEs for each of the Monte Carlo sets with the two lter settling times above and the MRSE for the entire simulation.) The hypersonic un-aided INS results are presented as Table 6.2 in a slightly di erent fashion. Without external aiding, the navigation lter propagates the in- ertial navigation states with the initialized uncompensated accelerometer and gyro errors. These errors cause the position, velocity, and attitude errors to grow steadily over time so that the mean RMS errors are less informative than for the aided cases. Instead, the RMS error states at the end of the simulation (at the 1,000 km down- range) are presented along with the maximum error encountered for the entire 1,000 simulation set. The navigation and tactical grade IMUs are presented as the top 217 Table 6.1: Steady State Error Versus Monte Carlo Set Size North / Pitch East / Roll Down / Yaw Set Size State Units RMS Cov. RMS Cov. RMS Cov. Pos. m 0.1929 0.2049 0.2311 0.2221 0.1145 0.1148 100 Vel. m/s 0.0044 0.0045 0.0041 0.0038 0.0016 0.0015 Att. 10 3 1.2851 1.2965 1.0427 0.9934 7.5255 7.4114 Pos. m 0.2055 0.2049 0.2229 0.2221 0.1144 0.1148 1,000 Vel. m/s 0.0045 0.0045 0.0038 0.0038 0.0016 0.0015 Att. 10 3 1.2945 1.2965 0.9892 0.9935 7.1992 7.4121 Pos. m 0.2039 0.2049 0.2220 0.2221 0.1148 0.1148 10,000 Vel. m/s 0.0045 0.0045 0.0038 0.0038 0.0015 0.0015 Att. 10 3 1.2885 1.2965 0.9982 0.9935 7.3265 7.4119 and bottom half of the table, respectively. 6.2 Monte Carlo Set Size Three Monte Carlo set sizes were compared to investigate the e ect of in- creasing or decreasing the number of simulation runs per a given INS con guration. The chosen test case was the Mach 6, high n variation trajectory INS/GGI system with navigation grade IMUs aided by a stabilized 0.001 E o gradiometer at 1 Hz. This con guration was picked because it yields the best INS/GGI performance and has an increased simulation duration versus the Mach 7 \Best" case presented in Sec. 6.3.1.1. The computed steady state error and lter estimate for each of the nine nav- igation states is summarized in Table 6.1 for Monte Carlo sets of 100, 1,000 and 218 10,000 simulations. The gure of merit that is used for this steady is how closely the Monte Carlo steady state errors predict the optimal lter performance since this estimates how closely the simulations capture all random processes of the simulation. Quantitatively, this term is RMSi \RMSi 1; (6.4) where RMSi is the ith state?s steady state RMS computed from the Monte Carlo set and\RMSi is the steady state lter estimate of the error. The quantity would be zero for an in nite number of random simulations assuming the lter is optimal. The value of\RMS for each state was chosen to be the average of the three steady state lter values, i.e. from the 100, 1,000, and 10,000 sets. Figure 6.1 plots the 100 and 1,000 set steady state lter error normalized by the 10,000 simulation set values for each of the nine navigation states ( ; ;h;vN;vE;vD; N; E; D). From this plot, it is apparent that the lter estimate error is essentially constant regardless of Monte Carlo set size. Figure 6.2 then plots the normalized steady state error RMS as calculated by Eq. (6.4). The 100 Monte Carlo set has the largest deviation from the lter estimate because there are not enough simulations to accurately capture a full Gaussian distribution of all the random states. The 1,000 set, on the other hand, has less than a 1% variation from the lter estimate for all states except the Eastern velocity (1.35%) and yaw angle, D (2.87%). And the 10,000 simulation set better captures only ve of the nine navigation states compared to the 1,000 simulation set. The mean-radial-spherical-error for the normalized position, velocity, and at- 219 Figure 6.1: Normalized Steady State Filter Error vs. Monte Carlo Set Size Figure 6.2: Normalized Steady State Error vs. Monte Carlo Set Size 220 Figure 6.3: Normalized Steady State MRSE vs. Monte Carlo Set Size titude states are plotted in Fig. 6.3 as a function of the Monte Carlo set size. The position and velocity errors are much better estimated when the number of simula- tions is increased from 100 to 1,000. However, increasing the set size from 1,000 to 10,000 has diminishing returns. The attitude error linearly decreases as the set size is increased by an order of magnitude. It should be noted that while increasing the number of simulations better captures all the randomness of the system, the computational e ort increases sub- stantially. For the hypersonic simulations, a 1,000 simulation Monte Carlo set took about 16{20 minutes to run on a dual processor 64-bit AMD 2.2 GHz Opteron 246 with 2 GB of RAM. The 10,000 simulation set took on the order of 3{4 hours to complete, and the 100 simulation set several minutes. Taking the computation time, number of Monte Carlo con gurations simulated, and the trends in Fig. 6.3 into ac- count, the nominal 1,000 simulation set size was deemed a good compromise for this 221 work. 6.3 Results Before presenting the Monte Carlo sensitivity results, several single simulation results will be shown to better understand the later discussions. The Monte Carlo errors and lter estimates are tabulated for the nine navigation states for the two settling times discussed above, and the columns of the tables are organized as follows. The North / Pitch columns constitute the crossrange, Northern velocity, and N (pitch) errors, the East / Roll columns similarly constitute the downrange, Eastern velocity, and E (roll) errors, and the Down / Yaw columns the altitude, Downward velocity, and yaw errors. In order to limit the scope of the sample simulation results presented, the \Best" or \Nominal" case for each INS aid, GGI and GPS, will be rst shown. Then one of the system design parameters will be deviated from this best or nominal case. The gradiometer cases are presented rst, followed by the GPS cases. The Monte Carlo steady state MRSE sensitivities to numerous parameters are then presented. These results are rst shown by comparing the main system design parameters (IMU quality and GGI type or GPS measurement) as a function of GGI noise of GPS update interval. Then, the sensitivity to Mach number and n variation (for the hypersonic cases) are shown. The two subsonic cases are discussed in the single simulation result sections. As motivation for the need of an external aid to the INS, the dead reckoning 222 inertial navigation results are listed in Table 6.2 for the mean and maximum RMS at the end of the hypersonic 1,000 km simulations. These results are only for the high gravitational gradient variation trajectories. The low n variation trajectory free-inertial errors are given in Tables E.2 and E.4 on pg. 358. As shown in Table 6.2, after the Mach 7 free inertial 1,000 km cruise, the position errors grow to 25 m in the horizatal and 65 m in the vertical for a navigation grade IMU with no initial position, velocity, or attitude errors. The maximum error for the free inertial Monte Carlo simulations are approximately 4 times as large. The tactical grade IMUs produce nearly 1 km horizontal errors and 700 m vertical error for a total position error of 1.5 km due to only IMU error sources. The Mach 6 cases produce dead reckoning errors since the simulation is run longer and the error growth is a function of time. The Mach 8 case is simulated for a shorter duration and therefore has lower free-inertial errors. To enable safe operation of such systems and to meet precision strike goals (on the order og 3 m),166 an external aid is a necessity. 6.3.1 Gravity Gradiometer Aided INS Table 6.3 summarizes the 162 hypersonic gravity gradiometer aided inertial navigation system con gurations tested in this work. The subsonic cases were only simulated along the high gravitational gradient variation trajectories with navigation grade IMUs and a stabilized GGI with 0.1, or 0.001 E o updates at 1 Hz. 223 Table 6.2: Dead Reckoning Navigation Accuracy after 1000 km Cruise, High n Variation Trajectory M1 North / Pitch East / Roll Down / Yaw IMU State Units RMS Max RMS Max RMS Max Mach 6 Pos. m 31.635 107.96 34.665 109.35 97.457 356.11 Nav. Vel. m/s 0.0925 0.2941 0.1081 0.3388 0.3635 1.3564 Att. 10 3 0.6222 2.0768 0.6595 2.3628 0.7135 2.6651 Mach 6 Pos. m 1428.9 4323.4 1452.3 5107.9 945.93 3586.6 Tac. Vel. m/s 6.9633 20.405 7.0990 24.955 3.7091 14.651 Att. 10 3 144.47 447.63 148.02 485.69 159.95 613.88 Mach 7 Pos. m 23.823 85.192 26.542 88.183 65.974 278.82 Nav. Vel. m/s 0.0817 0.2394 0.0965 0.3456 0.2787 0.9450 Att. 10 3 0.5744 2.4795 0.5459 2.2561 0.6129 1.9046 Mach 7 Pos. m 918.82 3468.8 952.09 3095.8 703.89 2878.8 Tac. Vel. m/s 5.0600 19.586 5.2833 16.942 3.1409 12.570 Att. 10 3 126.02 437.87 121.19 434.81 132.75 443.72 Mach 8 Pos. m 21.588 104.31 20.740 66.693 48.893 158.06 Nav. Vel. m/s 0.0770 0.2786 0.0821 0.2667 0.2323 0.7725 Att. 10 3 0.5110 1.8697 0.4941 1.6466 0.5351 1.6637 Mach 8 Pos. m 655.86 2338.8 680.45 2722.5 500.13 1771.4 Tac. Vel. m/s 4.1040 15.929 4.2188 15.004 2.5330 8.9636 Att. 10 3 113.97 458.36 113.18 417.27 116.21 408.02 224 Table 6.3: INS/GGI Monte Carlo Test Matrix Parameter Values # IMU Grade Navigation, Tactical (Stab. only) 2 Mach Number 6, 7, 8 3 Gradient Variation \High," \Low" 2 Instrument Noise 0.1, 0.01, 0.001 E o (1- ) 3 Instrument Type Stabilized, Strapdown 2 Data Rate 1, 5, 10 sec 3 6.3.1.1 Monte Carlo Results For the single simulation runs, only the Mach 7 cases will be shown since they represent the same trends as the other two hypersonic cases. The \best" INS/GGI system con guration is summarized in Table 6.4 along with the varied parameter values chosen to give a brief discussion of some of the system sensitivities. The subsonic 0.1 and 0.001 E o INS/GGI cases will be presented and discussed at the end of this subsection. Table 6.4: INS/GGI \Best" and O -Nominal Simulation Parameters IMU Trajectory Noise Type Data Rate \Best" Value Nav. High n Variation 0.001 E o Stabilized 1 sec Perturbed Value Tac. Low n Variation 0.1 E o Strapdown 10 sec 225 Hypersonic Cases Figure 6.4 plots a single \Best" case stabilized gradiometer simulation error and 1- lter envelope for the nine navigation states. As shown, the lter accurately predicts the errors for each state with this future-grade gradiometer. Compared to the free inertial nal position error, the covert GGI-aided INS reduces the steady state MRSE by a factor of 220 from 75.0 to 0.336 m. The steady state (500 km settling time) velocity error is also reduced remarkably to 0.0069 m/s, a factor of nearly 45 below the free-inertial case. The attitude states surprisingly increase in error from the unaided simula- tions. This result is somewhat misleading as it is actually due to the simulation formulation|not the INS/GGI lter performance. Because the simulations were conducted without any initial position, velocity, or attitude errors, the free-inertial attitude errors are governed primarily by the uncompensated gyro errors. The nav- igation grade gyros have a simulated turn-on bias of 0.003 =Hr, which therefore cause only a 0.4 10 3 attitude error after the 482 sec Mach 7 simulation; close to the values reported in Table 6.2. Had the free-inertial simulation introduced initial attitude errors, the nal errors would have grown much larger, and the improvement in attitude determination by GGI-aiding would be more obvious. Some trends that are apparent for all simulated gradiometer-aided systems can be identi ed by Table 6.5. First, the vertical position and velocity errors are typically one-half the horizontal errors. This performance characteristic is attributed to the fact that the vertical gravitational gradient, DD, is approximately twice that 226 0 100200300400 ?1 ?0.5 0 0.5 1 ? r N Position Errors, m 0 100200300400 ?1 ?0.5 0 0.5 1 ? r E 0 100200300400 ?1 ?0.5 0 0.5 1 ? r D Time, s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v N Velocity Errors, m/s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v E 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v D Time, s 0 100200300400 ?10 ?5 0 5 10 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?10 ?5 0 5 10 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.4: Sample \Best" Stabilized Gradiometer-Aided INS Simulation Table 6.5: \Best" Gradiometer-Aided INS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 1.2069 1.2692 0.3331 0.3469 0.4711 0.4770 100 km Vel. m/s 0.0124 0.4474 0.0151 0.4466 0.0103 0.4455 Att. 10 3 0.9076 23.138 0.6166 22.820 5.3159 27.662 Pos. m 0.2172 0.2172 0.2244 0.2259 0.1239 0.1229 500 km Vel. m/s 0.0054 0.0054 0.0039 0.0039 0.0019 0.0019 Att. 10 3 1.6298 1.6488 1.1060 1.0763 9.5646 9.7920 227 of NN or EE by Laplace?s constraint, Eq. (2.5) on pg. 43. Hence, variations in the inline horizontal gradients cause greater variations in the vertical component. Second, the tilt errors, N (pitch for Eastern ight) and E (roll), are reduced to an error oor of 0.001 while the yaw error dominates the total attitude error. This phenomenon occurs because the gradiometer acts as essentially a gravitational compass to hone in on the vertical gradient and reduce the tilt errors. Furthermore, due to Earth?s oblateness, the gravitational gradients are more sensitive to latitude than longitude variations so the E (roll) error has slightly better performance than the N (pitch) error. Figure 6.5 shows a single simulation using the \Best" case parameters but with the navigation grade IMUs replaced by a tactical grade suite. Table 6.6 lists the 1,000 Monte Carlo navigation results. For this case the lter again accurately estimates the navigation errors. The INS/GGI produces position errors only slightly greater than those of the \Best" case, and corresponds to a three order-of-magnitude improvement from the free inertial tactical grade IMU errors. Velocity errors are reduced by over two orders-of-magnitude. The attitude errors are only reduced by a factor of two, however this is again caused by the simulation formulation. The \Best" case parameter set is next own over the \Low" gravity gradient variation trajectory to investigate the sensitivity of the system on signal strength. As seen in Fig. 6.6 and Table 6.7, there is minimal change in navigation performance between the two chosen trajectories. The most apparent di erence between these two cases is that the lobe pattern of the lter 1- envelope is less pronounced com- pared to Fig. 6.4 and 6.5. The North (crossrange) position error is essentially the 228 0 100200300400 ?1 ?0.5 0 0.5 1 ? r N Position Errors, m 0 100200300400 ?1 ?0.5 0 0.5 1 ? r E 0 100200300400 ?1 ?0.5 0 0.5 1 ? r D Time, s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v N Velocity Errors, m/s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v E 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v D Time, s 0 100200300400 ?50 0 50 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?50 0 50 ? ? 0 100200300400 ?200 ?100 0 100 200 ? ? Time, s Figure 6.5: Sample Tactical Grade IMU Gradiometer-Aided INS Simulation Table 6.6: Tactical Grade IMU, Gradiometer-Aided INS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 1.2911 1.3136 0.4212 0.4264 0.5151 0.5080 100 km Vel. m/s 0.0176 0.4527 0.0220 0.4537 0.0129 0.4482 Att. 10 3 13.726 38.116 13.095 37.154 55.708 81.951 Pos. m 0.2991 0.2970 0.3701 0.3690 0.1777 0.1787 500 km Vel. m/s 0.0150 0.0148 0.0167 0.0166 0.0068 0.0068 Att. 10 3 24.626 28.610 23.492 26.878 100.19 107.51 229 0 100200300400 ?1 ?0.5 0 0.5 1 ? r N Position Errors, m 0 100200300400 ?1 ?0.5 0 0.5 1 ? r E 0 100200300400 ?1 ?0.5 0 0.5 1 ? r D Time, s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v N Velocity Errors, m/s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v E 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v D Time, s 0 100200300400 ?10 ?5 0 5 10 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?10 ?5 0 5 10 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.6: Sample \Low" Gravity Gradient Trajectory Gradiometer-Aided INS Simulation Table 6.7: \Low" Gravity Gradient Variation Gradiometer-Aided INS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5544 0.5456 0.7716 0.7762 0.3811 0.3899 100 km Vel. m/s 0.0115 0.4474 0.0032 0.4467 0.0253 0.4454 Att. 10 3 0.8298 23.039 0.6211 22.829 5.0592 27.503 Pos. m 0.2102 0.2092 0.2499 0.2488 0.1165 0.1176 500 km Vel. m/s 0.0054 0.0054 0.0041 0.0041 0.0018 0.0018 Att. 10 3 1.4896 1.4739 1.1139 1.0950 9.1022 9.5086 230 same for both trajectories, the East (downrange error) is about 10% higher for the \Low" trajectory, and the altitude error is surpisingly 6% lower for the \Low" tra- jectory. Again the vertical errors are approximately half those of the horizontal and the attitude errors are dominated by the yaw error. These results are quite promis- ing. Initially it was believed that the gradiometer-aided INS would only be viable over regions with large gradient variations. But from this analysis it is now believed that the INS/GGI navigation package can yield exceptional performance with min- imum sensitivity to the region of interest if the GGI noise oor can be reduced to 0.001 E o/pHz. A more complete investigation is performed and discussed in the following subsection that shows that signal variation does indeed a ect navigation performance for other con gurations and higher GGI noise levels. Figure 6.7 plots a sample simulation over the \High" gradient variation tra- jectory, but now with increased gradiometer noise. The noise was increased from 0.001 E o to 0.1 E o with updates still simulated at 1 Hz. This increased noise value is still more than an order of magnitude lower than any currently planned airborne gradiometer. However, simulations performed with a 1 E o simulated noise level showed negligible improvement over the free-inertial simulations, so a noise value of 0.1 E o was chosen to show the e ect of greater instrument noise on overall system performace. With the two order-of-magnitude increase in noise, the total steady state position error increased by a factor of 75 over the \Best" case INS/GGI. The velocity error increased by a factor of 40. The total attitude error is approximately the same regardless of noise level. Yet again, this result is slightly misleading. The tilt errors increased by a factor of 5 with the increased noise as one might expecet, 231 but the yaw error was halved over the low noise case. The reason for the decrease in yaw angle error is that the lter covariance was initially high enough that it never attempted to update the yaw estimate, thus allowing the yaw to propagate as if it were a free-inertial case with errors occurring from the gyro bias and a random walk from integrating its noise. As mentioned above, with the lack of initial position, ve- locity, and attitude errors, the attitude errors remain quite small without updates. If the initial lter covariance were tuned di erently, the yaw angle error would have likely increased over the \Best" case yaw error. The yaw error issue will be even more apparent in the sensitivity study results. Next, the \Best" case INS/GGI was simulated with 10 sec updates insead of the 1 sec updates in the other cases. By sampling at longer intervals, the lter processes fewer measurements over the cruise pro le and thus the lter requires more time to reach steady state. Moreover, the decreased update rate e ectively increases the noise in terms of its simulated power spectral density, see Eq. (6.5) on pg. 244. Figure 6.8 illustrates that the lter undergoes many saw-tooth error spikes between measurement updates in the initial portion of the 10 sec update INS/GGI simulation. After about halfway through the simulation the lter has converged to steady state position, velocity, and tilt errors. At this point, the lter begins to remove the yaw error until the simulation ends. The speed that the lter reaches steady state could be improved by additional lter tuning. Referring to Table 6.9, the position errors grew consistently by a factor of 3 over the 1 sec update results, however the total position error was still less than a meter (0.9789 m). The velocity errors for the 10 sec update case are about twice those of the nominal 1 Hz case 232 0 100200300400 ?40 ?20 0 20 40 ? r N Position Errors, m 0 100200300400 ?40 ?20 0 20 40 ? r E 0 100200300400 ?40 ?20 0 20 40 ? r D Time, s 0 100200300400 ?1 ?0.5 0 0.5 1 ? v N Velocity Errors, m/s 0 100200300400 ?1 ?0.5 0 0.5 1 ? v E 0 100200300400 ?1 ?0.5 0 0.5 1 ? v D Time, s 0 100200300400 ?50 0 50 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?50 0 50 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.7: Sample Increased Noise Gradiometer-Aided INS Simulation Table 6.8: Increased Noise Gradiometer-Aided INS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 8.6561 12.969 11.476 13.665 5.7341 9.5701 100 km Vel. m/s 0.0988 0.5453 0.1299 0.5654 0.0577 0.4940 Att. 10 3 3.6155 27.152 2.8556 26.061 2.9739 49.546 Pos. m 14.919 15.376 18.833 19.307 9.2399 9.3009 500 km Vel. m/s 0.1616 0.1815 0.2101 0.2178 0.0875 0.0891 Att. 10 3 6.5040 8.8750 5.1360 6.9104 5.3491 49.183 233 0 100200300400 ?2 ?1 0 1 2 ? r N Position Errors, m 0 100200300400 ?2 ?1 0 1 2 ? r E 0 100200300400 ?2 ?1 0 1 2 ? r D Time, s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v N Velocity Errors, m/s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v E 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v D Time, s 0 100200300400 ?20 ?10 0 10 20 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?20 ?10 0 10 20 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.8: Sample Stabilized Gradiometer-Aided INS Simulation with 10 sec Up- dates Table 6.9: Stabilized Gradiometer-Aided INS Case with 10 sec Updates Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 1.4780 1.5096 0.5849 0.5765 0.6223 0.6194 100 km Vel. m/s 0.0164 0.4525 0.0171 0.4490 0.0119 0.4473 Att. 10 3 1.2480 23.743 0.8191 23.099 9.8083 34.572 Pos. m 0.6152 0.6499 0.6601 0.6392 0.3796 0.3792 500 km Vel. m/s 0.0129 0.0146 0.0084 0.0082 0.0052 0.0052 Att. 10 3 2.2424 2.7375 1.4705 1.5795 17.651 22.230 234 (0.0163 vs. 0.0069 m/s), and similarly the attitude errors grew by a factor of 2 (17.85 vs. 9.765 ). The nal gradiometer-aided INS parameter to be perturbed is the use of a strapdown GGI instead of a stabilized GGI. The bene ts of a strapdown version of a gradiometer versus a stabilized version is similar to the tradeo s between strap- down and stabilized IMU sensors. A strapdown system provides for a smaller and, ideally, a mechanically less complicated sensor at the cost of additional computing requirements to address the angular rate and angular acceleration issues. Because angular rates and accelerations present themselves as false gravitational gradients, see Eq. (5.3) and (5.4) on pg. 185, all airborne gradiometers have been built with a stabilized platform in mind. If the angular accelerations and rates could be su - ciently estimated by either gyros (as assumed in this work) or by the gradiometer measurements itself (see Eq. (5.6) on pg. 186), a strapdown gradiometer could be built with reduced mass and volume. As explained next, the strapdown INS/GGI simulated here brings up unexpected computional issues. Figure 6.9 plots a representative strapdown gradiometer-aided INS simulation. It is quite apparent that in this con guration the lter diverges, most noteably in al- titude. This divergence is believed to be caused by numerical truncation error in the lter covariance propagation, Eq. (C.40) on pg. 324, and Kalman gain calculation, Eq. (C.49) on pg. 326: Pk+1 = kPk Tk +Qk Kk+1 = P k+1HTk+1 Hk+1P k+1HTk+1 +Rk+1 1: 235 0 100200300400 ?100 ?50 0 50 100 ? r N Position Errors, m 0 100200300400 ?40 ?20 0 20 ? r E 0 100200300400 ?6 ?4 ?2 0 2 ? r D Time, s 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v N Velocity Errors, m/s 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v E 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v D Time, s 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? ? 0 100200300400 ?20 ?10 0 10 20 ? ? Time, s Figure 6.9: Sample Strapdown Gradiometer-Aided INS Simulation Table 6.10: Strapdown Gradiometer-Aided INS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 7.0255 7.4693 6.5463 4.6380 2.2185 3.3934 100 km Vel. m/s 0.0464 0.4645 0.0450 0.4613 0.0108 0.4463 Att. 10 3 0.0349 0.0854 0.0483 0.1033 2.2739 23.039 Pos. m 12.617 5.4450 10.730 4.4720 2.6048 0.2349 500 km Vel. m/s 0.0735 0.0360 0.0669 0.0304 0.0095 0.0033 Att. 10 3 0.0285 0.0176 0.0468 0.0157 3.6020 1.5141 236 All real quantities in the FORTRAN simulations use double precision data types, however the gyro process noise variance in Qk and Pk is many orders-of-magnitude larger than the gradiometer measurement noise variance in Rk and therefore numer- ical truncation occurs in the Kalman gain calculation. This truncation essentially causes the lter to ignore some of the GGI measurement noise matrix, RGGI = diag( 2 L), and therefore the Kalman gain is not calculated optimally. For cases where the GGI noise is increased or the gyro noise is reduced so their variances are closer, no divergence exists. Furthermore, lter divergence occurs almost instantly when the GGI noise is decreased below 0.001 E o with navigation grade gyros. In order to alleviate this issue, higher numerical precision could be used in the simula- tion, and/or a square-root Kalman lter implementation could be chosen instead of the standard lter implemented in this research. This divergence issue also caused all strapdown gradiometer aided INS simulations with tactical grade IMUs to di- verge instantly. Regardless of the slow lter divergence, the strapdown gradiometer INS results are listed in Table 6.10. Surprisingly, the position errors are less than 17 m even with lter divergence. For the Monte Carlo sensitivities in the next sub- section, the actual diverging errors are presented along with an extrapolation of the converged lter results into the 0.001 E o GGI con gurations. Subsonic Cases A representative time history of the commercial aircraft INS/GGI is shown in Fig. 6.10 and the Monte Carlo results are listed in Table 6.11. This case uses the 237 0 200400600800 ?1 ?0.5 0 0.5 1 ? r N Position Errors, m 0 200400600800 ?1 ?0.5 0 0.5 1 ? r E 0 200400600800 ?1 ?0.5 0 0.5 1 ? r D Time, s 0 200400600800 ?0.02 ?0.01 0 0.01 0.02 ? v N Velocity Errors, m/s 0 200400600800 ?0.02 ?0.01 0 0.01 0.02 ? v E 0 200400600800 ?0.02 ?0.01 0 0.01 0.02 ? v D Time, s 0 200400600800 ?10 ?5 0 5 10 ? ? Attitude Errors, ? ? 10 ?3 0 200400600800 ?10 ?5 0 5 10 ? ? 0 200400600800 ?50 0 50 ? ? Time, s Figure 6.10: Sample Commercial Aircraft INS/GGI Simulation Table 6.11: Commercial Aircraft INS/GGI Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5459 0.5522 0.6386 0.6225 0.3091 0.3516 50 km Vel. m/s 0.0174 0.4462 0.0141 0.4465 1.4869 0.4451 Att. 10 3 0.5209 22.729 0.5003 22.722 7.4570 30.0509 Pos. m 0.1479 0.1476 0.2291 0.2252 0.0933 0.0923 250 km Vel. m/s 0.0031 0.0031 0.0037 0.0037 0.0011 0.0011 Att. 10 3 0.9338 0.9082 0.8968 0.8970 13.420 14.088 238 \best" parameters as listed in Table 6.4 with the only di erence being the reduced altitude of 10,000 m and East velocity of 250 m/s. The simulation is also run to a downrange of 500 km instead of 1,000 km. As shown in both Fig. 6.10 and Table 6.11, the lter is accurately estimating the simulation errors. Comparing this simulation with the Mach 7 \best" INS/GGI system shows sur- prisingly little change in navigation performance. The downrange and East velocity errors are practically identical. The crossrange error is about 2/3 the hypersonic case, and the altitude error is about 3/4. The crosstrack (North) and downward ve- locity errors are both approximately half those of the Mach 7 simulation. And lastly, all steady state errors are comparable. These results are encouraging because they show that a future-grade gradiometers can provide exceptional navigation aiding even if their size and mass were not reduced to missile-class sizes. Figure 6.11 and Table 6.12 show the results of the commercial aircraft case if its noise level were increased to 0.1 E o with a 1 Hz update rate. The navigation performance is similar to the hypersonic case shown in Fig. 6.7 and Table 6.8. These results give an estimate as what a nearer-future INS/GGI system?s navigation accuracy might be. It should be noted that the local terrain e ects have not been included in any of the simulations, so this INS/GGI navigation accuracy would likely be improved if more signal frequencies were included. The GGI-based survey mission results are presented in Fig. 6.12 and Table 6.13. This system is the closest to the current environment of airborne gradiometry. However, the simulations here are presented with space-grade noise levels which are three orders of magnitude lower than currently proposed airborne gradiometers. 239 0 200400600800 ?40 ?20 0 20 40 ? r N Position Errors, m 0 200400600800 ?40 ?20 0 20 40 ? r E 0 200400600800 ?20 ?10 0 10 20 ? r D Time, s 0 200400600800 ?1 ?0.5 0 0.5 1 ? v N Velocity Errors, m/s 0 200400600800 ?1 ?0.5 0 0.5 1 ? v E 0 200400600800 ?1 ?0.5 0 0.5 1 ? v D Time, s 0 200400600800 ?20 ?10 0 10 20 ? ? Attitude Errors, ? ? 10 ?3 0 200400600800 ?20 ?10 0 10 20 ? ? 0 200400600800 ?50 0 50 ? ? Time, s Figure 6.11: Sample Commercial Aircraft INS/GGI Simulation w/ Increased Noise Table 6.12: Commercial Aircraft INS/GGI Case w/ Increased Noise Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 5.7697 9.8692 8.3473 11.382 4.9747 8.7136 50 km Vel. m/s 0.0609 0.4909 0.0720 0.5040 2.9411 0.4786 Att. 10 3 1.1469 23.398 0.9617 23.486 5.0887 49.220 Pos. m 9.6523 9.7997 14.681 14.842 7.9068 7.7713 250 km Vel. m/s 0.0753 0.0835 0.1087 0.1072 0.0635 0.0614 Att. 10 3 2.0607 2.1133 1.7274 2.2715 9.1567 48.596 240 0 500 1000 ?1 ?0.5 0 0.5 1 ? r N Position Errors, m 0 500 1000 ?1 ?0.5 0 0.5 1 ? r E 0 500 1000 ?1 ?0.5 0 0.5 1 ? r D Time, s 0 500 1000 ?0.02 ?0.01 0 0.01 0.02 ? v N Velocity Errors, m/s 0 500 1000 ?0.02 ?0.01 0 0.01 0.02 ? v E 0 500 1000 ?0.02 ?0.01 0 0.01 0.02 ? v D Time, s 0 500 1000 ?10 ?5 0 5 10 ? ? Attitude Errors, ? ? 10 ?3 0 500 1000 ?10 ?5 0 5 10 ? ? 0 500 1000 ?50 0 50 ? ? Time, s Figure 6.12: Sample GGI Survey INS/GGI Simulation Table 6.13: GGI Survey INS/GGI Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5568 0.5740 0.7188 0.6438 0.3760 0.3352 30 km Vel. m/s 0.0569 0.4463 0.0357 0.4470 4.2667 0.4451 Att. 10 3 0.5459 22.722 0.5356 22.716 10.9640 32.003 Pos. m 0.1917 0.1869 0.4885 0.3345 0.1217 0.1106 150 km Vel. m/s 0.0034 0.0033 0.0086 0.0046 0.0013 0.0011 Att. 10 3 0.9788 0.8999 0.9601 0.8885 19.731 17.604 241 The simulation is also run for 300 km at a velocity of 40 m/s and a 100 m cruise altitude. From Table 6.13, this low altitude and velocity system has degraded perfor- mance in its downrange and alongtrack (East) velocity accuracy. Comparing the lter estimates of these states to the Monte Carlo simulation errors, it also appears that the lter may not be performing correctly since the simulation errors are no- ticeably higher than the lter?s estimates. The errors of these East states are also about twice those of the commercial aircraft and comparable scramjet cases. The other position errors are about 30% higher than the other subsonic case, and the North and Downward velocity errors are about the same. The case where the GGI noise is simulated at 0.1 E o at 1 Hz is shown in Fig. 6.13 and the Monte Carlo results are listed in Table 6.14. These results show that a future-grade airborne GGI aided INS can provide reasonable covert, passive navigation. If terrain e ects were included in the gravitational eld, these results would like be improved. The velocity accuracy is also improved over the comparable high-noise INS/GGI simulations because the truth velocity is decreased to only 40 m/s for the GGI-survey simulations. 6.3.1.2 Sensitivity Results This section investigates many of the sensitivities of future INS/GGI systems. The rst set of plots (Fig. 6.14 and 6.15) compares the performance of the grav- ity gradiometer instrument type and the e ect of the IMU grade on the system 242 0 500 1000 ?40 ?20 0 20 40 ? r N Position Errors, m 0 500 1000 ?40 ?20 0 20 40 ? r E 0 500 1000 ?20 ?10 0 10 20 ? r D Time, s 0 500 1000 ?0.5 0 0.5 ? v N Velocity Errors, m/s 0 500 1000 ?0.5 0 0.5 ? v E 0 500 1000 ?0.5 0 0.5 ? v D Time, s 0 500 1000 ?10 ?5 0 5 10 ? ? Attitude Errors, ? ? 10 ?3 0 500 1000 ?10 ?5 0 5 10 ? ? 0 500 1000 ?50 0 50 ? ? Time, s Figure 6.13: Sample GGI Survey INS/GGI Simulation w/ Increased Noise Table 6.14: GGI Survey INS/GGI Case w/ Increased Noise Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 6.3029 10.573 10.675 13.010 5.4411 9.1511 30 km Vel. m/s 0.0640 0.4813 0.0534 0.4812 5.3278 0.4674 Att. 10 3 0.6774 22.894 0.7255 23.130 9.3193 46.656 Pos. m 10.580 11.068 18.833 17.774 8.6461 8.5760 150 km Vel. m/s 0.0563 0.0664 0.0729 0.0661 0.0433 0.0413 Att. 10 3 1.2154 1.2094 1.3021 1.6341 16.771 43.982 243 performance. Only the steady state mean-radial-spherical-errors for the Mach 7 simulations are presented in this analysis as a baseline for the following plots. Fig- ures 6.16{6.21 then focus on one of the three INS/GGI system con gurations and investigates the sensitivities of speed/altitude on the system and the gravity gra- dient signal variations. Each of these gures are normalized by the Mach 7, high n variation trajectory cases that are presented in Fig. 6.14 and 6.15. Because the MRSE attitude errors from the Monte Carlo simulations were found to be some- what misleading due to the initial lter covariance matrix, the lter?s estimate of the attitude errors are also included in the results. The GGI update rate and noise are combined in terms of the sensor?s e ective power spectral density (PSD) in an e ort to reduce the results that are presented. Referring to Eq. (C.9) on pg. 313, the square root of the instrument?s PSD can be written as pq GGI = L p tGGI; (6.5) which has units of E ops, or equivalently E o/pHz (similar to the IMU noise spec- i cations). Therefore, increasing the time between instrument updates causes an e ective increase in the GGI noise by a factor ofp tGGI. All the INS/GGI results are plotted with this parameter as the abscissa (x-axis). Figure 6.14 plots the position and velocity MRSE for the nominal Mach 7 INS/GGI cases along the high gradient variation trajectory. Referring to part (a) of the gure, the stabilized GGI cases have improved position accuracy over the strapdown GGI case because the stabilized sensor does not estimate the gyro or 244 (a) (b) Figure 6.14: INS/GGI Steady State MRSEs for Mach 7, High n Variation, (a) Position (b) Velocity 245 attitude errors in order to observe the gravitational gradients, see Eq. (5.67) on pg. 210. The strapdown GGI has approximately 2.4{8.1 times the position MRSE compared to the navigation grade IMU / stabilized GGI system, with the largest sensitivity occurring at the lowest noise levels. For GGI noise <0.01 E o/pHz, the strapdown GGI lter begins to diverge, which causes increased error. The stabilized GGI with tactical grade IMUs produces position accuracy only 19{56% more than that of the stabilized GGI with navigation grade IMUs, with the largest sensitivity at the lowest GGI noise levels. Neither GGIs have direct observability of the velocity errors (Eq. (5.67) and Eq. (5.56) on pg. 207). Thus, the velocity error corrections are made by way of the position and attitude error observability. And since a stabilized GGI has better position accuracy but worse attitude accuracy, whereas the strapdown GGI has reduced position accuracy and improved attitude accuracy, the velocity errors of both sensors are somewhat similar. From Fig. 6.14 (b), the stabilized GGIs have superior velocity performance for the lowest GGI noise levels even with the non- diverging strapdown GGI extrapolation. At higher noise levels, the navigation grade INS/GGI systems are comparable. The tactical grade IMU / stabilized GGI system has approximately 1.7{2.8 times the velocity error of the comparable INS/GGI with navigation grade IMUs. And again, the largest error sensitivity occurs for the lowest GGI noises. The attitude errors for the three INS/GGI con gurations are shown in Fig. 6.15 with the actual steady state MRSE (a) and the lter estimated MRSE (b). Because of the attitude and gyro error observability, the strapdown GGI produces about 246 (a) (b) Figure 6.15: INS/GGI Steady State Attitude MRSEs for Mach 7, High n Variation, (a) Monte Carlo (b) Filter 247 an order of magnitude lower orientation error than the comparable stabilized GGI / navigation grade IMU case. Even for the low noise, diverging simulations, the strapdown GGI outperforms the stabilized GGI. The tactical grade IMU / stabilized GGI case has a steady 0.1 total attitude error for all simulated noise levels which is due to the yaw error not reaching steady state by the end of the simulation, as can be seen in Fig. 6.5. Had the lter been tuned di erently, these INS/GGI systems would have reached a steady state yaw error faster and would have most likely changed the results. Also, the chosen initial lter covariance caused the stabilized GGI with navigation grade IMUs to overestimate the attitude errors at high noise levels and thereby allowed the gyros to run without corrections, thus causing lower errors when pqGGI < 0.01 E o/pHz. Stabilized GGI, Navigation Grade IMU Figures 6.16 and 6.17 show the GGI/INS steady state MRSE sensitivities for the INS with navigation grade IMUs and a stabilized GGI aid. All the plot are normalized by the M1 = 7, high n variation cases shown in Fig. 6.14. The high gradient variation trajectories are plotted with solid lines and the low gradient trajectories are plotted with dashed lines. Furthermore, the Mach 6 cases use circles for data point markers, Mach 7 use asterisks, and Mach 8 uses squares. Referring to Fig. 6.16 (a), increasing speed and altitude (larger Mach number) increases the overall position error on average 5%. This performance degradation is due to two main factors. First, the faster trajectories travel the 1000 km range in 248 (a) (b) Figure 6.16: Normalized INS/GGI Steady State MRSE for Stabilized GGI w/ Nav. Grade IMUs, (a) Position (b) Velocity 249 a shorter time span so fewer measurements are made during a simulation. Second, the increase in cruise altitude reduces the magnitude of the gravitational gradient variations. And comparing the gradient variation simulations, the dashed lines, the position errors increase about 2% over the comparable high n variation cases. At the highest noise levels, the results are less sensitive to these e ects because the GGI is relatively ine ective as a map-matching INS aid. The velocity sensitivities follow the same trends as the position errors but have approximately twice the magnitude in their variations. Increasing or decreasing the cruise Mach number causes 7{24% change in velocity MRSE ( 15% on average). The lower signal variation results in about a 5% increase in the total error. Also, GGI noise levels near 0.02 E o/pHz are most sensitive to these changes. These results are quite promising because they show that a future GGI with a noise level of 0.001 E opHz is essentially insensitive to the magnitude of the signal variation so that it could be an e ective INS aid worldwide|not just over regions with high gradient variations. Figure 6.17 plots the attitude sensitivities for this con guration. Referring back to Fig. 6.15, the lter overestimates the attitude error for GGI noises greater than 0.01 E o. Therefore, the Monte Carlo error in part (a) of the Fig. 6.17 exhibits odd trends above this noise level. For the lower noise cases, the lter performs optimally and the trends in Fig. 6.17 (a) and (b) are similar. Referring to the l- ter estimated attitude MRSE, part (b), increasing the cruise velocity and altitude increases the attitude MRSE up to 30%. This sensitivity reduced as GGI noise increases and the gyros tend toward their dead-reckoning, free-inertial values. Sur- 250 (a) (b) Figure 6.17: Normalized INS/GGI Steady State Attitude MRSE for Stabilized GGI w/ Nav. Grade IMUs, (a) Monte Carlo (b) Filter 251 prisingly, the lower n variation trajectories reduce the attitude error by almost 5%. The cause of this improvement in attitude accuracy is unknown, but may be due to the gravity vector being more stable so that the attitude errors are less a ected by gravity errors. Stabilized GGI, Tactical Grade IMU Figure 6.18 plots the position and velocity sensitivities for the INS/GGI cases with tactical grade IMUs and a stabilized gradiometer. The position MRSE is less sensitive to Mach number and gradient variation changes compared to the previous navigation grade IMU / stabilized GGI system, especially at the lower and higher GGI noise levels. At the lowest GGI noise level, the position accuracy varies less than 2% from the nominal Mach 7, high gradient trajectory simulation. For GGI noise levels in the 0.01{0.1 E o/pHz range, an increase in Mach number causes the position MRSE to rise about 5%. The trajectories cause on average about a 1% change in the position MRSE. For the low GGI noise levels, the lower n variation trajectories produces lower errors, but forpqGGI 0.01 E o/pHz this trend reverses. The velocity sensitivities are similar to the navigation grade IMU / stabilized GGI trends. The primary di erence is that the tactical grade IMU system perfor- mance is almost entirely insensitive to speed / altitude or gradient signal variation at the lowest GGI noise levels. For a 0.001 E o/pHz stabilized GGI, there is less than a 1% change in the velocity MRSE. For a 0.01 E o/pHz sensor, the velocity MRSE has about 15% variation due to Mach number changes and about 5% sensitivity to 252 (a) (b) Figure 6.18: Normalized INS/GGI Steady State MRSE for Stabilized GGI w/ Tac. Grade IMUs, (a) Position (b) Velocity 253 signal variation. At the higher noise levels, the INS/GGI becomes less sensitive to these changes as it again starts to act more like a dead-reckoning, GGI-less INS. Comparing the Monte Carlo and lter-estimated attitude MRSE in Fig. 6.19, it is evident that the lter is working optimally. Also, these plots show that the attitude errors are insensitive to the trajectory?s gradient signal variation. According to the lter estimates, there is only a small increase in the error for high gravity eld variations at low GGI noise levels and the trend reverses at higher GGI noises. The Mach number a ects the attitude errors 5{10% with an increase in error as the speed decreases. This result is caused by the yaw error not reaching steady state by the end of the simulation (see Fig. 6.5). Therefore, since the lower Mach number cases run for a longer time, the yaw error increases to a higher value. If the initial lter covariance matrix were tuned di erently these results would likely be di erent. Navigation Grade IMU, Strapdown GGI The normalized MRSE for the strapdown GGI with navigation grade IMUs are plotted in Fig. 6.20 and 6.21. The trends below the 0.01 E o/pHz GGI noise levels will not be discussed in depth because they include the numerical divergence issues discussed on pg. 235. Furthermore, all strapdown GGI / tactical grade IMU systems diverged immediately due to the increased gyro noise of the tactical IMUs. Therefore only the navigation grade IMUs could be simulated with the strapdown gradiometer. As shown in Fig. 6.20, the strapdown GGI system is very sensitive to the 254 (a) (b) Figure 6.19: Normalized INS/GGI Steady State Attitude MRSE for Stabilized GGI w/ Tac. Grade IMUs, (a) Monte Carlo (b) Filter 255 (a) (b) Figure 6.20: Normalized INS/GGI Steady State MRSE for Strapdown GGI w/ Nav. Grade IMUs, (a) Position (b) Velocity 256 (a) (b) Figure 6.21: Normalized INS/GGI Steady State Attitude MRSE for Strapdown GGI w/ Nav. Grade IMUs, (a) Monte Carlo (b) Filter 257 gravitational gradient signal variation. At a GGI noise of 0.01 E o/pHz the lower n variation trajectories have over 2.5 times the position error and between 1.5{ 2.5 the velocity error as the corresponding high n variation trajectory cases. The navigation performance is less sensitive at high GGI noise levels because the GGI is less e cient as an INS aid. And at the two highest GGI noise levels, the low n variation cases produce lower position and velocity MRSE since the gravity errors contributing to the velocity errors are smaller. Increasing the Mach number is shown to degrade position and velocity accuracy by up to 20%. This trend is reversed for the position MRSE at high GGI noise levels. Figure 6.21 shows that the attitude errors for the strapdown GGI system are also quite sensitive to the gravitational gradient signal strength. At worse, the lower gradient variation trajectory doubles the total attitude error. The attitude sensitivity to Mach number is much lower, but follows some rather odd trends. The Monte Carlo errors show that increasing M1 increases the error for the low n variation cases with GGI noises <0.1 E o/pHz and the high n cases with GGI noise 0.1 E o. For the other cases, the Mach number sensitivity trends are reversed. It is unknown why these trends occur, but the magnitude of these sensitivities ( 10%) are large enough to warrant further investigation. 6.3.2 Global Positioning System Aided Navigation Like the INS/GGI results, the INS/GPS results are rst presented and dis- cussed in terms of single representative simulations and their detailed errors. The 258 Table 6.15: INS/GPS Nominal and O -Nominal Simulation Parameters IMU Trajectory Measurements Data Rate Nominal Value Nav. High n Variation and _ 1 sec Perturbed Value Tac. High n Variation -Only 1 sec Monte Carlo sensitivity results are shown and discussed afterwards. The nominal INS/GPS simulation set parameters are listed in Table 6.15 along with their per- turbed values. The full hypersonic INS/GPS text matrix used for the sensitivity analyses is summarized in Table 6.16. The subsonic INS/GPS simulations were only performed with navigation grade IMUs and GPS pseudorange and range-rate mea- surements at a 1 Hz update rate to provide a baseline to the INS/GGI simulations. The primary concern with an INS/GPS system is the update rate or blackout duration of the GPS receiver. This rate is a function of the receiver design and the vehicle dynamics. For the atmospheric, hypersonic cruise simulations, the update intervals were varied from 1 sec to 300 sec. The subsonic systems? sensitivity to update rate was not investigated since they are used as only a baseline comparison Table 6.16: INS/GPS Monte Carlo Test Matrix Parameter Values # IMU Grade Navigation, Tactical 2 Mach Number 6, 7, 8 3 Measurement Type Only, and _ 2 Data Rate 1, 10, [30:30:120], [180:60:300] sec 9 259 to the proposed future INS/GGI system. Also, all the INS/GPS simulations were performed over only the high gravitational gradient variation trajectory because the e ect of GPS visibility and GDOP on the two trajectories is small, as shown in Fig. D.3 on pg. 341 and Fig. D.4 on pg. 350. 6.3.2.1 Monte Carlo Results Hypersonic Cases A sample simulation of the nominal Global Positioning System case is given in Fig. 6.22. The simulation uses the nominal 24-satellite GPS constellation and simulates pseudorange and pseudorange rate measurements to aid a navigation- grade INS at Mach 7. As shown in Fig. 6.22, the INS/GPS lter is stable and quickly reaches steady state in the horizontal and tilt states. The altitude, vertical velocity, and yaw states are less observable to the lter and therefore take slightly longer to correct and reach steady state values. The horizontal position state error envelopes have a discontinuous jump at 72 sec because the number of visible satellites increases from 6 to 7. This increase in measurement observables causes the geometric dilution of precision (GDOP) to drop from 11 to 3.4 instanteously and the lter covariance to reduce accordingly. Compared to the \Best" gradiometer-aided INS case, the nominal GPS performance is approximately twice as good, see Table 6.5 and 6.17. The North (crossrange) position error is half that of the INS/GGI case, and the East (downrange) error is one-quarter the gradiometer case, but the altitude is much closer to the nav.-grade IMU / stabilized GGI scenario. The overall velocity 260 error is about half the INS/GGI error (0.0031 vs. 0.0069 m/s), as is the yaw error. The tilt errors are only slightly lower then the gradiometer-aided case. Figure 6.23 and Table 6.18 present results for the INS/GPS with tactical grade IMUs and both GPS measurements. The odd lter envelope towards the end of the simulation is due to two satellites going out of view, one at 407 sec and another at 426 sec. This tactical-grade case follows trends similar to the previous navigation grade case in relation to the comparable GGI cases. The tactical gradiometer crossrange position error is a factor of 2 larger than the current GPS case, the downrange error is a factor of 4 larger, and the altitude is only about 20% larger. The total velocity error for the tactical GPS is about half that of the tactical GGI case (0.0129 vs. 0.0234 m/s), and the total attitude error is comparable to the gradiometer case (0.0950 vs. 0.106 ). The last sample GPS case to be presented is the e ect of measurement observ- ables. If only pseudorange measurements (i.e. position information) were available, the performance is signi cantly degraded because of the lack of velocity and pre- cise position knowledge from the pseudorange rate measurements, see Eq. (D.20) on pg. 345. Referring to Fig. 6.24 and Table 6.19, the total position error increases to almost 10 times that of the nominal navigation grade INS/GPS case. The velocity error is a little less sensitive, and only increases by a factor of 7. The attitude error also increases, but only by a factor of 3 over the case when pseudorange rate measurements are also made. 261 0 100200300400 ?0.5 0 0.5 ? r N Position Errors, m 0 100200300400 ?0.5 0 0.5 ? r E 0 100200300400 ?0.5 0 0.5 ? r D Time, s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v N Velocity Errors, m/s 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v E 0 100200300400 ?0.02 ?0.01 0 0.01 0.02 ? v D Time, s 0 100200300400 ?50 0 50 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?50 0 50 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.22: Sample Nominal Global Positioning System Simulation Table 6.17: Nominal Global Positioning System Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5487 0.3908 0.2848 0.2363 0.6372 0.4076 100 km Vel. m/s 0.1394 0.1561 0.1140 0.1125 0.1143 0.1260 Att. 10 3 0.7448 22.969 0.5160 22.751 3.1799 25.347 Pos. m 0.1031 0.0953 0.0594 0.0540 0.1030 0.0869 500 km Vel. m/s 0.0024 0.0024 0.0016 0.0015 0.0015 0.0014 Att. 10 3 1.3350 1.3442 0.9228 0.9517 5.7198 5.6250 262 0 100200300400 ?0.4 ?0.2 0 0.2 0.4 ? r N Position Errors, m 0 100200300400 ?0.4 ?0.2 0 0.2 0.4 ? r E 0 100200300400 ?0.4 ?0.2 0 0.2 0.4 ? r D Time, s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v N Velocity Errors, m/s 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v E 0 100200300400 ?0.04 ?0.02 0 0.02 0.04 ? v D Time, s 0 100200300400 ?50 0 50 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?50 0 50 ? ? 0 100200300400 ?200 ?100 0 100 200 ? ? Time, s Figure 6.23: Sample Tactical Grade IMU Global Positioning System Simulation Table 6.18: Tactical Grade IMU Global Positioning System Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5405 0.4089 0.2773 0.2495 0.6456 0.4273 100 km Vel. m/s 0.1398 0.1562 0.1183 0.1126 0.1209 0.1271 Att. 10 3 13.439 37.976 12.768 37.120 49.469 81.184 Pos. m 0.1496 0.1402 0.0942 0.0862 0.1429 0.1288 500 km Vel. m/s 0.0091 0.0089 0.0071 0.0069 0.0058 0.0057 Att. 10 3 24.110 28.357 22.898 26.816 88.961 106.13 263 0 100200300400 ?3 ?2 ?1 0 1 2 3 ? r N Position Errors, m 0 100200300400 ?3 ?2 ?1 0 1 2 3 ? r E 0 100200300400 ?3 ?2 ?1 0 1 2 3 ? r D Time, s 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v N Velocity Errors, m/s 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v E 0 100200300400 ?0.2 ?0.1 0 0.1 0.2 ? v D Time, s 0 100200300400 ?50 0 50 ? ? Attitude Errors, ? ? 10 ?3 0 100200300400 ?50 0 50 ? ? 0 100200300400 ?50 0 50 ? ? Time, s Figure 6.24: Sample Pseudorange Only Global Positioning System Simulation Table 6.19: Pseudorange Only Global Positioning System Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 4.8386 2.2330 1.8196 1.3189 3.2582 1.6642 100 km Vel. m/s 0.0197 0.4545 0.0178 0.4480 0.0157 0.4490 Att. 10 3 1.2829 23.806 0.8624 23.166 10.358 35.401 Pos. m 1.0518 0.9264 0.4696 0.4216 0.8063 0.5913 500 km Vel. m/s 0.0182 0.0181 0.0071 0.0065 0.0112 0.0082 Att. 10 3 2.3054 2.8519 1.5483 1.6996 18.640 23.722 264 Subsonic Cases A commercial aircraft INS/GPS simulation is shown in Fig. 6.25 and the Monte Carlo set results are listed in Table 6.20. From the gure, the position, velocity, and attitude states reach steady-state conditions at or before the 250 km ( 400 sec) settling period. The position MRSE is almost half the low-noise INS/GGI simulation (0.1521 vs. 0.2281 m), with the crossrange (North) error being halved and the downrange (East) error being reduced to a third the INS/GGI value, see Table 6.11. The altitude error is approximately the same for both the INS/GPS and INS/GGI cases since the gradiometer-aiding system has improved performance in the vertical channel. The total velocity error for this GPS-aided case is 60% that of the INS/GGI (0.0029 vs. 0.0050 m/s). And again, the improvement is almost exclusively due to the reduced horizontal errors. The attitude error is similar to the INS/GGI simulation. The GGI survey INS/GPS case is presented in Fig. 6.26 as a sample time history and in Table 6.21 for the full 1,000 simulation Monte Carlo set. This low- altitude, low-velocity INS/GPS case has a remarkable improvement in the navigation accuracy over the comparable INS/GGI case shown on pg. 241. The overall position MRSE decreases by a factor of four (0.1441 vs. 0.5387 m) primarily due to the downrange error decreasing by a factor of 7. The di erence in performance of these two systems may be due to the shorter range of the simulation (300 km) which reduces the amount of variation of the gravitational gradient signal compared to the other cases (which end at 500 or 1,000 km). If the local terrain e ects were 265 0 200400600800 ?0.4 ?0.2 0 0.2 0.4 ? r N Position Errors, m 0 200400600800 ?0.4 ?0.2 0 0.2 0.4 ? r E 0 200400600800 ?0.4 ?0.2 0 0.2 0.4 ? r D Time, s 0 200400600800 ?0.01 ?0.005 0 0.005 0.01 ? v N Velocity Errors, m/s 0 200400600800 ?0.01 ?0.005 0 0.005 0.01 ? v E 0 200400600800 ?0.01 ?0.005 0 0.005 0.01 ? v D Time, s 0 200400600800 ?5 0 5 ? ? Attitude Errors, ? ? 10 ?3 0 200400600800 ?5 0 5 ? ? 0 200400600800 ?50 0 50 ? ? Time, s Figure 6.25: Sample Commercial Aircraft Global Positioning System Simulation Table 6.20: Commercial Aircraft INS/GPS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5966 0.3750 0.2899 0.2343 0.6597 0.3923 50 km Vel. m/s 0.1365 0.1526 0.1120 0.1092 2.4688 0.1233 Att. 10 3 0.5081 22.719 0.4887 22.716 7.7545 29.476 Pos. m 0.0999 0.0880 0.0640 0.0569 0.0952 0.0739 250 km Vel. m/s 0.0021 0.0020 0.0016 0.0015 0.0011 0.0010 Att. 10 3 0.9091 0.8916 0.8733 0.8849 13.955 13.054 266 included in the gravitational gradient signal model, the INS/GGI error should be reduced to a value closer to that of the INS/GGI case. The velocity error is about a third of the INS/GGI value (0.0028 vs. 0.0094 m/s) with the largest decrease in the alongtrack (East) component. And the tilt errors are about the same for both navigation aids. The INS/GPS yaw error, however, increases to twice the value of the INS/GGI error. 6.3.2.2 Sensitivity Results The INS/GPS sensitivities are presented in a similar fashion to the INS/GGI cases. First, the Mach 7 steady state MRSEs will be compared for INS/GPS con g- urations with either navigation or tactical grade IMUs and GPS pseudorange with and without pseudorange rate updates. After these results, the sensitivity of Mach number on the navigation performance is discussed for the four INS/GPS systems. Figure 6.27 plots the steady state Mach 7 position and velocity MRSE for the four INS/GPS systems simulated and update intervals from 1 to 300 sec. From part (a), the position error is strongly a function of GPS measurement. When pseudorange rate measurements are simulated, the position MRSE decreases on average by a factor 8 for the navigation grade IMU INS/GPS, and a factor of 7 for the tactical grade IMU system. The e ect of IMU quality on the position error is less severe. For the and _ simulations, the tactical grade IMUs increase the error 35%, and for the -only cases the error increases 25% for tGPS < 90 sec. The velocity errors exhibit similar trends, but with the noted di erence that 267 0 500 1000 ?0.4 ?0.2 0 0.2 0.4 ? r N Position Errors, m 0 500 1000 ?0.4 ?0.2 0 0.2 0.4 ? r E 0 500 1000 ?0.4 ?0.2 0 0.2 0.4 ? r D Time, s 0 500 1000 ?0.01 ?0.005 0 0.005 0.01 ? v N Velocity Errors, m/s 0 500 1000 ?0.01 ?0.005 0 0.005 0.01 ? v E 0 500 1000 ?0.01 ?0.005 0 0.005 0.01 ? v D Time, s 0 500 1000 ?5 0 5 ? ? Attitude Errors, ? ? 10 ?3 0 500 1000 ?5 0 5 ? ? 0 500 1000 ?50 0 50 ? ? Time, s Figure 6.26: Sample GGI Survey Global Positioning System Simulation Table 6.21: GGI Survey INS/GPS Case Settling North / Pitch East / Roll Down / Yaw Time State Units Error Cov. Error Cov. Error Cov. Pos. m 0.5980 0.3753 0.2791 0.2266 0.7167 0.3881 30 km Vel. m/s 0.1374 0.1537 0.1144 0.1114 5.6713 0.1260 Att. 10 3 0.5039 22.712 0.4992 22.710 21.861 31.739 Pos. m 0.0964 0.0861 0.0630 0.0556 0.0866 0.0692 150 km Vel. m/s 0.0021 0.0019 0.0016 0.0015 0.0010 0.0009 Att. 10 3 0.9015 0.8807 0.8921 0.8780 39.346 17.129 268 (a) (b) Figure 6.27: INS/GPS Steady State MRSE for Mach 7 Simulations, (a) Position (b) Velocity 269 the IMU quality causes larger variations. The increase in velocity error due to a tactical grade IMU is about 150% for the and _ simulations and 90% for the -only simulations. The sensitivity to measurement type is 400{590% for the navigation grade INS/GPS and 80{420% for the tactical grade con guration. There is a substantial increase in the position and velocity errors when the GPS update interval is greater than about 90 sec. The reason for this is that the GPS measurements are made so infrequent that the in- ight calibration of the IMU errors can not be performed as e ciently. For low tGPS, the GPS-aiding allows the INS to reduce the accelerometer and gyro bias and (to a lesser extent) scale factor errors so that between updates the dead reckoning navigation accuracy is improved. In other words, the INS/GPS in- ight calibration essentially improves the quality of the IMUs. The pseudorange-only / navigation grade IMU system does not follow this trend at long update intervals because the lter overestimates the attitude errors here, which causes the lter to perform suboptimally as seen in Fig. 6.28. Comparing the Monte Carlo and lter-estimated errors of Fig. 6.28, it is ap- parent that the INS/GPS is accurately predicting the errors for all cases except the navigation grade IMUs at high update intervals. For tGPS > 30 sec, the -only attitude errors start to be overestimated. The and _ navigation grade INS/GPS begins to perform suboptimally at tGPS 90 sec. Comparing these errors before these times, the attitude error increases 170{300% with a lack of pseudorange rate measurements. Because there is no direct observability of the attitude errors from the GPS measurements, the INS/GPS must use the estimated velocity vector to 270 (a) (b) Figure 6.28: INS/GPS Steady State Attitude MRSE for Mach 7 Simulations, (a) Monte Carlo (d) Filter 271 estimate the attitude states. So, when the velocity observability is decreased, i.e. no _ measurements, the performance is degraded. The tactical IMU systems both have an approximately constant 0.1 attitude error because the yaw error has not reached steady state by the end of the simulation. Furthermore, because of this poor tuning of the initial lter covariance of the yaw state, the attitude sensitivities to Mach number with not be presented. The total attitude MRSE is tabulated, however, in App. E for the interested reader. Navigation Grade IMU, Pseudorange & Pseudorange Rate The position and velocity sensitivities to Mach number are plotted in Fig. 6.29 for the navigation grade INS/GPS with pseudorange and range-rate measurements. The rst noticeable characteristic of these plots is the spike at 120 sec for the Mach 8 simulation. This spike is a result of the in- ight IMU calibration issue. Because the Mach 8 case is simulated for the shortest amount of time to reach its 1000 km range, fewer updates are made than for the other two cases. Therefore, the faster simulation transitions to higher errors at a lower tGPS because fewer overall measurements are made in the simulation. Neglecting this phenomenon, the position MRSE produces an 8% change in error for a change in Mach number. The velocity errors have an average sensitivity of 15% for changes in Mach number. 272 (a) (b) Figure 6.29: Normalized INS/GPS Steady State MRSE w/ & _ Measurements and Nav. Grade IMUs, (a) Position (b) Velocity 273 (a) (b) Figure 6.30: Normalized INS/GPS Steady State MRSE w/ & _ Measurements and Tac. Grade IMUs, (a) Position (b) Velocity 274 Tactical Grade IMU, Pseudorange & Pseudorange Rate The MRSE of the tactical grade INS with pseudorange and pseudorange rate measurements is plotted in Fig. 6.30. Like the navigation grade INS/GPS above, the sensitivities have spikes due to the transition from e cient calibration of the IMU errors to less e cient calibration. This results in the negative spike at tGPS = 90 sec for the Mach 6 case since the baseline (Mach 7) simulation has increased error occuring at an earlier update interval than the Mach 6 baseline. The positive spike at tGPS = 120 sec for the Mach 8 case follows the same reasoning. At the highest measurement intervals, the INS/GPS is essentially a dead-reckoning system. Prior to the two spikes mentioned, the position error increases about 5% for an increase or decrease in Mach number from the M1 = 7 baseline. The reason why both the Mach 6 and 8 cases increase error is unknown. The Mach 6 position errors may have increased because the yaw errors are greater than the Mach 7 or 8 cases which may act to o set the additional updates made. Conversely, the Mach 8 simulations have less updates over their mission duration, but the yaw error has less time to accumulate as well. The velocity errors follow the same trends but with reduced sensitivity to Mach number. Navigation Grade IMU, Pseudorange-Only The GPS pseudorange-only updates to a navigation grade IMU INS/GPS pro- duce unusual sensitivities to changes in Mach number, as shown in Fig. 6.31. The position and velocity errors typically increase for higher Mach numbers, but the 275 (a) (b) Figure 6.31: Normalized INS/GPS Steady State MRSE w/ Measurements and Nav. Grade IMUs, (a) Position (b) Velocity 276 Mach 8 case at the fastest data rate cause less position error than the corresponding Mach 7 simulation set. The cause of this outlier is unknown. The over estimation of the yaw errors in the lter cause a suboptimal Kalman gain calculation which may be partially responsible. On average, the Mach 6 position errors decrease about 5% from the baseline, and the Mach 8 errors increase around 5% when the lowest and highest update interval results are ignored. The velocity errors are much more sensitive to Mach number. The Mach 6 velocity MRSE drops 15% and the Mach 8 rises 10{20% for most cases. The longest tGPS results are almost dead reckoning case because so few measurements are made. Tactical Grade IMU, Pseudorange-Only The Mach number sensitivities of the tactical grade IMU, -only INS/GPS is less ambiguous than the preceding system. The lower quality IMUs cause the gyro and attitude to increase more rapidly so that the lter performs optimally even at the longest tGPS, unlike the previous nav.{grade con guration. The higher Mach numbers produce larger position and velocity errors because fewer GPS updates are processed during a simulation run. The position errors vary less than 10% for updates occurring at least every 30 seconds. For longer update intervals, the errors can increase over 20-57%. The velocity error sensitivities increase about 10{15% for the Mach 8 case and decrease about the same percentage when the Mach number is decreased to 6. 277 (a) (b) Figure 6.32: Normalized INS/GPS Steady State MRSE w/ Measurements and Tac. Grade IMUs, (a) Position (b) Velocity 278 6.4 Chapter Summary The results from the 1,000-set Monte Carlo INS/GGI and INS/GPS simula- tions are presented and discussed in this chapter. Section 6.1 discusses how the Monte Carlo simulations are performed using the elements of the previous chapters and rst four appendices. The following section studies the e ect of set size on how well the Monte Carlo simulations capture all the random processes of a sample INS/GGI simulation. It is shown in Sec. 6.2 that the lter-estimated errors are insensitive to Monte Carlo set size, and that the Monte Carlo derived errors capture the random processes more accurately as the set size is increased. However, increas- ing the Monte Carlo set size corresponds to an approximately linear increase in the computational expense with diminishing returns in the e ectiveness of capturing the random processes (see Fig. 6.3 on pg. 221). Section 6.3 then quanti es the navigation accuracy and sensitivities of the completely inertial, passive, covert INS/GGI and baseline INS/GPS systems. The INS/GGI results show surprisingly impressive sub-meter, INS/GPS-like total posi- tion error for a system with a space-grade stabilized GGI and current navigation grade IMUs for both hypersonic and subsonic missions. The INS/GGI also typically has half the vertical position and velocity errors compared to it horizontal errors because of the stronger signal variation in altitude. The hypersonic INS/GGI system sensitivities to noise level, Mach number and gravitational gradient variation are also thoroughly investigated through extensive Monte Carlo simulations in Sec. 6.3.1. The main conclusions from these analyses 279 are as follows: A stabilized GGI with navigation grade IMUs at Mach 7 over the high n variation trajectory produces a total position MRSE of 0.336 m when the GGI noise level is 0.001 E o at an update rate of 1 Hz. A strapdown GGI aided INS produces up to an order of magnitude increase in the position error of a comparable stabilized GGI, with the largest sensitivity occuring at the lowest noise levels. Conversely, the stabilized GGI produces attitude (i.e. orientation) errors that are an order of magnitude higher than the strapdown sensor. Reducing the quality of the IMUs with a stabilized GGI aided INS causes a 20{50% increase in position error and a 70{180% increase in velocity error. The strapdown GGI is also shown to cause lter divergence when estimating gyro noise that is much larger than its own instrument noise. Two solutions are suggested to combat the numerical truncation issue that is at the root of the divergence: 1. Higher precision data types may be used to compute the Kalman gain matrix. (Double precision is used exclusively in this work.) 2. A square-root Kalman lter may be implemented instead of the traditional lter used in this work. The premise is that a square-root lter uses standard deviations instead of variances in its calculations, so that the values of the gyro and GGI noises are more similar in magnitude and thus numerical truncation is less likely to occur. Table 6.22 summarizes the steady-state position MRSE as a function of GGI noise level through power law regressions. The conclusions resulting from these 280 Table 6.22: Hypersonic INS/GGI Postion MRSE (m) Sensitivity to GGI Noise GGI Type IMU n Var. Mach 6 Mach 7 Mach 8 Stabilized Nav. High 209pqGGI0:939 219pqGGI0:938 227pqGGI0:934 Stabilized Tac. High 244pqGGI0:903 250pqGGI0:902 258pqGGI0:904 Strapdown Nav. High 376pqGGI0:620 377pqGGI0:602 359pqGGI0:580 Stabilized Nav. Low 217pqGGI0:943 226pqGGI0:941 235pqGGI0:937 Stabilized Tac. Low 246pqGGI0:905 260pqGGI0:910 263pqGGI0:907 Strapdown Nav. Low 286pqGGI0:302 212pqGGI0:211 164pqGGI0:138 GGI noise level, pqGGI, is in E o/pHz units analyses are: Future improvements in GGI noise levels will lead to greater improvements in position error if the gradiometer is stabilized than if it is a strapdown sensor. This is shown in the exponential coe cients, which are the slopes of results such as Fig. 6.14 (a) on pg. 245. The main reason for the lower performance of the strapdown GGI/INS is its need to account for the IMU gyro errors. The stabilized GGI, navigation grade IMU system produces about a factor of 8.5 steady state position MRSE sensitivity to a 10 change in GGI noise level. The stabilized GGI, tactical grade IMU system is slightly less sensitive at about a factor of 8 for an order of magnitude change in pqGGI. The strapdown GGI/INS produces only a factor of 4 improvement in position MRSE with a ten-fold improvement in GGI noise level when ying over the high n variation trajectory. And the sensitivity is reduced further to a factor of 1.4{2 when the system is own over the low gradient variation trajectory 281 Table 6.23: Hypersonic INS/GGI Sensitivities to n Variation and Mach Number n Variation Mach Number GGI Type IMU Pos. Vel. Att. Pos. Vel. Att. Stabilized Nav. 1 1 1 1 2 2 Stabilized Tac. 1 1 1 1 2 2 Strapdown Nav. 3 3 3 2 2 1 Error Sensitivity: 1) <10%, 2) 10{100%, 3) >100% and pqGGI is varied by a factor of 10. The system speci c INS/GGI sensitivities are summarized in Table 6.23 in terms of the order of magnitude of the error variation. The primary conclusions are: The stabilized GGI aided INS is insensitive to changes in the gravitational gradient signal variation. The position, velocity, and attitude errors typically vary less than 10% between the high and low n trajectory results. The stabilized GGI aided INS position error is also insensitive to Mach number variations. However, the velocity and attitude errors are more sensitive and increase over 10% when the Mach number is increased or decreased. The strapdown GGI aided INS, however, is extremely sensitive to changes in the gravitational gradient signal. Position, velocity, and attitude errors can increase over 100% the high n trajectory values when the same con guration is own over the low gradient variation trajectory. The strapdown GGI aided INS is noticeably less sensitive to Mach number variations. The position and velocity errors can increase up to 20% when the 282 Mach number is increased by 1, and the attitude error is least sensitive and changes at most 10% because most of the strapdown GGI update information is used to correct attitude and gyro errors. Section 6.3.2 investigates the nominal integrated INS/GPS navigation perfor- mance. The Monte Carlo simulation results show sub-meter position errors for GPS update intervals of 30 seconds or faster when pseudorange and pseudorange rate measurements are available. However, the absence of pseudorange rate measure- ments produces almost an order of magnitude increase in the total position error. The e ect of IMU quality is much less severe, and position MRSE only increases about 30% when tactical grade IMUs are simulated. Compared to the INS/GGI results, the INS/GPS system with and _ updates at 1 Hz produces about half the position MRSE of a 0.001 E o stabilized GGI aided INS with 1 Hz updates, regardless of the IMU quality. Also, the INS/GPS system needs and _ measurements only once every 10 sec to produce the same position MRSE as the 0.001 E o stabilized GGI/INS with 1 Hz updates. The main unexpected result from the INS/GPS simulations is the large error growth when GPS updates occur less than every minute. The inability to reduce the IMU errors in- ight produces about an order of magnitude increase in position and velocity error when the GPS update interval increases from 60 to 90 seconds. 283 Chapter 7 Conclusions and Future Work This dissertation presents the rst complete open literature methodology, derivation, implementation, and simulation of a map-matching gravity gradiome- ter aided inertial navigation system. Gravity gradiometer aiding is particularly applicable to military applications where GNSS signals may be jammed, spoofed, or otherwise unavailable. A hypersonic atmospheric cruise missile was thus chosen as an ideal application since its rst use will be for a high speed cruise missile, and the high velocities and temperatures of ight may cause traditional aids to perform poorly or be impractical. Moreover, the increased velocity allows for greater gravita- tional variation between measurements, and the relatively high altitudes attenuate small terrain anomalies and uctuations. However, the current size ( 1 m3) and weight ( 250 kg without additional electronics cabinet) of current commercial air- borne GGIs is prohibitively large. Therefore, further research and development is required to reduce future generation gradiometers to the point where they may be integrated into size and weight constrained air vehicles. For the nearer future, two subsonic missions which can accomodate the mass and volume of current GGIs are also simulated. 284 7.1 Summary of Contributions The following are the primary contributions that this dissertation makes to the state of the art. The characterization of gravitational gradients for use as a map-matching nav- igation aid is performed for the rst time. A parametric analysis is presented to estimate when local terrain e ects may be neglected from a computed gravitational eld map. Improvements to an integrated INS/GGI using a Kalman lter approach are identi ed and implemented. A thorough methodology to determine the measurement of a strapdown / stabilized, stationary / rotating accelerometer, stationary / rotating gravity gradiometer instrument is derived. This work provides the rst linearized error derivation of a strapdown GGI that includes attitude errors and a means to convert tensor measurements from the body frame to the navigation frame. This work also derives a new formulation and linearized error equation for a stabilized GGI whose attitude and attitude rate errors are included in the sensor noise speci cation. This work is the rst to simulate an INS/GGI system at hypersonic speeds and altitudes. And the rst to simulate an airborne INS/GGI with space-grade GGI noise levels. The hypersonic INS/GGI sensitivities to noise level, Mach number, and gra- dient signal variation are investigated for the rst time. 285 The rst comprehensive study of a hypersonic INS/GPS system and it?s sen- sitivities to Mach number, IMU quality, GPS update interval, and available measurements is also performed. 7.2 Recommendations for Future Work Some areas of future work are proposed below pertaining to an integrated INS/GGI system. Other applications for gravity gradiometer instruments are dis- cussed afterwards along with a brief mention of the similarity of the current work to magnetometer-based map-matching navigation. The gravitational gradient eld could be characterized with local terrain e ects included. This analysis would be infeasible on a global scale, but could be useful over moderate regions of interest. The gravitational potential, acceleration, gradients, and possibly third order gradients could be augmented to the lter state vector. This would allow the lter to essentially update the state estimates using an optimal t of the gravity eld instead of a single point measurement as done in this work. This would, however, involve implementing a possibly complex linearized gravitational eld model.94,167{171 To alleviate the strapdown GGI divergence issue, a square-root Kalman lter or higher precision oating point operations may be implemented. The centripetal errors in the velocity error dynamics could be added. Also, the e ect of Earth?s oblateness could be included in the Jacobian from the gravity map coordinates to the navigation frame position states. 286 INS/GGI simulations could be performed that included local terrain e ects in the gravitational eld map. This is particularly important for lower altitude missions since the spherical harmonic model severely aliases the true signal content. The sensitivity of the INS/GGI system to gravitational eld errors in the onboard gravity map could be investigated. In such cases, the inclusion of a higher order method for interpolating the gravity eld data from the pre- computed map could be implemented and its bene t could be assessed.95,165 Gravity gradiometer based navigation is also viable for extraterrestrial appli- cations where navigation satellites are unavailable. Spherical harmonic models for the Moon, Mars, and Venus are available onliney, and can be used to assess the navigation performance on and around these bodies. An ideal extraterrestrial mis- sion would be to rst send a satellite with a gradiometer payload to perform high resolution gravity maps of the planetary bodies. Then, future gradiometer-equipped missions would not only be able to improve the gravity model, but eventually nav- igate using the gravitational gradient map and the tools described herein. Low temperature, exo-atmospheric applications are particularly appealing because gra- diometer instruments can yield higher precision due to reduced noise having to be ltered from vehicle motion and the improved mechanical stability at low tempera- tures. Some other extensions to this work could be the use of a gravity gradiometer instrument for obstacle avoidance. In theory, as a GGI-equipped system moves yhttp://pds-geosciences.wustl.edu/dataserv/index.htm 287 toward an obstacle it would measure an increase in the gravitational potential of the obstruction and could move to avoid collision. Similarly, a GGI system could traverse a corridor by avoiding the increased gravitational gradients as it nears one of the boundaries. One last application for a gravity gradiometer instrument that could have near- future impact is its use as a warning system. If a GGI were properly calibrated, it could be used to measure large masses moving toward or near the instrument. This could be implemented as a missile defense system if the sensor noise is low enough and the gravitational gradients caused by the incoming missile is large enough to be unambiguously detected. As one nal comment, because the magnetic potential eld possesses many of the same properties as the gravitational eld, the methodology presented within this work may be useful in magnetometer-based map-matching inertial navigation systems.172 288 Appendix A Global Gravitational Maps This appendix provides additional global gravitational gradient plots at various altitudes. The rst four sets of plots, Fig. A.1{A.4, illustrate the gradients as computed by the modi ed NGS/NOAA geopot97.v0.4.e.f program for all six gradients at 10 km and 100 km altitudes. (The gravitational gradient plots at the Earth?s surface are shown in Fig. 2.7 & 2.8 on pg. 61 & 63.) The inline gradients are plotted with a 20 E o variation from the global mean of the gradient component and the o -diagonal gradients are plotted with a 10 E o variation from their means. The second set of gures are the 5 5 grid point ( 220 220 km) moving- window standard deviation plots that quantify the variation of the gravitational gradient signal. All six components of the gradient tensor are shown at 0, 10, and 100 km altitudes in terms of log10(E o). Similar plots were computed for altitudes of 100 m, 1 km, and 500 km. These gures are not shown because they are qualitatively simular to Fig. A.5{A.9, and their magnitudes are summarized in Fig. 2.11 on pg. 66. 289 (a) (b) (c) Figure A.1: Inline Gravitational Gradients at 10 km (a) NN (b) EE (c) DD 290 (a) (b) (c) Figure A.2: O -Diagonal Gradients at 10 km (a) NE (b) ND (c) ED 291 (a) (b) (c) Figure A.3: Inline Gravitational Gradients at 100 km (a) NN (b) EE (c) DD 292 (a) (b) (c) Figure A.4: O -Diagonal Gradients at 100 km (a) NE (b) ND (c) ED 293 (a) (b) (c) Figure A.5: Inline n, log10(E o), at Surface (a) NN (b) EE (c) DD 294 (a) (b) (c) Figure A.6: O -Diagonal n, log10(E o), at Surface (a) NE (b) ND (c) ED 295 (a) (b) (c) Figure A.7: Inline n, log10(E o), at 10 km (a) NN (b) EE (c) DD 296 (a) (b) (c) Figure A.8: O -Diagonal n, log10(E o), at 10 km (a) NE (b) ND (c) ED 297 (a) (b) (c) Figure A.9: Inline n, log10(E o), at 100 km (a) NN (b) EE (c) DD 298 (a) (b) (c) Figure A.10: O -Diagonal n, log10(E o), at 100 km (a) NE (b) ND (c) ED 299 Appendix B Thrust Coe cient Curve Fits The thrust coe cient is calculated as a function of the inlet compression ratio, (A0=A1), at a given design Mach number, freestream Mach number, and angle of attack by the quadratic polynomial curve- t: CT;ref = T q1A0 ref = 8 >>< >>: a A 0 A1 2 +b A 0 A1 +c; if A 0 A1 A 0 A1 max 0; else (B.1) where the coe cients a, b, and c are de ned for the following con gurations: Base Area-to-Capture Area Ratio: (A5=A0) = 1, 2, 3, 4, 6, and 8. Equivalence Ratio: ER = 0.25, 0.50, 0.75, and 1.00 or ERmax to cause thermal choking. Freestream Mach Number: M1 = 4, 5, 6, 7, and 8. Combustor Expansion Ratio: (A4=A2) = 2, 3, and 4. The thrust coe cients were only curve t for combustor expansion ratios of 3 and 4. Tables B.1{B.6 list the curve t coe cients for the (A4=A2) = 4 scramjets and each of the (A5=A0), ER, and M1 con gurations along with the maximum contraction ratio, (A0=A1)max, values where the ts are valid. Table B.7 then lists the curve- t coe cients that are di erent for a combustor expansion ratio of 3. The (A0=A1)max values are the same as those in the (A4=A2) = 4 300 tables. The changes are primarily in the lower Mach number regime where the thrust coe cient is most sensitive to the combustor expansion. At the other conditions, the two combustor expansion ratios result in essentially the same thrust level so that the (A4=A2) = 4 ts are valid for the (A4=A2) = 3 designs. 301 Table B.1: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 1 ER M1 a b c (A0=A1)max 4 -0.49869841e-1 0.49430349 -0.82621644 5.34124 5 -0.68479882e-2 0.99125012e-1 -0.10943232 7.22688 0.25 6 -0.45458090e-2 0.66848095e-1 -0.10020729 8.36429 7 -0.67763083e-2 0.96744182e-1 -0.25636996 8.51539 8 -0.74915009e-2 0.98622305e-1 -0.27782115 7.95164 4 -0.76991209e-1 0.73788027 -1.0082527 5.29383 5 -0.89241822e-2 0.14105697 0.51239270e-2 7.21112 0.50 6 -0.71480649e-2 0.11498241 -0.93289771e-1 8.41051 7 -0.49207419e-2 0.80802043e-1 -0.97280841e-1 8.57179 8 -0.68767250e-2 0.10346300 -0.23251623 7.98609 4 -0.83692969e-1 0.80721960 -0.94895952 5.25097 5 -0.14337469e-1 0.21505580 0.16144186e-1 7.16958 0.75 6 -0.76123522e-2 0.13438755 -0.10170729e-1 8.35844 7 -0.38090162e-2 0.74719918e-1 0.35054487e-1 8.52622 8 -0.78741334e-2 0.12091532 -0.19482288 8.00242 4 -0.95907300e-1 0.96465505 -1.2327463 5.28320 1.00 5 -0.18608201e-1 0.26443796 0.75892441e-1 6.90048 or 6 -0.76161545e-2 0.14704879 0.67975813e-1 8.20316 ERmax 7 -0.52199660e-2 0.10007825 0.62896756e-1 8.53903 8 -0.76527114e-2 0.12160603 -0.12667565 7.94352 302 Table B.2: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 2 ER M1 a b c (A0=A1)max 4 -0.14723606e-1 0.15796242 0.51099570e-1 5.14437 5 -0.61912328e-2 0.81047984e-1 0.49456662e-1 7.44221 0.25 6 -0.41471505e-2 0.59300713e-1 -0.10809633e-2 8.30857 7 -0.32666482e-2 0.45973194e-1 -0.30536811e-1 8.52718 8 -0.37074063e-2 0.47618649e-1 -0.72715650e-1 7.97840 4 -0.31959459e-1 0.30944616 0.15479222 5.24959 5 -0.73712938e-2 0.11021194 0.23658088 7.19087 0.50 6 -0.54228444e-2 0.85381017e-1 0.12362564 8.35794 7 -0.42692735e-2 0.67897666e-1 0.46325748e-1 8.44007 8 -0.52866323e-2 0.73192206e-1 -0.35329442e-1 7.97164 4 -0.29631520e-1 0.29916407 0.48348068 5.28796 5 -0.93505382e-2 0.14449643 0.40529337 7.28516 0.75 6 -0.67267738e-2 0.10948393 0.25521763 8.41082 7 -0.49389437e-2 0.85813169e-1 0.13326497 8.53236 8 -0.50788323e-2 0.79375985e-1 0.46574283e-1 7.96868 4 -0.36879008e-1 0.37874472 0.51512400 5.28645 1.00 5 -0.14375675e-1 0.19316592 0.52549549 7.23138 or 6 -0.72949267e-2 0.12302538 0.36783282 8.36721 ERmax 7 -0.52483802e-2 0.93660064e-1 0.22848722 8.61187 8 -0.64503764e-2 0.99332058e-1 0.76087323e-1 8.16626 303 Table B.3: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 3 ER M1 a b c (A0=A1)max 4 -0.11146619e-1 0.12005949 0.13773469 5.24767 5 -0.34334492e-2 0.51767211e-1 0.10856734 7.21186 0.25 6 -0.28067197e-2 0.41974676e-1 0.39753387e-1 8.26063 7 -0.24956469e-2 0.35852152e-1 -0.26626258e-2 8.49547 8 -0.27291409e-2 0.35642461e-1 -0.38064314e-1 7.89869 4 -0.22256065e-1 0.21882103 0.38402274 5.26673 5 -0.73835782e-2 0.10376249 0.31873785 7.22574 0.50 6 -0.51619733e-2 0.77800273e-1 0.18304066 8.26729 7 -0.36059536e-2 0.57763705e-1 0.10704738 8.52998 8 -0.41643449e-2 0.59162080e-1 0.27658310e-1 7.85842 4 -0.20878524e-1 0.21415689 0.74281116 5.19420 5 -0.79771064e-2 0.12288176 0.52224229 7.21105 0.75 6 -0.70450758e-2 0.10813795 0.29998903 8.24834 7 -0.50225198e-2 0.79620912e-1 0.20958715 8.49802 8 -0.55665969e-2 0.81268786e-1 0.80020500e-1 7.95805 4 -0.33413460e-1 0.33764979 0.72563305 5.21640 1.00 5 -0.14096373e-1 0.18317335 0.64149865 7.13521 or 6 -0.54601094e-2 0.95317851e-1 0.51840686 8.23574 ERmax 7 -0.63256723e-2 0.10099417 0.27526047 8.46407 8 -0.59622853e-2 0.88468574e-1 0.17026105 7.97723 304 Table B.4: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 4 ER M1 a b c (A0=A1)max 4 -0.11735694e-1 0.12017128 0.10247929 5.15165 5 -0.39596504e-2 0.54340433e-1 0.85155807e-1 7.18347 0.25 6 -0.27058573e-2 0.39683807e-1 0.34540011e-1 8.18063 7 -0.22989753e-2 0.32597252e-1 0.12484620e-2 8.54976 8 -0.32100665e-2 0.39236480e-1 -0.46205128e-1 7.84728 4 -0.26173887e-1 0.24413133 0.32844282 5.19329 5 -0.73612348e-2 0.10218527 0.29417668 7.09964 0.50 6 -0.38172068e-2 0.61641187e-1 0.22316541 8.16017 7 -0.36021941e-2 0.55258603e-1 0.12305535 8.45813 8 -0.35772045e-2 0.50520919e-1 0.59264004e-1 7.94291 4 -0.22675460e-1 0.22457780 0.72504452 5.25161 5 -0.65538781e-2 0.10185956 0.57817918 7.19310 0.50 6 -0.41582314e-2 0.71822435e-1 0.41713740 8.26676 7 -0.53155034e-2 0.81314672e-1 0.21679152 8.52047 8 -0.47039122e-2 0.68863229e-1 0.13574411 8.02259 4 -0.27615043e-1 0.28026860 0.87356171 5.22439 1.00 5 -0.93552788e-2 0.13103816 0.77145709 7.33966 or 6 -0.61425042e-2 0.98500147e-1 0.52539729 8.21213 ERmax 7 -0.39304571e-2 0.70793960e-1 0.36916636 8.49806 8 -0.59267952e-2 0.87243640e-1 0.18333127 7.94496 305 Table B.5: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 6 ER M1 a b c (A0=A1)max 4 -0.12529136e-1 0.12206898 -0.14968492e-1 5.27252 5 -0.34291034e-2 0.44775234e-1 0.50857793e-1 7.23918 0.25 6 -0.17723672e-2 0.27413365e-1 0.27566069e-1 8.32048 7 -0.22768270e-2 0.31015937e-1 -0.20237576e-1 8.49832 8 -0.29900442e-2 0.36468051e-1 -0.53566511e-1 7.95358 4 -0.17822279e-1 0.16317851 0.43083852 5.26965 5 -0.41700614e-2 0.63750064e-1 0.36001300 7.23207 0.50 6 -0.36742344e-2 0.55665557e-1 0.22172396 8.36120 7 -0.28725371e-2 0.45126210e-1 0.14099377 8.51603 8 -0.41147649e-2 0.53997072e-1 0.54505152e-1 8.01917 4 -0.73027373e-2 0.85815793e-1 0.96204392 4.89482 5 -0.71594565e-2 0.10308161 0.56938533 7.16355 0.75 6 -0.32037931e-2 0.58435989e-1 0.44438378 8.29234 7 -0.33599537e-2 0.57076361e-1 0.28317578 8.52126 8 -0.47533838e-2 0.67321569e-1 0.14667716 7.94179 4 -0.24566725e-1 0.24658399 0.90865333 5.25089 1.00 5 -0.12409402e-1 0.15827602 0.69721794 7.13662 or 6 -0.50926776e-2 0.85637217e-1 0.55995518 8.27444 ERmax 7 -0.37227053e-2 0.64248349e-1 0.40919142 8.38862 8 -0.41636836e-2 0.64323415e-1 0.27021240 7.93855 306 Table B.6: Thrust Coe cient Curve-Fits, (A4=A2) = 4, (A5=A0) = 8 ER M1 a b c (A0=A1)max 4 -0.81368756e-2 0.83305787e-1 -0.78812864e-1 5.26415 5 -0.24273038e-2 0.37205281e-1 -0.28679517e-1 7.08101 0.25 6 -0.27358687e-2 0.37034942e-1 -0.49023916e-1 8.25361 7 -0.21923976e-2 0.29228660e-1 -0.52145408e-1 8.56251 8 -0.18622312e-2 0.23863216e-1 -0.60198579e-1 7.91689 4 -0.15535548e-1 0.15076497 0.29289183 5.25000 5 -0.49321370e-2 0.69684983e-1 0.27020668 7.14472 0.50 6 -0.28437544e-2 0.45463429e-1 0.19738206 8.24510 7 -0.22132714e-2 0.35625016e-1 0.12808787 8.57162 8 -0.30407192e-2 0.41748329e-1 0.54903349e-1 7.95809 4 -0.20904792e-1 0.19666099 0.63291311 5.26652 5 -0.43906829e-2 0.68583118e-1 0.60015736 7.04946 0.75 6 -0.32898975e-2 0.55867996e-1 0.42090689 8.23280 7 -0.33136575e-2 0.54277481e-1 0.26759355 8.51059 8 -0.37348158e-2 0.54625293e-1 0.16373794 7.95528 4 -0.31442963e-1 0.32812950 0.63112734 5.56779 1.00 5 -0.10250625e-1 0.13090870 0.72675794 7.16425 or 6 -0.33573140e-2 0.62001513e-1 0.59820948 8.34410 ERmax 7 -0.39744570e-2 0.63993267e-1 0.41216060 8.55414 8 -0.42656742e-2 0.64204486e-1 0.26598943 8.03373 307 Table B.7: Thrust Coe cient Curve-Fit Corrections when (A4=A2) = 3 (A0=A1) ER M1 a b c 1 0.25 4 -0.20378066e-1 0.21893979 -0.17174159 5 -0.61046225e-2 0.91499066e-1 -0.68965469e-1 0.50 4 -0.33391996e-1 0.34025251 -0.83416244e-1 5 -0.92850269e-2 0.14476685 0.19757637e-1 6 -0.33391996e-1 0.34025251 -0.83416244e-1 7 -0.92850269e-2 0.14476685 0.19757637e-1 0.75 4 -0.37944022e-1 0.41943896 -0.83230792e-1 5 -0.14127845e-1 0.20846118 0.71733448e-1 6 -0.37944022e-1 0.41943896 -0.83230792e-1 1.00 or ERmax 4 -0.58856104e-1 0.63578795 -0.40942939 5 -0.20862734e-1 0.27915649 0.77373924e-1 6 -0.91072104e-2 0.15896258 0.76438585e-1 2 0.25 4 -0.14047642e-1 0.14979141 0.85859973e-1 0.50 4 -0.29743597e-1 0.28007699 0.25672616 5 -0.62390492e-2 0.98860848e-1 0.28015185 0.75 4 -0.21746194e-1 0.24603850 0.59721737 5 -0.92268944e-2 0.13712409 0.45986089 1.00 or ERmax 4 -0.29013543e-1 0.32770909 0.63977507 3 0.50 4 -0.26227989e-1 0.24948865 0.34706481 0.75 4 -0.21193622e-1 0.23135170 0.70716066 5 -0.81680616e-2 0.12046957 0.55572720 1.00 or ERmax 4 -0.26329484e-1 0.29179985 0.82637599 4 0.75 4 -0.15057494e-1 0.17001544 0.84237894 5 -0.62919768e-2 0.94912554e-1 0.62820260 1.00 or ERmax 4 -0.31547564e-1 0.31857302 0.81193267 6 0.50 4 -0.18223755e-1 0.16971610 0.41201838 0.75 4 -0.90162669e-2 0.11241800 0.90180647 1.00 or ERmax 4 -0.18108766e-1 0.19886729 1.0167907 8 1.00 4 -0.24920003e-1 0.26817736 0.78188375 308 Appendix C Extended Kalman Filter Model This chapter describes the process of ltering and the derivation of the Kalman lter gain to minimize the error of a dynamical system. The rst section provides a de nition for ltering and assumptions used in the following sections. The next section brie y discusses the Wiener lter as a segue into the importance of Kalman?s work in the early 1960?s. The third section derives the the Kalman lter with a preface on linear system dynamics. The last section describes the extended Kalman lter, the linearization of the system dynamics, and a summary of the implemented model. C.1 Filtering Assumptions The problem of accurately estimating the states of a system can be broken into three categories.1,9 The rst, and the focus of this chapter, is ltering where the goal is to estimate the states at the current time with information up to and including the current time. The second category is prediction where the goal is to estimate the states at a future time. And the last category is that of smoothing where the estimated states at a given time of interest is based on information before 309 Figure C.1: Extended Kalman Filter, from Ref. [9] and after this time epoch. Filtering is the main concern for navigation systems because one wishes to continually have an accurate estimate of the states of the system with minimal computational load to allow real time processing. A number of reasonable assumptions must be made pertaining to the sys- tem modeling in order to make the mathematics tractable and ultimately derive the optimal Kalman gain.1,9 The rst assumption will be that the system can be modeled by linear dynamics. This assumption will be used through the majority of the derivations but will be extended to include nonlinear systems whose dynamics have been linearized about the current state estimate (i.e. an extended Kalman lter as shown in Fig. C.1 from Brown and Hwang,9 pg. 344). The other main assumption is that the random processes and measurement noises are driven by un- correlated white Gaussian variables. This assumption is based on the Central Limit Theorem of statistics which states that the summation of a set of random variables 310 with nite variance will tend to a normal, Gaussian distribution.Since the process and measurement noises are typically small variations about a true unbiased (i.e., zero-mean) value, this assumption holds well. Also, the assumption that the noises are uncorrelated is reasonable since each noise is typically produced by a di erent sensor or system. And furthermore, the white noise assumption holds as long as we concern ourself with a nite band of system frequencies which is su ciently less than the highest system frequencies. C.2 Random Processes This section provides several de nitions used in the derivation of the Kalman lter gain. First, the probability density function is de ned as (Jekeli,1 pg. 166): Z b a fx(x)dx P(a xk b) (C.1) Or in words, this function de nes the probability that a random variable, xk, will have a particular value in the interval [a;b]. The mean, x, or expectation, E[x], of a random variable is de ned as its rst moment (Jekeli,1 pg. 168 and Brown and Hwang,9 pg. 26): x = E[x] Z 1 1 xfx(x)dx: (C.2) The variance, or degree that the random number?s value deviates from the mean is de ned as the second moment: 2x E[(xk x)2] = E[x2k] 2x (C.3) = Z 1 1 (x x)2fx(x)dx: 311 The standard deviation is commonly used in place of the variance and is de ned as the square-root of the variance, x. The last de nition is that of the covariance between two random variables xk and yk: cov(xk;yk) E[(xk x)(yk y)] = E[xkyk] x y; (C.4) where E[xy] = R1 1R1 1xyf(x;y)dxdy and f(x;y) is the joint density function. If the two random variables are uncorrelated, cov(xk;yk) = 0; ) E[xkyk] = x y: (C.5) For a vector of random variables, x, the vector mean is de ned like before: x = E[x]: (C.6) The covariance matrix of a random vector with itself is de ned as Px = cov(x;x) = 0 BB BB BB @ cov(x1;x1) cov(x1;xN) ... ... ... cov(xN;x1) cov(xN;xN) 1 CC CC CC A ; (C.7) where x1 denotes the rst random variable and xN denotes the last random variable in the vector x. Also, because cov(xi;xj) = cov(xj;xi), the covariance matrix is symmetric (Px = PTx ). And furthermore, the diagonal elements of the covariance matrix are equal to the variances of the random variable vector. Following the assumptions given in Sec. C.1, zero-mean white Gaussian ran- dom processes will be used to model the forcing noises in the system dynamics. A Gaussian or normal process is de ned by the probability density function (Brown 312 and Hwang,9 pg. 25{26): fN(x) = 1 x p2 exp 12 2 x (x x)2 : (C.8) As shown, the Gaussian distribution is de ned completely by its mean and variance. This distribution can then be specialized to a zero-mean white noise process, denote as w(t) or v(t). By de nition, the noise will have a mean of zero. The variance, however, is harder to quantify. White noise is characterized by having zero covari- ance over any non-zero time interval (Jekeli,1 pg. 177), meaning the variance is only de ned at an in nitesimal period of time. This un-physical de nition is mathemat- ically realized using the Dirac delta function, and it can be shown that the variance of a white noise process is (Jekeli,1 pg. 179): 2w = qw= t; (C.9) where qw is the amplitude of the power spectral density, which is constant for all frequencies and the impetus for the process being labeled white. The next subsections de ne three stochastic models that will be used in the accelerometer, gyro, and GPS receiver clock error models. C.2.1 Random Constant A random constant is usually used to model a random bias or any other con- stant that does not have a predetermined value. The di erential equation that describes a random constant is _x(t) = 0; x(t0) = x0; (C.10) 313 where the initial condition (x0) has a given variance ( 2x) and zero mean. C.2.2 Random Walk A random walk is a process whose value varies, or \walks," stochastically over time. This process is modeled as an integrated white Gaussian noise, w(t), with a mean of zero and a known variance ( 2w): _x(t) = w(t); x(t0) = 0: (C.11) The initial value of the random walk process is assumed to be zero. However, if the system being modeled has a random initial condition, the system may be modeled as the combination of a random walk and a random constant. C.2.3 First Order Gauss-Markov Process A Gauss-Markov process is a random process whose characteristics are corre- lated from one time to another. A rst order Gauss-Markov is de ned by the linear di erential equation: _x(t) = x(t) +w(t); x(t0) = 0; (C.12) where is a time constant that describes the level of correlation. For low , the correlation is reduced so that the signal begins to resemble a white noise process. The gyro noise is modeled in this manner so that its contribution to the angular velocity error can be accounted for in the strapdown gravity gradiometer instrument measurement updates, see Sec. 5.2.1. 314 For this work, the rst order Gauss-Markov process is discretely modeled by xk+1 = exp ( t)xk +wk; (C.13) where = 2:146=(0:5 t) is the prescribed time constant,94 and the discrete white process variance is (Brown and Hwang,9 pg. 124) 2gm = (qw= t) [1 exp( 2 t)]: (C.14) C.3 Linear Dynamic Systems An n-dimensional linear system is one whose dynamics can be written as _x(t) = A(t)x(t) +B(t)u(t) +M(t)w(t) (C.15a) y(t) = C(t)x(t) +D(t)u(t) +N(t) (t); (C.15b) where x(t) is the n-dimensional continuous state vector of the system, u(t) is the command control input to the the system, y(t) is the output measurement vector, and w(t) and (t) are the continuous random process and measurement noises, respectively. The matrices on the right hand side of the equations may be time varying, and essentially map the current states, controls, and noises into the state derivatives and measurements. The general theory of linear dynamic systems can be found in references such as Chen.173 The discussion here will now be focused to the ltering problem at hand. With the assumptions that there are no explicit control inputs and the noises are mapped directly to the state derivatives and measurements, the linear dynamics 315 can be rewritten as _x(t) = F(t)x(t) + w(t) (C.16a) y(t) = H(t)x(t) + (t): (C.16b) A change in notation has also been performed to coincide with typical ltering terminology. To derive the discrete form of the linear dynamics, we rst solve for x(t). The general solution to a di erential equation is the summation of its homogeneous and particular solutions. The homogeneous solution to Eq. (C.16a), i.e. _xH(t) = F(t)xH(t); w(t) = 0 (C.17) is given as xH(t) = (t;t0)xH(t0); (C.18) where (t;t0) is the state transition matrix from time t0 to t. The particular solution is found by assuming the form xP(t) = (t;t0) P(t); and xP(t0) = 0; (C.19) where vP(t) is a time varying forcing vector that will be found. Substitution of the particular solution into the dynamic equation yields _ (t;t0) P(t) + (t;t0) _ P(t) = F(t) (t;t0) P + w(t): (C.20) With the properties that the state transition matrix is by de nition invertible and its time derivative is:1,173 _ (t;t0) = F(t) (t;t0); (C.21) 316 one can solve for _ P(t) and integrate to get P(t) = Z t t0 1( ;t0)w( )d : (C.22) Substituting P(t) back into Eq. (C.19), adding the particular and homogeneous solutions together, and using the properties that 1( ;t0) = (t0; ); and (C.23) (t;t0) (t0; ) = (t; ); (C.24) the general solution to Eq. (C.16a) is found to be x(t) = (t;t0)x(t0) + Z t t0 (t; )w( )d : (C.25) Applying the solution and Eq. (C.16b) to two discrete times t = tk+1 and t0 = tk, the discrete linear dynamics are then xk+1 = kxk + wk (C.26a) yk+1 = Hk+1xk+1 + k+1; (C.26b) where the subscripts refer to the time of interest (tk or tk+1), and k (tk+1;tk) (C.27) wk Z tk+1 tk (t; )w( )d : (C.28) For \small" time increments, the state transition matrix tends to the identity matrix so that the discrete process noise can be approximated as wk Z tk+1 tk w( )d w(tk) t: (C.29) 317 C.4 Wiener Filter Filtering has its roots in the eld of electronics.9 The goal there, and in some degree universally, is to produce the greatest signal to noise separation when the characteristics of both processes are known. In electronics, this was usually implemented as a way to keep a certain range of frequencies and lter out all others, as in the removal of noise from an amplitude or frequency modulated radio signal. Norbert Wiener, during World War II, addressed this signal to noise lter problem and produced the Wiener lter, rst published in 1949.174 With the as- sumption that the signal and noise were random processes with known characteris- tics, the Wiener lter solved for the optimal lter weighting function to minimize the mean-square error. A limitation to the Wiener lter is its assumption that the signal and noise are both \noise-like" processes. In many cases, the signal is partially deterministic and hence the Wiener lter is no longer optimal. Also, the Wiener lter assumes a single-input-single-output (SISO) system, which greatly restricts its usefulness. The complementary Wiener lter attempts to extend the Wiener lter for use in multiple- input-multiple-output (MIMO) systems, however the noise-like assumption persists. The last major hindrance in the Wiener lter is its lack of recursion, especially when formulated for a discrete system. The discrete Wiener lter solves for the optimal weight factors (i.e., gains) for a state at a given time by using all the measurements prior to and including the current time. When many measurements are made or the state is estimated at many times, the Wiener lter solution becomes computationally 318 infeasible. To alleviate some of the burden, only a set of measurements can be used for each state estimate, but this then limits the lter knowledge in calculating the optimal estimate. C.5 Extended Kalman Filter Rudolf E. Kalman in 1960 and 1961 published two papers that solved the major problems relevant in the Wiener lter by formulating the problem in a state- space manner.24,25 Kalman did not assume that both the signal and its corrupting source were noise-like. Instead, it was only assumed that they were driven by un- correlated process and measurement noises. The state-space derivation removed the SISO assumption of the Wiener lter so that any number of inputs and outputs could be modeled. And arguably the largest contribution of Kalman?s lter was its use of recursion. Now, only the previous states and covariances were needed to estimate the current state and covariance, which allowed for a much more e cient implementation. The only major restriction left in Kalman?s formulation was the assumption of linear dynamics. Since most system dynamics are actually modeled with nonlinear relations, the Kalman lter will not optimally produce the best state estimates. One common solution to this problem is to linearize the system about a given state trajectory. If a nominal trajectory is chosen, the Kalman gains can be computed o ine and the implementation is referred to as a Linear Kalman Filter. The major downfall of this method is that the guidance law must keep the true trajectory near 319 the pre-computed nominal one or the lter will diverge. However, is the system is continuously linearized with the current estimated states, an Extended Kalman Filter (EKF) is implemented. This typically results in better performance than the Linear Kalman lter since the estimates are usually closer to the true values than the nominal trajectory. This work uses the EKF implementation for this reason, and will be explained in the next sections. It should be noted, though, that the EKF is slightly riskier because divergence will result when the errors between the true and estimated trajectories grow too large, as shown with some of the strapdown GGI aided INS simulations. The following three subsections derive the various components of the Extended Kalman Filter. The rst subsection explains the process of linearizing the true nonlinear equations with respect to the current states. The next subsection de nes an error as used in this dissertation and the Kalman lter propagation and update equations. And the last section summarizes the initialization, propagation, and update of the Extended Kalman Filter. C.5.1 System Linearization The Kalman lter was originally derived to give the optimal gain for the blend- ing of noisy measurements into corrections for the estimated states given a linear dynamical system. Unfortunately, most system dynamics are nonlinear (i.e., the navigation equations in Sec. 4.3). One popular way to still use the Kalman lter methods is to linearize the nonlinear system dynamics. The linearization is a good 320 approximation of the true system dynamics as long as the higher order terms and time increment are su ciently \small". For a generic nonlinear system, the dynamical equations are _x(t) = f (x(t);u(t);t) + w(t) (C.30a) y(t) = h (x(t);t) + (t); (C.30b) where f is the nonlinear time rate of change of the states and y is the nonlinear measurement. Taking a Taylor series expansion of the dynamics about the estimated states, assuming u = 0, and temporarily dropping the explicit time dependence notation, gives _x = f (bx) + @f @x x=bx (x bx) + 12 @2f @x2 x=bx (x bx)2 + + w (C.31a) y = h (bx) + @h @x x=bx (x bx) + 12 @2h @x2 x bx (x bx)2 + + ; (C.31b) where x is the vector of truth states and bx is the vector of estimated states. Ne- glecting the second and higher order terms, the linearized dynamics are _x(t) f(bx(t);t) +F(t) (x(t) bx(t)) + w(t) (C.32a) y(t) h(bx(t);t) +H(t) (x(t) bx(t)) + (t); (C.32b) where the linearized dynamic equations and measurement matrices are F(t) @f(t) @x(t) x(t)=bx(t) (C.33a) H(t) @h(t) @x(t) x(t)=bx(t) : (C.33b) 321 If the estimated nonlinear state dynamics and measurement are de ned as _bx(t) f(bx(t);t) (C.34a) by(t) h(bx(t);t); (C.34b) and an error state is de ned as the perturbation between the estimated state and the true state, see Eq. (C.38), the linear dynamics become _bx(t) _x(t) = _x(t) = F(t) x(t) w(t) (C.35a) byk+1 yk+1 = y(t) = H(t) x(t) (t): (C.35b) And the discrete error dynamics are (from Sec. C.3) xk+1 = k xk wk (C.36a) byk+1 yk+1 = yk+1 = Hk+1 xk+1 k+1: (C.36b) The state transition matrix is often calculated using a rst order Taylor series ex- pansion of the matrix exponential of the linearized dynamics, F, matrix: k = eFk t I +Fk t; (C.37) where it has been assumed that Fk is constant over the time interval. Unfortunately, this approximation is poor for the exponential in the Gauss-Markov process. Jekeli addressed this problem by increasing the series truncation to 30 terms.94 This approach was implemented initially, but the run times were drastically increased. It was then found that setting the gyro noise portion of the state transition matrix to diag(exp( t)) was as e ective as the higher order series but without the added computational burden. 322 The linearized state dynamics matrix, F(t), is summarized in Sec. 4.4.4 for the inertial navigation states, Sec. 4.5.1 for the IMU states, and in Sec. D.4 for the GPS receiver clock states. The linearized measurement matrix, H(t), is derived in Sec. 5.2.1 and 5.2.2 for gravity gradiometer instrument aiding and Sec. D.3.2 and D.3.3 for the global positioning system aiding. C.5.2 Discrete Kalman Filter The derivation of the discrete Kalman lter propagation and update equations will liberally use the error states, which are de ned as x bx x; (C.38) where x is a small perturbation (or error) from the true state, x, and bx is the lter- estimated state. It should be noted that this de nition of the error is not universal. Many papers de ne the error as the true value minus the estimated value; however, in this work all errors will be consistently de ned as those in Eq. (C.38). These error states are also assumed to propagate in time according to the discrete linear system dynamics of Eq. (C.36a). The error covariance matrix is also of signi cant importance in the Kalman lter equations because it provides the lter a current estimate of all the system errors. This covariance matrix at time tk is de ned as Pk cov( xk; xk) (C.39) = E[( xk x)( xk x)T] = E[( xk)( xk)T]; 323 where the last equality holds assuming the errors have zero-mean, which is a result of the forcing white process noise having zero mean. The propagation of the error covariance is found by substituting in the linear error dynamics with the assumption that the process noise and error states are uncorrelated so E[ xkwk] = 0. Therefore, Pk+1 = E[( xk+1)( xk+1)T] = E[( k x wk)( k x wk)T] = kE[ xk xTk] Tk E[wk xtk] Tk kE[ xkwTk ] +E[wkwTk ] = kPk Tk +Qk; (C.40) where the discrete process noise covariance is de ned as Qk E[wkwTk ]: (C.41) Without external measurements, Eq. (C.36a) and (C.40) can be propagated deter- ministically from their initial conditions to yield the error and covariance at any time in the future. Only knowledge of the discrete error process noise, Qk, and the error state transition matrix, k, are required for the calculations. When external measurements are collected, for example by a GGI or GPS, a feedback loop is used to blend the noisy measurements with the current state estimates to produce the best updated estimate of the state vector. The derivation that follows minimizes the mean square error the of the state vector, and then is used to update the whole estimated states. Starting with the measurement update feedback form: x+k+1 = x k+1 +Kk+1 (yk+1 byk+1) = x k+1 +Kk+1 yk+1; (C.42) where the superscript \+" means the updated or a posteriori error state, \ " is the a 324 priori error, and Kk+1 is the Kalman gain matrix to be derived. Then, substituting the discrete measurements, Eq. (C.36b), yields x+k+1 = x k+1 Kk+1 Hk+1 x k+1 k+1 = (I Kk+1Hk+1) x k+1 +Kk+1 k+1: (C.43) The a posteriori error state is next substituted back into the de nition of the error covariance matrix to get the updated error covariance: P+k+1 = E ( x+k+1)( x+k+1)T = (I Kk+1Hk+1)E ( x k+1)( x k+1)T (I Kk+1Hk+1)T + (I Kk+1Hk+1)E ( x k+1)( k+1)T KTk+1 + Kk+1E ( k+1)( x k+1)T (I Kk+1Hk+1)T + Kk+1E ( k+1)( k+1)T KTk+1: (C.44) Assuming the errors and measurement noises are uncorrelated, E ( x )( )T = 0, and the updated covariance matrix is now simply P+k+1 = (I Kk+1Hk+1)P k+1 (I Kk+1Hk+1)T +Kk+1Rk+1KTk+1; (C.45) where P k+1 is the a priori error covariance and the measurement noise covariance is de ned as Rk+1 E k+1 Tk+1 : (C.46) The updated covariance in Eq. (C.45) is also referred to as the Joseph Form. Other forms can be derived by substituting the Kalman gain into the Joseph form that are less computationally expensive. However, the Joseph form is used in this research 325 because it has better numerical properties (Brown and Hwang,9 pg. 261). Also, after each iteration, the covariance matrix is recalculated by Pk+1 = 12 Pk+1 +PTk+1 (C.47) to reenforce the symmetric property to hold. Like the Wiener lter, the goal of the Kalman lter is to minimize the mean- square error of the states by optimally blending the measurement with the estimated states. Mathematically, the Kalman gain is found by minimizing the trace (sum of diagonal elements) of the error covariance matrix since this is the sum of the error variances, see Eq. (C.7) on pg. 312. It can be shown that (Brown and Hwang,9 pg. 217): d trace P+k+1 d Kk+1 = 2 H k+1P k+1 T + 2K k+1 H k+1P k+1HTk+1 +Rk+1 : (C.48) Setting this derivative equal to zero and solving for the Kalman gain matrix gives Kk+1 = P k+1HTk+1 Hk+1P k+1HTk+1 +Rk+1 1; (C.49) which is the optimal gain used in Eq. (C.42). While the discussion throughout this section has focused on the error states, what we are truly concerned with is how the \whole" state estimates are updated us- ing the noisy measurement information and the Kalman gain. Using the error update equation, Eq. (C.42) and substituting the de nition of the error terms, Eq. (C.38): bx+k+1 x+k+1 = bx k+1 x k+1 +Kk+1 (yk+1 byk+1): (C.50) Since xk+1 is the true state at time tk+1, its a priori and a posteriori values are 326 equivalent, so the estimated states are updated using bx+k+1 = bx k+1 +Kk+1 (yk+1 byk+1): (C.51) C.5.3 Summary The Extended Kalman lter algorithm is summarized in this section. The summary is separated into the three major EKF components. Initial Conditions The Extended Kalman lter is initialized with an estimated and truth state vector and rotation matrix. The initial error covariance is typically de ned as a diagonal matrix whose elements correspond to the initial state error variances. x(t0) = x0 (C.52) bx(t0) = bx0 (C.53) Cnb (t0) = Cnb;0 (C.54) bCnb (t0) = bCnb;0 (C.55) P(t0) = P0 (C.56) Propagation The nonlinear system dynamics are used to numerically integrate the truth and estimated state vectors and the rotation matrices using the simulated accelerometer and gyro measurements, see Ch. 4 and Sec. 3.3.2. The rotation matrix is integrated 327 using its equivalent quaternion as discussed in Sec. 4.3.3.1, and the GPS receiver dynamics are integrated according to Sec. D.4. The INS states use a fourth-order Runge-Kutta algorithm (Sec. 4.3.3.2) to numerically integrate the states at a con- stant rate of 20 Hz. At each time step the system dynamics are also linearized about the current state estimates and the error covariance is propagated using the error state transition matrix as discussed at the end of Sec. C.5.1. The IMU process noises are derived in Sec. 4.5.2, and the GPS clock process noises are explained in Sec. D.4. _x(tk) = f (x (tk)) + w(tk) (C.57) _q(tk) = 12Aqq (C.58) _bx(tk) = f (bx (tk)) (C.59) _bq(tk) = 1 2 bAqbq (C.60) Fk = @f @x x=bx(tk) (C.61) k = I +Fk t (C.62) k;gyro noise = diag(e t) (C.63) Pk+1 = kPk Tk +Qk (C.64) Measurement Update When an external measurement is made using the GGI or GPS, the linearized measurement matrix is calculated and the Kalman gain is computed. The estimated states are then updated using the Kalman gain and the residual of the noisy truth measurement and the estimated measurement. All measurements are simulated 328 using nonlinear equations, see Eq. (5.44), (5.57), (D.14), and (D.19). And their linearized errors are given in Eq. (5.56), (5.67), (D.15), and (D.20) respectively. The estimated rotation matrix is also updated using the calculated rotation angle errors, and then these states are reset to zero, as explained in Sec. 4.4.2. yk+1 = h(xk+1) + k+1 (C.65) byk+1 = h(bxk+1) (C.66) Hk+1 = @h @x x=bxk+1 (C.67) Kk+1 = Pk+1HTk+1 Hk+1Pk+1HTk+1 +Rk+1 1 (C.68) bxk+1 = bxk+1 +Kk+1 (yk+1 byk+1) (C.69) bCnb = (I + n) bCnb (C.70) n = 0 (C.71) P+k+1 = (I Kk+1Hk+1)Pk+1 (I Kk+1Hk+1)T +Kk+1Rk+1KTk+1 (C.72) 329 Appendix D Global Positioning System Model The nominal United States of America?s twenty-four satellite Global Position- ing System (GPS) was modeled and integrated with the inertial navigation system (INS) to provide baseline navigation performance for various GPS dropouts and measurements. This chapter describes the modeling of the nominal constellation and the determination of each satellite vehicle?s (SV) position and velocity as a function of time. The simulated measurements and their Kalman lter models are described in Sec. D.3 along with the visibility test to determine which SVs provide measurements and the importance of the geometric dilution of precision. The user?s GPS receiver clock model is presented in the following section with the simulated noise values. Lastly, Sec. D.5 summarizes the Fortran module implementation of this chapter and its use in the overall simulation. D.1 GPS Satellite Constellation The position and velocity of each satellite in the nominal 24-satellite GPS constellation can be calculated by the following orbital parameters. The nominal design values are an eccentricity eGPS = 0:00, inclination iGPS = 55 , and semimajor 330 axis aGPS = 26;561:75km. In reality, the eccentricity is generally less than 0.02 (Parkinson and Spilker,7 pg. 179), thus for modeling purposes it will be assumed that the eccentricity is always zero . Also, due to the assumption of a circular orbit, the semimajor axis is equivalent to the semiminor axis, and orbital radius is therefore constant. It should be noted that the stated orbital radius above has been corrected for Earth?s bulk oblateness (Parkinson and Spilker,7 pg. 178{181). Each SV?s position and velocity can then be determined by these assumed nominal values, a value of the Earth?s gravitational constant, each SV?s initial right ascension of the ascending node and argument of latitude at a given reference time, and the time o set between the current time and the initial reference time. Earth?s gravitational constant is modeled as the more accurate 1984 World Geodetic System (WGS84) value, GM = 3:986004418 1014 m3=s2, than the original GPS value of GM = 3:9860050 1014 m3=s2 (Ref. [113], pg. 3-3). According to the WGS84 report,113 the GPS Operational Control Segment began using the improved value during the fall of 1994 and removed a radial bias of 1.3 m for the orbit estimators. In an e ort to maintain consistency between previous and future GPS receivers, the GPS interface control document175 continues to use the original, less accurate gravitational constant. However, for the purposes of simulating the GPS satellite vehicles? (SV) position and velocity, the current and more accurate WGS84 gravitational value is used. The orbital period of the nominal GPS constellation can be found from the 331 Table D.1: Simulated GPS Parameters Parameter Name Symbol Value Units Semimajor Axis aGPS 26,561.75 km Eccentricity eGPS 0.00 | Inclination iGPS 55.0 Earth?s Gravitational Constant GM 3.986 004 418 1014 m3/s2 Orbital Period PGPS 43,082.015 s Initial Reference Time t0 Midnight, July 1, 1993 | Earth?s Rotation Rate !e 7.292 115 1467 10 5 rad/s Speed of Light c 2.99792458 108 m/s orbital radius and Earth?s gravitational constant by115 PGPS = 2 q a3GPS=GM = 43;082:015s; (D.1) using the values of aGPS and GM above. The GPS period is approximately one half a sidereal day = 0:5 (2 )=!e = 43;082:050s. The minimal di erence between the two values is due to the correction of Earth?s oblateness in the value of aGPS, and truncation error with the GPS de nition175 of . The parameters used for the simu- lated GPS module are summarized in Table D.1 which includes the initial reference time and the ICD-GPS-200175 values of Earth?s rotation, !e, (which is equivalent to the International Astronomical Union value stated in WGS84 reference113) and speed of light, c. These parameters are constant for each satellite in the constella- tion, and the remainder of this section explains the two parameters that uniquely de ne a speci c SV?s position and velocity. 332 The twenty-four satellites of the GPS-24 constellation are divided into six orbit planes with four satellite vehicles each. The orbit planes are de ned by six right ascensions of the ascending node, GPS, which are the angles in the equatorial plane measured positively from the Earth-Centered-Inertial frame?s x-axis (the mean vernal equinox) to the location of the ascending node. The ascending node is de ned as the point on the equatorial plane at which the satellite crosses the equator from South to North (Vallado,115 pg. 107). The right ascension of the ascending node is constant in the ECI frame, and the GPS constellation planes are equispaced by 60 starting at 32.847 as tabulated in Table D.2 (from Parkinson and Spilker,7 pg. 181). For each GPS, four satellites are \phased" by having various mean anomalies, MGPS, that have been optimized to minimize the e ects of a single SV failure on the total system performance.7 For the idealized circular inclined SV orbit, the mean anomaly is equivalent to the true anomaly, GPS, and the argument of latitude, uGPS (Vallado,115 pg. 108{111). The argument of latitude will be used for the remainder of this dissertation, but it should be noted that the actual GPS signal broadcast includes information to correct for deviations from this ideal scenario. For these more realistic situations, these three orbital angles are not equal and the method to calculate the SV position is more complicated (see ICD-GPS-200,175 pg. 98{100 or Parkinson and Spilker,7 pg. 138y). For both the idealized and realistic cases, the mean anomaly varies linearly in time, and continuing with the assumption of the yIt should be noted that on pg. 138 of Parkinson and Spilker, there is a slight typo in the calculation of \yk." The correct equation is yk = x0k sin k +y0k cosik cos k, not yk = y0k sin k + y0k cosik cos k. 333 Table D.2: GPS-24 Satellite Constellation, from Ref. [7] SV ID GPS, uGPS(t0), SV ID GPS, uGPS(t0), 1 A3 272.847 11.676 13 D1 92.847 135.226 2 A4 272.847 41.806 14 D4 92.847 167.356 3 A2 272.847 161.786 15 D2 92.847 265.446 4 A1 272.847 268.126 16 D3 92.847 35.156 5 B1 332.847 80.956 17 E1 152.847 197.046 6 B2 332.847 173.336 18 E2 152.847 302.596 7 B4 332.847 204.376 19 E4 152.847 333.686 8 B3 332.847 309.976 20 E3 152.847 66.066 9 C1 32.847 111.876 21 F1 212.847 238.886 10 C4 32.847 241.556 22 F2 212.847 345.226 11 C3 32.847 339.666 23 F3 212.847 105.206 12 C2 32.847 11.796 24 F4 212.847 135.346 nominal circular inclined orbit the argument of latitude does as well. The value of the argument of latitude for the jth SV at a given time is uGPS;j(t) = uGPS;j(t0) + (t t0) s GM a3GPS = uGPS;j(t0) + (t t0) 2 PGPS; (D.2) where uGPS is in radians, and t is the current time in relation to t0 (de ned as midnight, July 1, 1993) (Parkinson and Spilker,7 pg. 138, 180). The arguments of latitude for the nominal reference constellation at the reference time are tabulated in Table D.2 (from Parkinson and Spilker,7 pg. 181). 334 D.2 GPS Satellite Vehicle Position and Velocity The Global Positioning System?s satellite vehicle position and velocity are rst computed in each satellite?s orbital plane and then rotated into the Earth-Centered- Earth-Fixed (ECEF) coordinate frame. The Kalman lter later accounts for the transformation from the ECEF frame into the navigation frame through appropriate transformation matrices. The Perifocal coordinate system (PQW) is a satellite-based frame with its origin at the Earth?s center. The 1-axis (bP in Fig. D.1) typically points toward the orbit perigee, but because the orbit is circular the perigee is unde ned and the x-axis is thus de ned to point toward GPS;j, i.e., the mean vernal equinox of the jth SV orbit. The 2-axis (bQ) is in the orbital plane and 90 from GPS;j in the direction of satellite motion. The 3-axis (cW) completes the right hand coordinate system and is out of the orbit plane so that there is no position or velocity component in this Figure D.1: Perifocal Coordinate System, from Ref. [115] 335 direction (Vallado,115 pg. 161{162). The jth GPS satellite?s position and velocity as a function of the true anomaly and orbit semi-parameter, pGPS, is (Vallado,115 pg. 122{125) rPQWjj = p GPS cos( GPS;j(t)) 1 +eGPS cos( GPS;j(t)); pGPS sin( GPS;j(t)) 1 +eGPS cos( GPS;j(t)); 0 (D.3) vPQWjj = s GM pGPS sin( GPS;j(t)); s GM pGPS e GPS + cos( GPS;j(t)) ;0!:(D.4) For the assumed circular inclined orbit, several simpli cations can be made to the expressions above. First, the semi-parameter is equal to the orbit radius and semi- major axis since eGPS = 0:00.115 Second, the true anomaly can be replaced by the argument of latitude as explained on page 333. The circular inclined orbit position and velocity of the jth SV in its PQW coordinate frame is now rPQWjj = aGPS cos(uGPS;j(t)); aGPS sin(uGPS;j(t)); 0 (D.5) vPQWjj = rGM aGPS sin(uGPS;j(t)); rGM aGPS cos(uGPS;j(t)); 0 ! : (D.6) The PQW to ECEF coordinate transformation generally consists of four ro- tations. The rst three rotations transform the position and velocity vectors to the ECI frame, and the last rotation transforms from the ECI to the ECEF frame. The general PQW to ECI transformation is (Vallado,115 pg. 173) CiPQWj = R3( GPS;j)R1( iGPS)R3( !GPS;j); (D.7) where !GPS;j is the argument of perigee of the jth SV. For a circular orbit, this !GPS;j is unde ned and thus set to zero.115 The ECEF to ECI transformation is a single rotation due to Earth?s spin: Cei = R3 !e(t t0) ; 336 repeated from Eq. (4.20). The rotations about Earth?s spin axis can be combined, and the total PQW to ECEF transformation for the jth SV is now CePQWj = R3 GPS;j +!e(t t0) R1( iGPS) (D.8) = R3( j(t))R1( iGPS) = 0 BB BB BB @ cos( j(t)) sin( j(t)) cos(iGPS) sin( j(t)) sin(iGPS) sin( j(t)) cos( j(t)) cos(iGPS) cos( j(t)) sin(iGPS) 0 sin(iGPS) cos(iGPS) 1 CC CC CC A ; using the de nitions of the rotation matrices from Eq. (4.8) and de ning the short- hand notation of j(t) GPS;j !e(t t0). The position and velocity for the GPS SVs can now be found in the ECEF frame. Multiplying Eq. (D.8) by Eq. (D.5) yields the position of the jth satellite: rej = aGPS 0 BB BB BB @ cos( j) cos(uGPS;j) sin( j) cos(iGPS) sin(uGPS;j) sin( j) cos(uGPS;j) + cos( j) cos(iGPS) sin(uGPS;j) sin(iGPS) sin(uGPS;j) 1 CC CC CC A ; (D.9) where the explicit time dependency of j and uGPS;j has been dropped for brevity. The jth satellite velocity is similarly found from Eq. (D.6) and (D.8) to be vej = rGM aGPS 0 BB BB BB @ cos( j) sin(uGPS;j) sin( j) cos(iGPS) cos(uGPS;j) sin( j) sin(uGPS;j) + cos( j) cos(iGPS) cos(uGPS;j) sin(iGPS) cos(uGPS;j) 1 CC CC CC A : (D.10) D.3 GPS Measurements This section describes the simulated GPS measurement observables. The rst subsection explains the method to calculate which satellites are visible to the user. 337 The next subsection de nes the primary code-based GPS measurement denoted as pseudorange. The following subsection de nes a carrier-phase based measurement equivalent to the time rate of change of the pseudorange which is used to yield velocity and precise position information. And the nal subsection de nes and discusses the important geometric dilution of precision quantity. D.3.1 Visibility Test The GPS satellites only present information to the user when their broadcast line-of-sight to the user is unobstructed. The primary obstruction source is the Earth, i.e., when the user and the satellite are on opposite sides of the Earth. Local terrain, buildings, and an assortment of other features may also obstruct the GPS signal between the user and a given satellite vehicle. Furthermore, satellites with low elevation angles relative to the user transmit farther through the atmosphere than SVs with high elevation angles, thereby causing increased error e ects associated with the ionosphere and troposphere. For these reasons, GPS receivers typically ignore SVs below a minimum elevation angle, Emin. Determining when a SV is \in-view" and able to produce measurement information is therefore important to correctly simulate the GPS constellation?s usefulness, and is the subject of this subsection. According to Parkinson and Spilker7 pg. 183, \Each GPS satellite broadcasts to the Earth with an antenna coverage pattern that somewhat exceeds the angle GPS = 13.87 subtended by the Earth." The satellite half-angle, GPS, is also a 338 Figure D.2: GPS Visibility Angles, From Ref. [7] function of minimum elevation angle (Parkinson and Spilker,7 pg. 184): GPS = sin 1 a e aGPS cos (Emin) ; (D.11) where ae = 6;378;137:0 m is the semimajor axis of Earth, given in Table 4.1 on page 145. The Earth half-angle, GPS, is the maximum angle between the user and satel- lite where a satellite is still visible. Referencing Fig. D.2 from Parkinson and Spilker,7 pg. 183, and using the fact that the sum of angles in a triangle is , the Earth half- angle can be found from GPS + ( =2 +Emin) + GPS = to be GPS = 2 Emin GPS: (D.12) It should be noted that this expression and Fig. D.2 both assume that the altitude of the user is much less than Earth?s radius so that GPS intersects the user?s position. 339 This is a reasonable assumption for the current work because the nominal altitudes are 25 km, so that altitude is approximately 0.4% the Earth?s semimajor axis. The actual angle between the user, re, and a given satellite, rej, (in ECEF coordinates) with the Earth?s center as the vertex can be found from the dot product of the two vectors: GPS;j = cos 1 re re j jjrejjjjrejjj : (D.13) The visibility test is then to compute GPS;j for all 24 SVs at the given epoch, and if GPS;j GPS, the jth GPS satellite is visible to the user. (The user?s position in ECEF coordinates is calculated using Eq. (4.24) and the current n-frame position states, and the rej calculation was given in the previous section). In this research the altitudes of the hypersonic scramjet simulations are rela- tively high ( 25 km) so that local terrain and building obstructions are neglected. Therefore, a zero-degree elevation limit is used when determining the visibility of a GPS satellite. This results in a satellite half-angle, GPS, equal to 13.89 and an Earth half-angle, GPS, of 76.11 . (For lower altitudes, it is common to use a 5 elevation angle7 and thus the reduced GPS = 13:84 and GPS = 71:16 .) The percentage of satellite vehicles visible is shown in Fig. D.3. The gure was obtained by propagating the GPS constellation over two periods ( 24 hours) using 100,000 time steps. The user?s position was held constant at the initial longitude, latitude, and altitude for the Mach 6 cases (See Table D.3). The Mach 7 and 8 cases produced higher altitudes, but negligibly di erent results than the Mach 6 cases. As shown in the gure, typically 8 or 9 satellites are visible. Ten satellites are 340 Figure D.3: Simulated GPS Satellite Visibility somewhat common, however 11 satellites and less than 8 satellites are quite rare. Furthermore, there were no cases when less than 6 or more than 11 satellites were in view. The sensitivity to user position is also moderately small as shown by the similar trends for the two assumed user positions. The \High Gravity Gradients" cases are more focused at the 8 or 9 SV range whereas the \Low" cases are more likely to have 10 visible satellites. D.3.2 Pseudorange Pseudorange is the the principal measurement observable produced by the Global Positioning System. The term \pseudo"-range is used because the measure- ment is comprised of the range between the satellite and user with an additional error 341 Table D.3: User Position for GPS Satellite Visibility Analysis Case Longitude, Latitude, Altitude, m High Gravity Gradients 45.0 -113.0 22043.8 Low Gravity Gradients 38.0 -100.0 22043.8 due to the user?s receiver clock bias. This clock bias between the user?s on-board receiver and the satellite constellation time is the largest error source in determining the true range between user and satellite, and is therefore explicitly determined in the measurement. Other sources of error, such as the atmosphere, ephemeris data, satellite clock bias, multipath, and receiver noise are accounted for in the User- Equivalent-Range-Error (UERE) whose error budget is tabulated at the end of this subsection. The pseudorange measurement, using the user?s position transformed into ECEF coordinates, re, and the jth satellite vehicle, is nominally j =jjrej rejj+cbu; (D.14) where the speed of light is taken as c = 2:99792458 108 m/s (ICD-GPS-200,175 pg. 89), and bu is the user?s GPS receiver clock bias. The Kalman Filter measurements are calculated by a small perturbation analy- sis. Neglecting broadcast, atmospheric, multipath, and receiver errors, pseudorange is a function of only position and clock bias. The small perturbation of the jth pseudorange measurement is then j = @ j @re @re @rn rn +c bu + ; (D.15) 342 where rn is the user position error in the n-frame, bu is the user clock bias error, and is a white noise process that captures the additional uncompensated error sources. The partial derivatives of Eq. (D.14) with respect to user?s ECEF position are @ j @re = re re j T jjrej rejj; (D.16) which is equivalent to the transpose of the jth satellite to user line-of-sight unit vector. The position Jacobian from the navigation frame to the ECEF frame is found by taking the partial derivatives of the ECEF position vector with respect to the navigation frame variables. Recalling Eq. (4.24) on page 144: 0 BB BB BB @ re1 re2 re3 1 CC CC CC A = 0 BB BB BB @ (Ne +h) cos cos (Ne +h) cos sin (Ne(1 e2) +h) sin 1 CC CC CC A ; the Jacobian is then found to be @re @rn = 0 BB BB BB @ @re1=@ @re1=@ @re1=@h @re2=@ @re2=@ @re2=@h @re3=@ @re3=@ @re3=@h 1 CC CC CC A (D.17) = 0 BB BB BB @ (Ne +h) sin cos (Ne +h) cos sin cos cos (Ne +h) sin sin (Ne +h) cos cos cos sin (Ne (1 e2) +h) cos 0 sin 1 CC CC CC A ; where it has been assumed that the change in radius of curvature, Ne, due to latitude, , is negligible (Jekeli,1 pg. 154). Also, the ECEF position vector is independent of 343 Table D.4: Precise Positioning System Error Model, P/Y Code, from Ref. [7] One-Sigma Error, m Error Source Bias Random Total Ephemeris Data 2.1 0.0 2.1 Satellite Clock 2.0 0.7 2.1 Ionosphere 1.0 0.7 1.2 Troposphere 0.5 0.5 0.7 Multipath 1.0 1.0 1.4 Receiver Measurement 0.5 0.2 0.5 RMS User Equivalent Range Error (UERE) 3.3 1.5 3.6 Filtered RMS UERE 3.3 0.4 3.3 the navigation frame velocity vector, vn, so this Jacobian has been omitted in the pseudorange linear perturbation in Eq. (D.15). Lastly, the uncompensated errors, , are modeled as a white noise process with a standard deviation of 3.6 meters, which corresponds to the un ltered root- mean-square (RMS) UERE. The individual error sources are given in Table D.4 (from Parkinson and Spilker,7 pg. 483). Parkinson and Spilker,7 Jekeli,1 and Farrell and Barth2 provide further discussions on the cause of the pseudorange errors as well as some ways to model and reduce them. D.3.3 Pseudorange Rate The user?s GPS receiver can measure the frequency shift of the carrier wave from the nominal broadcast values. The observed frequency di ers due to Doppler shifts produced by satellite and user motion as well as frequency drift (bias time rate 344 of change) of the satellite and user clocks. The idealized Doppler shift caused by the velocity di erence in the jth satellite and user along their line of sight is (Parkinson and Spilker,7 pg. 411) Dj = ve j v e c re j r e jjrej rejj fcarrier; (D.18) where the user?s ECEF velocity is found by ve = Cenvn and Cen is given in Eq. (4.21) on page 143. The nominal carrier frequency has been denoted fcarrier. For the Global Positioning System two signals are transmitted at di erent frequencies: L1 = 1575:42 MHz and L2 = 1227:60 MHz. The Doppler shift can be converted into a pseudorange rate (also known as carrier phase Doppler) measurement by scaling the idealized Doppler by the speed of light and the carrier frequency, and including the user?s clock bias rate. Math- ematically, the jth pseudorange rate measurement is then (Parkinson and Spilker,7 pg. 411) _ j = Dj cf carrier +c_bu = vej ve re j r e jjrej rejj +c_bu: (D.19) The pseudorange rate Kalman Filter measurements are found by a small per- turbation analysis similar to the pseudorange measurements above. Again, neglect- ing the broadcast, atmospheric, multipath, and receiver errors, the pseudorange rate measurement is a function of only user position, velocity, and clock bias rate. Therefore the small perturbation of the jth measurement is of the form _ j = @ _ j @re @re @rn + @ _ j @ve @ve @rn rn+ @ _ j @ve @ve @vn vn+c _bu+ _ ; (D.20) 345 where vn is the n-frame user velocity error, _bu is the error of the user clock bias rate, and _ is a white noise process that accounts for the additional uncompensated errors. As explained on page 344, the ECEF position is independent of the n-frame velocities, so this Jacobian is a zero-matrix and has been omitted in the linearized pseudorange rate equation above. Parkinson and Spilker7 neglect the position error terms in the pseudorange rate measurement above by implicitly assuming that the line of sight vector error is negligible. This assumption was rst implemented, however the lter simulation quickly diverged making it evident that this was an invalid assumption for the hypersonic velocities simulated. After including these terms, the lter performed as expected. The partial derivatives of the pseudorange rate with respect to ECEF position and velocity can be found to be @ _ j @re = bve ve j T jjrej brejj + ve j bv e re j br e jjrej brejj3 bre j r e T (D.21) @ _ j @ve = bre re j T jjrej brejj: (D.22) The Jacobians of the ECEF velocity vector are found by taking the partial derivatives of ve = Cenvn = 0 BB BB BB @ (sin cos )vN (sin )vE (cos cos )vD (sin sin )vN + (cos )vE (cos sin )vD (cos )vN (sin )vD 1 CC CC CC A (D.23) with respect to the navigation position and velocity states. The n-frame position to 346 ECEF velocity Jacobian is then @ve @rn = 0 BB BB BB @ @vN=@ @vN=@ @vN=@h @vE=@ @vE=@ @vE=@h @vD=@ @vD=@ @vD=@h 1 CC CC CC A (D.24) = 0 BB BB @ (cos cos )vN + (sin cos )vD (sin sin )vN (cos )vE + (cos sin )vD 0 (cos sin )vN + (sin sin )vD (sin cos )vN (sin )vE (cos cos )vD 0 (sin )vN (cos )vD 0 0 1 CC CC A ; and the n-frame to ECEF velocity Jacobian is equal to theCen transformation matrix: @ve @vn = Cen = 0 BB BB BB @ sin cos sin cos cos sin sin cos cos sin cos 0 sin 1 CC CC CC A : (D.25) The position transformation Jacobian in Eq. (D.20) is given in Eq. (D.17). Because the author was unable to nd a de nitive pseudorange rate error budget, the total uncompensated error is estimated using the following assumptions. The pseudorange rate error, _ , is modeled as a white noise process with a standard deviation of 0.20 m/s. This value is estimated assuming that the user?s GPS receiver can reliably measure the Doppler shift to within 1 Hz. The Doppler measurement is then scaled by c=L1 to get a corresponding pseudorange rate error of 0.1903 m/s which is rounded to the above simulated value. Moreover, according to pg. 1-6 of the NAVSTAR GPS User Equipment Introduction,176 the Precise Positioning System \receivers can achieve 0.2 metres per second 3-D velocity accuracy, but this is somewhat dependent on receiver design." Therefore, while the pseudorange rate error has been estimated less elegantly than the pseudorange UERE, it is deemed 347 suitable for this baseline simulation analysis and provides the desirable excellent integrated navigation performance. D.3.4 Geometric Dilution of Precision The e ective error of the GPS measurements depend heavily on the orienta- tion of the visible satellite-to-user line of sight vectors. The Geometric Dilution of Precision (GDOP) is a quantitative measurement of this phenomenon, and the sensed error at a given time is essentially the nominal error value multiplied by the GDOP. The Geometric Dilution of Precision can be de ned by starting with the pseu- dorange error perturbation, Eq. (D.15), and keeping the user position vector in ECEF coordinates so that j = @ j @re re +c bu + : (D.26) For N visible satellites and pseudorange measurements and rearranging slightly, = 0 BB BB BB @ @ 1=@re 1 ... ... @ N=@re 1 1 CC CC CC A 0 BB @ re c bu 1 CC A G xG; (D.27) where G is the GPS geometry matrix (because the partial derivatives are equal to the unit vector from the satellite to the user) and xG is the vector of position errors (including user clock bias in terms of meters). The position errors can then be found by taking the pseudo-inverse of G: xG = GTG 1GT ( ): (D.28) 348 The covariance of the position error (and clock bias) is then E ( xG) xTG = GTG 1GTE h ( ) ( )T i G GTG 1; (D.29) where the geometry matrix has been pulled out of the expectation operator because it contains no random component. Assuming the pseudorange perturbation and noise are uncorrelated between measurements and their variance is a constant, the expectation term can be rewritten as 2I, where 2 is the constant variance and I is a 4 4 identity matrix. The position covariance can now be simpli ed to E ( xG) xTG = 2 GTG 1; (D.30) where GTG 1 is \the matrix of multipliers of ranging variance to give position variance." (Parkinson and Spilker,7 pg. 474) The individual components of the matrix yield the dilutions of precision (DOPs) along its diagonal: GTG 1 = 0 BB BB BB BB BB @ (re1 DOP)2 covariance terms (re2 DOP)2 (re3 DOP)2 covariance terms (Time DOP)2 1 CC CC CC CC CC A : (D.31) The GDOP is the total RMS of the DOPs and is calculated by taking the square root of the trace of GTG 1. Furthermore, the position dilution of precision (PDOP) can be found be taking the square root of the sums of the rst three diagonal components, and the time DOP is the square root of the fourth diagonal element. The GDOP is important as it states that for a given ranging error, the e ective position (including clock bias) error is proportionally greater by the given GDOP 349 0 5 10 15 20 0 5 10 15 Low Gradients GPS GDOP and Number of Visible SVs 0 5 10 15 20 0 5 10 15 High Gradients Time, hr # Visible SVs # Visible SVs GDOP GDOP Figure D.4: Simulated GPS Geometric Dilution of Precision value, as shown by Eq. (D.30). The GDOP for the two initial simulation positions given in Table D.3 over the course of two GPS orbits is shown in Fig. D.4. The number of visible satellites are also plotted to show the e ect of a satellite going in and out of view. The GDOP spikes around four hours are characteristic of periods of poor satellite geometry even though seven satellites are visible. This results in approximately an order of magnitude greater position error than other simulation periods where the GDOP is much closer to unity. Furthermore, this underscores the need to simulate the GPS constellation at di erent times in their orbits to quantify the e ect of time on navigation performance. 350 D.4 GPS Receiver Error Model Two states are used to model the user?s GPS receiver clock bias, bu, and bias rate (drift), _bu. These states both vary due to phase and frequency uctuations in the receiver?s clock oscillator or the atomic frequency standard.1,2,7,9 These variations are modeled by white noise processes with phase and frequency power spectral densities (S and Sf) estimated from the user?s clock Allan variance parameters. The clock?s dynamic model is therefore written as d dt 0 BB @ bu _bu 1 CC A = 0 BB @ 0 1 0 0 1 CC A 0 BB @ bu _bu 1 CC A+ 0 BB @ f 1 CC A: (D.32) The state transition matrix for the GPS receiver clock can be exactly calculated by its rst-order Taylor series expansion: u = 0 BB @ 1 t 0 1 1 CC A; (D.33) and used to solve for the discrete process noise matrix (Farrell and Barth,2 pg. 152): Qk;u = 0 BB @ S t+ 13Sf t3 12Sf t2 1 2Sf t 2 Sf t 1 CC A; (D.34) where t is the sampling period. The simulated values for the power spectral densities are then found by tting the rst component of Qk;u with the Allan clock error variance (Brown and Hwang,9 pg. 430) 1 2h0 t+ 2h 1 t 2 + 2 3 2h 2 t3; (D.35) at two sampling periods with the method of least-squares. The h terms above are the Allan variance parameters associated with the noise of a given clock type as 351 Table D.5: Simulated GPS Receiver Clock Parameters Allan Variance Parameters Power Spectral Densities h0 h 1 h 2 S ;s2=s Sf;s2=s3 2 10 19 7 10 21 2 10 20 1:00693069 10 19 4:03101008 10 19 a function of sampling time. The least-squares formulation is then (Farrell and Barth,2 pg. 153y) 0 B@ S Sf 1 CA = 1 t1 t32 t2 t31 0 B@ t32 t31 3 t2 3 t1 1 CA 0 B@ t1 t21 t31 t2 t22 t32 1 CA 0 BB BB @ h0=2 2h 1 (2=3) 2h 2 1 CC CC A ; (D.36) where t1 and t2 are the sampling periods to t around. The simulated PSDs t to sampling periods of t1 = 1=20s and t2 = 5s are given in Table D.5 along with the Allan variance parameters for the assumed temperature-compensated crystal (from Brown and Hwang,9 pg. 431). For consistency with standard GPS terminology, the actual clock states used in the simulation are scaled by the speed of light to states in terms of position and velocity instead of time bias and frequency drift. In order to correctly account for the scaling, the driving noise power spectral densities are multiplied by the speed of yThere is an error in the equation in Farrell and Barth. The coe cient to the h 2 term should be 23 2; the reference omitted the square on . 352 light squared. Therefore, the clock states are implemented by d dt 0 BB @ cbu c_bu 1 CC A = 0 BB @ 0 1 0 0 1 CC A 0 BB @ cbu c_bu 1 CC A+ 0 BB @ c c f 1 CC A _xu = Fuxu + u; (D.37) where the scaled driving noises have variances of 2 = c 2S t = 0:18099684 hm s i2 (D.38) 2f = c 2Sf t = 0:72457824 hm s2 i2 ; (D.39) assuming the integration time step, t = 1=20s. The clock noise states, u, are simulated using a random Gaussian distribution and the variances above. The clock states, xu, are then numerically integrated as follows. The clock bias state, cbu, uses a trapezoidal integration for the clock bias rate term and Euler integration for the noise term, so that (cbu)k+1 = (cbu)k + 12 (c_bu)k+1 + (c_bu)k t+ (c )k t: (D.40) The clock bias rate uses Eulerian integration: (c_bu)k+1 = (c_bu)k + (c f)k: (D.41) And substituting the new clock bias rate into the clock bias integration equation: (cbu)k+1 = (cbu)k + (c_bu)k t+ (c f)k t 2 2 + (c )k t: (D.42) D.5 Summary This section summarizes how this chapter is implemented into the overall Fortran simulation. The GPS update rate is rst input by the program user. Then 353 for each simulation run, the main Fortran program randomly initializes a time o set so that the GPS constellation is arbitrarily positioned at a point in its orbit. The truth clock states are randomly initialized with variances of (15 m)2 and (0.5 m/s)2, and the initial lter covariance matrix, P(t0), uses the same values at the diagonal elements corresponding to these states. Also, the estimated clock states are both initialized to zero. While no measurements are being made, the truth and estimated clock states are numerically integrated according to Eq. (D.41) and (D.42) using the scaled noise variances in Eq. (D.38) and (D.39). The Kalman lter covariance matrices are propagated using the state transition matrix of Eq. (D.33) and the process noise matrix of Eq. (D.34) multiplied by the speed of light squared (because of the unit conversion explained at the end of the last section). When a GPS measurement is to be made, the Fortran GPS module rst cal- culates the position of each satellite at the current simulation time plus additional time o set using Eq. (D.9). The user?s truth and estimated ECEF position states are then calculated by Eq. (4.24) on page 144 or 343 using the current navigation frame truth and estimated position states, respectively. The visibility of each SV is determined by the discussion in Sec. D.3.1, particularly Eq. (D.13) using a zero- degree minimum elevation mask. Then, for each visibile satellite, the GDOP is calculated according to Sec. D.3.4 using the estimated ECEF user position states, and the velocities are calculated by Eq. (D.10). The truth and estimated pseudoranges and (optionally) pseudorange rates are determined by Eq. (D.14) and Eq. (D.19) for each visible SV. The residuals are next 354 calculated by residualj = ( j + ) b j; (D.43) where the rst term on the right hand side is the simulated noisy jth pseudorange measured by the GPS receiver, and b j is the jth estimated pseudorange. The pseudo- range rate residuals are similarly calculated. The variances for each measurement are (3.6m)2 and (0.20 m/s)2. Lastly, the Kalman lter measurement matrices, HGPS, are constructed using Eq. (D.15){(D.17) for pseudorange, and Eq.(D.20){(D.22), (D.17), and (D.24){(D.25) for pseudorange rate. The measurement noise matrices, RGPS, are diagonal matrices with the speci ed noise variances. The GPS module then passes the Kalman lter matrices and the measurement residuals to the main Fortran program which uses the Kalman lter subroutines to improve the estimated state vector according to App. C. 355 Appendix E Additional Monte Carlo Results This appendix supplements the results presented in Ch. 6 with additional hypersonic Monte Carlo simulations which were used in the sensitivity analyses of the aforementioned chapter. Each table in this appendix lists the mean radial spherical error (MRSE) for the position, velocity, and attitude states after three settling distances: 0 km, which is the MRSE value for the entire simulation duration; 100 km, which is the MRSE that neglects the initial lter transients; and 500 km, which approximates the steady state cruise errors. It should be noted that the tactical grade IMU cases do not typically reach a steady state yaw error by the 500 km settling distance, so the results for the tac. grade IMU simulations are not quite indicative of the true steady state attitude MRSE. (See Ch. 6 for further discussion.) The current appendix is order as follows. Section E.1 summarizes the dead reckoning (free-inertial) hypersonic results. Section E.2 then lists the MRSE for the gravity gradiometer aided inertial navigation system simulations. And lastly, Sec. E.3 presents the baseline INS/Global Positioning System navigation results. 356 E.1 Dead Reckoning Results The mean radial spherical error (MRSE) results for the 1,000 km dead reckon- ing (free-inertial) Monte Carlo simulations are shown in Tables E.1{E.4. The rst two tables list the navigation grade IMU error accumulation due to the accelerom- eter, gyro, and gravity errors from over the high and low gravitational gradient variation trajectories. Tables E.3 and E.4 then summarize the tactical grade IMU dead reckoning errors for both the high and low variation trajectories. Each table includes the Mach 6, 7, and 8 scramjet results computed from the 1,000-simulation Monte Carlo sets. Table E.1: Dead Reckoning: Navigation Grade IMUs, High Variation Settling Distance: 0 km 100 km 500 km M State Units Error Cov. Error Cov. Error Cov. Pos. m 37.603 616.54 40.535 679.67 64.539 1219.3 6 Vel. m/s 0.1592 3.1474 0.1722 3.3051 0.2821 4.6106 Att. 10 3 0.4693 83.459 0.5044 83.129 0.9010 80.446 Pos. m 27.853 477.67 29.801 526.11 45.862 942.84 7 Vel. m/s 0.1304 2.8758 0.1404 3.0039 0.2249 4.0617 Att. 10 3 0.4142 84.335 0.4446 84.100 0.7936 82.147 Pos. m 23.795 385.35 25.155 424.08 36.456 759.37 8 Vel. m/s 0.1147 2.6615 0.1229 2.7658 0.1921 3.6260 Att. 10 3 0.3731 84.927 0.3999 84.757 0.7132 83.308 357 Table E.2: Dead Reckoning: Navigation Grade IMUs, Low Variation Settling Distance: 0 km 100 km 500 km M State Units Error Cov. Error Cov. Error Cov. Pos. m 40.337 616.09 43.365 679.13 68.483 1217.9 6 Vel. m/s 0.1730 3.1446 0.1852 3.3020 0.2860 4.6051 Att. 10 3 0.4625 83.348 0.4968 83.005 0.8874 80.213 Pos. m 32.497 476.91 34.637 525.26 52.539 940.96 7 Vel. m/s 0.1519 2.8693 0.1618 2.9967 0.2434 4.0491 Att. 10 3 0.4177 84.253 0.4485 84.008 0.8005 81.978 Pos. m 26.361 384.56 27.756 423.20 39.596 757.25 8 Vel. m/s 0.1272 2.6523 0.1346 2.7557 0.1948 3.6080 Att. 10 3 0.3725 84.860 0.3992 84.682 0.7117 83.171 Table E.3: Dead Reckoning: Tactical Grade IMUs, High Variation Settling Distance: 0 km 100 km 500 km M State Units Error Cov. Error Cov. Error Cov. Pos. m 587.03 858.47 650.87 948.43 1163.03 1703.1 6 Vel. m/s 3.3720 4.9110 3.7191 5.2612 6.6611 8.1173 Att. 10 3 101.24 150.76 110.94 157.74 199.56 214.69 Pos. m 402.25 624.50 445.58 689.21 793.21 1236.3 7 Vel. m/s 2.5937 4.1264 2.8577 4.3905 5.1105 6.5430 Att. 10 3 84.806 138.47 92.930 144.13 167.13 190.17 Pos. m 291.01 481.03 322.09 530.36 571.58 950.66 8 Vel. m/s 2.1124 3.5942 2.3261 3.7997 4.1555 5.4744 Att. 10 3 76.327 129.38 83.637 134.05 150.43 172.03 358 Table E.4: Dead Reckoning: Tactical Grade IMUs, Low Variation Settling Distance: 0 km 100 km 500 km M State Units Error Cov. Error Cov. Error Cov. Pos. m 600.86 857.51 665.92 947.31 1187.9 1700.7 6 Vel. m/s 3.4138 4.9042 3.7649 5.2536 6.7456 8.1035 Att. 10 3 100.97 150.64 110.64 157.61 199.01 214.44 Pos. m 406.42 623.41 449.95 688.00 798.90 1233.8 7 Vel. m/s 2.6216 4.1164 2.8885 4.3796 5.1666 6.5234 Att. 10 3 86.438 138.38 94.729 144.03 170.36 189.99 Pos. m 294.77 479.94 325.96 529.16 576.10 948.03 8 Vel. m/s 2.1049 3.5823 2.3168 3.7866 4.1368 5.4505 Att. 10 3 75.739 129.30 83.000 133.97 149.25 171.88 E.2 Gravity Gradiometer Aided Navigation The hypersonic gravity gradiometer aided INS MRSE results are listed in Tables E.5{E.22 for the two GGI types, two simulated IMU speci cations, two variation trajectories, and three cruise Mach numbers. Each table includes the posi- tion, velocity, and attitude MRSEs for the three settling distances, three simulated GGI noise level standard deviations ( L = 0.001, 0.01, and 0.1 E o; 1 E o 10 9 s 2), and three update rates ( t = 1, 5, and 10 sec). Tables E.5{E.7 on pg. 361{363 present the stabilized gradiometer, navigation grade inertial measurement unit INS results over the high gravitational gradient variation trajectory for the three simulated Mach numbers (6, 7, and 8). Tables E.8{E.10 on pg. 364{366 then present the same stabilized GGI, nav. grade IMU 359 con guration results as simulated over the low variation trajectory. Tables E.11{E.16 on pg. 367{372 next list the stabilized GGI, tactical grade IMU Monte Carlo simulation results. The rst three tables correspond to the high gravitational gradient variation trajectory simulations at the three hypersonic ve- locities, and the latter three tables correspond to the low variation trajectories. Lastly, Tables E.17{E.22 summarize the simulated strapdown GGI, naviga- tion grade IMU con guration?s hypersonic navigation performance on pg. 373{378. Tables E.17{E.19 are for the high variation trajectories, and Tables E.20{E.22 are for the low variation trajectories with each table representing a single cruise Mach number. Inertial navigation systems consisting of a strapdown GGI and tactical grade IMUs were simulated, but diverged almost instantly because of the numerical trun- cation issues discussed in Ch. 6. Therefore, full Monte Carlo simulations were not performed for this INS/GGI con guration. 360 Table E.5: INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.2531 1.2989 1.3047 1.3507 0.3240 0.3233 1 Vel. m/s 0.0329 0.7149 0.0218 0.7730 0.0061 0.0061 Att. 10 3 4.6307 44.066 4.1034 41.700 7.3813 7.5899 .001 Pos. m 1.4827 1.5706 1.4659 1.5293 0.6709 0.6764 5 Vel. m/s 0.0375 0.7315 0.0241 0.7756 0.0111 0.0113 Att. 10 3 6.7807 46.714 6.6872 43.694 12.032 12.702 .001 Pos. m 1.6852 1.8337 1.6328 1.6632 0.9219 0.9355 10 Vel. m/s 0.0378 0.7475 0.0258 0.7775 0.0145 0.0151 Att. 10 3 7.9106 48.414 8.1111 45.189 14.595 16.373 .01 Pos. m 4.2590 6.3569 4.0783 6.3627 2.7380 2.7931 1 Vel. m/s 0.0714 0.7733 0.0378 0.7900 0.0353 0.0384 Att. 10 3 11.114 54.886 11.960 51.867 21.525 31.614 .01 Pos. m 6.2098 8.4138 5.7123 7.9871 5.8824 5.9503 5 Vel. m/s 0.0946 0.8129 0.0543 0.8082 0.0649 0.0714 Att. 10 3 9.6797 59.181 10.526 56.611 18.943 41.637 .01 Pos. m 7.5477 9.8941 6.9249 9.1992 8.2130 8.2657 10 Vel. m/s 0.1063 0.8357 0.0668 0.8206 0.0870 0.0938 Att. 10 3 8.2927 60.526 9.0137 58.113 16.220 44.641 .1 Pos. m 15.355 21.463 14.725 20.229 24.307 24.691 1 Vel. m/s 0.1882 0.9465 0.1468 0.9010 0.2315 0.2438 Att. 10 3 5.2273 63.162 5.5696 61.019 10.019 49.365 .1 Pos. m 28.506 36.364 29.453 35.596 50.802 52.640 5 Vel. m/s 0.2869 1.0802 0.2663 1.0240 0.4466 0.4801 Att. 10 3 5.9907 64.981 6.4128 62.957 11.536 51.142 .1 Pos. m 38.230 46.576 40.543 46.613 70.711 72.572 10 Vel. m/s 0.3520 1.1616 0.3477 1.1072 0.5939 0.6404 Att. 10 3 7.3783 66.145 7.9951 64.196 14.385 52.246 361 Table E.6: INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.2858 1.3474 1.3377 1.3996 0.3360 0.3366 1 Vel. m/s 0.0344 0.7177 0.0221 0.7734 0.0069 0.0069 Att. 10 3 5.8014 45.380 5.4279 42.677 9.7653 9.9880 .001 Pos. m 1.5576 1.6357 1.5396 1.5892 0.7060 0.7114 5 Vel. m/s 0.0388 0.7362 0.0245 0.7766 0.0125 0.0132 Att. 10 3 8.3911 49.012 8.5468 45.838 15.379 17.710 .001 Pos. m 1.7653 1.9214 1.7070 1.7306 0.9789 0.9873 10 Vel. m/s 0.0388 0.7543 0.0265 0.7788 0.0163 0.0175 Att. 10 3 9.4599 51.125 9.9213 47.880 17.853 22.453 .01 Pos. m 4.3398 6.4797 4.1528 6.4685 2.9415 2.9683 1 Vel. m/s 0.0765 0.7823 0.0406 0.7936 0.0405 0.0446 Att. 10 3 10.628 57.784 11.490 55.045 20.678 38.059 .01 Pos. m 6.4176 8.6453 5.9515 8.2043 6.2902 6.3438 5 Vel. m/s 0.1030 0.8267 0.0609 0.8152 0.0778 0.0838 Att. 10 3 7.8673 61.192 8.5309 58.844 15.350 45.735 .01 Pos. m 7.8314 10.207 7.2554 9.5085 8.6661 8.8306 10 Vel. m/s 0.1166 0.8526 0.0760 0.8303 0.1037 0.1114 Att. 10 3 6.3584 62.100 6.8687 59.851 12.358 47.571 .1 Pos. m 15.801 22.149 15.476 21.131 25.742 26.376 1 Vel. m/s 0.2099 0.9780 0.1731 0.9279 0.2791 0.2972 Att. 10 3 5.1509 64.289 5.4837 62.220 9.8638 50.453 .1 Pos. m 29.654 37.527 31.209 37.315 54.014 55.778 5 Vel. m/s 0.3237 1.1297 0.3150 1.0749 0.5359 0.5817 Att. 10 3 7.9576 66.730 8.6592 64.820 15.580 52.815 .1 Pos. m 38.213 47.951 41.029 48.672 71.806 76.289 10 Vel. m/s 0.3784 1.2194 0.3855 1.1693 0.6626 0.7628 Att. 10 3 9.4384 68.321 10.346 66.528 18.617 54.658 362 Table E.7: INS/GGI: Stabilized GGI, Nav. Grade IMUs, High Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 0.9765 0.9817 0.9767 0.9747 0.3499 0.3538 1 Vel. m/s 0.0371 0.7211 0.0224 0.7740 0.0077 0.0079 Att. 10 3 7.2219 47.151 7.0175 44.118 12.628 13.345 .001 Pos. m 1.2578 1.3065 1.1955 1.1997 0.7512 0.7591 5 Vel. m/s 0.0399 0.7399 0.0257 0.7780 0.0143 0.0154 Att. 10 3 9.7615 51.306 10.104 48.138 18.186 22.706 .001 Pos. m 1.4293 1.6283 1.3431 1.3650 1.0406 1.0566 10 Vel. m/s 0.0385 0.7599 0.0276 0.7806 0.0188 0.0204 Att. 10 3 10.532 53.342 11.159 50.227 20.084 27.343 .01 Pos. m 5.1153 6.7624 4.9641 6.6903 3.1325 3.1783 1 Vel. m/s 0.0841 0.7954 0.0452 0.7984 0.0486 0.0527 Att. 10 3 9.5473 60.050 10.348 57.546 18.625 42.745 .01 Pos. m 7.3771 9.1563 6.9826 8.6860 6.7539 6.8215 5 Vel. m/s 0.1146 0.8438 0.0718 0.8251 0.0964 0.1022 Att. 10 3 6.4039 62.521 6.9014 60.300 12.418 47.928 .01 Pos. m 8.8600 10.834 8.3882 10.138 9.2846 9.4656 10 Vel. m/s 0.1297 0.8716 0.0895 0.8430 0.1279 0.1357 Att. 10 3 5.4356 63.176 5.8384 61.017 10.504 49.014 .1 Pos. m 15.837 22.842 15.785 22.040 27.348 27.852 1 Vel. m/s 0.2330 1.0090 0.2029 0.9559 0.3341 0.3531 Att. 10 3 6.2879 65.533 6.7847 63.539 12.207 51.516 .1 Pos. m 29.388 38.489 31.256 38.732 55.213 58.116 5 Vel. m/s 0.3458 1.1746 0.3447 1.1221 0.5902 0.6742 Att. 10 3 9.9194 68.776 10.882 67.018 19.585 55.184 .1 Pos. m 38.383 48.932 41.542 50.181 73.711 78.760 10 Vel. m/s 0.3973 1.2677 0.4110 1.2219 0.7093 0.8633 Att. 10 3 11.307 70.738 12.459 69.141 22.424 57.892 363 Table E.8: INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 0.9714 0.9920 0.9788 0.9958 0.3229 0.3243 1 Vel. m/s 0.0365 0.7155 0.0282 0.7730 0.0062 0.0062 Att. 10 3 4.5738 44.072 4.0177 41.595 7.2266 7.3473 .001 Pos. m 1.2545 1.2819 1.2038 1.1952 0.6869 0.6849 5 Vel. m/s 0.0403 0.7308 0.0311 0.7758 0.0114 0.0116 Att. 10 3 6.4881 46.504 6.3177 43.469 11.367 12.205 .001 Pos. m 1.3947 1.5562 1.3151 1.3428 0.9489 0.9516 10 Vel. m/s 0.0396 0.7465 0.0317 0.7778 0.0150 0.0155 Att. 10 3 7.6265 47.951 7.7610 44.766 13.965 15.426 .01 Pos. m 5.0184 6.6512 4.9109 6.6363 2.8394 2.8668 1 Vel. m/s 0.0747 0.7762 0.0430 0.7912 0.0374 0.0407 Att. 10 3 10.464 54.686 11.261 51.621 20.265 31.215 .01 Pos. m 7.0229 8.8622 6.6195 8.4535 6.0551 6.1688 5 Vel. m/s 0.0991 0.8159 0.0602 0.8115 0.0713 0.0777 Att. 10 3 9.9498 58.936 10.823 56.336 19.477 41.163 .01 Pos. m 8.4670 10.418 7.9533 9.7743 8.4144 8.5652 10 Vel. m/s 0.1100 0.8386 0.0715 0.8250 0.0933 0.1024 Att. 10 3 8.6197 60.312 9.3712 57.873 16.863 44.206 .1 Pos. m 15.121 22.028 14.486 20.920 24.961 25.409 1 Vel. m/s 0.1949 0.9523 0.1532 0.9094 0.2473 0.2615 Att. 10 3 5.4494 63.235 5.8115 61.097 10.454 49.428 .1 Pos. m 28.994 37.422 30.018 36.815 52.933 54.276 5 Vel. m/s 0.3040 1.0947 0.2839 1.0414 0.4873 0.5165 Att. 10 3 6.5818 65.271 7.0718 63.266 12.722 51.407 .1 Pos. m 38.482 47.937 40.868 48.158 72.500 74.775 10 Vel. m/s 0.3655 1.1804 0.3613 1.1292 0.6282 0.6857 Att. 10 3 7.9138 66.544 8.5915 64.624 15.458 52.666 364 Table E.9: INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 0.9950 1.0033 1.0017 1.0036 0.3399 0.3389 1 Vel. m/s 0.0376 0.7182 0.0280 0.7734 0.0070 0.0071 Att. 10 3 5.5955 45.355 5.1643 42.526 9.2903 9.6843 .001 Pos. m 1.2616 1.3115 1.2074 1.2176 0.7194 0.7258 5 Vel. m/s 0.0416 0.7353 0.0316 0.7768 0.0132 0.0137 Att. 10 3 8.1191 48.665 8.2355 45.461 14.818 16.952 .001 Pos. m 1.4334 1.6102 1.3542 1.3762 0.9989 1.0116 10 Vel. m/s 0.0408 0.7533 0.0326 0.7792 0.0172 0.0183 Att. 10 3 9.2658 50.461 9.6888 47.214 17.434 21.079 .01 Pos. m 5.1651 6.7707 5.0479 6.7403 2.9773 3.0544 1 Vel. m/s 0.0806 0.7858 0.0454 0.7952 0.0439 0.0477 Att. 10 3 10.829 57.510 11.748 54.729 21.141 37.610 .01 Pos. m 7.2413 9.0953 6.8573 8.6759 6.4223 6.5768 5 Vel. m/s 0.1082 0.8303 0.0666 0.8193 0.0849 0.0921 Att. 10 3 8.1863 60.982 8.8848 58.610 15.986 45.397 .01 Pos. m 8.6145 10.732 8.1227 10.086 8.8989 9.1391 10 Vel. m/s 0.1203 0.8560 0.0806 0.8354 0.1126 0.1221 Att. 10 3 7.0102 61.951 7.5971 59.686 13.668 47.321 .1 Pos. m 15.799 22.741 15.455 21.829 26.713 27.117 1 Vel. m/s 0.2188 0.9855 0.1811 0.9378 0.3013 0.3182 Att. 10 3 5.5495 64.425 5.9248 62.364 10.657 50.595 .1 Pos. m 29.544 38.573 31.115 38.497 54.941 57.351 5 Vel. m/s 0.3347 1.1456 0.3255 1.0935 0.5645 0.6203 Att. 10 3 8.5104 67.140 9.2760 65.260 16.689 53.292 .1 Pos. m 39.089 49.246 41.985 50.124 74.497 78.333 10 Vel. m/s 0.3961 1.2381 0.4023 1.1909 0.7016 0.8068 Att. 10 3 10.072 68.833 11.047 67.080 19.878 55.332 365 Table E.10: INS/GGI: Stabilized GGI, Nav. Grade IMUs, Low Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 0.9724 1.0150 0.9729 1.0124 0.3558 0.3551 1 Vel. m/s 0.0399 0.7210 0.0294 0.7740 0.0078 0.0081 Att. 10 3 6.9399 46.973 6.7099 43.908 12.072 12.997 .001 Pos. m 1.2827 1.3400 1.2251 1.2400 0.7600 0.7656 5 Vel. m/s 0.0425 0.7398 0.0316 0.7780 0.0150 0.0158 Att. 10 3 9.6522 51.051 10.012 47.842 18.017 22.279 .001 Pos. m 1.4856 1.6611 1.4073 1.4078 1.0604 1.0679 10 Vel. m/s 0.0423 0.7599 0.0339 0.7808 0.0195 0.0211 Att. 10 3 10.302 53.044 10.919 49.892 19.650 26.877 .01 Pos. m 5.2330 6.8807 5.1124 6.8358 3.1665 3.2274 1 Vel. m/s 0.0868 0.7954 0.0495 0.7993 0.0510 0.0550 Att. 10 3 9.6770 59.749 10.508 57.216 18.910 42.382 .01 Pos. m 7.4060 9.3081 7.0380 8.8836 6.9301 6.9591 5 Vel. m/s 0.1176 0.8449 0.0750 0.8275 0.1021 0.1076 Att. 10 3 6.6443 62.350 7.1724 60.117 12.904 47.810 .01 Pos. m 8.8994 11.014 8.4681 10.377 9.5483 9.6757 10 Vel. m/s 0.1316 0.8736 0.0922 0.8465 0.1349 0.1435 Att. 10 3 5.6786 63.071 6.1073 60.907 10.986 48.978 .1 Pos. m 16.125 23.341 16.089 22.626 27.854 28.605 1 Vel. m/s 0.2393 1.0178 0.2079 0.9669 0.3505 0.3766 Att. 10 3 6.6706 65.743 7.2092 63.766 12.970 51.780 .1 Pos. m 30.430 39.384 32.424 39.743 57.274 59.608 5 Vel. m/s 0.3569 1.1904 0.3559 1.1404 0.6186 0.7120 Att. 10 3 10.495 69.294 11.525 67.579 20.739 55.911 .1 Pos. m 39.661 50.042 42.962 51.426 76.241 80.662 10 Vel. m/s 0.4089 1.2843 0.4231 1.2409 0.7403 0.9024 Att. 10 3 12.008 71.324 13.240 69.779 23.827 58.794 366 Table E.11: INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.3231 1.3759 1.3761 1.4306 0.5057 0.5057 1 Vel. m/s 0.0431 0.7247 0.0312 0.7821 0.0233 0.0232 Att. 10 3 58.730 100.11 63.449 103.28 114.10 125.99 .001 Pos. m 1.6710 1.7042 1.6602 1.6651 0.9825 0.9875 5 Vel. m/s 0.0520 0.7448 0.0370 0.7877 0.0342 0.0344 Att. 10 3 58.215 101.20 63.016 104.17 113.32 127.66 .001 Pos. m 1.8600 2.0077 1.8059 1.8369 1.3408 1.3292 10 Vel. m/s 0.0559 0.7628 0.0414 0.7911 0.0413 0.0413 Att. 10 3 59.311 101.76 64.269 104.62 115.57 128.51 .01 Pos. m 4.6339 6.6840 4.4683 6.7035 3.6301 3.6291 1 Vel. m/s 0.0981 0.7963 0.0636 0.8121 0.0830 0.0828 Att. 10 3 62.651 103.48 68.178 106.20 122.61 131.39 .01 Pos. m 6.9845 9.1623 6.5602 8.7890 7.6677 7.6952 5 Vel. m/s 0.1402 0.8523 0.1025 0.8487 0.1524 0.1541 Att. 10 3 62.818 104.72 68.467 107.45 123.13 133.40 .01 Pos. m 8.6750 10.989 8.1695 10.386 10.687 10.718 10 Vel. m/s 0.1656 0.8883 0.1303 0.8759 0.2031 0.2065 Att. 10 3 60.505 105.34 65.957 108.10 118.61 134.27 .1 Pos. m 18.629 24.653 18.384 23.769 30.930 31.226 1 Vel. m/s 0.3263 1.0775 0.2985 1.0442 0.5054 0.5243 Att. 10 3 66.331 109.15 72.332 112.11 130.08 139.33 .1 Pos. m 35.344 42.309 37.024 42.181 64.462 64.438 5 Vel. m/s 0.5182 1.2996 0.5190 1.2660 0.9017 0.9308 Att. 10 3 68.834 113.91 75.068 117.22 134.99 146.39 .1 Pos. m 46.740 54.223 49.943 55.065 87.681 87.634 10 Vel. m/s 0.6310 1.4263 0.6500 1.3996 1.1383 1.1723 Att. 10 3 72.038 116.32 78.571 119.83 141.29 150.25 367 Table E.12: INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.3945 1.4171 1.4525 1.4715 0.5079 0.5063 1 Vel. m/s 0.0442 0.7271 0.0310 0.7821 0.0234 0.0233 Att. 10 3 54.575 95.312 58.850 97.720 105.81 114.45 .001 Pos. m 1.7128 1.7546 1.6992 1.7097 1.0013 0.9952 5 Vel. m/s 0.0536 0.7489 0.0376 0.7879 0.0351 0.0349 Att. 10 3 52.251 96.237 56.443 98.378 101.48 115.70 .001 Pos. m 1.9323 2.0790 1.8793 1.8879 1.3612 1.3495 10 Vel. m/s 0.0567 0.7690 0.0422 0.7918 0.0429 0.0425 Att. 10 3 53.352 96.701 57.830 98.717 103.98 116.33 .01 Pos. m 4.6671 6.8126 4.4894 6.8206 3.8253 3.8064 1 Vel. m/s 0.1059 0.8066 0.0704 0.8178 0.0948 0.0938 Att. 10 3 52.631 98.023 57.232 99.924 102.90 118.41 .01 Pos. m 7.2763 9.4384 6.8759 9.0638 8.1560 8.1496 5 Vel. m/s 0.1565 0.8724 0.1190 0.8637 0.1811 0.1825 Att. 10 3 55.267 99.266 60.220 101.20 108.27 120.15 .01 Pos. m 8.8874 11.345 8.4168 10.752 11.084 11.325 10 Vel. m/s 0.1844 0.9137 0.1494 0.8959 0.2371 0.2441 Att. 10 3 56.193 100.12 61.253 102.10 110.13 121.30 .1 Pos. m 18.950 25.100 18.977 24.406 32.064 32.304 1 Vel. m/s 0.3545 1.1144 0.3320 1.0779 0.5655 0.5829 Att. 10 3 59.555 105.03 64.935 107.36 116.75 128.42 .1 Pos. m 34.952 42.736 37.007 43.067 64.393 65.950 5 Vel. m/s 0.5373 1.3390 0.5458 1.3060 0.9509 0.9969 Att. 10 3 64.645 109.99 70.547 112.73 126.84 136.45 .1 Pos. m 46.599 54.648 50.244 56.055 88.272 89.338 10 Vel. m/s 0.6591 1.4677 0.6889 1.4434 1.2076 1.2444 Att. 10 3 65.068 112.29 71.030 115.23 127.71 140.38 368 Table E.13: INS/GGI: Stabilized GGI, Tac. Grade IMUs, High Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.0238 1.0623 1.0246 1.0605 0.5124 0.5089 1 Vel. m/s 0.0466 0.7303 0.0310 0.7823 0.0236 0.0235 Att. 10 3 47.931 91.931 51.553 93.689 92.692 106.02 .001 Pos. m 1.3676 1.4381 1.3118 1.3403 1.0111 1.0140 5 Vel. m/s 0.0533 0.7520 0.0375 0.7888 0.0360 0.0363 Att. 10 3 46.739 92.644 50.427 94.161 90.664 106.86 .001 Pos. m 1.6187 1.7986 1.5422 1.5468 1.3841 1.3869 10 Vel. m/s 0.0559 0.7741 0.0426 0.7933 0.0452 0.0452 Att. 10 3 46.451 92.978 50.213 94.397 90.274 107.27 .01 Pos. m 5.5411 7.1543 5.4245 7.1197 4.0157 4.0148 1 Vel. m/s 0.1151 0.8221 0.0783 0.8260 0.1086 0.1095 Att. 10 3 48.273 94.297 52.529 95.620 94.449 109.11 .01 Pos. m 8.2062 9.9848 7.8933 9.6023 8.4953 8.5383 5 Vel. m/s 0.1702 0.8939 0.1327 0.8791 0.2071 0.2112 Att. 10 3 49.109 95.930 53.496 97.341 96.178 111.38 .01 Pos. m 10.014 11.969 9.6483 11.396 11.678 11.788 10 Vel. m/s 0.2009 0.9370 0.1670 0.9139 0.2687 0.2768 Att. 10 3 50.440 97.020 54.978 98.505 98.849 112.94 .1 Pos. m 18.562 25.474 18.772 24.956 32.705 33.025 1 Vel. m/s 0.3676 1.1412 0.3495 1.1016 0.5985 0.6217 Att. 10 3 55.383 102.44 60.408 104.36 108.61 121.44 .1 Pos. m 35.079 43.085 37.546 43.795 66.500 66.991 5 Vel. m/s 0.5603 1.3682 0.5772 1.3357 1.0080 1.0448 Att. 10 3 61.055 107.07 66.696 109.41 119.93 129.40 .1 Pos. m 45.921 54.914 49.862 56.770 88.660 90.360 10 Vel. m/s 0.6795 1.4980 0.7169 1.4758 1.2592 1.2972 Att. 10 3 62.306 109.20 68.049 111.75 122.37 133.22 369 Table E.14: INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.0699 1.0826 1.0847 1.0931 0.5017 0.4988 1 Vel. m/s 0.0454 0.7250 0.0346 0.7820 0.0232 0.0232 Att. 10 3 60.112 99.734 64.988 102.82 116.86 125.15 .001 Pos. m 1.3865 1.4356 1.3410 1.3593 0.9764 0.9792 5 Vel. m/s 0.0528 0.7438 0.0395 0.7876 0.0345 0.0345 Att. 10 3 60.145 100.79 65.136 103.72 117.13 126.84 .001 Pos. m 1.6137 1.7503 1.5440 1.5489 1.3260 1.3213 10 Vel. m/s 0.0550 0.7616 0.0423 0.7912 0.0418 0.0416 Att. 10 3 58.692 101.30 63.577 104.14 114.32 127.64 .01 Pos. m 5.4416 7.0280 5.3717 7.0429 3.6341 3.6588 1 Vel. m/s 0.0990 0.7994 0.0641 0.8136 0.0859 0.0865 Att. 10 3 59.232 103.39 64.438 106.07 115.88 131.18 .01 Pos. m 7.7770 9.6410 7.4369 9.3079 7.7802 7.7959 5 Vel. m/s 0.1430 0.8560 0.1044 0.8528 0.1622 0.1637 Att. 10 3 62.767 104.77 68.392 107.50 122.99 133.44 .01 Pos. m 9.5300 11.519 9.1229 10.989 10.744 10.847 10 Vel. m/s 0.1688 0.8919 0.1331 0.8809 0.2140 0.2184 Att. 10 3 63.951 105.47 69.727 108.23 125.39 134.41 .1 Pos. m 18.174 25.055 17.891 24.280 31.132 31.584 1 Vel. m/s 0.3316 1.0843 0.3032 1.0536 0.5239 0.5446 Att. 10 3 67.543 109.53 73.669 112.52 132.48 139.87 .1 Pos. m 34.651 42.976 36.255 42.964 64.141 65.278 5 Vel. m/s 0.5227 1.3113 0.5219 1.2803 0.9159 0.9604 Att. 10 3 69.363 114.36 75.664 117.71 136.06 147.11 .1 Pos. m 46.324 55.103 49.517 56.075 88.024 88.882 10 Vel. m/s 0.6384 1.4419 0.6577 1.4180 1.1609 1.2093 Att. 10 3 73.264 116.78 79.936 120.32 143.75 151.00 370 Table E.15: INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.0732 1.0880 1.0834 1.0943 0.4997 0.5013 1 Vel. m/s 0.0477 0.7275 0.0355 0.7820 0.0232 0.0233 Att. 10 3 51.086 95.050 54.992 97.384 98.865 113.87 .001 Pos. m 1.4101 1.4508 1.3611 1.3663 0.9988 0.9921 5 Vel. m/s 0.0545 0.7478 0.0404 0.7880 0.0357 0.0353 Att. 10 3 53.171 95.955 57.448 98.075 103.28 115.19 .001 Pos. m 1.6341 1.7867 1.5639 1.5639 1.3495 1.3486 10 Vel. m/s 0.0568 0.7677 0.0439 0.7920 0.0436 0.0433 Att. 10 3 52.543 96.380 56.876 98.407 102.25 115.81 .01 Pos. m 5.4598 7.1478 5.3621 7.1511 3.8303 3.8534 1 Vel. m/s 0.1067 0.8106 0.0702 0.8201 0.0991 0.0994 Att. 10 3 54.200 98.032 58.986 99.926 106.05 118.43 .01 Pos. m 8.0740 9.9015 7.7839 9.5654 8.2502 8.2545 5 Vel. m/s 0.1573 0.8765 0.1196 0.8681 0.1901 0.1931 Att. 10 3 56.157 99.459 61.202 101.40 110.03 120.44 .01 Pos. m 9.8845 11.855 9.5303 11.330 11.448 11.452 10 Vel. m/s 0.1884 0.9172 0.1539 0.9009 0.2520 0.2560 Att. 10 3 54.527 100.37 59.405 102.37 106.80 121.67 .1 Pos. m 18.711 25.506 18.694 24.897 32.553 32.651 1 Vel. m/s 0.3616 1.1203 0.3386 1.0859 0.5864 0.6004 Att. 10 3 60.411 105.44 65.873 107.80 118.43 129.08 .1 Pos. m 35.586 43.425 37.718 43.852 66.796 66.820 5 Vel. m/s 0.5581 1.3507 0.5672 1.3200 0.9971 1.0257 Att. 10 3 64.907 110.41 70.837 113.19 127.36 137.18 .1 Pos. m 46.713 55.545 50.404 57.067 89.629 90.614 10 Vel. m/s 0.6673 1.4833 0.6965 1.4615 1.2289 1.2810 Att. 10 3 65.971 112.72 72.032 115.71 129.51 141.17 371 Table E.16: INS/GGI: Stabilized GGI, Tac. Grade IMUs, Low Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 1.0823 1.0935 1.0908 1.0963 0.5094 0.5050 1 Vel. m/s 0.0494 0.7300 0.0360 0.7822 0.0237 0.0236 Att. 10 3 48.022 91.710 51.664 93.444 92.878 105.66 .001 Pos. m 1.4272 1.4685 1.3787 1.3774 1.0121 1.0120 5 Vel. m/s 0.0552 0.7518 0.0405 0.7888 0.0366 0.0368 Att. 10 3 46.707 92.485 50.373 93.985 90.554 106.63 .001 Pos. m 1.6427 1.8272 1.5719 1.5855 1.3873 1.3873 10 Vel. m/s 0.0574 0.7741 0.0446 0.7935 0.0460 0.0462 Att. 10 3 45.922 92.848 49.678 94.259 89.300 107.11 .01 Pos. m 5.5815 7.2673 5.4870 7.2597 4.0456 4.0461 1 Vel. m/s 0.1162 0.8229 0.0785 0.8277 0.1133 0.1140 Att. 10 3 48.178 94.329 52.430 95.657 94.254 109.21 .01 Pos. m 8.3029 10.109 8.0238 9.7708 8.5573 8.6143 5 Vel. m/s 0.1709 0.8958 0.1324 0.8823 0.2126 0.2192 Att. 10 3 51.206 96.131 55.814 97.562 100.34 111.73 .01 Pos. m 10.059 12.101 9.7358 11.582 11.811 11.891 10 Vel. m/s 0.2032 0.9393 0.1690 0.9177 0.2798 0.2862 Att. 10 3 50.756 97.294 55.301 98.806 99.414 113.41 .1 Pos. m 18.877 25.778 19.119 25.325 33.298 33.380 1 Vel. m/s 0.3786 1.1467 0.3604 1.1089 0.6249 0.6378 Att. 10 3 56.281 102.83 61.392 104.79 110.37 122.13 .1 Pos. m 35.560 43.677 38.063 44.471 67.381 67.894 5 Vel. m/s 0.5732 1.3803 0.5904 1.3499 1.0368 1.0740 Att. 10 3 60.758 107.48 66.361 109.87 119.31 130.16 .1 Pos. m 46.409 55.697 50.392 57.653 89.584 91.632 10 Vel. m/s 0.6804 1.5131 0.7173 1.4931 1.2654 1.3322 Att. 10 3 64.644 109.66 70.643 112.26 127.02 134.09 372 Table E.17: INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 11.729 10.620 10.244 9.5024 17.445 7.2003 1 Vel. m/s 0.1392 0.7848 0.0637 0.7904 0.0957 0.0435 Att. 10 3 4.5092 23.133 2.2844 22.997 3.5618 1.4451 .001 Pos. m 14.505 12.739 13.552 11.453 23.442 10.885 5 Vel. m/s 0.1354 0.8057 0.0734 0.7985 0.1127 0.0595 Att. 10 3 4.9722 23.969 3.1423 23.489 5.0903 2.3304 .001 Pos. m 15.450 13.848 14.979 12.437 26.155 12.697 10 Vel. m/s 0.1229 0.8203 0.0758 0.8021 0.1152 0.0662 Att. 10 3 4.8798 24.509 3.4646 23.734 5.7539 2.7714 .01 Pos. m 13.394 18.565 13.347 17.384 22.260 20.979 1 Vel. m/s 0.1071 0.8422 0.0651 0.8154 0.0873 0.0882 Att. 10 3 3.6818 24.929 2.5823 24.229 3.6387 3.6432 .01 Pos. m 19.559 25.050 20.161 24.114 34.501 33.108 5 Vel. m/s 0.1217 0.8730 0.0808 0.8311 0.1161 0.1142 Att. 10 3 3.9487 25.629 2.8725 24.541 4.0807 4.2050 .01 Pos. m 24.451 29.499 25.619 28.813 44.376 41.580 10 Vel. m/s 0.1317 0.8901 0.0945 0.8416 0.1391 0.1328 Att. 10 3 4.0718 25.992 3.0613 24.696 4.4083 4.4844 .1 Pos. m 45.886 54.123 49.582 55.404 88.854 88.671 1 Vel. m/s 0.1820 0.9656 0.1570 0.8985 0.2519 0.2387 Att. 10 3 4.7328 27.550 3.6868 25.701 6.0294 6.2251 .1 Pos. m 77.306 89.088 84.900 94.007 152.44 158.22 5 Vel. m/s 0.2413 1.0449 0.2381 0.9754 0.3985 0.3887 Att. 10 3 6.1010 29.794 5.5657 27.791 9.4547 9.9866 .1 Pos. m 92.553 111.81 102.00 119.20 183.19 203.57 10 Vel. m/s 0.2709 1.0917 0.2767 1.0251 0.4683 0.4854 Att. 10 3 6.7348 31.167 6.4908 29.228 11.145 12.574 373 Table E.18: INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 11.104 10.476 9.8557 9.4242 16.766 7.0500 1 Vel. m/s 0.1413 0.7926 0.0656 0.7923 0.0999 0.0473 Att. 10 3 4.3016 23.080 2.2747 23.040 3.6024 1.5143 .001 Pos. m 13.900 12.470 13.257 11.288 22.912 10.536 5 Vel. m/s 0.1368 0.8140 0.0761 0.8004 0.1170 0.0630 Att. 10 3 4.8303 23.947 3.2593 23.545 5.3360 2.4235 .001 Pos. m 15.110 13.582 15.007 12.291 26.152 12.367 10 Vel. m/s 0.1236 0.8296 0.0795 0.8042 0.1228 0.0699 Att. 10 3 4.3971 24.516 3.2534 23.769 5.3480 2.8284 .01 Pos. m 13.988 18.801 14.209 17.863 23.748 21.796 1 Vel. m/s 0.1110 0.8534 0.0694 0.8200 0.0955 0.0964 Att. 10 3 3.5192 24.753 2.4831 24.156 3.5028 3.5073 .01 Pos. m 20.658 26.176 21.625 25.643 37.226 35.859 5 Vel. m/s 0.1292 0.8886 0.0905 0.8404 0.1330 0.1323 Att. 10 3 3.8057 25.499 2.7287 24.466 3.9343 4.0662 .01 Pos. m 25.999 31.048 27.575 30.847 47.922 45.251 10 Vel. m/s 0.1423 0.9081 0.1088 0.8538 0.1672 0.1569 Att. 10 3 3.9337 25.916 2.9311 24.647 4.2702 4.3911 .1 Pos. m 47.962 57.258 52.234 59.408 93.650 95.971 1 Vel. m/s 0.1960 0.9921 0.1789 0.9232 0.2935 0.2901 Att. 10 3 4.9006 27.748 3.8088 25.888 6.2745 6.5618 .1 Pos. m 78.696 95.330 86.691 101.56 155.67 171.91 5 Vel. m/s 0.2675 1.0880 0.2723 1.0217 0.4604 0.4854 Att. 10 3 6.1600 30.271 5.6886 28.297 9.6896 10.899 .1 Pos. m 93.744 118.30 103.55 127.03 185.99 217.69 10 Vel. m/s 0.3018 1.1446 0.3159 1.0832 0.5398 0.6034 Att. 10 3 7.0573 31.695 6.9041 29.799 11.894 13.602 374 Table E.19: INS/GGI: Strapdown GGI, Nav. Grade IMUs, High Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 12.367 11.018 11.476 9.9472 19.885 8.0657 1 Vel. m/s 0.1544 0.8020 0.0853 0.7962 0.1172 0.0552 Att. 10 3 4.4648 23.126 2.8837 23.246 4.2404 1.9055 .001 Pos. m 15.171 13.361 14.918 12.306 26.187 12.310 5 Vel. m/s 0.1518 0.8221 0.0964 0.8066 0.1360 0.0755 Att. 10 3 4.2822 23.834 3.1523 23.709 4.7159 2.7391 .001 Pos. m 16.482 14.483 16.675 13.419 29.393 14.311 10 Vel. m/s 0.1408 0.8363 0.0994 0.8111 0.1406 0.0838 Att. 10 3 3.8930 24.307 3.0941 23.857 4.6375 3.0061 .01 Pos. m 13.895 19.934 14.363 19.224 24.855 23.918 1 Vel. m/s 0.1185 0.8676 0.0780 0.8294 0.1070 0.1142 Att. 10 3 3.5846 24.570 2.5018 24.104 3.2687 3.4106 .01 Pos. m 21.052 27.579 22.285 27.402 39.203 38.684 5 Vel. m/s 0.1406 0.9051 0.1037 0.8532 0.1532 0.1569 Att. 10 3 4.0411 25.220 2.8674 24.415 3.8765 3.9705 .01 Pos. m 25.159 32.570 26.891 32.803 47.607 48.427 10 Vel. m/s 0.1488 0.9256 0.1182 0.8683 0.1796 0.1855 Att. 10 3 4.0596 25.603 2.9650 24.615 4.1312 4.3304 .1 Pos. m 49.816 60.139 54.530 63.026 98.044 102.57 1 Vel. m/s 0.2150 1.0186 0.2047 0.9496 0.3394 0.3466 Att. 10 3 5.0269 27.976 3.9617 26.101 6.5785 6.9411 .1 Pos. m 78.404 99.729 86.566 106.91 155.69 181.59 5 Vel. m/s 0.2929 1.1317 0.3047 1.0695 0.5201 0.5838 Att. 10 3 6.2973 30.657 5.8815 28.707 10.090 11.632 .1 Pos. m 88.244 121.32 97.586 130.84 175.51 224.55 10 Vel. m/s 0.3205 1.1950 0.3393 1.1390 0.5813 0.7134 Att. 10 3 6.6646 32.032 6.4966 30.163 11.191 14.253 375 Table E.20: INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 6 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 16.736 12.817 15.795 11.646 27.817 10.743 1 Vel. m/s 0.1507 0.7932 0.0747 0.7985 0.1204 0.0606 Att. 10 3 5.5260 23.231 3.4075 23.336 4.8853 2.1038 .001 Pos. m 27.762 17.487 28.071 16.522 50.143 19.522 5 Vel. m/s 0.1597 0.8131 0.0956 0.8088 0.1578 0.0814 Att. 10 3 5.7841 24.086 4.0349 23.907 6.0346 3.1310 .001 Pos. m 34.850 20.555 36.120 19.789 64.642 25.399 10 Vel. m/s 0.1630 0.8266 0.1109 0.8145 0.1859 0.0926 Att. 10 3 5.7427 24.532 4.2648 24.113 6.2382 3.5023 .01 Pos. m 30.791 37.114 32.988 37.826 58.910 57.459 1 Vel. m/s 0.1341 0.8678 0.1011 0.8449 0.1597 0.1522 Att. 10 3 4.3583 25.368 3.4604 24.846 4.7248 4.7618 .01 Pos. m 48.175 55.893 52.358 58.402 93.832 94.657 5 Vel. m/s 0.1744 0.9180 0.1472 0.8837 0.2413 0.2300 Att. 10 3 5.5147 26.683 4.6128 25.968 6.8237 6.7813 .01 Pos. m 58.321 67.103 63.606 70.744 114.09 116.99 10 Vel. m/s 0.1971 0.9460 0.1738 0.9070 0.2891 0.2790 Att. 10 3 6.0139 27.524 5.1380 26.720 7.7761 8.1360 .1 Pos. m 90.808 122.70 99.781 132.15 179.53 227.86 1 Vel. m/s 0.2657 1.0718 0.2551 1.0193 0.4308 0.5219 Att. 10 3 7.5158 31.939 6.6880 30.654 11.441 15.149 .1 Pos. m 89.802 166.73 98.804 180.82 177.73 315.34 5 Vel. m/s 0.2597 1.1657 0.2557 1.1131 0.4297 0.7051 Att. 10 3 6.8928 35.035 6.3846 33.646 10.840 20.532 .1 Pos. m 84.666 180.80 93.196 196.29 167.60 342.94 10 Vel. m/s 0.2499 1.1996 0.2496 1.1481 0.4171 0.7629 Att. 10 3 6.1561 35.914 5.8084 34.522 9.7851 22.109 376 Table E.21: INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 7 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 17.857 13.173 17.279 12.152 30.535 11.657 1 Vel. m/s 0.1582 0.8009 0.0815 0.8010 0.1328 0.0658 Att. 10 3 5.3710 23.212 3.4249 23.407 5.0142 2.2169 .001 Pos. m 30.343 18.481 31.280 17.788 55.909 21.800 5 Vel. m/s 0.1744 0.8235 0.1121 0.8139 0.1878 0.0920 Att. 10 3 5.4181 24.008 3.8370 23.915 5.7317 3.1305 .001 Pos. m 38.831 21.978 41.050 21.551 73.584 28.572 10 Vel. m/s 0.1809 0.8397 0.1361 0.8217 0.2304 0.1075 Att. 10 3 5.1474 24.483 4.0197 24.106 6.1252 3.4762 .01 Pos. m 33.584 39.647 36.310 40.899 64.936 63.044 1 Vel. m/s 0.1514 0.8903 0.1219 0.8623 0.1985 0.1888 Att. 10 3 4.3859 25.354 3.4727 24.957 4.9292 4.9571 .01 Pos. m 50.680 59.498 55.327 62.732 99.201 102.57 5 Vel. m/s 0.1975 0.9501 0.1743 0.9115 0.2903 0.2912 Att. 10 3 5.5170 26.881 4.5459 26.269 6.8538 7.3190 .01 Pos. m 60.548 71.466 66.332 75.954 118.99 126.52 10 Vel. m/s 0.2215 0.9833 0.2055 0.9407 0.3466 0.3547 Att. 10 3 6.1351 27.847 5.2935 27.128 8.1956 8.8647 .1 Pos. m 82.158 124.57 90.409 134.71 162.65 232.54 1 Vel. m/s 0.2727 1.1192 0.2668 1.0679 0.4533 0.6295 Att. 10 3 7.1033 32.266 6.2155 30.964 10.633 15.707 .1 Pos. m 75.095 158.09 82.684 171.70 148.69 298.79 5 Vel. m/s 0.2520 1.2072 0.2515 1.1576 0.4228 0.7945 Att. 10 3 6.0438 34.694 5.5046 33.233 9.2896 19.790 .1 Pos. m 70.082 168.51 77.204 183.11 138.77 319.00 10 Vel. m/s 0.2397 1.2406 0.2429 1.1926 0.4062 0.8464 Att. 10 3 5.5213 35.355 5.1550 33.877 8.6051 20.949 377 Table E.22: INS/GGI: Strapdown GGI, Nav. Grade IMUs, Low Var., Mach 8 L Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. .001 Pos. m 18.948 13.630 18.783 12.758 33.302 12.747 1 Vel. m/s 0.1662 0.8087 0.0916 0.8044 0.1511 0.0728 Att. 10 3 5.0291 23.172 3.1762 23.456 4.7050 2.2858 .001 Pos. m 34.199 19.541 35.953 19.108 64.368 24.178 5 Vel. m/s 0.1949 0.8348 0.1389 0.8209 0.2360 0.1063 Att. 10 3 5.2052 23.933 3.7671 23.920 5.6605 3.1211 .001 Pos. m 41.521 23.393 44.352 23.283 79.512 31.693 10 Vel. m/s 0.2018 0.8537 0.1625 0.8310 0.2786 0.1269 Att. 10 3 5.1076 24.458 4.1260 24.115 6.2961 3.4723 .01 Pos. m 35.201 41.704 38.255 43.401 68.468 67.616 1 Vel. m/s 0.1685 0.9128 0.1419 0.8810 0.2335 0.2290 Att. 10 3 4.4146 25.392 3.4656 25.089 5.1002 5.1848 .01 Pos. m 53.276 62.397 58.394 66.225 104.72 108.98 5 Vel. m/s 0.2217 0.9813 0.2049 0.9398 0.3467 0.3551 Att. 10 3 5.7894 27.100 4.7768 26.549 7.3772 7.8135 .01 Pos. m 62.383 74.728 68.537 79.877 122.99 133.72 10 Vel. m/s 0.2482 1.0187 0.2381 0.9741 0.4062 0.4310 Att. 10 3 6.2644 28.147 5.4253 27.464 8.6084 9.4613 .1 Pos. m 71.488 122.86 78.717 133.15 141.60 229.75 1 Vel. m/s 0.2664 1.1578 0.2629 1.1083 0.4454 0.7168 Att. 10 3 6.3558 32.340 5.4502 30.981 9.2523 15.735 .1 Pos. m 61.617 148.17 67.863 161.01 121.98 279.42 5 Vel. m/s 0.2389 1.2399 0.2409 1.1930 0.4037 0.8612 Att. 10 3 5.3044 34.249 4.7899 32.707 7.9991 18.841 .1 Pos. m 62.495 156.54 68.916 170.15 123.83 295.48 10 Vel. m/s 0.2423 1.2741 0.2490 1.2292 0.4180 0.9109 Att. 10 3 5.1465 34.808 4.8173 33.251 7.9939 19.819 378 E.3 Global Positioning System Aided Navigation The baseline hypersonic INS/GPS Monte Carlo simulation results are summa- rized in Tables E.23{E.34. Tables E.23{E.25 list the MRSE results for the Mach 6, 7, and 8 cases with navigation grade IMUs and GPS pseudorange, , and pseudorange rate, _ , updates at intervals of 1 to 300 seconds. Tables E.26{E.28 on pg. 383-385 summarize the tactical grade IMU, and _ results as GPS updates are simulated every 1{300 seconds. Tables E.29{E.34 list the pseudorange-only INS/GPS results for the hypersonic Monte Carlo simulations. The rst half of the tables, on pg. 386{388, are performed with the simulated navigation grade IMU speci cations. And Tables E.32{E.34 on pg. 389{391 list the tactical grade IMU, pseudorange-only INS simulations for the Mach 6, 7, and 8 cases. All INS/GPS simulations are performed over the high gravitational gradient variation trajectories since the low variation trajectories produce essentially the same GPS visibility and geometric dilution of precision (GDOP) values as the high trajectories, as shown in Sec. D.3. Therefore, the INS/GPS navigation performance should be nearly identical between the two trajectories. 379 Table E.23: INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 6 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 1.0153 0.5806 1.0771 0.6039 0.1663 0.1357 1 Vel. m/s 0.1927 0.2094 0.2074 0.2259 0.0032 0.0029 Att. 10 3 3.5271 41.958 3.0019 40.874 5.3976 5.3941 Pos. m 1.2715 0.7988 1.2421 0.7429 0.4642 0.3948 10 Vel. m/s 0.2113 0.2222 0.2175 0.2287 0.0072 0.0065 Att. 10 3 4.5543 43.681 4.2191 41.671 7.5889 7.5228 Pos. m 1.6828 1.3727 1.3366 0.9194 0.9005 0.6841 30 Vel. m/s 0.2223 0.2472 0.2120 0.2308 0.0122 0.0102 Att. 10 3 5.5208 46.314 5.4406 42.562 9.7880 9.8346 Pos. m 2.4643 2.5036 1.3077 1.0611 1.1323 0.9879 60 Vel. m/s 0.2375 0.2713 0.2128 0.2327 0.0151 0.0143 Att. 10 3 6.0396 47.918 6.6964 43.683 12.048 12.652 Pos. m 2.8265 2.6694 1.6654 1.2651 1.5420 1.3510 90 Vel. m/s 0.2465 0.2706 0.2213 0.2325 0.0200 0.0189 Att. 10 3 7.1277 48.894 7.9054 44.791 14.225 15.355 Pos. m 3.1126 3.0220 1.9793 1.6316 2.0355 2.0056 120 Vel. m/s 0.2494 0.2793 0.2244 0.2413 0.0248 0.0285 Att. 10 3 5.6479 51.756 6.2606 48.050 11.264 22.974 Pos. m 11.106 12.080 10.916 11.746 18.190 20.310 180 Vel. m/s 0.3209 0.3696 0.3052 0.3414 0.1851 0.2174 Att. 10 3 4.5634 61.997 5.0503 59.655 9.0826 46.199 Pos. m 23.191 23.167 24.354 24.053 42.451 42.454 240 Vel. m/s 0.4246 0.4606 0.4203 0.4427 0.3931 0.4001 Att. 10 3 5.8967 64.220 6.5311 62.127 11.747 50.222 Pos. m 31.562 42.041 33.561 45.032 59.069 80.218 300 Vel. m/s 0.4825 0.5798 0.4822 0.5754 0.4828 0.6434 Att. 10 3 5.5397 65.586 6.1345 63.591 11.033 51.624 380 Table E.24: INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 7 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 0.8455 0.5903 0.8878 0.6121 0.1574 0.1398 1 Vel. m/s 0.1983 0.2137 0.2133 0.2300 0.0033 0.0031 Att. 10 3 3.8905 42.456 3.3065 41.081 5.9456 5.8611 Pos. m 1.3752 0.8313 1.3505 0.7700 0.5407 0.4261 10 Vel. m/s 0.2039 0.2237 0.2076 0.2283 0.0090 0.0076 Att. 10 3 5.4510 44.839 5.1901 42.500 9.3365 9.5509 Pos. m 1.5499 1.3799 1.1710 0.9195 0.8136 0.7112 30 Vel. m/s 0.2244 0.2480 0.2124 0.2302 0.0130 0.0119 Att. 10 3 6.5897 47.657 6.8312 43.927 12.290 13.102 Pos. m 2.4149 2.2778 1.4855 1.1235 1.2897 1.0805 60 Vel. m/s 0.2342 0.2650 0.2103 0.2298 0.0189 0.0171 Att. 10 3 7.4678 49.486 8.2850 45.462 14.908 16.781 Pos. m 2.9581 2.6011 1.9840 1.4404 1.8732 1.6322 90 Vel. m/s 0.2573 0.2777 0.2316 0.2422 0.0255 0.0249 Att. 10 3 7.7110 51.502 8.5540 47.755 15.391 22.059 Pos. m 3.0919 2.8461 2.1925 1.7505 2.3539 2.2219 120 Vel. m/s 0.2482 0.2752 0.2239 0.2408 0.0300 0.0347 Att. 10 3 5.4358 55.547 6.0259 52.369 10.840 32.277 Pos. m 14.377 14.707 14.725 14.959 25.043 26.091 180 Vel. m/s 0.3767 0.4082 0.3659 0.3880 0.2840 0.3053 Att. 10 3 5.4406 64.566 6.0259 62.500 10.837 50.581 Pos. m 26.145 25.994 27.859 27.499 48.990 48.660 240 Vel. m/s 0.4703 0.4977 0.4720 0.4878 0.4897 0.4863 Att. 10 3 6.6106 65.302 7.3276 63.290 13.180 51.365 Pos. m 29.752 64.824 31.769 70.655 55.436 126.34 300 Vel. m/s 0.4479 0.7566 0.4454 0.7759 0.4270 1.0132 Att. 10 3 4.7688 69.240 5.2797 67.513 9.4946 55.742 381 Table E.25: INS/GPS: & _ Updates, Nav. Grade IMUs, Mach 8 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 0.9622 0.5905 1.0141 0.6103 0.1789 0.1487 1 Vel. m/s 0.1968 0.2104 0.2108 0.2256 0.0038 0.0035 Att. 10 3 4.4477 43.108 3.8104 41.461 6.8537 6.7323 Pos. m 1.1600 0.8466 1.1092 0.7793 0.5429 0.4498 10 Vel. m/s 0.2114 0.2272 0.2148 0.2301 0.0097 0.0087 Att. 10 3 6.7862 46.227 6.6718 43.646 12.005 12.208 Pos. m 1.9290 1.4572 1.5304 0.9671 0.9382 0.7807 30 Vel. m/s 0.2315 0.2504 0.2165 0.2303 0.0158 0.0140 Att. 10 3 7.8858 49.351 8.4014 45.692 15.119 17.080 Pos. m 2.5532 2.2038 1.7487 1.2356 1.4404 1.2615 60 Vel. m/s 0.2454 0.2686 0.2206 0.2354 0.0222 0.0214 Att. 10 3 9.1541 51.822 10.158 48.112 18.282 22.583 Pos. m 2.8408 2.4777 2.0405 1.5070 1.9100 1.7485 90 Vel. m/s 0.2515 0.2789 0.2268 0.2455 0.0286 0.0294 Att. 10 3 7.2581 54.453 8.0517 51.108 14.489 29.156 Pos. m 5.1623 5.4951 4.6559 4.8938 7.1414 7.9608 120 Vel. m/s 0.2785 0.3165 0.2585 0.2881 0.0998 0.1173 Att. 10 3 4.4391 59.660 4.9161 56.974 8.8425 40.150 Pos. m 16.510 16.777 17.233 17.465 29.480 30.614 180 Vel. m/s 0.4178 0.4437 0.4113 0.4302 0.3642 0.3831 Att. 10 3 6.7129 65.682 7.4417 63.686 13.388 51.515 Pos. m 22.716 36.251 24.158 39.086 42.094 69.547 240 Vel. m/s 0.4401 0.6058 0.4381 0.6099 0.4248 0.7061 Att. 10 3 5.8799 68.180 6.5165 66.368 11.722 54.344 Pos. m 32.998 85.686 35.535 94.040 62.54 168.49 300 Vel. m/s 0.4395 0.9194 0.4357 0.9598 0.4049 1.3491 Att. 10 3 3.7842 73.128 4.1873 71.742 7.5293 61.532 382 Table E.26: INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 6 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 1.1096 0.6226 1.1786 0.6472 0.2357 0.2116 1 Vel. m/s 0.2057 0.2196 0.2207 0.2361 0.0130 0.0127 Att. 10 3 58.597 98.863 63.113 102.35 113.49 124.20 Pos. m 1.4541 0.8889 1.4315 0.8332 0.6415 0.5557 10 Vel. m/s 0.2182 0.2332 0.2227 0.2385 0.0220 0.0214 Att. 10 3 58.819 99.965 63.534 103.22 114.25 125.83 Pos. m 1.8339 1.5308 1.4421 1.0466 1.0820 0.9521 30 Vel. m/s 0.2380 0.2600 0.2224 0.2398 0.0316 0.0304 Att. 10 3 59.788 101.18 64.886 103.82 116.68 126.97 Pos. m 2.8754 2.8610 1.6789 1.3774 1.6798 1.5718 60 Vel. m/s 0.2575 0.2877 0.2274 0.2446 0.0453 0.0447 Att. 10 3 59.393 102.18 64.760 104.56 116.46 128.37 Pos. m 3.5836 3.4358 2.4028 1.9909 2.8881 2.6856 90 Vel. m/s 0.2745 0.3019 0.2447 0.2595 0.0680 0.0666 Att. 10 3 59.042 102.98 64.373 105.47 115.76 130.08 Pos. m 7.5166 7.5455 6.8036 6.6463 10.945 11.152 120 Vel. m/s 0.3491 0.3749 0.3268 0.3411 0.2208 0.2262 Att. 10 3 61.312 105.51 66.868 108.18 120.25 133.84 Pos. m 31.672 31.541 33.758 33.299 59.538 59.114 180 Vel. m/s 0.6227 0.6566 0.6355 0.6551 0.7996 0.7968 Att. 10 3 65.757 110.82 71.701 113.89 128.94 141.69 Pos. m 53.853 50.643 58.373 54.497 103.74 97.221 240 Vel. m/s 0.8192 0.8214 0.8535 0.8381 1.1879 1.1220 Att. 10 3 70.645 112.98 77.071 116.21 138.60 144.95 Pos. m 79.024 83.950 86.273 91.462 153.84 163.74 300 Vel. m/s 0.9895 1.0497 1.0406 1.0904 1.5111 1.5624 Att. 10 3 71.162 114.82 77.631 118.20 139.61 147.89 383 Table E.27: INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 7 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 0.8448 0.6190 0.8864 0.6419 0.2273 0.2090 1 Vel. m/s 0.2051 0.2153 0.2195 0.2307 0.0129 0.0127 Att. 10 3 49.379 94.176 52.828 97.010 94.972 113.08 Pos. m 1.2727 0.8790 1.2359 0.8162 0.6082 0.5476 10 Vel. m/s 0.2143 0.2328 0.2170 0.2361 0.0220 0.0214 Att. 10 3 51.267 95.147 55.186 97.660 99.220 114.31 Pos. m 1.8464 1.5253 1.4645 1.0484 1.0640 0.9437 30 Vel. m/s 0.2338 0.2616 0.2181 0.2408 0.0317 0.0303 Att. 10 3 52.478 96.194 56.933 98.105 102.36 115.15 Pos. m 2.9395 2.6398 1.9315 1.4369 1.9255 1.6422 60 Vel. m/s 0.2665 0.2880 0.2366 0.2490 0.0492 0.0462 Att. 10 3 53.318 97.030 58.145 98.723 104.54 116.33 Pos. m 4.4633 4.2712 3.6240 3.2581 5.1615 5.0088 90 Vel. m/s 0.2981 0.3256 0.2724 0.2902 0.1213 0.1211 Att. 10 3 55.153 98.446 60.167 100.25 108.18 118.52 Pos. m 8.0701 8.0768 7.6707 7.5506 12.515 12.803 120 Vel. m/s 0.3752 0.3991 0.3581 0.3726 0.2738 0.2811 Att. 10 3 57.211 101.13 62.404 103.15 112.20 122.64 Pos. m 31.716 31.481 33.962 33.502 59.872 59.422 180 Vel. m/s 0.6650 0.6949 0.6823 0.7016 0.8758 0.8694 Att. 10 3 60.609 106.55 66.094 109.00 118.83 130.79 Pos. m 57.174 57.189 62.255 62.091 110.92 110.89 240 Vel. m/s 0.8701 0.8962 0.9099 0.9262 1.2780 1.2801 Att. 10 3 63.780 109.24 69.620 111.91 125.18 135.23 Pos. m 78.038 96.760 85.400 106.03 152.13 189.96 300 Vel. m/s 1.0128 1.1543 1.0684 1.2122 1.5615 1.7881 Att. 10 3 63.888 110.70 69.728 113.51 125.37 137.68 384 Table E.28: INS/GPS: & _ Updates, Tac. Grade IMUs, Mach 8 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 1.0085 0.6247 1.0643 0.6462 0.2400 0.2145 1 Vel. m/s 0.2000 0.2160 0.2132 0.2308 0.0130 0.0127 Att. 10 3 45.699 90.855 48.730 93.182 87.610 105.07 Pos. m 1.2069 0.9133 1.1569 0.8446 0.6479 0.5656 10 Vel. m/s 0.2170 0.2353 0.2186 0.2368 0.0227 0.0219 Att. 10 3 48.046 91.764 51.656 93.694 92.880 105.99 Pos. m 1.9185 1.5571 1.5177 1.0597 1.1516 0.9738 30 Vel. m/s 0.2358 0.2607 0.2184 0.2384 0.0332 0.0316 Att. 10 3 47.854 92.678 51.902 94.035 93.323 106.61 Pos. m 2.9568 2.5538 2.1535 1.5688 2.1568 1.8963 60 Vel. m/s 0.2609 0.2874 0.2340 0.2519 0.0568 0.0532 Att. 10 3 46.984 93.332 51.224 94.517 92.097 107.46 Pos. m 4.8598 4.7534 4.2319 4.0276 6.3613 6.4023 90 Vel. m/s 0.3241 0.3447 0.3021 0.3148 0.1637 0.1679 Att. 10 3 47.948 95.610 52.266 96.965 93.973 110.74 Pos. m 12.869 12.827 13.156 13.006 22.368 22.565 120 Vel. m/s 0.4536 0.4793 0.4473 0.4650 0.4362 0.4427 Att. 10 3 52.335 99.249 57.059 100.89 102.59 116.28 Pos. m 31.637 32.036 34.038 34.329 59.816 60.900 180 Vel. m/s 0.6769 0.7170 0.6973 0.7292 0.8950 0.9147 Att. 10 3 58.800 103.74 64.210 105.78 115.46 123.63 Pos. m 51.035 57.997 55.520 63.195 98.385 112.88 240 Vel. m/s 0.8548 0.9264 0.8920 0.9628 1.2266 1.3421 Att. 10 3 58.527 105.76 63.888 107.97 114.87 127.11 Pos. m 89.218 118.92 97.933 130.88 174.56 234.72 300 Vel. m/s 1.1044 1.3274 1.1693 1.4075 1.7276 2.1370 Att. 10 3 61.095 109.35 66.738 111.91 120.01 133.51 385 Table E.29: INS/GPS: Updates, Nav. Grade IMUs, Mach 6 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 5.7337 2.9529 5.9940 2.9678 1.2628 1.0801 1 Vel. m/s 0.0550 0.7373 0.0283 0.7786 0.0184 0.0175 Att. 10 3 8.5429 48.760 8.6979 45.839 15.652 17.935 Pos. m 6.9961 4.3938 6.8666 4.0566 3.4967 3.0628 10 Vel. m/s 0.0747 0.7764 0.0429 0.7915 0.0452 0.0428 Att. 10 3 11.746 55.214 12.588 52.394 22.654 32.738 Pos. m 9.0135 6.6522 8.7520 5.3034 6.2100 5.2876 30 Vel. m/s 0.0718 0.8354 0.0566 0.8039 0.0712 0.0661 Att. 10 3 11.261 58.193 12.480 55.419 22.461 39.118 Pos. m 10.070 10.867 9.8697 6.6449 8.5678 7.7020 60 Vel. m/s 0.0671 0.9110 0.0697 0.8158 0.0946 0.0885 Att. 10 3 10.182 59.675 11.301 57.091 20.339 42.503 Pos. m 11.880 11.895 11.811 7.8020 11.550 9.7868 90 Vel. m/s 0.0777 0.9195 0.0815 0.8253 0.1157 0.1065 Att. 10 3 8.7932 60.582 9.7577 58.121 17.559 44.546 Pos. m 12.780 13.097 12.770 9.1506 12.993 12.245 120 Vel. m/s 0.0852 0.9304 0.0898 0.8374 0.1308 0.1294 Att. 10 3 6.7070 61.278 7.4391 58.907 13.385 46.002 Pos. m 15.024 39.844 15.360 39.181 18.405 66.673 180 Vel. m/s 0.0938 1.1736 0.0991 1.1091 0.1461 0.6587 Att. 10 3 1.9838 66.883 2.1874 64.952 3.9310 52.433 Pos. m 17.499 87.837 18.049 92.682 22.807 163.05 240 Vel. m/s 0.1029 1.5607 0.1094 1.5430 0.1662 1.4802 Att. 10 3 1.0613 73.096 1.1632 71.708 2.0874 61.543 Pos. m 18.788 132.14 19.601 141.79 26.505 251.38 300 Vel. m/s 0.1143 1.8582 0.1222 1.8749 0.1906 2.0820 Att. 10 3 0.9091 76.269 0.9939 75.185 1.7829 66.917 386 Table E.30: INS/GPS: Updates, Nav. Grade IMUs, Mach 7 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 5.8780 3.0802 6.1105 3.0814 1.4060 1.1771 1 Vel. m/s 0.0607 0.7435 0.0309 0.7803 0.0225 0.0209 Att. 10 3 10.077 51.500 10.472 48.545 18.846 23.953 Pos. m 7.5515 4.6390 7.4026 4.2807 3.9170 3.4009 10 Vel. m/s 0.0809 0.7870 0.0469 0.7959 0.0533 0.0506 Att. 10 3 11.414 58.079 12.276 55.469 22.093 38.894 Pos. m 8.9762 6.7441 8.7676 5.3157 6.5078 5.4300 30 Vel. m/s 0.0754 0.8500 0.0642 0.8084 0.0842 0.0738 Att. 10 3 9.7306 60.325 10.777 57.827 19.395 43.674 Pos. m 10.696 10.257 10.500 6.7629 9.1676 7.9058 60 Vel. m/s 0.0726 0.9153 0.0761 0.8219 0.1068 0.0992 Att. 10 3 8.3040 61.522 9.2145 59.171 16.580 46.260 Pos. m 11.768 11.502 11.754 8.1620 11.880 10.523 90 Vel. m/s 0.0866 0.9281 0.0915 0.8362 0.1337 0.1263 Att. 10 3 6.6073 62.122 7.3284 59.842 13.185 47.372 Pos. m 12.517 12.513 12.633 9.2943 13.807 12.525 120 Vel. m/s 0.0942 0.9362 0.1001 0.8452 0.1499 0.1431 Att. 10 3 4.6282 62.793 5.1280 60.603 9.2241 48.869 Pos. m 14.746 49.744 15.037 50.993 17.627 87.982 180 Vel. m/s 0.0947 1.3228 0.1005 1.2777 0.1498 0.9802 Att. 10 3 1.3737 71.404 1.5122 69.863 2.7158 58.892 Pos. m 16.743 100.78 17.325 107.78 22.295 190.25 240 Vel. m/s 0.1037 1.7541 0.1105 1.7606 0.1686 1.8798 Att. 10 3 0.9397 78.138 1.0295 77.243 1.8465 70.280 Pos. m 16.041 130.31 16.582 140.24 21.235 248.31 300 Vel. m/s 0.1002 1.8757 0.1067 1.8937 0.1628 2.0901 Att. 10 3 0.7743 78.392 0.8452 77.523 1.5148 70.751 387 Table E.31: INS/GPS: Updates, Nav. Grade IMUs, Mach 8 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 5.4744 3.0245 5.6716 3.0270 1.3097 1.1960 1 Vel. m/s 0.0615 0.7475 0.0300 0.7816 0.0227 0.0230 Att. 10 3 11.075 54.109 11.620 51.247 20.914 29.566 Pos. m 7.4604 4.6138 7.3223 4.2442 3.9691 3.4374 10 Vel. m/s 0.0849 0.7951 0.0505 0.7989 0.0591 0.0554 Att. 10 3 10.051 60.153 10.799 57.719 19.437 43.045 Pos. m 8.4087 7.1502 8.2802 5.6041 6.9087 5.8526 30 Vel. m/s 0.0760 0.8676 0.0697 0.8147 0.0947 0.0848 Att. 10 3 7.7684 61.939 8.5997 59.631 15.476 46.718 Pos. m 10.739 10.128 10.679 7.2131 10.395 8.7704 60 Vel. m/s 0.0844 0.9238 0.0893 0.8322 0.1307 0.1179 Att. 10 3 6.4071 62.810 7.1061 60.596 12.787 48.352 Pos. m 12.428 11.411 12.452 8.6520 12.854 11.310 90 Vel. m/s 0.0999 0.9378 0.1065 0.8479 0.1616 0.1478 Att. 10 3 5.4215 63.282 6.0101 61.114 10.813 49.081 Pos. m 13.734 18.903 13.899 17.039 15.428 26.525 120 Vel. m/s 0.1013 1.0309 0.1079 0.9516 0.1639 0.3476 Att. 10 3 3.5153 65.912 3.8913 63.921 6.9989 51.541 Pos. m 14.781 56.458 15.156 59.002 18.387 102.41 180 Vel. m/s 0.0956 1.4501 0.1015 1.4210 0.1517 1.2464 Att. 10 3 1.2162 75.676 1.3381 74.534 2.4027 65.857 Pos. m 16.037 92.621 16.362 98.999 19.229 174.17 240 Vel. m/s 0.0974 1.7381 0.1036 1.7417 0.1558 1.8198 Att. 10 3 0.8875 79.727 0.9721 78.997 1.7437 73.227 Pos. m 15.870 146.70 16.209 158.94 19.259 282.02 300 Vel. m/s 0.0950 1.9308 0.1009 1.9545 0.1515 2.1790 Att. 10 3 0.6715 80.379 0.7320 79.718 1.3110 74.468 388 Table E.32: INS/GPS: Updates, Tac. Grade IMUs, Mach 6 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 6.1444 3.1146 6.4395 3.1375 1.5785 1.3797 1 Vel. m/s 0.0686 0.7505 0.0406 0.7904 0.0410 0.0395 Att. 10 3 62.797 101.82 68.052 104.80 122.38 128.86 Pos. m 8.2561 4.8787 8.1887 4.5618 4.8311 3.9098 10 Vel. m/s 0.1030 0.7989 0.0692 0.8123 0.0932 0.0824 Att. 10 3 63.846 103.53 69.427 106.34 124.86 131.67 Pos. m 9.6738 7.2846 9.4505 5.9720 7.4679 6.5207 30 Vel. m/s 0.1157 0.8672 0.0961 0.8343 0.1417 0.1251 Att. 10 3 62.667 104.40 68.388 107.05 122.99 132.84 Pos. m 12.057 12.015 11.835 7.9115 11.373 10.045 60 Vel. m/s 0.1387 0.9558 0.1274 0.8630 0.1981 0.1814 Att. 10 3 61.470 105.02 67.058 107.71 120.59 133.77 Pos. m 14.147 14.070 14.140 10.227 15.503 14.184 90 Vel. m/s 0.1719 0.9867 0.1637 0.8975 0.2630 0.2495 Att. 10 3 62.182 105.74 67.832 108.47 121.98 134.75 Pos. m 19.468 20.116 20.134 17.015 26.984 26.531 120 Vel. m/s 0.2616 1.0839 0.2642 1.0062 0.4443 0.4605 Att. 10 3 68.238 109.02 74.466 111.97 133.92 139.11 Pos. m 63.737 83.014 69.541 86.989 117.59 152.62 180 Vel. m/s 0.7959 1.7888 0.8576 1.7928 1.5118 1.9036 Att. 10 3 75.996 123.86 82.949 128.04 149.17 163.18 Pos. m 118.44 154.82 130.21 166.68 225.95 295.97 240 Vel. m/s 1.2884 2.4213 1.4044 2.4955 2.4943 3.1559 Att. 10 3 85.052 132.23 92.955 137.23 167.18 178.59 Pos. m 189.50 241.57 209.15 263.08 367.88 469.52 300 Vel. m/s 1.8557 3.0311 2.0348 3.1736 3.6318 4.3763 Att. 10 3 94.237 140.22 103.11 146.06 185.46 193.93 389 Table E.33: INS/GPS: Updates, Tac. Grade IMUs, Mach 7 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 6.4903 3.1435 6.7685 3.1578 1.6236 1.4001 1 Vel. m/s 0.0737 0.7550 0.0421 0.7912 0.0430 0.0411 Att. 10 3 52.241 96.691 56.462 98.835 101.51 116.56 Pos. m 8.2089 4.8668 8.1591 4.5423 4.8222 3.9436 10 Vel. m/s 0.1098 0.8080 0.0757 0.8168 0.1050 0.0909 Att. 10 3 55.871 97.965 60.713 99.925 109.16 118.43 Pos. m 10.186 7.6940 10.046 6.3402 8.2014 7.1209 30 Vel. m/s 0.1240 0.8879 0.1088 0.8468 0.1659 0.1490 Att. 10 3 53.376 98.784 58.185 100.66 104.61 119.47 Pos. m 12.129 11.778 12.058 8.4607 12.403 10.988 60 Vel. m/s 0.1532 0.9708 0.1469 0.8818 0.2331 0.2168 Att. 10 3 55.166 99.552 60.160 101.47 108.16 120.51 Pos. m 15.173 15.034 15.516 12.096 19.259 17.571 90 Vel. m/s 0.2107 1.0263 0.2113 0.9437 0.3485 0.3347 Att. 10 3 56.485 101.30 61.602 103.34 110.76 122.89 Pos. m 20.581 20.547 21.493 18.289 29.872 28.842 120 Vel. m/s 0.3058 1.1246 0.3168 1.0536 0.5385 0.5486 Att. 10 3 60.353 105.50 65.813 107.85 118.33 129.11 Pos. m 65.260 84.052 71.076 88.821 118.68 155.76 180 Vel. m/s 0.8604 1.8885 0.9319 1.9045 1.6439 2.0863 Att. 10 3 71.762 120.57 78.416 124.32 141.01 155.38 Pos. m 133.05 173.27 146.43 187.94 254.47 334.23 240 Vel. m/s 1.4506 2.6366 1.5884 2.7361 2.8272 3.5799 Att. 10 3 81.341 130.68 89.037 135.49 160.12 174.78 Pos. m 125.72 189.31 138.37 205.70 240.63 366.09 300 Vel. m/s 1.4457 2.6940 1.5824 2.7992 2.8152 3.6867 Att. 10 3 80.152 130.08 87.724 134.83 157.76 173.61 390 Table E.34: INS/GPS: Updates, Tac. Grade IMUs, Mach 8 Settling Distance: 0 km 100 km 500 km t State Units Error Cov. Error Cov. Error Cov. Pos. m 5.8145 3.1507 6.0425 3.1574 1.6854 1.4473 1 Vel. m/s 0.0758 0.7596 0.0439 0.7927 0.0467 0.0435 Att. 10 3 48.233 92.998 52.090 94.511 93.657 107.48 Pos. m 8.5636 4.9282 8.4419 4.5952 4.7636 4.0728 10 Vel. m/s 0.1161 0.8187 0.0815 0.8230 0.1148 0.1020 Att. 10 3 49.537 94.103 53.826 95.458 96.778 108.90 Pos. m 9.7680 7.8781 9.6873 6.4150 8.4260 7.3441 30 Vel. m/s 0.1301 0.9080 0.1203 0.8574 0.1856 0.1680 Att. 10 3 50.373 95.009 54.931 96.339 98.771 110.05 Pos. m 13.291 11.998 13.338 9.2909 14.277 12.469 60 Vel. m/s 0.1788 0.9936 0.1783 0.9085 0.2898 0.2645 Att. 10 3 50.010 96.389 54.510 97.812 98.006 112.00 Pos. m 15.867 15.156 16.248 12.812 20.017 18.872 90 Vel. m/s 0.2371 1.0539 0.2434 0.9758 0.4060 0.3918 Att. 10 3 53.753 98.838 58.626 100.45 105.41 115.64 Pos. m 26.862 29.905 28.437 29.216 41.732 48.458 120 Vel. m/s 0.4015 1.2765 0.4254 1.2239 0.7347 0.8485 Att. 10 3 57.305 105.31 62.547 107.48 112.46 126.30 Pos. m 65.493 86.432 71.395 92.009 119.38 161.56 180 Vel. m/s 0.8905 1.9625 0.9686 1.9871 1.7118 2.2229 Att. 10 3 67.304 117.58 73.609 120.98 132.37 148.90 Pos. m 92.039 135.98 100.91 147.03 172.71 260.57 240 Vel. m/s 1.1728 2.3885 1.2817 2.4605 2.2738 3.0722 Att. 10 3 71.857 123.08 78.670 127.07 141.48 159.58 Pos. m 105.18 185.25 115.63 201.73 200.01 359.01 300 Vel. m/s 1.2368 2.5613 1.3534 2.6521 2.4029 3.4112 Att. 10 3 70.362 123.12 77.023 127.11 138.52 159.66 391 BIBLIOGRAPHY [1] Jekeli, C., Inertial Navigation Systems with Geodetic Applications, Walter de Gruyter, New York, 2001. 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