ABSTRACT
Title of dissertation: NONLINEAR CONTROL DESIGN
TECHNIQUES FOR PRECISION
FORMATION FLYING
AT LAGRANGE POINTS
Richard J. Luquette, Doctor of Philosophy, 2006
Dissertation directed by: Associate Professor Robert M. Sanner
Department of Aerospace Engineering
Precision spacecraft formation flying is an enabling technology for a variety
of proposed space-based observatories, such as NASA?s Terrestrial Planet Finder
(TPF), the Micro-Arcsecond X-Ray Imaging Mission (MAXIM), and Stellar
Imager (SI). This research specifically examines the precision formation flying
control architecture, characterizing the relative performance of linear and nonlinear
controllers. Controller design is based on a 6DOF control architecture, characteristic
of precision formation flying control. In an effort to minimize the influence of
design parameters in the comparison, analysis employs ?equivalent? controller
gains, and incorporates an integrator in the linear control design. Controller
performance is evaluated through various simulations designed to reflect a realistic
space environment. The simulation architecture includes a full gravitational model
and solar pressure effects. Spacecraft model properties are based on realistic mission
design parameters. Control actuators are modeled as a fixed set of thrusters for both
translation and attitude control. Analysis includes impact on controller performance
due to omitted dynamics in the model (gravitational sources and solar pressure) and
model uncertainty (mass properties, thruster placement and thruster alignment).
Linearized equations of relative motion are derived for spacecraft operating
in the context of the Restricted Three Body Problem. Linearization is performed
with respect to a reference spacecraft within the formation. Analysis demonstrates
robust stability for the Linear Quadratic Regulator controller design based on the
linearized dynamics.
Nonlinear controllers are developed based on Lyapunov analysis, including
both non-adaptive and adaptive designs. While the linear controller demonstrates
greater robustness to model uncertainty, both nonlinear controllers exhibit superior
performance. The adaptive controller provides the best performance. As a key
feature, the adaptive controller design requires only relative navigation knowledge.
Analysis demonstrates the ability of the nonlinear controller to compensate
for unknown dynamics and model uncertainty. Results exhibit the potential of a
nonlinear adaptive architecture for improving controller performance. Nonlinear
adaptive control is a viable strategy for meeting the extreme control requirements
associated with formation flying missions like MAXIM and Stellar Imager. Mission
specific analysis from a systems perspective is required to determine the best
controller design.
NONLINEAR CONTROL DESIGN TECHNIQUES FOR
PRECISION FORMATION FLYING AT LAGRANGE POINTS
by
Richard J. Luquette
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2006
Advisory Committee:
Associate Professor Robert M. Sanner, Chair/Advisor
Professor Darryll J. Pines
Professor Norman M. Wereley
Professor Emeritus William W. Adams
Dr. F. Landis Markley
Dedicated In Memory Of:
Parents
Norman A. and Helen J. Luquette
Brothers
Norman A. Luquette, Jr.
Robert A. Luquette
William E. Luquette
Brother-in-Law
Lawrence Kime
Father-in-Law
Edward Baclawski
ii
Acknowledgments
While the contributions of this research reflect many personally invested hours
in analysis and study, great credit is attributed to those who supported me. As I
reflect on the experience, a number of people emerge as key players, deserving of
my personal recognition and gratitude.
I thank my advisor, Professor Rob Sanner for his persistent encouragement and
guidance. I thank each member of my dissertation committee, Professor Darryll
Pines, Professor Norman Wereley, Professor William Adams, and Dr. F. Landis
Markley of NASA/Goddard Space Flight Center.
I owe gratitude to many of my colleagues at NASA?s Goddard Space Flight
Center. Notably, Dr. Jesse Leitner, both colleague and friend, supported me as an
encourager and enabler. He also served as a technical advisor and reviewer, and
contributed his knowledge and experience with formation flying technology to help
ensure practical relevance of this research. John VanEepoel spent careful hours
supporting editorial review.
My involvement in youth ministry through my local church and the nation-
wide Spoke Folk Ministries blessed me with an extensive network of people, who
served both to encourage and inspire. The names are too many to list, but I?m
indebted to each member of this community.
Family was another important contributor to this accomplishment. Most
importantly my parents served a critical role throughout my life as nurturers,
supporters and encouragers. Unfortunately, the were only able to witness the
iii
beginnings of this endeavor. Their deaths prevent them from joining the graduation
celebration.
My colleague, fellow graduate student and friend, Julie Thienel (more formally
Dr. Julie Thienel!!!) has been central throughout my graduate experience. We
endured hours of classroom lectures, challenging assignments, and seemingly endless
research. We served and encouraged each other through long years of study, and
emerged as everlasting friends.
My greatest debt is owed to my wife, Debby, for her continual support,
encouragement and tolerance through our years of marriage.
As a closing, I consider my life threaded with constants and variables. The
variables are people and a changing environment. Certain people have an enduring
presence in my life, others come and go. However, people change through life. The
principal constant in my life is my faith. Therefore, I end with a prayer of thanks
to the God who equipped me with the academic skills and persistence required to
complete this effort, and who provided my supporters, highlighted above.
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Table of Contents
List of Tables ix
List of Figures xi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Motivation...Looking to the Stars . . . . . . . . . . . . . . . . 1
1.1.2 Mechanism...Spacecraft Precision Formation Flying . . . . . . 2
1.1.3 Context...Libration Point Missions . . . . . . . . . . . . . . . 3
1.1.4 Focus...Formation Control . . . . . . . . . . . . . . . . . . . . 6
1.2 Research Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 A Benchmark Problem . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Generalized Architecture . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3.1 Assertion: . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3.2 Analysis: . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Precision Formation Flying Missions . . . . . . . . . . . . . . . . . . 11
1.4 Formation Flying - Current Technology Assessment . . . . . . . . . . 14
1.4.1 Current Control Technology . . . . . . . . . . . . . . . . . . . 15
1.5 Contributions and Conclusions . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 Linearized Relative Dynamics . . . . . . . . . . . . . . . . . . 21
1.5.3 6DOF Control Design . . . . . . . . . . . . . . . . . . . . . . 22
1.5.3.1 Linear Control Design . . . . . . . . . . . . . . . . . 23
1.5.3.2 Nonlinear Control Design . . . . . . . . . . . . . . . 24
1.5.3.3 Adaptation . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.3.4 Implementation Issues . . . . . . . . . . . . . . . . . 25
1.5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Synopsis of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6.1 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Spacecraft Dynamics 30
2.1 Orbital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Restricted Three Body Problem . . . . . . . . . . . . . . . . . 31
2.1.1.1 Libration Points . . . . . . . . . . . . . . . . . . . . 34
2.1.1.2 Stability of Libration Points . . . . . . . . . . . . . . 35
2.1.2 Dynamics of Relative Motion in the Restricted Three Body
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.2.1 Dynamics in Rotational Frame/Validation . . . . . . 42
2.1.2.2 Linearization with Leader at a Collinear Libration
Point . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.2.3 Linearization with a Single Primary Mass . . . . . . 46
v
2.1.3 Dynamics of the Earth/Moon-Sun System . . . . . . . . . . . 47
2.1.3.1 Unperturbed Model . . . . . . . . . . . . . . . . . . 48
2.1.3.2 Perturbed Model . . . . . . . . . . . . . . . . . . . . 49
2.2 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.1 Dynamics, rigid body - no wheels . . . . . . . . . . . . . . . . 53
2.2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.3 Attitude Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.4 Linearized Equations of Motion . . . . . . . . . . . . . . . . . 54
2.3 Coupled Orbit and Attitude Dynamics . . . . . . . . . . . . . . . . . 55
3 Linear and Nonlinear Nonadaptive Control Theory and Design 58
3.1 Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1.1 State-Space Model . . . . . . . . . . . . . . . . . . . 59
3.1.1.2 Linear System Control Theory . . . . . . . . . . . . 60
3.1.2 Lyapunov Based Nonlinear Control . . . . . . . . . . . . . . . 62
3.1.2.1 Dynamics Model . . . . . . . . . . . . . . . . . . . . 62
3.1.2.2 Control Design Theory . . . . . . . . . . . . . . . . . 63
3.2 Formation Flying at L2, Linear Control Design . . . . . . . . . . . . . 65
3.2.1 Linearized Dynamics . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1.1 Linearized Translation Dynamics . . . . . . . . . . . 65
3.2.2 Linear Translation Control . . . . . . . . . . . . . . . . . . . . 66
3.2.2.1 PID Control Design . . . . . . . . . . . . . . . . . . 67
3.2.2.2 Linear Controller Design . . . . . . . . . . . . . . . . 68
3.2.2.3 Robustness of Time Invariant Assumption . . . . . . 70
3.2.2.4 Robustness of Double Integrator Dynamic Model
Assumption . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.2.5 Lyapunov Based Robustness Analysis . . . . . . . . . 74
3.2.3 Linear Attitude Control . . . . . . . . . . . . . . . . . . . . . 77
3.2.3.1 Linearized Attitude Dynamics . . . . . . . . . . . . . 77
3.2.3.2 Linear Attitude Control . . . . . . . . . . . . . . . . 77
3.3 Formation Flying at L2, Nonlinear Control Design . . . . . . . . . . . 78
3.3.1 Summary of Spacecraft Dynamics . . . . . . . . . . . . . . . . 78
3.3.1.1 Translation . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.1.2 Attitude Dynamics . . . . . . . . . . . . . . . . . . . 79
3.3.1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.2.1 Translation . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.2.2 Attitude . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 Combined Translation/Attitude Control Law . . . . . . . . . . . . . . 82
4 Simulation Results and Analysis for Linear and Nonadaptive Control 85
4.1 Simulation Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Dynamic Environment . . . . . . . . . . . . . . . . . . . . . . 85
vi
4.1.2 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.2.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.3 Spacecraft Model . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.3.1 Leader Spacecraft . . . . . . . . . . . . . . . . . . . . 88
4.1.3.2 Follower Spacecraft . . . . . . . . . . . . . . . . . . . 89
4.1.4 Thruster Biasing . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Scenario 1 ? Distant Formation, 100 Kilometer Range . . . . . . . . . 96
4.4 Scenario 2 ? Close Formation, 50 Meter Range . . . . . . . . . . . . . 102
4.5 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5.1 Compensation for Solar Pressure . . . . . . . . . . . . . . . . 107
4.5.2 Compensation for Mass Property Uncertainty . . . . . . . . . 110
4.5.3 Compensation for Thruster Misalignment/Misplacement . . . 113
4.5.4 Compensation for Solar Pressure, Mass Properties, Thruster
Misalignment/Misplacement . . . . . . . . . . . . . . . . . . . 116
4.5.5 Performance Impact of Time Invariant Dynamics Model . . . 119
4.5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5 Adaptive Control System Design 122
5.1 Model Uncertainty and Unmodeled Dynamics . . . . . . . . . . . . . 122
5.1.1 Spacecraft Parameter Uncertainty . . . . . . . . . . . . . . . . 123
5.1.2 Unmodeled Dynamics . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Nonlinear Adaptive Control Theory . . . . . . . . . . . . . . . . . . . 124
5.2.1 Parameter/Model Uncertainty . . . . . . . . . . . . . . . . . . 125
5.2.2 Actuation Uncertainty . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Parametrization for Nonlinear Adaptive Formation Control . . . . . . 129
5.3.1 Linear Parametrization for Translation Control . . . . . . . . 129
5.3.2 Linear Parametrization for Attitude Control . . . . . . . . . . 134
5.3.3 Linear Parametrization for Thruster Performance and
Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4 Parametrization for Adaptive Translation without Absolute Navigation137
5.5 Combined Translation/Attitude Adaptive Control . . . . . . . . . . . 140
6 Model Uncertainty Effects: Simulation Results and Analysis 142
6.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.1 Compensation for Solar Pressure . . . . . . . . . . . . . . . . 146
6.2.2 Compensation for Mass Property Uncertainty . . . . . . . . . 150
6.2.3 Compensation for Thruster Misalignment/Misplacement . . . 157
6.2.4 Compensation for Solar Pressure, Mass Properties, Thruster
Alignment/Position Uncertainty . . . . . . . . . . . . . . . . . 163
6.2.5 Performance Impact of Time Invariant Dynamics Model . . . 168
6.3 Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
vii
7 Summary and Conclusions 172
7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.2 Results Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.4 Concluding Remark ? The End and a Beginning . . . . . . . . . . . . 177
Bibliography 179
viii
List of Tables
4.1 Simulation Model Thruster Firing Direction and Location . . . . . . 89
4.2 Scenario 1 ? Controller Tracking Performance Summary . . . . . . . . 98
4.3 Scenario 1 ? Controller Fuel Performance Summary . . . . . . . . . . 101
4.4 Scenario 2 ? Controller Tracking Performance Summary . . . . . . . . 105
4.5 Scenario 2 ? Controller Fuel Performance Summary . . . . . . . . . . 106
4.6 Scenario 1 with Solar Perturbation ? Controller Tracking Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Scenario 1 with Mass Property Uncertainty ? Controller Tracking
Performance Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.8 Scenario 1 with Thruster Alignment Uncertainty ? Controller
Tracking Performance Summary . . . . . . . . . . . . . . . . . . . . . 116
4.9 Scenario 1 with Solar Pressure, Mass Properties, Thruster
Alignment/Position Uncertainty ? Controller Tracking Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.10 Scenario 1, 28 days after Epoch ? Linear Controller Tracking
Performance Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.11 Scenario 1 ? Linear Controller Fuel Performance Summary . . . . . . 120
6.1 Scenario 1 with No Perturbations ? Nonlinear Controller Tracking
Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2 Scenario 1 with Solar Pressure ? Nonlinear Controller Tracking
Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Scenario 1 with Solar Pressure ? Controller Fuel Performance Summary149
6.4 Scenario 1 with Mass Uncertainty ? Nonlinear Controller Tracking
Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5 Scenario 1 with Mass Uncertainty ? Controller Fuel Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.6 Scenario 1 with Thruster Misalignment/Performance Effects ?
Nonlinear Controller Tracking Performance Comparison . . . . . . . . 162
ix
6.7 Scenario 1 with Thruster Misalignment/Performance Effects ?
Controller Fuel Performance Summary . . . . . . . . . . . . . . . . . 162
6.8 Scenario 1 with All Perturbations ? Nonlinear Controller Tracking
Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.9 Scenario 1 with All Perturbations ? Controller Fuel Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.10 Scenario 1, 28 days after Epoch ? Nonlinear Adaptive Controller
Tracking Performance Summary . . . . . . . . . . . . . . . . . . . . . 168
6.11 Scenario 1 ? Nonlinear Adaptive Controller Fuel Performance Summary169
6.12 Scenario 1 with No Perturbations ? Controller Tracking Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.13 Scenario 1 ? Linear Controller Fuel Performance Summary . . . . . . 171
7.1 Scenario 1 with All Perturbations ? Controller Tracking Performance
Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
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List of Figures
1.1 Rotating Frame for Two Body System . . . . . . . . . . . . . . . . . . . 4
1.2 Lagrange/Libration Points in Earth/Moon-Sun Rotating Frame . . . 5
1.3 Three Spacecraft Formation Design for a Large Aperture Telescope
Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Two Spacecraft Formation Operating in the Earth/Moon-Sun Rotating
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 General Restricted Three Body Problem, Plane of Rotation . . . . . . . . 32
2.2 Circular Restricted Three Body Problem . . . . . . . . . . . . . . . . . 33
2.3 Libration Point Locations for Circular Restricted Three Body Problem . . 35
2.4 Two Spacecraft Orbiting in the RTBP Frame . . . . . . . . . . . . . . . 39
2.5 Two Spacecraft Orbiting in the Earth/Moon - Sun Rotating Frame . . . . 48
3.1 Closed-Loop Control Block Diagram . . . . . . . . . . . . . . . . . . 69
3.2 Closed-Loop Gain, all-channels, for GCL(s) . . . . . . . . . . . . . . 70
3.3 Maximum Singular Value Comparison: Inverse Model
Uncertainty 1?(?) and Closed-Loop Gain . . . . . . . . . . . . . . . . 72
3.4 Maximum Singular Value Comparison: Inverse Model
Uncertainty, 1?(?), for a Double Integrator Dynamics Model
and Closed-Loop Gain, GCL(s) . . . . . . . . . . . . . . . . . . . . . 74
4.1 Simulation Model of Follower Spacecraft Depicting Thruster Layout . 88
4.2 Linear Translation Control: Closed-Loop Gain, all-channels . . . . . . 95
4.3 Linear Attitude Control: Closed-Loop Gain, all-channels . . . . . . . 95
4.4 Profiles of Desired Leader/Follower Range along Separation Vector
(Inertial X-axis), and Ideal (Perfect Tracking) Control . . . . . . . . . 97
4.5 Desired Slew Angle for Follower and Ideal (Perfect Tracking) Torque
Profile about Rotation Axis . . . . . . . . . . . . . . . . . . . . . . . 97
xi
4.6 Range Error and Translation Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . . . 99
4.7 Attitude Error and Torque Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . . . 100
4.8 Instantaneous Thrust Level and Fuel Consumption for Linear (solid)
and Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . 101
4.9 Desired Leader/Follower Range and Ideal (Perfect Tracking) Control
Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.10 Range Error and Translation Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . . . 104
4.11 Attitude Error and Torque Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . . . 105
4.12 Instantaneous Thrust Level and Fuel Consumption for Linear (solid)
and Nonlinear(dashed) Control Algorithms . . . . . . . . . . . . . . . 106
4.13 Range Error and Translation Control Profile for Linear Controller,
Perturbed with Solar Pressure (solid) and Unperturbed (dashed) . . . 108
4.14 Range Error and Translation Control Profile for Nonlinear Controller,
Perturbed with Solar Pressure (solid) and Unperturbed (dashed) . . . 109
4.15 Range Error Profile Comparison with Mass Uncertainty Effects
(solid) and without (dashed) for Linear (top) and Nonlinear (bottom)
Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.16 Attitude Error Profile Comparison with Mass Uncertainty Effects
(solid) and without (dashed) for Linear (top) and Nonlinear (bottom)
Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.17 Range Error Profile Comparison with Thruster
Misalignment/Performance Effects (solid) and without (dashed) for
Linear (top) and Nonlinear (bottom) Control Algorithms . . . . . . . 114
4.18 Attitude Error Profile Comparison with Thruster
Misalignment/Performance Effects (solid) and without (dashed) for
Linear (top) and Nonlinear (bottom) Control Algorithms . . . . . . . 115
4.19 Range Error Profile Comparison with All Perturbation Effects (solid)
and without (dashed) for Linear (top) and Nonlinear (bottom)
Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xii
4.20 Attitude Error Profile Comparison with All Perturbation Effects
(solid) and without (dashed) for Linear (top) and Nonlinear (bottom)
Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1 Variation in True Earth Moon Gravity Field Experienced at L2 . . . 131
6.1 Translation/Attitude Error Profile Comparison with No Perturbation
Effects for Nonlinear Nonadaptive (solid) and Adaptive (dashed)
Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Range Error Profile, Scenario 1 with Solar Pressure Effects,
Comparison with Baseline (solid), and Nonlinear Nonadaptive
(top/dashed) and Adaptive (bottom/dashed) Control Algorithms . . 147
6.3 Attitude Error Profiles (Identical), Scenario 1 with Solar
Pressure Effects, Comparison with Baseline (solid), and Nonlinear
Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.4 Range Error Profile Comparison, Scenario 1 with Mass Uncertainty
Effects, Comparison with Baseline (solid), and Nonlinear
Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.5 Attitude Error Profile Comparison, Scenario 1 with Mass
Uncertainty Effects, Comparison with Baseline (solid), and Nonlinear
Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.6 Nonlinear Adaptive Controller Performance Improvement for Second
Run with Updated Mass Property Estimates. Position Tracking
(top), Attitude Tracking (bottom) with Updated Parameters (solid),
Original Simulation (dashed) . . . . . . . . . . . . . . . . . . . . . . . 155
6.7 Range Error Profile Comparison, Scenario 1 with Thruster
Misalignment/Performance Effects, Comparison with Baseline
(solid), and Nonlinear Nonadaptive (top/dashed) and Adaptive
(bottom/dashed) Control Algorithms . . . . . . . . . . . . . . . . . . 159
6.8 Attitude Error Profile Comparison, Scenario 1 with Thruster
Misalignment/Performance Effects, Comparison with Baseline
(solid), and Nonlinear Nonadaptive (top/dashed) and Adaptive
(bottom/dashed) Control Algorithms . . . . . . . . . . . . . . . . . . 160
xiii
6.9 Nonlinear Adaptive Controller Performance Improvement for
Second Run with Parameter Estimate Update for Thruster
Misalignment/Performance Effects. Position Tracking (top),
Attitude Tracking (bottom) with Updated Parameters (solid),
Original Simulation (dashed) . . . . . . . . . . . . . . . . . . . . . . . 161
6.10 Range Error Profile Comparison, Scenario 1 with All Perturbation
Effects, Comparison with Baseline (solid), and Nonlinear
Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.11 Attitude Error Profile Comparison, Scenario 1 with All Perturbation
Effects, Comparison with Baseline (solid), and Nonlinear
Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.12 Nonlinear Adaptive Controller Performance Improvement for Second
Run with Parameter Estimate Update for All Perturbation Effects.
Position Tracking (top), Attitude Tracking (bottom) with Updated
Parameters (solid), Original Simulation (dashed) . . . . . . . . . . . . 166
xiv
Chapter 1
Introduction
1.1 Background
?The first science in the modern sense that grew in the Mediterranean
civilisation was astronomy...The rudiments of astronomy exist in all cultures, and
were evidently important in the concerns of early people all over the world.?
From Jacob Bronowski, The Ascent of Man [4]
1.1.1 Motivation...Looking to the Stars
Our continued quest to understand the universe is evident in our pursuit of
technologies to enhance our observational capability. Space-based observatories,
such as the Hubble Space Telescope, provide dramatically superior image quality
in comparison to Earth-based observatories. Looking to the future, Distributed
Spacecraft System (DSS) technologies will enable higher resolution imagery and
interferometry, robust and redundant fault-tolerant architectures, and complex
networks dispersed over clusters of satellites.
1
1.1.2 Mechanism...Spacecraft Precision Formation Flying
A Distributed Spacecraft System (DSS) is a collection of two or more spacecraft
functioning to fulfill a shared or common objective. As a subset of the DSS
architecture, formation flying missions add the requirement to maintain a relative
position and/or orientation with respect to each other, or a common target.
The term precision formation flying implies a requirement for continuous control
(normally implemented in discrete time) to maintain the formation within the design
specification. Control system designs for precision formation flying missions will vary
based on the dynamic environment. For example, the dynamic environment for low
Earth orbit differs significantly from that experienced in deep space.
Numerous formation flying missions are under development to enhance our
remote sensing capability, including the Terrestrial Planet Finder, the Micro-
Arcsecond X-Ray Imaging Mission, and the Stellar Imager. Although currently
conceived as single spacecraft, Constellation-X was evaluated as a formation flying
mission, and is used to characterize another mission profile. A description of each
of these missions is deferred to Section 1.3.
The success of each of these missions depends on varied sets of enabling
technologies, with a precision formation flying requirement critical to each. Specific
formation definition and control requirements are mission dependent. These
missions are ordered by their range of demands for formation flying technology.
TPF requires position control at the level of centimeters over a separation distance
of a kilometer. As a formation flying concept, Constellation-X requires millimeter
2
level control over a separation distance of 50 meters. MAXIM and SI, the
most challenging, require micron to nanometer level control oversub-kilometer to
kilometer ranges. The spacecraft control requirement for each of these missions
is driven by optical performance requirements. Combining spacecraft control with
active optical control serves to reduce the spacecraft position control requirements
listed above.
1.1.3 Context...Libration Point Missions
Due to the large spacecraft separations and tight control requirements
discussed above, the low Earth orbit environment is generally unfavorable for
precision formation flying due to the high fuel cost associated with overcoming
the local gravitational gradient. Therefore, TPF, Constellation-X, MAXIM, and SI
are currently envisioned to orbit the Earth/Moon-Sun, L2, libration point. This
region of space is characterized by a benign gravitational gradient, compared to
those experienced in a local Earth orbit.
Libration points are defined within the context of a rotating reference frame
defined by two large masses rotating about their common center of mass, Figure
1.1. A libration point represents a location within the rotating frame at which the
dynamical forces due to gravity and rotation are in equilibrium. These solutions, also
referred to as Lagrange points, were first identified by Lagrange in 1772, as published
in his prize memoir, Essai sur le Probl?eme des Trois Corps, [46]. In the absence
of perturbing forces, an object placed at any one of these locations remains fixed in
3
the rotating frame. Libration point locations for the Earth/Moon-Sun system are
identified in Figure 1.2.
Libration points L1, L2 and L3, termed collinear, are unstable. The stability
of the equilateral points, L4 and L5, is dependent on the mass ratio, ?, of the two
primary masses. The points are unstable if 0.03852 < ? < 0.96148, otherwise they
are stable. For the Earth/Moon-Sun system with ? = 0.000003, L4 and L5 are
stable [68]. A detailed discussion of libration points, including stability properties,
appears in the next chapter, Section 2.1.
While L1 and L2 are naturally unstable, it is possible to stabilize an
orbit about either point with reasonable fuel cost. The regions surrounding
L1 and L2 provide desirable locations for certain missions. Both regions are
within reasonable proximity of Earth, easing the challenges of accessibility and
communications between Earth stations and the spacecraft. As previously noted,
Figure 1.1: Rotating Frame for Two Body System
4
Figure 1.2: Lagrange/Libration Points in Earth/Moon-Sun Rotating Frame
fuel economy is a critical design consideration for space missions. The regions
surrounding L1 and L2 provide the important advantage of a shallow gravity
gradient, compared to the gravity field experienced by an Earth orbiting mission.
A shallow gravity gradient supports the goal of minimizing fuel requirements while
achieving formation performance criteria. Simultaneous reduction in solar and Earth
interference provides the advantage of L2 over L1 for a mission designed to probe
deep space.
5
1.1.4 Focus...Formation Control
From a systems perspective precision formation flying requires a high level of
autonomy with an enabling subset of component technologies. Defined in the context
of closed-loop control, the components generally align with metrology, estimation,
control design and actuation.
Most of these technologies exist in some form. Formation flying demands
refinement of each technology to a higher level of performance. Detailed exploration
of each technology and the associated development challenges is beyond the scope of
this effort. However, it is important to recognize interdependencies with formation
flying technology development.
The precision formation flying control problem is uniquely defined for various
mission classes, due to differences in the dynamic environment, principally the
gravity field. Motivated by mission concepts for TPF, Constellation-X, MAXIM
and SI, this research effort focuses on the precision formation control algorithm
design problem within the context of the restricted three body problem. Research
objectives are detailed in the next section.
1.2 Research Focus
This research was originally designed to address the specific problem of
precision formation control in the vicinity of the Earth/Moon-Sun L2, libration
point. However, the analysis framework generalizes to other trajectories within the
context of the Earth/Moon-Sun gravity field, as well as other RTBP rotating frames.
6
The theoretical construct for linear and nonlinear control algorithms is presented
in Chapters 3 and 5. A benchmark problem, discussed below, provides a relevant
scenario for demonstrating the theory. Modeling and analysis in Chapters 4 and 6,
conform to the benchmark problem for libration point missions, presented in [5].
1.2.1 A Benchmark Problem
The general problem is to design a control algorithm for the Follower to track
a desired trajectory relative to a Leader spacecraft in the context of a virtual space-
based observatory. This new class of missions defines the extreme requirements for
precision formation control.
The benchmark problem, based on a realistic mission profile, defines a generic
formation design problem for a space-based telescope with a 20 spacecraft sparse
aperture aligned with a distant detector spacecraft. The specifications permit
independent control of each aperture spacecraft with respect to the detector
spacecraft in a Leader/Follower configuration. The Leader/Follower architecture,
applied in this research, allows modeling the formation with three spacecraft, a
Leader and two Followers, Figure 1.3. One Follower, the Freeflyer, maintains a
trajectory in close proximity to the Leader. The second Follower, the Detector,
maintains a distant trajectory. The two Follower formation facilitates examination
of control strategies for both short and long separation distances.
The three spacecraft, Leader/Follower formation is stationed in the vicinity
of L2 in the Earth/Moon-Sun rotating frame. Each Follower is required to
7
Figure 1.3: Three Spacecraft Formation Design for a Large Aperture Telescope
Mission
Figure 1.4: Two Spacecraft Formation Operating in the Earth/Moon-Sun Rotating Frame
8
maintain a predetermined trajectory (position and attitude) relative to the Leader,
effectively defining two distinct formations with the Leader as a common spacecraft.
Figure 1.4 depicts a typical Leader/Follower formation. The Earth/Moon system
is modeled as a combined mass located at the system center of mass to facilitate
control design. The Earth and Moon are treated as separate bodies for simulations.
The position and relative motion of the Earth/Moon barycenter about the Sun
defines the rotating frame. The Leader maintains a planned ballistic trajectory
with periodic station-keeping maneuvers, and a predetermined attitude trajectory.
Attitude control on the Leader is accomplished with reaction wheels to avoid
perturbing its orbit trajectory. Each Follower tracks a specified separation and
attitude trajectory relative to the Leader. Measurement data provides the relative
position and velocity between the two spacecraft. Each spacecraft is equipped
to measure attitude, referenced to an inertial coordinate frame. Each Follower is
equipped with fixed thrusters to serve as actuators for both translation and attitude
maneuvers. Thrusters are fully throttleable. Detailed spacecraft and trajectory
specifications for the benchmark problem are included in the modeling discussion in
Section 4.1.
1.2.2 Generalized Architecture
As introduced, the control algorithm applied to the benchmark problem is
based on a generalized architecture within the context of the restricted three body
problem. While motivated by mission scenarios similar to the benchmark problem,
9
the theoretical framework is constructed about the dynamics for a generic restricted
three body problem (RTBP), Section 2.1. The control strategies are formulated with
the general RTBP dynamics and simplifying assumptions, such as a slow moving
trajectory with the RTBP rotating frame. The result is a generalized architecture for
defining formation flying control laws, and an analysis framework for characterizing
their performance.
1.2.3 Thesis
The goal of this research is to provide a fair performance comparison of linear
and nonlinear control strategies applied to the precision formation flying problem
to test the following assertion through analysis.
1.2.3.1 Assertion:
Linear control may prove adequate to meet formation flying requirements
with less stringent performance specifications. However, a linear control design
provides inadequate compensation to overcome perturbations due to unmodeled
nonlinearities in the system dynamics necessary to achieve the strict performance
requirements for precision formation flying. The control performance specifications
for MAXIM and SI, require nonlinear adaptive techniques applied to the six-degree
of freedom (6DOF) control problem with coupled attitude and orbit dynamics.
10
1.2.3.2 Analysis:
Analysis provides the mechanism for testing the thesis. Three basic control
designs are developed: Linear, Nonlinear (based on Lyapunov theory), and
Nonlinear Adaptive (also Lyapunov based). Control gains for each design are
chosen to be ?equivalent?, allowing fair comparison of the performance. Tracking
performance is assessed with simulations. The MATLAB based simulation employs a
rigid body model for attitude dynamics, a realistic gravity model based on planetary
ephemeris data, and a model for solar pressure. Controller robustness to model
uncertainty is demonstrated through analysis and simulation.
1.3 Precision Formation Flying Missions
As discussed earlier, a principal goal for precision formation flying is to support
advanced astronomy missions. Several of these missions are highlighted here.
The Terrestrial Planet Finder (TPF) is designed to detect and characterize
remote planets from formation through various stages of development. Science
goals include spectrographic analysis of planetary atmospheres, seeking to identify
planetary bodies with the capacity to support life. TPF aims to locate tiny, faint
planets around distant stars, requiring suppression of the glare from the parent star
by a factor of a million or better. This level of suppression will enable imaging
of planetary systems as far away as 50 light years. With a resolution a hundred
times finer than the Hubble Space Telescope, TPF also allows the study of the
black hole at the center of the Milky Way and other exciting phenomena in the
11
universe. As one of several design concepts TPF is proposed as an interferometer
composed of a four-element linear array of a spacecraft formation in orbit about
the Earth/Moon-Sun L2 libration point. Target dependent, spacecraft separations
range 75-1000 meters with a control tolerance on the order of centimeters. The
attitude control requirement for the individual spacecraft is on the order of a few
arcminutes. [35, 48, 50, 61]
The Constellation-X Observatory (Con-X) will enable scientists to investigate
black holes, Einstein?s Theory of General Relativity, galaxy formation, the evolution
of the Universe on the largest scales, the recycling of matter and energy, and
the nature of ?dark matter?. The current concept for Con-X is a single
spacecraft. However, several alternative design concepts have been considered for the
observatory. One envisions a formation of two spacecraft acting as a virtual X-ray
telescope. Based on this design, Con-X will orbit the Earth/Moon-Sun L2 libration
point. The two spacecraft will maintain a 50 meter separation with a control
tolerance on the order of millimeters. The formation will remain inertially fixed
during observations. Repointing maneuvers will occur approximately twice a day
over a period of one hour with a nominal slew angle of 60 degrees. The attitude
control requirement for the individual spacecraft is on the order of arcseconds.[10]
The Micro-Arcsecond X-Ray Imaging Mission (MAXIM) is an X-Ray
observatory, designed to image the event horizon of a black hole. With light
trapped by its strong gravitational field, direct imaging of a black hole is not
possible. However, it is possible to image the event horizon, a region surrounding the
12
black hole characterized by strong X-Ray emissions. The current MAXIM concept
envisions a central optics hub spacecraft, surrounded by a fleet of 32 spacecraft,
forming a 500-1000 meter diameter ring. The fleet has even angular spacing in
the optical plane with a random radial distribution. The array forms an X-Ray
lens, focused on a detector spacecraft, located 20,000 kilometers from the optics
hub/ring array along the line of sight. The position control requirement for the
spacecraft optics forming the ring array about the hub is on the order of nanometers.
The spacecraft position control could be reduced with active optics. The detector
spacecraft is required to maintain the 20,000 kilometer separation range to within
10 meters while holding the position error off the line of sight to a tolerance on the
order of microns.
MAXIM Pathfinder, a proposed predecessor mission to the full MAXIM
mission, is designed to demonstrate the feasibility of space-based X-ray
interferometry for astronomical applications. The Pathfinder mission concept limits
the ring array to six spacecraft, and operates with relaxed, extremely challenging,
control tolerances at a reduced range of 450 kilometers. The attitude control
requirement for these missions is on the order of arcseconds. Pathfinder is not
currently supported as part of the MAXIM mission development, but provides a
good design profile for analysis. [8, 19, 20, 21, 22, 37, 44, 55]
Stellar Imager (SI) provides a space-based UV-optical Fizeau Interferometer
with an angular resolution of 100 micro-arcseconds. SI is designed to study the
various effects of stellar magnetic fields, the field generating dynamos, and the
13
internal structure and dynamics of the associated stars. Ultimately the science goal
is to achieve the best-possible forecasting of solar activity on time scales ranging
up to decades, combined with understanding the impact of stellar magnetic activity
on astrobiology and life in the universe. SI consists of a reconfigurable array of
10-30 one-meter class ?mirrorsats?, forming a 0.5 kilometer primary mirror, focused
on a hub (detector) spacecraft separated along the line of sight by ?5 kilometers.
Maintenance of the range between the ?mirrorsats? and the hub requires control to
the level of nanometers. Spacecraft attitude control requirements are on the order
of 10?s of micro-arcseconds. [6, 7, 20, 56]
1.4 Formation Flying - Current Technology Assessment
Precision formation control is a topic of active research. EO-1, launched in
November 2000, is NASA?s first mission to successfully demonstrate autonomous
formation flying [15]. EO-1?s orbit was designed to lag LANDSAT-7?s ground track
by one minute with a tolerance of +/- 6 seconds, equivalent to a 450 kilometer
along track separation with a tolerance of 85 kilometers. EO-1 represents a basic
form of formation flying with orbit control implemented as discrete maneuvers,
approximately every three weeks. The EO-1 formation flying experiment was
conducted during the period, January-July 2001, and November 2001. While EO-
1?s baseline mission was completed in November 2001 the spacecraft continues to
support Earth science.
Another indicator of the importance of formation flying development is the
14
near term, proposed ST9 mission concept. ST9 is a technology demonstration
mission funded under NASA?s New Millennium Program. Precision Formation
Flying is one of five technologies competing for the ST9 flight opportunity. In
contrast to EO-1 the ST-9 precision formation flying mission concept is conceived
to demonstrate continuous control. Details for the ST-9 mission concept are not
publicly available. [47]
The following subsection highlights recent technology developments associated
with precision formation flying.
1.4.1 Current Control Technology
In 1994, Egeland and Godhaven [14] published an adaptive attitude control
law for a rigid spacecraft, providing the framework for the attitude portion of the
control law presented here. Conventional attitude control systems are based on
linear models. This design is based on the nonlinear attitude dynamics and provides
an adaptive mechanism to autotune the controller performance. Their work builds
on earlier research by Wen and Kreutz-Delgado (1991) [66], Slotine and DiBenedetto
(1990) [57], and Crouch (1984) [11].
The general problem of 6DOF (position/attitude) vehicle control is addressed
by Fossen and Sagatun (1991) [16], and reappears in [17]. Their work focuses on
the problem of 6DOF control of underwater vehicles. However, the framework is
applicable to the 6DOF control problem for spacecraft formation flying.
The design for relative position control is based on an analysis of the
15
relative equations of motion, derived from basic astrodynamics [1, 2, 68], and
adaptive nonlinear control [58]. Related work on this topic is presented in
[3, 12, 25, 32, 45, 59, 65].
Wang, Hadaegh and Lau (1999) [65] explored the problem of synchronized,
formation rotation and attitude control. The analysis demonstrates coordinated
simultaneous control of relative position and attitude within a formation.
However, the spacecraft dynamic model assumes a gravity-free and disturbance-free
environment.
deQueiroz, Kapila and Yan (2000) [12] propose a relative position adaptive
control law for a formation in a circular orbit. Adaptation is applied to compensate
for uncertainty/slow variation in spacecraft mass properties, disturbance forces and
gravity. Their work builds on earlier research by Kapila, Sparks, Buffington and
Yan (2000) [32] and by Vassar and Sherwood (1985) [63]. The formation design
includes two spacecraft in a Leader/Follower configuration. The Leader follows a
ballistic trajectory while the Follower is controlled to orbit the Leader in a circular
path with a 100 meter radius. Simulated results employ a geostationary Earth
orbit and artificially assume a fixed disturbance force, rather than employing a high
fidelity dynamic simulator. The analysis only addresses relative position control
with perfect actuation. Spacecraft attitude effects are not considered.
Hamilton, Folta and Carpenter (2002) [26] examined the problem of relative
position control for a formation stationed in the vicinity of the Earth/Moon-Sun
L2 libration point. Their control design is based on linearized dynamics extracted
16
from Generator, a high fidelity simulator developed at Purdue University. While the
linearized dynamics matrix is time-varying the design treats it as constant, applying
periodic updates. Since Generator is computationally intensive, a typical mission
scenario would require periodic uploading/updating of the control algorithm based
on tracking data. Ground-based tracking data analysis would estimate an updated
spacecraft state. The updated state would be fed to Generator, then Generator
would update the state-space expression for the linearized dynamics. Finally, the
new gain matrix, computed using linear control design, would be uploaded to
the spacecraft. Their design includes a Kalman filter to examine the impact of
measurement noise. Attitude dynamics are not addressed. Perfect actuation is
assumed.
Marchand and Howell (2005) [43], also exploited Generator to study natural
formations and control strategies for formations in the vicinity of a libration point.
Their analysis compared the application of nonlinear and linear control designs.
The nonlinear designs utilized feedback linearization techniques. As in [26], the
linear design is based on the linear dynamics matrix computed with Generator. The
technique employed for feedback linearization requires knowledge of the spacecraft
state. Attitude dynamics are not addressed. Perfect actuation is assumed. Results
demonstrated equivalent performance based on tracking error and fuel consumption.
The analysis did not assess the impact of unmodeled dynamics and modeling errors
on tracking performance.
In 2003 Gurfil, Idan and Kasdin (2003) [25] proposed an adaptive control
17
algorithm for deep-space formation flying. The approach uses feedback linearization,
applies control methods for linear systems, and then adds a neural network to
compensate for unmodeled dynamics. The equations of relative motion, the basis of
the control law design, assume circular-restricted three body problem (CRTBP)
dynamics. The feedback linearization mechanism requires the spacecraft state
relative to the reference frame origin, the Earth. The analysis only addresses relative
position control, and assumes perfect actuation.
Related research includes the study of the control architecture for large
spacecraft formations. Mesbahi and Hadaegh (2001) [45] examine the application
of logic-based switching in combination with elementary graph theory linear matrix
inequalities to define a framework for implementing control of large formations. The
design assumes linear dynamics. Beard, Lawton and Hadaegh (2001) [3] study the
issue of coordinated control for large spacecraft formations. Their study assumes a
gravity-free and disturbance-free environment.
Finally in recent literature, Hsiao and Scheeres (2005) [27] examine control
strategies for stabilizing natural relative spacecraft motion for formations orbiting
libration points. The study is conducted in the context of the circular restricted
three body problem. The control strategy employs pole placement to locate the
poles of the stabilized linearized dynamics along the imaginary axis.
Smith and Hadeagh (2005) [59] present a method of formation control using
switched topologies. A centralized control topology is designed based on all relative
spacecraft measurements within the formation. Analysis shows the equivalence of
18
distributed local relative control topologies when redundant relative measurements
are unavailable. Switching between the equivalent topologies is applied on individual
spacecraft based on available measurements. The analysis assumes a gravity and
disturbance free environment.
Vaddi, Alfriend, Vadali and Sengupta (2005) [62] develop a control strategy
for a two spacecraft formation orbiting a central body. The strategy utilizes orbital
element differences. Analysis shows the solution is fuel-optimal and maintains
homogenous fuel consumption between spacecraft. Their results correlate with
similar numerical optimization studies.
Sengupta and Vadali (2005) [54] present an orbit transfer/formation control
algorithm for an Earth orbiting spacecraft. Their approach employs Lyapunov
analysis with Euler parameters to characterize the orbit.
Based on this review of current literature, several key observations are made
regarding formation flying technology.
Cited works:
? Focus on problem specific formation control in either Earth orbit or in the
vicinity of the Earth/Moon-Sun L2 libration point. Lack of a common
framework makes comparing results difficult.
? Employ numerical analysis tools, such as Generator, to model RTBP
dynamics.
? Assumed perfect actuation and knowledge of spacecraft parameters.
19
? Focus on translation control as a 3DOF problem without considering attitude
maneuvers.
? Neglect modeling uncertainty associated with dynamics, mass properties, and
actuator performance.
? Require knowledge of absolute spacecraft states to compute control.
The next section identifies specific contributions of this research.
1.5 Contributions and Conclusions
Based on noted observations from the literature review, this research is directed
toward development of a generic 6DOF (coupled translation/attitude) approach to
the formation flying control problem within the context of the RTBP. Also, adopting
the benchmark problem framework from [5] facilitates future comparative analysis.
The following discussion traces the development of this research, highlighting
relevant publications and contributions.
1.5.1 Preliminary Analysis
The initial effort examined the application of nonlinear adaptive control
techniques to track a prescribed trajectory about the Earth-Moon L2 point, reference
[38]. The adaptation mechanism compensated for unmodeled dynamics (solar
pressure and gravity). The control design, based on the Circular Restricted Three
Body Problem dynamics, performed successfully despite modeling errors associated
20
with gravitational dynamics. Analysis demonstrated the gravitational influence
of the sun is non-negligible within the Earth-Moon system. In fact the angular
momentum of the Earth-Moon system varies in direction and magnitude due to
the sun?s gravity field. Also, the lunar orbit is slightly elliptic with an eccentricity
of 0.055. While perhaps an unlikely application of continuous control, this step
established a foundation for building the control architecture for precision formation
flying.
Next, reference [39] studied nonlinear adaptive methods applied to formation
control. In this case the formation was stationed near the Earth/Moon-Sun
L2 point. As with the previous case, adaptation compensated for unmodeled
dynamics associated with gravity and solar pressure. The control design assumed
perfect actuation independent of the spacecraft attitude.
Finally, reference [40], developed the 6DOF formation control architecture,
combining translation and attitude control. Adaptation compensated for uncertain
mass properties in addition to the dynamic modeling errors associated with gravity
and solar pressure.
1.5.2 Linearized Relative Dynamics
Linear control theory offers the designer with a rich heritage of tools. However,
application of the tools requires an equivalent linear expression of the true nonlinear
system dynamics. As noted, prior work employs numerical models to compute
the dynamics relative to a reference spacecraft, or applied the linearized dynamics
21
about the libration point. Numerical models for relative linearized dynamics
are computationally expensive. Linearized dynamics about a libration point are
easily computed with analytic methods, but do not accurately reflect the linearized
dynamics of the formation [41].
Reference [41] represents a key contribution through development of an
analytic expression of the linearized dynamics about a Leader spacecraft. Prior
work commonly views relative dynamics in terms of a RTBP rotating frame when
modeling system behavior near a libration point. This perspective unnecessarily
constrains the problem. To maintain generality linearization is performed using
inertial coordinates with restricted three body problem dynamics. Details are
presented in Section 2.1.2. While time-varying, an important feature of the solution
is its applicability for any trajectory of the reference spacecraft. Therefore, the
linearized model can be applied for missions stationed in the vicinity of any libration
point or alternative trajectories such as an Earth drift away orbit. Reference [42]
demonstrates the utility of the analytic linear model in a linear control design for
the MAXIM mission.
1.5.3 6DOF Control Design
Spacecraft position and attitude control are typically treated as independent
problems. The approach is appropriate for single spacecraft. Precision formation
flying implies a requirement for continuous control. While unforced spacecraft
translation and attitude dynamics are uncoupled, continuous thruster firing
22
potentially induces a coupling action. Using a common set of actuators (thrusters),
the design approach couples the translation and attitude control problem into a
single 6DOF architecture. The coupling action is particularly important if the
physical characteristics of the thrusters are uncertain, i.e. pointing and placement.
This issue is particularly important for the nonlinear adaptive control design, and
is discussed in detail in Sections 5.2.2 and 5.5.
1.5.3.1 Linear Control Design
The Linear Quadratic Regulator (LQR) is a common technique for linear
control design, providing robust and optimal performance. However, an LQR design
requires time invariant dynamics. Recall, the linearized relative dynamics for the
formation flying problem are time varying. Therefore, a gain scheduling strategy is
required which assumes time invariant dynamics over regular intervals. A robustness
analysis in Section 3.2.2.3 shows the dynamics allow a time invariant characterization
with LQR gain updates computed every two weeks. The two week time period is
specific for the selected design scenario (benchmark problem). A mission specific
robustness analysis must be performed to determine the appropriate interval for
gain scheduling.
Standard linearization of the rigid body attitude dynamics is applied to
extend the design to 6DOF control. Chapter 3 reviews linear control design theory
and develops the linear control design structure for the formation flying problem.
23
Chapter 4 presents specific details on the controller design for the benchmark
problem and simulation results.
1.5.3.2 Nonlinear Control Design
The stated thesis goal is to present a comparative analysis of linear and
nonlinear control designs. The specific nonlinear control approach is a Lyapunov
based design. Both nonadaptive and adaptive formulations are considered. Design
theory details appear in Chapter 3 (nonadaptive) and Chapter 5 (adaptive). An
important requirement for performance comparison is the specification of nonlinear
controller gains that are ?equivalent? to the linear design. As noted, linear gains
are computed using LQR. The definition of equivalent gains is explained in Sections
4.2 and 6.1.
1.5.3.3 Adaptation
The adaptive mechanism of the nonlinear controller represents a set of
integrators structured according to the system dynamics. The structure enables
systematic compensation for model uncertainty, resulting in improved performance.
The adaptive nonlinear controller design compensates for unmodeled dynamics
associated with gravity and solar pressure, mass property uncertainty, and actuator
(thruster) performance uncertainty.
24
1.5.3.4 Implementation Issues
Precision formation flying implies autonomous spacecraft control, minimizing
the interface with ground based control networks. Implementation of a linear control
design requires knowledge of the absolute reference spacecraft state to compute the
linearized equations of motion. Similarly control designs presented in the literature
review require absolute spacecraft state knowledge. Absolute spacecraft knowledge
requires ground based tracking and analysis. Therefore these control designs are not
fully autonomous.
Another important contribution is the development of a nonlinear adaptive
design that requires only relative state measurements. The design is limited by
general assumptions, such as slow moving trajectories within the RTBP frame.
However, the assumptions are consistent with profiles for proposed formation flying
missions. The nonlinear nonadaptive controller also depends on absolute state
knowledge. The adaptive design compensates for unmodeled gravitational and solar
dynamics, and uncertainty in mass properties and thruster performance.
1.5.4 Simulation Results
Simulation results for the linear and nonadaptive nonlinear controllers are
presented in Chapter 4. Nonlinear adaptive controller simulation results appear in
Chapter 6. Controller performance is characterized with and without unmodeled
dynamics and uncertain spacecraft parameters.
The simulation is based on realistic spacecraft parameters and employs a
25
full gravity model based on planetary ephemeris data. The control cycle is set
at one Hertz, typical for attitude control systems. Perhaps one unrealistic aspect
of the simulation is the duration of the reconfiguration maneuvers. Typically these
maneuvers would occur over much longer periods of time, reducing the required fuel.
However, the compressed maneuver periods provide a consistent scenario to assess
controller performance, and provide the benefit of reduced computational time for
the simulation.
The linear controller is robust to dynamics and mass uncertainty. However,
performance degrades when confronted with actuator performance uncertainty. In
contrast the nonadaptive nonlinear controller provides orders of magnitude better
tracking control in the absence of model uncertainty. Each component of model
uncertainty degrades tracking performance. However, the nonlinear nonadaptive
controller outperforms the linear controller based on maximum and mean tracking
error.
The nonlinear adaptive controller exhibits the best overall tracking
characteristic. Simulation results in Chapter 6 reflect equivalent performance for the
nonadaptive and adaptive controller designs in the absence of modeling uncertainty,
both orders of magnitude better than the linear control design. While performance
degraded, the adaptive controller demonstrates the ability to compensate for
unmodeled dynamics and uncertainties. For the case including all uncertainties, the
adaptive controller outperformed the nonadaptive design by two orders of magnitude
based on mean error for both translation and attitude tracking.
26
1.6 Synopsis of Contributions
While not conclusive, this analysis supports the thesis that nonlinear adaptive
control methods, as compared to tradition linear designs, are advantageous in the
development of precision formation flying algorithms. Ultimately the best controller
design is mission specific, based on many factors including the trajectory profile,
spacecraft parameters, actuators and performance requirements.
The following list highlights the contributions and unique features of this work
mentioned above.
? Derived analytic expression for linearized relative spacecraft dynamics within
a RTBP framework
? Eliminates need for numerical models of relative dynamics
? Time-varying expression valid for any unforced trajectory of the reference
spacecraft, not constrained to vicinity of libration point
? Analytic model provides a tool to understand the natural relative
dynamics without the tedium of numerical simulation
? Coupled, 6DOF translation/attitude dynamics model
? Accounts for dynamic coupling associated with actuation (thrusters)
? Linear versus Nonlinear Control Performance Comparison
? Based on ?equivalent? gains
27
? Benchmark problem simulation framework, facilitates comparison with
other research
? Stable, real-time adaptive controller development
? Compensates for mass property uncertainty
? Compensates for thruster performance uncertainty
? Compensates for unmodeled dynamics (gravity and solar pressure)
? Modified adaptive control law
? Removes need for absolute position information
1.6.1 List of Publications
Luquette, R. J. and Sanner, R. M., ?A Nonlinear Approach to Spacecraft
Trajectory Control in the Vicinity of a Libration Point,? Proceedings of the Flight
Mechanics Symposium, Goddard Space Flight Center, June 2001.
Luquette, R. J. and Sanner, R. M., ?A Nonlinear Approach to Spacecraft
Formation Control in the Vicinity of a Collinear Libration Point,? Proceedings of
the AAS/AIAA Astrodynamics Specialist Conference, July 2001, Paper No. AAS
01-330.
Luquette, R. J. and Sanner, R. M., ?A Nonlinear, Six-Degree of Freedom,
Precision Formation Control Algorithm, Based on Restricted Three Body
Dynamics,? 26th Annual AAS Guidance and Control Conference, February 2003,
Paper No. AAS 03-007.
28
Luquette, R. J. and Sanner, R. M., ?Linear State-Space Representation of the
Dynamics of Relative Motion, Based on Restricted Three Body Dynamics,? AIAA
Guidance, Navigation and Control Conference, August 2004, Paper No. AIAA-
2004-4783.
Luquette, R. J., Sanner, R. M., Leitner, J. and Gendreau, K. C., ?Formation
Control for the MAXIM Mission,? 2nd International Symposium on Formation
Flying, September 2004.
29
Chapter 2
Spacecraft Dynamics
This chapter reviews spacecraft orbital and attitude dynamics to provide the
context for the control design discussion in the following chapters. Section 2.1
presents the orbital dynamics with specific emphasis on the Restricted Three Body
Problem. The discussion includes linearized equations of motion about Libration
points and stability analysis. Section 2.2 discusses rigid body spacecraft kinematics
and dynamics. Section 2.3 discusses dynamic coupling between the orbital and
attitude dynamics.
2.1 Orbital Dynamics
As discussed in Chapter 1, the region surrounding the Earth/Moon-
Sun L2 point provides a favorable dynamic environment for precision formation
flying missions designed to probe deep space. Orbital dynamics in this region, as
well as near other libration points, are governed by gravity and solar pressure, plus
thruster action. Principal gravitational sources are the Sun and the Earth/Moon
system. For control design the Earth/Moon system is treated as a combined mass
located at the system barycenter. The spacecraft are comparably small such that
their mutual gravitational interaction is neglected. References for spacecraft orbital
dynamics include: [1, 2, 46, 68].
30
The dynamics associated with the gravity field of the Earth/Moon-Sun is
appropriately modeled by the Restricted Three Body Problem (RTBP). This section
begins with a review of the RTBP, followed by a discussion of the dynamics of relative
motion with the context of the RTBP.
2.1.1 Restricted Three Body Problem
The Restricted Three Body Problem (RTBP) examines the behavior of an
infinitesimal test mass in the combined gravitational field of two finite masses
orbiting their common center of mass, Figure 2.1. The description and analysis
of this problem appear in many texts. Reference [68] is the principal source for
this discussion. In the most general form of the problem the two primary masses
(primaries) follow elliptical trajectories. Analysis is greatly simplified by assuming
circular trajectories of the primaries about their barycenter. Euler originally
formulated the Circular Restricted Three Body Problem (CRTBP) in 1772.
The structure of the CRTBP is depicted in Figure 2.2. The coordinate system
is constructed so the X-axis passes through the center of each mass with the larger
mass located in the positive X-direction. The Z-axis is in the direction of angular
velocity of the two primary masses about the barycenter. The Y-axis completes the
triad forming a right handed coordinate frame.
The equations of motion in inertial coordinates for the test mass within the
context of CRTBP dynamics is expressed as:
m?r = ??1m r1||r
1||
3 ??2m
r2
||r2||3
31
Figure 2.1: General Restricted Three Body Problem, Plane of Rotation
Where:
?i = GMi, Gravitational Parameter for Primary Mass, Mi
ri ? Infinitesimal Mass Position from Primary Mass, Mi
m ? Mass of Infinitesimal Test Mass/Spacecraft
Transforming the equations of motion into the rotating frame of the CRTBP
provides a natural framework for evaluating the dynamic behavior of the test mass.
?X ?2n ?Y ?n2X = ??1(X?D1)||r
1||3
? ?2(X+D2)||r
2||3
?Y + 2n ?X ?n2Y = ? ?1Y||r
1||3
? ?2Y||r
2||3
?Z = ? ?1Z||r
1||3
? ?2Z||r
2||3
(2.1)
32
Figure 2.2: Circular Restricted Three Body Problem
Where:
D ? Distance Between Primaries, Constant
Di ? Distance of Primary Mass, Mi, from the Barycenter, Constant
n = radicalbig(?1 +?2)/D3, Angular Velocity of Primaries about Barycenter
R = Xi+Yj +Zk, Location of Infinitesimal Test Mass/Spacecraft
Introducing non-dimensional parameters facilitates analysis and maintains
generalized results. Non-dimensional parameters are defined such that D, (M1+M2),
and n have unity values. Defining ? = M2M
1+M2
, the additional parameters assume
the following values:
?1 = (1??) ?2 = ?
D1 = ? D2 = (1??)
33
Using the non-dimensional parameters, Equation 2.1 is expressed as:
?X ?2 ?Y ?X = ?(1??)(X??)||r
1||3
? ?(X+1??)||r
2||3
?Y + 2 ?X ?Y = ?(1??)Y||r
1||3
? ?Y||r
2||3
?Z = ?(1??)Z||r
1||3
? ?Z||r
2||3
(2.2)
With: ||r1|| = radicalbig(X ??)2 +Y 2 +Z2, and ||r2|| = radicalbig(X + 1??)2 +Y 2 +Z2.
2.1.1.1 Libration Points
The dynamics expressed in Equation 2.2 yield natural equilibrium points,
generally termed Libration Points, also Lagrangian Points. The equilibrium point
locations are computed by setting the derivative terms in Equation 2.2 to zero, and
solving for X, Y and Z. The defining equations for the equilibrium points are:
X = (1??)(X??)||r
1||3
+ ?(X+1??)||r
2||3
(a)
Y = (1??)Y||r
1||3
+ ?Y||r
2||3
(b)
0 = (1??)Z||r
1||3
+ ?Z||r
2||3
(c)
(2.3)
Equation 2.3c yields, Z = 0. Hence all the equilibrium points lie in the X?Y
plane. The location of these points are depicted in Figure 2.3. The points are
divided into two groups. The collinear points L1, L2 and L3 lie along the X-
axis. The location of the collinear points is governed by the value of ?, determined
by solving Equation 2.3a with Y = Z = 0. The equation does not lend itself
to an analytic solution, so numerical methods are required. The Triangular (also,
Equilateral) Points, L4 and L5, are at the apex of an equilateral triangle with the
34
Figure 2.3: Libration Point Locations for Circular Restricted Three Body Problem
base defined as the line between M1 and M2. As geometry governs the location of
the Triangular Points, their locations are readily determined as: [(?? 12),?
?3
2 ,0].
2.1.1.2 Stability of Libration Points
Stability properties of the Libration Points are determined by examining the
characteristics of the linearized equations of motion about each point. Denoting the
location of an equilibrium point as: [X0,Y0,Z0], the coordinates of the test mass are
defined as: [X0 +x,Y0 +y,Z0 +z]. The linearized equations of motion relative to a
Libration Point are given by:
35
?x?2?y ?x = ?
braceleftBig
(1??)
bracketleftBig
1
||R1||3 ?3
(X0??)2
||R1||5
bracketrightBig
+?
bracketleftBig
1
||R2||3 ?3
(X0+1??)2
||R2||5
bracketrightBigbracerightBig
x
+
braceleftBig
3(1??)(X0??)Y0||R
1||5
+ 3?(X0+1??)Y0||R
2||5
bracerightBig
y
?y + 2?x?y =
braceleftBig
3(1??)(X0??)Y0||R
1||5
+ 3?(X0+1??)Y0||R
2||5
bracerightBig
x
?
braceleftBig
(1??)
bracketleftBig
1
||R1||3 ?3
(X0??)2
||R1||5
bracketrightBig
+?
bracketleftBig
1
||R2||3 ?3
(X0+1??)2
||R2||5
bracketrightBigbracerightBig
y
?z = ?
braceleftBig
(1??)
||R1||5 ?
?
||R2||5
bracerightBig
z
(2.4)
With: ||R1|| = radicalbig(X0 ??)2 +Y 20 , and ||R2|| = radicalbig(X0 + 1??)2 +Y 20 .
Inspection of Equation 2.4 reveals coupling in the X-Y plane motion. The
motion along the Z-axis is uncoupled from the X-Y motion. If excited, Z-axis
dynamics at each point exhibit undamped harmonic motion with a frequency of
?Z =
radicalBig
(1??)
||R1||5 ?
?
||R2||5. Understanding the natural motion in the X-Y plane requires
separate analysis for the Triangular Points and the Collinear Points.
The stability of the Triangular Points is evaluated by substituting their
location, [(?? 12),?
?3
2 ,0], into Equation 2.4, yielding the matrix form of equations
for in-plane motion:
?? = A
T ?
(2.5)
Where:
? =
?
??
??
??
??
??
?
x
y
?x
?y
?
??
??
??
??
??
?
; AT =
?
??
??
??
??
??
?
0 0 1 0
0 0 0 1
3
4
3?3
2 (??
1
2) 0 2
3?3
2 (??
1
2)
9
4 ?2 0
?
??
??
??
??
??
?
36
Stability properties are determined by the roots of the characteristic equation:
det(? I?AT) = 0, which reduces to: ?4 +?2 + 274 ?(1??) = 0
The roots, given by: ? = ?
radicalBig
?1?
?
1?27?(1??)
2 , are purely imaginary for ? ? 0.033852
or? ? 0.96148. For 0.033852 < ? < 0.96148, the roots are complex pairs with one set
in the right hand plane (unstable), and the other set in the left hand plane (stable).
The Triangular Points are stable for both the Earth-Moon system (? = 0.01215)
and the Sun - Earth/Moon system (? = 3.0404e?006).
The location of the Collinear Points is expressed as: [X0,0,0]. Substituting
the coordinates into Equation 2.4, yields:
?? = A
C ?
(2.6)
Where:
? =
?
??
??
??
??
??
?
x
y
?x
?y
?
??
??
??
??
??
?
; AC =
?
??
??
??
??
??
?
0 0 1 0
0 0 0 1
(2? + 1) 0 0 2
0 ?(? ?1) ?2 0
?
??
??
??
??
??
?
Where: ? = (1??)(X
0??)3
+ ?(X
0+1??)3
The characteristic equation is:
det(? I?AC) = 0, expressed as: ?4 ?(? ?2)?2 ?(2? + 1)(? ?1) = 0
The roots, given by: ? = ?
radicalBig
(??2)?
?
(??2)2+4(2?+1)(??1)
2 , consist of two purely
imaginary roots and two real roots, symmetric about the origin. Hence, the Collinear
Points are unstable for all values of ?.
37
The orbit of the Earth/Moon barycenter about the sun is nearly circular with
an eccentricity of 0.0167. Therefore, the CRTBP provides a reasonable model of the
dynamics associated with the Libration Points for the Sun/Earth-Moon System. As
previously stated, reference [68] is the principal source for the material related to
the CRTBP.
2.1.2 Dynamics of Relative Motion in the Restricted Three Body
Problem
A typical two spacecraft formation is depicted in Figure 2.4. The spacecraft are
designated Leader and Follower. In this scenario, the Leader spacecraft is intended
to follow a ballistic trajectory with infrequent control for orbit maintenance. Control
is only applied to the Follower spacecraft to maintain a specified trajectory relative
to the Leader spacecraft.
A linear model of the relative dynamics sets the framework for linear control
design. Libration points represent natural locations for linearizing the dynamics
with the context of the RTBP. The linearized dynamics relative to a libration point
are presented in the previous section as part of the stability analysis. However,
applying this form of the linearized dynamics to the problem of spacecraft relative
motion limits the validity and utility of the model to regions within close proximity
of a libration point. Linearizing the dynamics about a reference spacecraft provides
a generalized solution, applicable to any trajectory within the context of the RTBP.
The development begins with the nonlinear equations of relative motion.
38
Figure 2.4: Two Spacecraft Orbiting in the RTBP Frame
The equations of relative motion in inertial coordinates are obtained by
differencing the equations of motion for the Follower and Leader.
?rF = ??1 r1F||r
1F ||3
??2 r2F||r
2F ||3
+uthrust,F
?rL = ??1 r1L||r
1L||3
??2 r2L||r
2L||3
?x = (??1 r1F||r
1F ||3
??2 r2F||r
2F ||3
+uthrust,F)?(??1 r1L||r
1L||3
??2 r2L||r
2L||3
)
= ?
braceleftBig
?1
||r1F ||3 +
?2
||r2F ||3
bracerightBig
x??1
braceleftBig
1
||r1F ||3 ?
1
||r1L||3
bracerightBig
r1L
??2
braceleftBig
1
||r2F ||3 ?
1
||r2L||3
bracerightBig
r2L +uthrust,F
(2.7)
Equation 2.7 represents the full nonlinear dynamics of relative motion. As
discussed, linearization of the relative dynamics about the Leader (reference)
spacecraft position sets the framework for linear control design. The
Leader/Follower separation is assumed to be much smaller than the distance of
39
the Leader (or Follower) to either of the primary masses, ||x|| << ||r1L|| and
||x|| << ||r2L||. As the first step, several terms on the right hand side of this
expression are examined, beginning with the term
braceleftBig
1
||r1F ||3 ?
1
||r1L||3
bracerightBig
braceleftBig
1
||r1F ||3 ?
1
||r1L||3
bracerightBig
=
braceleftBig
1
||r1L+x||3 ?
1
||r1L||3
bracerightBig
=
braceleftBig
1
[(r1L+x)?(r1L+x)]3/2 ?
1
||r1L||3
bracerightBig
=
braceleftBig
[(||r1L||2 +||x||2) + 2 (r1L ?x)]?3/2 ?||r1L||?3
bracerightBig
=
braceleftBiggbracketleftbigg
(1 +
parenleftBig
||x||
||r1L||
parenrightBig2
+ 2 (r1L?x)||r
1L||2
bracketrightbigg?3/2
?1
bracerightBigg
||r1L||?3
(2.8)
As assumed ||x|| << ||r1L||,
1 +
parenleftBig
||x||
||r1L||
parenrightBig2
+ 2 (r1L?x)||r
1L||2
? 1 + 2 (r1L?x)||r
1L||2
(2.9)
Substitute Equation 2.9 in the final expression of Equation 2.8, then apply
binomial expansion to first order.
braceleftBig
1
||r1F ||3 ?
1
||r1L||3
bracerightBig
?
braceleftbiggbracketleftBig
1 + 2 (r1L?x)||r
1L||2
bracketrightBig?3/2
?1
bracerightbigg
||r1L||?3
=
braceleftBigbracketleftBig
1 + (?32)
parenleftBig
2 (r1L?x)||r
1L||2
parenrightBig
+H.O.T.
bracketrightBig
?1
bracerightBig
||r1L||?3
??3 (r1L ?x)||r1L||?5
(2.10)
Likewise:
braceleftBig
1
||r2F ||3 ?
1
||r2L||3
bracerightBig
??3 (r2L ?x)||r2L||?5 (2.11)
40
Combining Equations 2.10 and 2.11 yields.
braceleftBig
?1
||r1F ||3 +
?2
||r2F ||3
bracerightBig
? ?1||r
1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBig (2.12)
Substituting Equations 2.10, 2.11 and 2.12 into Equation 2.7.
?x ??
braceleftBig
?1
||r1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBigbracerightBig
x
+
braceleftBig
3 ?1||r
1L||5
(r1L ?x)
bracerightBig
r1L +
braceleftBig
3 ?2||r
2L||5
(r2L ?x)
bracerightBig
r2L
+uthrust,F
(2.13)
Note: (r1L ?x) r1L = r1L (r1LT x) = [r1L r1LT]x
Rewrite Equation 2.13 as:
?x ??
braceleftBig
?1
||r1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBigbracerightBig
x
+3 ?1||r
1L||5
braceleftbig[r
1L r1L
T] xbracerightbig+ 3 ?2
||r2L||5
braceleftbig[r
2L r2L
T] xbracerightbig
+uthrust,F
=
braceleftBig
?
parenleftBig
?1
||r1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBigparenrightBig
I3
+3 ?1||r
1L||5
[r1L r1LT] + 3 ?2||r
2L||5
[r2L r2LT]
bracerightBig
x
+uthrust,F
(2.14)
In summary the linearized dynamics are expressed as:
?x = I?(t) x+uthrust,F (2.15)
Where:
I?(t) =
braceleftBig
?
parenleftBig
?1
||r1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBigparenrightBig
I3
+3 ?1||r
1L||5
[r1L r1LT] + 3 ?2||r
2L||5
[r2L r2LT]
bracerightBig
=
braceleftBig
?
parenleftBig
?1
||r1L||3
bracketleftBig
1? 3 (r1L?x)||r
1L||2
bracketrightBig
+ ?2||r
2L||3
bracketleftBig
1? 3 (r2L?x)||r
2L||2
bracketrightBigparenrightBig
I3
+ 3 ?1||r
1L||3
[e1L e1LT] + 3 ?2||r
2L||3
[e2L e2LT]
bracerightBig
41
Note: e1L and e2L denote unit vectors along r1L and r2L, respectively.
Finally, recalling the assumption, ||x|| << ||r1L|| and ||x|| << ||r2L||,
Equation 2.15 becomes:
I?(t) =
braceleftBig
?
parenleftBig
?1
||r1L||3 +
?2
||r2L||3
parenrightBig
I3 + 3 ?1||r
1L||3
[e1L e1LT]
+ 3 ?2||r
2L||3
[e2L e2LT]
bracerightBig (2.16)
Note: The time variation in I?(t) are due to the relatively slow variation in the
location of the Leader spacecraft relative to the two primaries of the RTBP.
For convenience, the expression for I?(t) is consolidated in terms of coefficients
c1(t) and c2(t).
I?(t) = braceleftbig?(c
1(t) +c2(t)) I3 + 3 c1(t) [e1L(t) e1L(t)T] + 3 c2(t) [e2L(t) e2L(t)T]
bracerightbig
c1(t) = ?1||r1L(t)||?3
c2(t) = ?2||r2L(t)||?3
(2.17)
The linearized dynamics (inertial coordinates) in matrix form are:
?
??
?
?x
?x
?
??
? =
?
??
?
0 I3
I?(t) 0
?
??
?
?
??
?
x
?x
?
??
?+
?
??
?
0
I3
?
??
?uthrust,F (2.18)
2.1.2.1 Dynamics in Rotational Frame/Validation
It is instructive to compare the general derivation above with the (simpler)
known linearized solutions for the CRTBP at a Libration Point. If valid, evaluating
Equation 2.18 with the Leader located at one of the equilibrium points must yield
42
the same linearized dynamics. Expressing the dynamics in the rotating CRTBP
frame is the first step.
Let DRI(t) express the transformation of a vector from the inertial frame
to the RTBP rotating frame. Assuming all terms are differentiable, the following
expressions relate position, velocity and acceleration between the two frames.
Ix = D
RI(t)
T Rx
I ?x = ?D
RI(t)
T Rx+D
RI(t)
T R ?x
I?x = ?D
RI(t)
T Rx+ 2 ?D
RI(t)
T R ?x+D
RI(t)
T R?x
(2.19)
Equation 2.19 requires the kinematics of DRI(t), expressed as:
?D
RI(t) = ?[n?]DRI(t)
?D
RI(t) = ?[?n?]DRI(t) + [n?][n?]DRI(t)
(2.20)
primenprime represents the angular rate of the rotating frame, depicted in Figure 2.4.
[n?] is the skew symmetric matrix formed by the vector, n, defined so that the
expression [x?]y, is the cross product equivalent of: x? y.
43
Combining Equations 2.18 and 2.19:
?
??
?
R ?x
R?x
?
??
? =
?
??
?
0 I3
{R?(t)?DRI(t) ?DRI(t)T} {?2 DRI(t) ?DRI(t)T}
?
??
?
?
??
?
Rx
R ?x
?
??
?
+
?
??
?
0
I3
?
??
?
Ru
thrust,F
Where:
R?(t) = {D
RI(t)
I?(t)D
RI(t)
T}
= ?
parenleftBig
?1
||r1L||3 +
?2
||r2L||3
parenrightBig
I3 + 3 ?1||r
1L||3
[Re1L Re1LT]
+ 3 ?2||r
2L||3
[Re2L Re2LT]
(2.21)
Combining Equations 2.20 and 2.21, and noting [n?]T = ?[n?]:
?
??
?
R ?x
R?x
?
??
? =
?
??
?
0 I3
{R?(t) + [?n?]?[n?][n?]} ?2 [n?]
?
??
?
?
??
?
Rx
R ?x
?
??
?
+
?
??
?
0
I3
?
??
?
Ru
thrust,F
(2.22)
For the CRTBP, ?n = 0, simplifying Equation 2.22:
?
??
?
R ?x
R?x
?
??
? =
?
??
?
0 I3
{R?(t)?[n?][n?]} ?2 [n?]
?
??
?
?
??
?
Rx
R ?x
?
??
?+
?
??
?
0
I3
?
??
?
Ru
thrust,F (2.23)
44
R?(t) = ?
parenleftBig
?1
||r1L||3 +
?2
||r2L||3
parenrightBig
I3 + 3 ?1||r
1L||3
[Re1L Re1LT] + 3 ?2||r
2L||3
[Re2L Re2LT]
2.1.2.2 Linearization with Leader at a Collinear Libration Point
Using the dimensionless parameters of the CRTBP, the Leader is placed at one
of the Collinear Points, ?1 = (1??) and ?2 = ?. Also, the unit vectors,Re1L and
Re
2L, lie along the X-axis. The coordinates are [X0,0,0]. Ranges to the primary
masses are: ||r1L|| = X0 ?? and ||r2L|| = X0 + 1??. The angular rate, primenprime, has a
value of unity, and is directed along the Z-axis. Therefore:
[Re1L Re1LT] = [Re2L Re2LT] =
?
??
??
1 0 0
0 0 0
0 0 0
?
??
??;[n?][n?] =
?
??
??
?1 0 0
0 ?1 0
0 0 0
?
??
??
? ?
parenleftBig
?1
||r1L||3 +
?2
||r2L||3
parenrightBig
=
parenleftBig
(1??)
(X0??)3 +
?
(X0+1??)3
parenrightBig
Substituting the expressions into Equation 2.23, and neglecting the control
force, yields:
?
??
??
??
??
??
??
??
??
??
?
?x
?y
?z
?x
?y
?z
?
??
??
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
??
??
?
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
(2? + 1) 0 0 0 2 0
0 ?(? ?1) 0 ?2 0 0
0 0 ?(? ?1) 0 0 0
?
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??
??
??
??
??
?
x
y
z
?x
?y
?z
?
??
??
??
??
??
??
??
??
??
?
(2.24)
45
The result, Equation 2.24, is identical to the linearized dynamics at the
Collinear Libration Points, expressed in Equation 2.6 for the X-Y plane motion, and
Equation 2.4 for the Z-axis motion. A similar analysis demonstrates the equivalence
of the linearized equations of motion near the Triangular Libration Points.
2.1.2.3 Linearization with a Single Primary Mass
Another interesting simplification verifies that the general linearized equations
of relative motion, Equations 2.17 and 2.18, collapse, as a special case, to the well
known Hill?s (Chlohessy-Wiltshire) equations of relative motion. Hill?s equations
are linearized equations of motion about a reference mass in a circular orbit about
a single, central mass. The coordinate frame is defined with the X-axis in the
radial direction, the Y-Axis along the instantaneous velocity vector, and the Z-axis
completes the triad of the right-handed coordinate frame. Equation 2.23 provides
an appropriate starting point, repeated here for reference.
?
??
?
R ?x
R?x
?
??
? =
?
??
?
0 I3
{R?(t)?[n?][n?]} ?2 [n?]
?
??
?
?
??
?
Rx
R ?x
?
??
?+
?
??
?
0
I3
?
??
?
Ru
thrust,F
R?(t) = ?
parenleftBig
?1
||r1L||3 +
?2
||r2L||3
parenrightBig
I3 + 3 ?1||r
1L||3
[Re1L Re1LT] + 3 ?2||r
2L||3
[Re2L Re2LT]
46
Transform the equation by setting ?2 = 0, and define primenprime as the orbital rate,
n =
radicalBig ?
1
||r1L||3. Then:
?
??
?
R ?x
R?x
?
??
? =
?
??
?
0 I3
{R?(t)?[n?][n?]} ?2 [n?]
?
??
?
?
??
?
Rx
R ?x
?
??
?+
?
??
?
0
I3
?
??
?
Ru
thrust,F
R?(t) = ?n2 I
3 + 3n2 [
Re
1L
Re
1L
T]
With:
[Re1L Re1LT] =
?
??
??
1 0 0
0 0 0
0 0 0
?
??
??;[n?] =
?
??
??
0 ?n 0
n 0 0
0 0 0
?
??
??;[n?][n?] =
?
??
??
?n2 0 0
0 ?n2 0
0 0 0
?
??
??
Consolidating the expression yields Hill?s (Chlohessy-Wiltshire) Equations, as
expected.
?
??
??
??
??
??
??
??
??
??
?
?x
?y
?z
?x
?y
?z
?
??
??
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
??
??
?
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
3n2 0 0 0 2n 0
0 0 0 ?2n 0 0
0 0 ?n2 0 0 0
?
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??
??
??
??
??
?
x
y
z
?x
?y
?z
?
??
??
??
??
??
??
??
??
??
?
(2.25)
2.1.3 Dynamics of the Earth/Moon-Sun System
As introduced at the beginning of this section, the goal is to model the
dynamics of a two spacecraft formation operating in the gravity environment of
47
the Earth/Moon-Sun, Figure 2.5. The Leader spacecraft is assumed to follow a
ballistic trajectory with infrequent control for orbit maintenance. Control is applied
to the Follower spacecraft to maintain a specified trajectory relative to the Leader
spacecraft.
Figure 2.5: Two Spacecraft Orbiting in the Earth/Moon - Sun Rotating Frame
2.1.3.1 Unperturbed Model
The principal gravitational sources are the Earth/Moon system and the Sun.
Perturbing forces, introduced by differential solar pressure and other gravitational
sources, are neglected for the moment. Based on reference vectors shown in
Figure 2.5 and Equation 2.7, the relative dynamics of the Follower spacecraft
referenced to the Leader are given by:
48
?x = ?{ ?em||r
EF ||3
+ ?s||r
SF ||3
}x??em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
??s{ 1||r
SF ||3
? 1||r
SL||3
}rSL +uthrust,F
(2.26)
Where:
r## ? Position vectors, depicted in Figure 2.5
?em ? Gravitational Parameter for Earth/Moon
?s ? Gravitational Parameter for Sun
uthrust,F ? External control force per unit mass applied to Follower
spacecraft
Equation (2.26) provides an exact expression of the dynamics of unperturbed
relative motion between the Follower and Leader spacecraft.
2.1.3.2 Perturbed Model
As mentioned, the dynamics represented by Equation (2.26) are perturbed by
forces associated with differential solar pressure and other gravitational sources.
Solar radiation generates a net force and torque on a spacecraft dependent on
the spacecraft geometry and surface reflectivity. In the most general case the net
force and torque are dependent on the spacecraft attitude. However for this analysis
the center of solar pressure is assumed to be aligned with the spacecraft center of
mass, resulting in zero net torque. Thus, a detailed discussion of solar torque is
omitted. Also, the spacecraft profile, and thus the solar pressure force, are assumed
attitude independent. The net force per unit mass generated by solar pressure is
49
expressed as:
fSolar = FSolarS/C Mass = Cr S/C AreaS/C Mass (SolarFlux)eSun to S/C (2.27)
Where:
Cr ? Spacecraft Coefficient of Reflectivity,
1 - perfectly absorbing, 2 - perfectly reflective
(SolarFlux) ? Solar flux at spacecraft position, (4.5?10?6N/m2)/D2,
D - distance from Sun in astronomical units.
eSun to S/C ? Unit vector point from the Sun to the S/C position
The differential acceleration due to solar pressure of the Follower spacecraft
relative to the Leader is expressed as:
?fsolar = fsolar,Follower ?fsolar,Leader (2.28)
The other principal perturbing force is generated by unmodeled gravitational
sources, i.e. the other planets. The full n-body gravitational model for relative
motion is expressed as:
?rF = ??1 r1F||r
1F ||3
??2 r2F||r
2F ||3
?summationtextni=3 ?i riF||r
iF ||3
+uthrust,F
?rL = ??1 r1L||r
1L||3
??2 r2L||r
2L||3
?summationtextni=3 ?i riL||r
iL||3
?x = ??1( r1F||r
1F ||3
? r1L||r
1L||3
)??2( r2F||r
2F ||3
? r2L||r
2L||3
)
?summationtextni=3 ?i( riF||r
iF ||3
? riL||r
iL||3
) +uthrust,F
(2.29)
50
Comparing Equations 2.7 and 2.29, the unmodeled differential gravitational
acceleration is given by:
?fpert = ?summationtextni=0 ?i( riF||r
iF ||3
? riL||r
iL||3
) (2.30)
Alternatively, the linearized model can be extended to include the full n-
body gravitational model. Following the derivation of Equations 2.15 and 2.16
the linearized dynamics for the n-body problem are expressed as:
?x = ?summationtextni=1 ?i( riF||r
iF ||3
? riL||r
iL||3
) +uthrust,F
= ?summationtextni=1 ?i
braceleftBig
( riF||r
iF ||3
? riL||r
iF ||3
)
bracerightBig
?summationtextni=1
braceleftBig
?i( riL||r
iL||3
? riL||r
iF ||3
)
bracerightBig
+uthrust,F
= ?summationtextni=1 ?i
braceleftBig
1
||riF ||3
bracerightBig
x?summationtextni=1
braceleftBig
?i( 1||r
iL||3
? 1||r
iF ||3
) riL
bracerightBig
+uthrust,F
? I?(t) x+uthrust,F
(2.31)
Where:
I?(t) = ?summationtextn
i=1
braceleftBig
?i
||riL||3
bracerightBig
I3 + 3summationtextni=1
braceleftBig
?i
||riL||3 [eiL eiL
T]
bracerightBig
Referring ahead to Section 5.3, the magnitude of the third body gravitational
effects are small compared to the gravity field of the two primary masses. Therefore,
due to the computational expense associated with directly incorporating n-body
effects in the linearized model, these effects are treated as a perturbation in the
dynamic model.
51
The unmodeled effects of solar pressure and gravitational perturbations in
Equation 2.7, yields modified dynamics expressed as:
?x = ?{ ?em||r
EF ||3
+ ?s||r
SF ||3
}x??em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
??s{ 1||r
SF ||3
? 1||r
SL||3
}rSL +?fsolar +?fpert +uthrust,F
(2.32)
The nature and magnitude of effects associated with differential solar pressure
and unmodeled gravity sources is further discussed in Section 5.1.
Equation 2.32 provides an exact expression of the dynamics of relative motion
between the Follower and Leader spacecraft. From Equations 2.17 and 2.18, the
linearized dynamics in matrix form are:
?
??
?
?x
?x
?
??
? =
?
??
?
0 I3
I?(t) 0
?
??
?
?
??
?
x
?x
?
??
?+
?
??
?
0
I3
?
??
?(uthrust,F) +Perturbations
Pertubations =
?
??
?
0
I3
?
??
?(?fsolar +?fpert)
(2.33)
2.2 Attitude Dynamics
Spacecraft attitude dynamics and control is the topic of many texts, including
[67, 68]. This section provides a general review of rigid body dynamics for
completeness.
52
2.2.1 Dynamics, rigid body - no wheels
The rotational dynamics of a rigid body spacecraft without reaction wheels is
given by:
HR ???[HR ??]? = ? (2.34)
Where:
? ? True Spacecraft Angular Rate
HR ? Spacecraft Moment of Inertia, constant
? ? Control torque applied to spacecraft
2.2.2 Kinematics
The kinematic equation of motion for a rotating body is given by:
?q = 12 Q(q) ?aug (2.35)
Where:
q= [??]T ? True Spacecraft Attitude Quaternion
? ? Vector Component of Attitude Quaternion
? ? Scalar Component of Attitude Quaternion
?aug= [? 0]T ? Augmented Rate Vector, Matches Quaternion Dimension
Q(q) =
?
??
?
? I3 + [??] ?
??T ?
?
??
?
53
2.2.3 Attitude Error
Attitude error is the difference between a true orientation and an actual
orientation of a body. Computing the attitude error is not as simple as differencing
the quaternions representing the true versus desired attitude. The computation
requires quaternion multiplication, ?q = [????]T = qcirclemultiplytextqd?1, evaluated as:
?q = Q(qd)T q (2.36)
2.2.4 Linearized Equations of Motion
With small body rates and attitude errors, the equations of motion can be
expressed in a linear form.
For ? ? 0, the term [HR ??] ? is neglected as a second order effect,
simplifying Equation 2.34 as:
? = HR ???[HR ??]? ?HR ?? (2.37)
The next goal is to reduce the kinematic expression of the attitude error to a
linear form.
From Equation 2.35 the attitude error kinematics are:
??q = 1
2 Q(?q) ??aug
54
Assuming the attitude error is small, the magnitude of the vector component
of ?q is small, ||??|| << 1. Also, the scalar component is approximately one, ?? ?
1. Therefore, for small attitude errors Q(?q) ? I4, allowing approximation of the
attitude kinematics as:
??q ? 12 ??aug = 12 {?aug ? ?d,aug} (2.38)
Equivalently:
??? ? 12 ?? = 12 {? ? ?d}
??? ? 0
(2.39)
Under the stated assumptions of small rates and small attitude errors,
combining Equations 2.37 and 2.39 yields the linearized equations for the error
dynamics of rotational motion:
?
??
?
???
???
?
??
? =
?
??
?
0 12 I3
0 0
?
??
?
?
??
?
??
??
?
??
?+
?
??
?
0
HR?1
?
??
? ? (2.40)
2.3 Coupled Orbit and Attitude Dynamics
Control implementation potentially introduces coupling between orbit and
attitude dynamics. While an ideal design allows independent orbit and attitude
control, real systems experience coupling due to imperfect actuation. For this
discussion actuator selection is limited to thrusters, fixed within the body frame
of the spacecraft with fully throttleable output. Net thrust and torque on the
55
spacecraft based on thruster output is expressed as:
bU =
?
??
?
R(q) (Iuthrust,F)
?
?
??
? = Bthrust Fthrust (2.41)
Where:
bU ? Combined, translation and rotation control in body coordinates
R(q) ? Transformation from inertial to body coordinates
Bthrust ? Control sensitivity matrix, defined in Equation 2.42
Fthrust ? Thruster output, [f1,f2,...,fn]T
Bthrust =
?
??
?
t1 t2 ??? ti ??? tn
d1 ?t1 d2 ?t2 ??? di ?ti ??? dn ?tn
?
??
? (2.42)
Where:
ti ? Unit vector in pointing direction of ith thruster, expressed in
S/C body coordinates
di ? Position vector of ith thruster relative to the S/C center of mass,
expressed in S/C body coordinates
Given a desired, bU , computation of the corresponding spacecraft thruster
output, Fthrust, is required. With a sufficient quantity of properly placed thrusters
the control sensitivity matrix, Bthrust, will have rank six, and thus have a pseudo
inverse.
56
The inverse mapping of Equation 2.41 from Bthrust to Fthrust is expressed as:
Fthrust = BthrustT(Bthrust BthrustT)?1 bU (2.43)
In general, Equation 2.43 yields a thrust vector, Fthrust, with negative
values. However, implementation constrains each element ofFthrust be non-negative,
assuming fixed thrusters. Therefore, it is necessary to bias the thruster output to
guarantee non-negative thrust commands. The topic of thruster biasing is deferred
to section 3.4.
57
Chapter 3
Linear and Nonlinear Nonadaptive Control Theory and Design
Building on the foundation of the dynamics for precision formation flying in
Chapter 2, this chapter develops control strategies based both on classical linear
control and nonlinear Lyapunov based control theory. As mentioned, the goal is to
design a coupled six-degree of freedom (6DOF) control law, addressing both orbital
trajectory control and attitude control. Section 3.1 reviews the theoretical basis for
various control design methods. Control strategies for formation flying are developed
in Sections 3.2 and 3.3. Section 3.2 develops a control algorithm based on linear
control theory. Section 3.3 develops a control algorithm based on nonlinear control
theory. Control strategies are designed separately for orbit and attitude dynamics.
Section 3.4 discusses implementing the combined control laws with a common set
of actuators (thrusters).
3.1 Control Theory
This section summarizes basic concepts of both linear and nonlinear control
theory. The summary is not exhaustive, nor is it possible to acknowledge the volumes
of material published on this topic. Basic texts on linear control theory include:
[9, 31, 49, 51, 69]. Texts on the control of nonlinear systems include [29, 33, 58].
58
3.1.1 Linear Systems
Although real systems manifest nonlinear dynamics, control engineers
commonly base designs on linearized dynamics, taking advantage of the strong
heritage and rich set of design methods and analysis tools available through linear
systems theory. The discussion begins with linear dynamic models followed by a
brief discussion of linear control design techniques.
3.1.1.1 State-Space Model
For the general case a dynamic system is represented by the following state
space model [51]:
?x = f(x,u,t), x(t0) = x0 (3.1)
y = h(x,u,t) (3.2)
Equation (3.1) is the system state equation. Equation (3.2) is the system output
equation.
In the case f(x,u,t) and h(x,u,t) are linear functions, the state space model
has the form:
?x = A(t) x+B(t) u, x(t0) = x0 (3.3)
y = C(t) x+D(t) u (3.4)
Real physical systems are most accurately represented by state space models
with the form of Equations (3.1) and (3.2). However, the control designer, seeking
the advantage of the extensive set of linear control design tools, often prefers a
59
linear model in the form of Equations (3.3) and (3.4). Therefore, it is necessary
to linearize Equations (3.1) and (3.2) about a desired operating point, generating
a system model with the form of Equations (3.3) and (3.4). A full discussion of
this procedure is provided in [51]. Assuming the system has an equilibrium point
for x = 0 and u = 0, i.e. ?x = f(0,0,t) = 0, choose x = 0 and u = 0 as
a nominal operating state. Form A(t), B(t), C(t) and D(t) with the following
Jacobian matrices:
A(t) = ?f(x,u,t)?x
vextendsinglevextendsingle
vextendsingle
x=0, u=0
B(t) = ?f(x,u,t)?u
vextendsinglevextendsingle
vextendsingle
x=0, u=0
(3.5)
C(t) = ?h(x,u,t)?x
vextendsinglevextendsingle
vextendsingle
x=0, u=0
D(t) = ?h(x,u,t)?u
vextendsinglevextendsingle
vextendsingle
x=0, u=0
(3.6)
As a further simplification, time dependence of the coefficient matrices is removed
by assuming a nominal value, i.e. A = A(t0), B = B(t0), C = C(t0), D = D(t0).
A system with constant coefficients is termed autonomous.
3.1.1.2 Linear System Control Theory
The state space model for an autonomous (time invariant) linear system is
expressed as:
?x = A x+B u, x(t0) = x0 (3.7)
y = C x+D u (3.8)
A, B, C, D are constant.
Numerous complete texts discuss control design methods for linear time
invariant (LTI) systems. For this research one method is sufficient to define a
60
performance baseline for evaluating nonlinear control techniques. The method
selected for this discussion is an optimal control technique, linear quadratic control.
Again, much is published on this topic. References [13, 31] are the principle sources
for this discussion.
The Linear Quadratic Regulator (LQR) design specifies the control input that
minimizes a quadratic cost function. Given a time invariant system modeled by
Equations (3.7) and (3.8), the cost function is defined as:
Jcost =
Tintegraldisplay
0
(?xTQ?x + uTRu)dt (3.9)
The LQR method specifies the control input, u, that minimizes Equation
(3.9), based on the design parameters: Q and R. These matrices serve as weighting
factors for desired system performance. The matrix Q weights the tracking error,
?x. The matrix R weights the control effort. Both, R and Q are symmetric, positive
definite matrices. The solution specifies u as:
u = ?K ?x, K = R?1 BT M (3.10)
Where M solves the following algebraic matrix Riccati equation:
M A+AT M ?M B R?1 BT M = ?Q (3.11)
The existence of the solution depends on the characteristics of the system
model. Specifically, the system must be stabilizable and detectable.[13] A system
is stabilizable and detectable if all unstable modes are controllable and observable.
Controllability implies that a control input can be specified to drive the system to
any state within a finite time. Observability implies the system output contains
61
sufficient information to determine the state of the system. Formal definition of
these terms appear in most of the cited texts on linear control theory.
3.1.2 Lyapunov Based Nonlinear Control
Nonlinear control strategy exploits the natural form of the system dynamics,
derived from either Hamiltonian or Euler-Lagrange dynamics. Lyapunov based
nonlinear control generates a linear response to the error dynamics while capitalizing
on the natural form of system dynamics. Lyapunov theory provides the framework
for analyzing the system response and stability for the control design. For a detailed
discussion of this method refer to [58].
3.1.2.1 Dynamics Model
The dynamics and kinematics for most physical systems can be expressed in
the following form, based on Hamiltonian or Euler-Lagrange dynamics.
Dynamics: H(?) ?? +C(?,?) ? +E(?,?) = u (3.12)
Kinematics: ?? = J(?) ? (3.13)
Where:
? - General Position Variable
? - General Velocity Variable
H(?)- Mass properties, H(?)= H(?)T> 0, ? ?
C(?,?)- Rotational Forces
E(?,?)- Environmental/Damping Forces
u- Input forces/torques
62
H(?) and C(?,?) are subject to the constraint [ ?H(?)?2 C(?,?)] is skew, which
is a consequence of the law of energy conservation.
With the simplifying assumption, ?? = ?, combine Equations 3.12 and 3.13:
H(?) ?? +C(?, ??) ?? +E(?, ??) = u (3.14)
3.1.2.2 Control Design Theory
The control design goal is to specify u, such that the system tracks a desired
trajectory, ?d(t). The following metrics are defined:
?? = (? - ?d), position tracking error
?- Design parameter, specified so ?= ?T> 0
??r(t) = ??d(t) - ? ??, reference velocity
s(t) = ??? + ? ?? = ?? - ??r, error metric
Then the control law is specified as:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)?KDs
With the design parameter, KD = KDT > 0
(3.15)
Equating Equations 3.14 and 3.15 yields:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)?KDs = H(?) ??+C(?, ??) ??+E(?, ??)
? H(?)[??? ??r] +C(?, ??)[ ??? ??r] +KDs = 0
? H(?) ?s+C(?, ??)s+KDs = 0
? H(?) ?s = ?[C(?, ??)+KD]s (3.16)
The closed-loop dynamics, Equation 3.16, drive the tracking error, ??, to zero.
63
Before proceeding with the proof, it is beneficial to recall the following corollary to
Barbalat?s Lemma [58].
If a scalar function V(x,t) satisfies the following conditions, then
?V(x,t) ? 0, as t ??
? V(x,t) is lower bounded
? ?V(x,t) is negative semidefinite
? ?V(x,t) is uniformly continuous in time
Note: The third condition is met if ?V(x,t) is bounded
The proof that ?? ? 0 is based on Lyapunov analysis [58]. Consider the
following Lyapunov function, and its derivative.
Define: V(s,t) = 12sTH(?)s? 0 (3.17)
? ?V(s,t) = sTH(?)?s+ 12sT ?H(?)s
Substitute: H(?) ?s = ?[C(?, ??)+KD]s, from Equation 3.16
? ?V(s,t) = ?sT[C(?, ??)+KD]s+ 12sT ?H(?)s
? ?V(s,t) = ?sTKDs+ 12sT[ ?H(?)?2 C(?, ??)]s
[ ?H(?)?2 C(?, ??)] is skew ? 12sT[ ?H(?)?2 C(?, ??)]s = 0
? ?V(s,t) = ?sTKDs < 0 ? snegationslash= 0, given: KD = KDT > 0 (3.18)
Since H(?) is positive definite, V(s,t), Equation ??. ?V(s,t) is negative
semidefinite, Equation 3.18. ?(t) and ?d(t) are assumed at least twice differentiable.
Therefore, ?V(s,t) is uniformly continuous in time. Thus, Barbalat?s Lemma
64
guarantees ?V(s,t) ? 0 as t ? 0, implying s(t) ? 0 as t ? 0. To prove stable
tracking, consider ??(t) as the output of the stable linear system, s(t) = ???(t)+? ??(t),
with s(t) as the input. It follows that s(t) ? 0 implies ??(t) ? 0, concluding the
proof. The control specified in Equation 3.15 guarantees global asymptotic tracking
stability for a system modeled by Equation 3.14. A similar procedure and proof
applies for certain systems with the more general form of kinematics expressed in
Equation 3.13.
This approach assumes perfect knowledge of the system dynamics and perfect
measurement of the system state, ?(t) and ??(t). This ideal is not encountered in
the control of real systems. Compensating for uncertainty in the system dynamics
is the topic of the next chapter.
3.2 Formation Flying at L2, Linear Control Design
3.2.1 Linearized Dynamics
This subsection summarizes the linearized dynamics of relative motion
associated with the formation flying problem, as developed in chapter 2.
3.2.1.1 Linearized Translation Dynamics
Neglecting disturbances, Equation 2.18 represents the linearized dynamics of
relative motion in regions governed by Restricted Three Body Problem dynamics.
65
The equation, applicable to the region surrounding a libration point, is expressed in
matrix form as:
?? = IA(t) ? +B (u
thrust,F ?uthrust,L)
(3.19)
Where:
? =
?
??
?
x
?x
?
??
?;
IA(t) =
?
??
?
0 I3
I?(t) 0
?
??
?; B =
?
??
?
0
I3
?
??
?
The thrust, uthrust,L, command for the Leader spacecraft is included for
completeness, but is considered zero. The expression for I?(t) is:
I?(t) = braceleftbig?(c
1 +c2) I3 + 3 c1 [eEL(t) eEL(t)T] + 3 c2 [eSL(t) eSL(t)T]
bracerightbig
c1 = ?em||rEL(t)||?3
c2 = ?s||rSL(t)||?3
3.2.2 Linear Translation Control
Equation 3.19 provides the basic translational dynamics for a linear control
system. However, there are two important issues to address in the design process.
First, the dynamics, although linear, are time varying. Can an LQR design, based on
linear time-invariant systems, be successfully implemented? Second, the dynamics
are linearized about the Leader spacecraft position, so the position/set point for the
Follower is offset from the system equilibrium point (Leader position). Therefore, the
controller must compensate for the gravity gradient between the spacecraft positions
in addition to other disturbances. The issues are addressed in reverse order. A
66
controller design, including gravity gradient compensation, is developed based on
the time invariant assumption. The robustness of the design is tested against the
true time varying nature of the dynamics. Note that the control design assumes
accurate knowledge of the relative position and velocity of the formation spacecraft.
Also, implementation of gain scheduling requires periodic update of the absolute
position of the Leader spacecraft.
3.2.2.1 PID Control Design
The controller is required to counteract the gravity gradient associated with
a setpoint offset between the Leader and Follower positions. Based on the mission
profiles discussed in Chapter 1, the offset is expected to remain fixed in inertial
coordinates during science observations. The thrust required to maintain the
offset position will follow a slowly changing dynamic profile. Therefore, adding a
sufficiently fast integrator to the control algorithm provides effective compensation
for the required offset thrust, as well as other steady state disturbances.
Including an integrator in the control design is accomplished by augmented
error dynamics evaluated at t0:
??
aug = Aaug ?aug +Baug (uthrust,F)
? = Caug ?aug
(3.20)
67
Where:
?aug =
?
??
?
v
?
?
??
? =
?
??
??
??
?
v
?x
??x
?
??
??
??
?
; Aaug =
?
??
??
??
?
0 I3 0
0 0 I3
0 I?(t0) 0
?
??
??
??
?
; Baug =
?
??
??
??
?
0
0
I3
?
??
??
??
?
3.2.2.2 Linear Controller Design
Using the time invariant augmented dynamics, the control law development
follows the Linear Quadratic Regulator (LQR) design, as discussed in Section 3.1.1.2.
The cost function is defined as:
Jcost =
Tintegraldisplay
0
braceleftbig [?
aug]
T Q?
aug + [uthrust,F]
T R u
thrust,F
bracerightbig dt (3.21)
The solution which minimizes Jcost is specified uthrust,F as:
uthrust,F = ?K ?aug, K = R?1 BaugT M (3.22)
Where M solves the following algebraic matrix Riccati equation:
M Aaug +AaugT M ?M Baug R?1 BaugT M = ?Q (3.23)
Following classical linear feedback control conventions, the negative of the
tracking error is defined as the input for the controller dynamics. The Follower
control is defined as the output. The controller dynamics and output equation are
expressed as:
?v = Acntrl v +Bcntrl (??)
uthrust,F = Ccntrl v +Dcntrl (??)
(3.24)
68
Where:
Acntrl = [ 0 ]; Bcntrl = [ I3 0 ]
Ccntrl = ?K
?
??
?
I3
06?3
?
??
? Dcntrl = K
?
??
?
03?6
I6
?
??
?
The compensator transfer function is expressed as:
Gc(s) = Ccntrl (sI)?1 Bcntrl +Dcntrl = K
?
??
??
??
?
?(s I3)?1 03?3
I3 03?3
03?3 I3
?
??
??
??
?
(3.25)
Define the plant model transfer function, Gm(s), based on Equation 3.19 with
A(t) evaluated at t0. An explicit expression for Gm(s) is derived in the next
section, reference equation 3.29. Based on the block diagram in Figure 3.1, the
closed-loop transfer function with the input/output defined as position vectors is
expressed as:
GCL(s) = [I3 0] {Gm(s)Gc(s)(I +Gm(s)Gc(s))?1} [I3 sI3]T (3.26)
Figure 3.1: Closed-Loop Control Block Diagram
69
Using the specific design parameters discussed in Chapter 5, the control law yields
the closed-loop gain profile similar to that in Figure 3.2. Note: the profile is identical
for all channels.
Figure 3.2: Closed-Loop Gain, all-channels, for GCL(s)
3.2.2.3 Robustness of Time Invariant Assumption
The time invariant assumption introduces a modeling error, expressed as a
multiplicative uncertainty in terms of the nominal plant model defined at t0, Gm(s);
the plant uncertainty, ?(s,t); and the time dependent plant model, G(s,t).
G(s,t) = (I +?(s,t)) Gm(s) (3.27)
70
Based on Equation 3.19, G(s,t) is evaluated as:
G(s,t) = C (sI?A(t))?1 B
=
bracketleftbigg
I3 0
bracketrightbigg
?
??
?
?
??
?
sI ?I3
?I?(t) sI
?
??
?
?
??
?
?1 ?
??
?
0
I3
?
??
?
=
bracketleftbigg
I3 0
bracketrightbigg
?
??
?
s(s2 I? I?(t))?1 (s2 I? I?(t))?1
I?(t)(s2 I? I?(t))?1 s(s2 I? I?(t))?1
?
??
?
?
??
?
0
I3
?
??
?
= (s2 I? I?(t))?1
(3.28)
Likewise, the nominal plant employed in the compensator design is expressed
as:
Gm(s) = (s2 I? I?(t0))?1 (3.29)
The modeling uncertainty is computed as:
?(s,t) = G(s,t)Gm(s)?1 ?I
= (s2 I? I?(t))?1 (s2 I? I?(t0))?I
(3.30)
Paraphrasing Theorem 2.4.4 of reference [24]: Given a plant modeled as
Equation 3.27 with a stabilizing controller Gc(s), the closed-loop system is well-
posed and internally stable for all ?(s,t) with 1?(?) > ?(GCL), provided ?(s,t) is
a rational transfer function such that G(s,t) and Gm(s) have the same number
of poles in the closed-right-half plane. GCL is the closed-loop gain of the nominal
plant from Equation 3.26.
71
Figure 3.3: Maximum Singular Value Comparison: Inverse Model Uncertainty 1?(?)
and Closed-Loop Gain
Based on the mission profile and control design presented in Chapter 5, Figure
3.3 superimposes the various gain profiles for 1?(?) with the closed-loop gain from
Figure 3.2. ?(?) is evaluated after holding the plant model, Gm(s), fixed for 1
day, 7 days, 14 days, and 28 days, as shown. The plot demonstrates the condition,
1
?(?) > ?(GCL), is met for all three cases for frequencies above 10
?6 rad/sec. For the
14 day case the gain at the lowest frequencies is approximately 0.77. The condition
is violated in all cases for a frequency of ? 4 ? 10?7 rad/sec. This frequency
represents the natural harmonic motion of the formation. For frequencies less than
10?7 rad/sec, the condition, 1?(?) > ?(GCL), is met for 14 days without a plant
model update, representing an upper bound on the required update interval.
In a strict sense the time invariant assumption does not guarantee a stable
72
closed-loop system. The problem is associated with the natural harmonic motion
between the Leader and Follower spacecraft. Examination of Equation 3.30 reveals
that G(s,t) is unbounded when s2 matches the eigenvalues of I?(t). The natural
motion has a period of approximately 6 months for missions stationed near the
Earth/Moon-Sun L2 point. Although the analysis is insufficient to guarantee
robustness against uncertainty, the results suggest the performance of the controller
would benefit with gain scheduled updates within a 15 day period. The time
interval between required updates is mission dependent. Mission specific analysis
is required to determine appropriate update intervals. Further analysis in Section
3.2.2.5 demonstrates the controller is robustly stable.
3.2.2.4 Robustness of Double Integrator Dynamic Model Assumption
As discussed in Section 1.4, references [45, 59, 65] employ double integrator
dynamics, i.e. zero gravity, for translation control design. Using the multiplicative
model, the plant uncertainty is depicted in Figure 3.4, evaluated at the end of 1 day
of propagation. The plant uncertainty profile is nearly identical when computed
after 0.1 days through 28 days of propagation. The condition, 1?(?) > ?(GCL), is
met only for frequencies above ? 6?10?7 rad sec?1. Therefore, as above, analysis
does not guarantee robust stability. However, the results show inclusion of the
gravity model in the control design improves system robustness.
73
Figure 3.4: Maximum Singular Value Comparison: Inverse Model Uncertainty, 1?(?),
for a Double Integrator Dynamics Model and Closed-Loop Gain, GCL(s)
3.2.2.5 Lyapunov Based Robustness Analysis
Analysis in Sections 3.2.2.3 and 3.2.2.4 shows the robustness criteria is met
for frequencies above the natural harmonic motion of the spacecraft formation.
However, the test fails for frequencies near (and below for some cases) the natural
system harmonic. Lyapunov theory provides an alternate test for robustness.
The closed-loop dynamics for the formation control problem are expressed in
terms of the augmented dynamics and controller design presented in Sections 3.2.2.1
and 3.2.2.2. The closed-loop dynamics are:
??
aug(t) = Aaug(t) ?aug(t)?Baug K ?aug(t)
(3.31)
Define ?Aaug(t) ?Aaug(t)?Aaug(t0), then rewrite Equation 3.31 as:
74
??
aug(t) = Aaug(t0) ?aug(t)?Baug K ?aug(t) +
?A
aug(t) ?aug(t)
= ACL(t0) ?aug(t) + ?Aaug(t) ?aug(t)
=
bracketleftBig
ACL(t0) + ?Aaug(t)
bracketrightBig
?aug(t)
(3.32)
Where: ACL(t0) ?Aaug(t0) ?Baug K
Robust stability is established with Lyapunov analysis. Define the
Lyapunov function V(?aug,t) = ?augT P ?aug with P = PT > 0, such that
P = PT > 0 satisfies:
ACL(t0)TP +PACL(t0) = ?I9 (3.33)
Equation 3.32 is asymptotically stable if the derivative of the Lyapunov
function is strictly negative. In fact, based on the quadratic nature of the
defined V(?aug,t), exponential stability is established by Theorem 4.10 from [33], if
?V(?
aug,t) < ?k||?aug||
2, where k is a positive constant.
Compute the Lyapunov function derivative.
?V(?
aug,t) =
??
aug
T P ?
aug +?aug
T P ??
aug
(3.34)
Combine Equations 3.32 and 3.34.
?V(?
aug,t) = ?aug
T
bracketleftBig
ACL(t0) + ?Aaug(t)
bracketrightBig
T P ?
aug
+?augT P
bracketleftBig
ACL(t0) + ?Aaug(t)
bracketrightBig
?aug
= ?augTbracketleftbigACL(t0)TP +PACL(t0)bracketrightbig?aug
+?augT
bracketleftBig ?
Aaug(t)TP +P ?Aaug(t)
bracketrightBig
?aug
(3.35)
Combine Equations 3.33 and 3.35.
75
?V(?
aug,t) = ??aug
T ?
aug +?aug
T
bracketleftBig ?
Aaug(t)TP +P ?Aaug(t)
bracketrightBig
?aug
= ?||?aug||2 + 2 ?augT P ?Aaug(t) ?aug
(3.36)
From the Cauchy-Schwartz Inequality:
?augT P ?Aaug(t) ?aug ? ||?aug||2 ||P|||| ?Aaug(t)|| (3.37)
Combine Equations 3.36 and 3.37.
?V(?
aug,t) ??||?aug||
2 + 2 ||?
aug||
2 ||P|||| ?A
aug(t)||
= ?||?aug||2 + 2 ||?aug||2 ??{P} ??{ ?Aaug(t)}
=
parenleftBig
?1 + 2 ??{P} ??{ ?Aaug(t)}
parenrightBig
||?aug||2
(3.38)
Where ??{P} represents the maximum singular value of P. Therefore,
2 ??{P} ??{ ?Aaug(t)} < 1 ? ?V(?aug,t0) < 0. Based on the benchmark
mission scenario, defined in Chapter 4,
bracketleftBig
2 ??{P} ??{ ?Aaug(t)}
bracketrightBig
? [0 1.5 ? 10?4]
for t ? [0 60 days]. Therefore, the controller design is robustly stable against
variations in Aaug(t).
The controller design based on double integrator dynamics exhibits similar
robustness characteristics with the variations in
bracketleftBig
2 ??{P} ??{ ?Aaug(t)}
bracketrightBig
in the range
[9?10?5 1.5?10?4] throughout the same 60-day period.
Based on this analysis, the Lyapunov function derivative is upper bounded by
?k||?aug||2 with 0 < k ? .999. Therefore, Equation 3.31 is exponentially stable.
Note, this analysis supports the validity of formulating a controller design based
on double integrator dynamics. However, it is important to consider compensation
76
for the real gravity gradient between spacecraft, which is accomplished with the
integrator component of the PID controller.
3.2.3 Linear Attitude Control
3.2.3.1 Linearized Attitude Dynamics
From Chapter 2, linearized attitude dynamics are developed with assumed
small body rates and small attitude errors. Assuming the output is equivalent to
the attitude state, from Equation 2.40 the dynamics are expressed in the form of
Equations 3.7 and 3.8 with:
A =
?
??
?
0 12 I3
0 0
?
??
?; B =
?
??
?
0
HR?1
?
??
?; C = I6; D = 0 (3.39)
3.2.3.2 Linear Attitude Control
The attitude control law is designed using the LQR method described in
Section 3.1.1.2 with the values ofAandB shown above. The control law is expressed
as:
? = ?K
?
??
?
?q
??
?
??
? with: K = R
?1 BT M (3.40)
77
Note: The attitude control design does not include an integrator based on the
assumed absence of disturbing torques. An integrator would serve to compensate
for a constant disturbance torque.
3.3 Formation Flying at L2, Nonlinear Control Design
Considering the available nonlinear control design techniques, the Lyapunov
based approach discussed in Section 3.1.2 is selected.
3.3.1 Summary of Spacecraft Dynamics
This subsection summarizes the linearized dynamics of relative motion
associated with the formation flying problem, as developed in chapter 2.
3.3.1.1 Translation
From Equation 2.7 the relative motion (translation) of two spacecraft operating
in the vicinity of a libration point is expressed as:
?x = ?{ ?em||r
EF ||3
+ ?s||r
SF ||3
}x??em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
??s{ 1||r
SF ||3
? 1||r
SL||3
}rSL +?fsolar +?fpert +uthrust,F
(3.41)
78
3.3.1.2 Attitude Dynamics
The rotational dynamics of a rigid body spacecraft without reaction wheels is
given by:
HR ???[HR ??]? = ? (3.42)
Where:
? ? True Spacecraft Angular Rate
HR ? Spacecraft Moment of Inertia, constant
3.3.1.3 Kinematics
The kinematic equation of motion for a rotating body is given by:
?q = 12
?
??
?
? I3 + [??]
??T
?
??
? ? (3.43)
Where:
q= [??]T ? True Spacecraft Attitude Quaternion
? ? Vector Component of Attitude Quaternion
? ? Scalar Component of Attitude Quaternion
3.3.2 Control Design
The control strategy is developed based on Lyapunov theory. Implementation
requires knowledge of the true (inertial reference) state of the Follower spacecraft,
including the absolute position of each spacecraft, the relative velocity, and the
79
spacecraft attitude and rate. As a minimum, the absolute position of the Leader
spacecraft is required. Additional state information may be required in the definition
of the desired state of the Follower. The control design assumes the Leader spacecraft
is following a ballistic trajectory, meaning there is no applied thrust. Also, the design
neglects unmodeled disturbances, and assumes perfect knowledge of the spacecraft
mass properties.
3.3.2.1 Translation
Comparing the form of Equation 3.12, and the dynamics expressed in Equation
3.41, provides:
? = x
? = ?x
H(?) = I3
C(?,?) = 0
E(?,?) = ?{ ?em||r
EF ||3
+ ?s||r
SF ||3
}???em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
??s{ 1||r
SF ||3
? 1||r
SL||3
}rSL
The control design follows from Equation 3.15, and is specified as:
u = I3 ?xr ?{ ?em||r
EF ||3
+ ?s||r
SF ||3
}x??em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
??s{ 1||r
SF ||3
? 1||r
SL||3
}rSL ?KDs
(3.44)
80
KD ? Design parameter, specified so KD = KDT > 0
?x = (x ? xd), position tracking error
?xr(t) = ?xd(t)?? ?x, reference velocity
? ? Design parameter, specified so? = ?T > 0
s(t) = ??x + ? ?x = ?x ? ?xr, error metric
3.3.2.2 Attitude
A globally stable control scheme based on Lyapunov control theory is provided
by [14]. The nonadaptive case is discussed in Section 3.1.2 of this document. The
adaptive case is discussed in Section 5.3.2 of this document. For the nonadaptive
case the control law is defined as:
? = HR ??r ?[HR ??]?r ?KRsR (3.45)
Where:
qd= [?d ?d]T ? Desired Spacecraft Attitude Quaternion
?q ? [?? ??]T = qcirclemultiplytextqd?1, error quaternion
?d ? Desired Spacecraft Angular Rate
?r ? (?d ??R ??), reference angular rate
sR ? (???r), angular rate error metric
?R ? Design parameter, constant, symmetric, positive definite matrix
KR ? Design parameter, symmetric, uniformly positive definite matrix
81
3.4 Combined Translation/Attitude Control Law
As discussed in Section 2.3, with proper thruster placement the control
sensitivity matrix, Bthrust, will have a pseudo inverse.
The desired thruster commands are computed using Equations 2.41 and 2.43,
repeated here for reference:
Fthrust = BthrustT(Bthrust BthrustT)?1 bU
Where:
bU =
?
??
?
R(q) (Iuthrust,F)
?
?
??
?
Assuming Bthrust represents the exact performance of the thrusters, this
algorithm generates thruster commands for the desired net thrust and torque for
translation and rotation control. However, the computation does not ensure the
thruster commands are non-negative, necessary for implementation. This problem
is overcome by biasing the thruster output with a command that generates zero
net thrust and torque. The bias command is designed such that BthrustFbias = 0,
which implies Fbias lies in the null space of Bthrust. Successful implementation of
this strategy requires a thruster configuration which meets two criteria.
First, as previously noted, thruster placement must result in a control
sensitivity matrix, Bthrust, with a pseudo inverse. For this coupled six degree of
freedom control application, Bthrust is required to have rank six.
Second, there must exist scalable thruster output levels such that the net force
and torque on the spacecraft is zero. As a simple example consider a configuration
82
with pairs of thrusters placed about a spacecraft. Simultaneous firing of each
thruster within a pair will generate zero net torque and force on the spacecraft. In
a more general design thrusters may be paired with different locations or configured
in larger subgroups. The important design consideration allows simultaneous firing
of the entire group and/or subgroups of thrusters with zero net force and torque on
the spacecraft. As noted above, these commands lie in the null space of Bthrust. The
set of all firing configurations, Fbias, that satisfy BthrustFbias = 0, can be reduced
to a set of basis vectors that spans the null space of Bthrust. The null space is
also determined as the space spanned by the eigenvectors associated with a zero
eigenvalue for the matrix, BthrustT Bthrust.
83
Formally stated, the requirements are:
1) (Bthrust BthrustT) is invertible.
2) There exists a basis, (v1,v2,...,vm) for the null space of Bthrust
such that the elements of each basis vector, vi, are non-negative. i.e. vij ? 0
Note: The basis vectors, vi, represent a logical thruster grouping for biasing
thrust commands.
As a final comment, selection of Fbias requires consideration of fuel efficiency.
The bias with minimum ||Fbias|| which generates a vector (Fthrust +Fbias) with all
non-negative terms is generally best.
84
Chapter 4
Simulation Results and Analysis for Linear and Nonadaptive Control
This chapter presents simulation results characterizing the performance of the
linear and nonlinear control algorithms presented in Chapter 3. Section 4.1 details
the simulation architecture. Section 4.2 discusses the scenario specific control design.
Section 4.3 presents simulation results for the distant formation with a nominal 100
kilometer range. Section 4.4 presents results for the close formation with a nominal
50 meter range. Section 4.5 discusses the effects of perturbations on the performance
of the linear and nonlinear methods.
4.1 Simulation Architecture
The simulation is designed to model a realistic dynamic environment for testing
and comparing the performance of various formation flying control strategies. The
simulation is constructed in MATLAB. Details of the simulation architecture follow.
4.1.1 Dynamic Environment
The dynamic environment model is limited to gravity and solar pressure. The
gravity field is generated using point mass models for the Sun, Earth, Moon and the
other planets. Higher order gravity models are not required due to the sufficiently
large range between the formation and each gravity source. Position vectors are
85
defined in the standard inertial, Earth centered J2000 coordinate system. The
positions of the Sun, Moon and other planets are computed from DE200 planetary
ephemeris data. The DE200 ephemeris file is maintained and published by NASA?s
Jet Propulsion Laboratory (JPL). Ephemeris data is stored as sets of Chebychev
polynomial coefficients associated with defined time intervals. Time intervals are
specified in Ephemeris Time. The process steps for computing position data are:
1) Convert spacecraft time to Ephemeris Time, 2) Find the corresponding set
of Chebychev coefficients for the Sun, Moon, and each planet, and 3) Evaluate
polynomials based on the computed Ephemeris Time. DE200 position data is
specified in J2000 coordinates with the origin at the solar system barycenter.
Therefore, it is necessary to transform the position data to an Earth centered
coordinate frame. The DE200 file also contains the astronomical data used to
generate the ephemeris data, including the gravitational parameters for each body.
The gravity field contributions for all planets are included in the simulation for
completeness, recognizing some have negligible influence. [30]
Solar pressure effects are included, except for simulated results presented in
Section 4.3 and 4.4, and as noted. The solar pressure model is based on a mean solar
energy flux of 1358 watts/meter2 at one astronomical unit (AU). The astronomical
unit definition is based on the nominal distance between the Earth and Sun. The
radiation flux at the spacecraft position is determined using an inverse-square rule
based on the spacecraft to Sun range. The force exerted on each spacecraft is aligned
along the Sun-spacecraft range vector, and is proportional to the cross-sectional
86
area/mass ratio, specified for each spacecraft. The cross-sectional area is assumed
attitude independent. Solar pressure is assumed torque-free, i.e., the center of solar
pressure is aligned through the spacecraft center of mass.
4.1.2 Propagation
Each spacecraft position is propagated using a fourth-order Runga-Kutta
numerical integrator with a step size of 1 second. Propagation is based on the full
nonlinear equations of motion, including gravity, solar pressure and commanded
thrust. The gravitational acceleration is determined as the summation of the
contributions of each celestial body, computed using the inverse-square law for
point masses. Earth-centered J2000 coordinates represent an accelerated frame.
Therefore, the total acceleration is biased with the acceleration of the Earth relative
to the Sun, Moon and other planets.
4.1.2.1 Validation
The simulation model was validated against commercial software designed
to model spacecraft dynamics. The specific tools were Satellite Tool Kit [52] and
Freeflyer [18]. Simulations with common scenarios produced highly correlated results
between this design and the commercial products. The comparison did not include
closed-loop control, due to limitations in the commercial products. The algorithm
for interpreting the JPL DE200 file is a simplified version of code developed by this
author for flight qualified, ground system software.
87
4.1.3 Spacecraft Model
The spacecraft model is depicted in Figure 4.1. The propulsion system is
configured with three sets of four thrusters. Each set of four provides thrust along a
single body axis, and torque along a perpendicular body axis, such that thrust and
torque authority are available along each of the three body axes.
Figure 4.1: Simulation Model of Follower Spacecraft Depicting Thruster Layout
4.1.3.1 Leader Spacecraft
The Leader spacecraft is modeled with a pure ballistic trajectory, i.e. no
control. The inertial mass properties are set to match the Follower spacecraft,
but have no influence on the simulation results. The Leader has a mass of 1100
kilograms. The cross-sectional area for determining solar pressure is 6 meters2 with
a reflectivity of 1.4.
88
4.1.3.2 Follower Spacecraft
The Follower mass is 2200 kilograms. The cross-sectional area for determining
solar pressure is 8 meters2 with a reflectivity of 1.4. Thruster positions and
orientations are depicted Figure 4.1 and detailed in Table 4.1.
The inertia matrix for the Follower is given by:
HR =
?
??
??
??
?
200 10 5
10 300 15
5 15 200
?
??
??
??
?
kg m2 (4.1)
Table 4.1: Simulation Model Thruster Firing Direction and Location
Thruster Firing Thruster Location (meters)
Number Direction X-Axis Y-Axis Z-Axis
F1 -Y-axis 0 +0.5 -0.5
F2 -Y-axis 0 +0.5 +0.5
F3 +Y-axis 0 -0.5 -0.5
F4 +Y-axis 0 -0.5 +0.5
F5 -X-axis +0.5 -0.5 0
F6 -X-axis +0.5 +0.5 0
F7 +X-axis -0.5 -0.5 0
F8 +X-axis -0.5 +0.5 0
F9 -Z-axis -0.5 0 +0.5
F10 -Z-axis +0.5 0 +0.5
F11 +Z-axis -0.5 0 -0.5
F12 +Z-axis +0.5 0 -0.5
89
4.1.4 Thruster Biasing
As discussed in Section 3.4, the thruster layout must meet two criteria. For
this design the control sensitivity matrix is defined as:
Bthrust =
?
??
??
??
??
??
??
??
??
??
?
0 0 0 0 ?1 ?1 1 1 0 0 0 0
?1 ?1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 ?1 ?1 1 1
?0.5 0.5 0.5 ?0.5 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 ?0.5 0.5 0.5 ?0.5
0 0 0 0 ?0.5 0.5 0.5 ?0.5 0 0 0 0
?
??
??
??
??
??
??
??
??
??
?
It is straight forward to show (Bthrust BthrustT) is invertible.
One set of conforming basis vectors for Fbias is:
[ 1 0 1 0 0 0 0 0 0 0 0 0 ]T
[ 0 1 0 1 0 0 0 0 0 0 0 0 ]T
[ 0 0 0 0 1 0 1 0 0 0 0 0 ]T
[ 0 0 0 0 0 1 0 1 0 0 0 0 ]T
[ 0 0 0 0 0 0 0 0 1 0 1 0 ]T
[ 0 0 0 0 0 0 0 0 0 1 0 1 ]T
Therefore the selected thruster configuration is suitable for implementing full
6DOF control.
90
4.1.5 Initial Conditions
The simulation is based on the benchmark mission defined in [5]. Analysis
considers two scenarios to represent a close formation (50 meter separation) versus
a distant formation (100 kilometers). The nominal formation is initialized in
a stabilizable orbit about the L2 point in the Earth/Moon-Sun system with
a transverse (Y-axis) amplitude of 300,000 kilometers, and a normal (Z-axis)
amplitude of 200,000 kilometers. The initial state of the Leader is defined as:
Epoch: October 1, 2004, 12 noon, (UTC)
Position (km, J2000 coordinates, Earth Equator):
[1404758.1805532565, 103765.03812730288, 262972.11578260816]
Velocity (km/sec, J2000 coordinates, Earth Equator):
[-.063208398706927252, .36421914258386690, .16244834559859272]
The Follower is initialized with the same velocity vector as the Leader with
the position vector offset from the Leader position. The offset is scenario specific.
4.2 Control Design
Linear control gains are computed using MATLAB?s lqr(ALQR,B,Q,R)
routine. The algorithm determines the gain which minimizes the cost function
discussed in Section 3.1.1.2. ALQR is constructed with diagonal partitions
representing the translational and attitude dynamics, expressed in Equations 3.19
and 3.39, respectively.
91
The structure appears as:
ALQR =
?
??
?
Atranslation 0
0 Aattitude
?
??
?
Atranslation is computed from Equation 3.20, evaluated with initial state
properties. With Q and R defined as diagonal matrices, the LQR design generates
a gain matrix with five submatrices, structured as:
KLQR =
?
??
??
??
??
??
??
??
?
Ki,trans 0 0 0 0
0 Kp,trans 0 0 0
0 0 Kd,trans 0 0
0 0 0 Kp,att 0
0 0 0 0 Kd,att
?
??
??
??
??
??
??
??
?
The indices i, p and d designate the integral, proportional and derivative gains,
respectively.
The gains for the nonlinear controller, Equation 3.44, are computed to match
the linear control gains, minimizing gain selection as a factor in the comparative
analysis. Recall the nonlinear controller includes the term KDs = KD( ??? + ???).
In the nonlinear controller KD is the equivalent derivative gain, and KD? is the
equivalent proportional gain. Therefore, the nonlinear gains are computed as:
KD,trans = Kd,trans; ?T = [Kd,trans]?1Kp,trans
KD,att = Kd,att; ?R = [Kd,att]?1Kp,att
92
Note: There is no equivalent integrator gain, Ki, in the nonadaptive nonlinear
controller design. Excluding perturbations, the nonadaptive nonlinear control design
incorporates all the known dynamics, eliminating the need for an integrator feature.
Linear control integrator gain equivalence with the adaptive nonlinear control design
is discussed in Chapter 6.
The values of Q and R used in the LQR design are:
Q =
?
??
??
??
??
??
??
??
?
Qtrans,I I3 0 0 0 0
0 Qtrans,P I3 0 0 0
0 0 Qtrans,D I3 0 0
0 0 0 Qatt,P I3 0
0 0 0 0 Qatt,D I3
?
??
??
??
??
??
??
??
?
R =
?
??
?
Rtrans I3 0
0 Ratt I3
?
??
?
Where:
Qtrans,I = 10?4; Qtrans,P = 1; Qtrans,D = 1; Qatt,P = 103; Qatt,D = 103
Rtrans = 1; Ratt = 1
The specific selection of the values for Q and R is based on a series of trial
simulations. Considering the diagonal elements of Q: Qtrans,I, Qtrans,P and Qtrans,D
are the weighting on integrated position, position and velocity errors in translation,
93
respectively; and Qatt,P and Qatt,D are the weighting on the attitude error and rate
error, respectively. For R, Rtrans is the weighting on the translation control; and
Ratt is the weighting on the attitude control. Position/velocity error weighting,
Qtrans,P and Qtrans,D, balance the influence of observed errors in determining the
translation control. Likewise, attitude/rate error weighting, Qatt,P and Qatt,D,
balances the influence on the torque command. Weighting on the integrated
translation error, Qtrans,P, is selected to provide reasonable convergence and tracking
of the bias force due to gravity and solar pressure.
The LQR design algorithm is based on the values of A and B from the
system dynamics model. For this design coupling of the translation and attitude
dynamics is introduced only through B. However, with the specific thruster
design, translation/attitude coupling action has been eliminated. Translation and
attitude coupling occurs only if the thrusters are misaligned. From an LQR design
perspective uncoupled translation/attitude dynamics (ideal) implies the relative
weighting between the translation and attitude components of Q and R, has no
effect on the computed value of KLQR.
The closed-loop gain for the linear control design is presented in Section 3.2.2.2.
As discussed the system is considered well-posed and robustly stable provided the
control law is updated within a 14 day period.
The closed-loop gain profiles are shown in Figures 4.2 and 4.3.
94
Figure 4.2: Linear Translation Control: Closed-Loop Gain, all-channels
Figure 4.3: Linear Attitude Control: Closed-Loop Gain, all-channels
95
4.3 Scenario 1 ? Distant Formation, 100 Kilometer Range
For the distant formation the Follower position is initialized with a 95 kilometer
separation along the inertial X-axis. During the first segment of the simulation,
control is applied to maintain the relative position and attitude at the initial
values. Subsequently, three maneuvers are commanded: pure translation, pure
attitude, combined translation and attitude. Translation maneuvers are along the
direction of the initial range vector. Attitude maneuvers are rotations about a
common Euler axis. The translation and attitude maneuver profiles are depicted in
Figures 4.4 and 4.5, respectively. Excluding solar pressure effects, the ideal (perfect
tracking) control effort, based on the design scenario, is included in Figures 4.4
and 4.5. Table 4.3 includes the ideal fuel requirement. Solar pressure effects are
neglected in the simulation to demonstrate the unperturbed performance of each
controller. The linear controller employs the integrator to compensate for unmodeled
gravity sources. In contrast the nonlinear controller incorporates all gravity sources
controller design. The sequence of maneuvers are detailed in the table below.
COMMANDED MANEUVER SEQUENCE
Simulation Time Commanded Maneuver
0?5 minutes: Maintain Steady State
5?65 minutes: Increase Range to 100 km
65?75 minutes: Maintain Steady State
75?95 minutes: 90 Degree Attitude Slew
95?105 minutes: Maintain Steady State
105?165 minutes: Decrease Range to 90 km
-90 Degree Attitude Slew
165?175 minutes: Maintain Steady State
96
Figure 4.4: Profiles of Desired Leader/Follower Range along Separation Vector
(Inertial X-axis), and Ideal (Perfect Tracking) Control
Figure 4.5: Desired Slew Angle for Follower and Ideal (Perfect Tracking) Torque
Profile about Rotation Axis
97
The simulation demonstrates the performance advantage of the nonlinear
control design. Figure 4.6 depicts the translational trajectory tracking performance.
The linear controller exhibits a transient response at the beginning of each interval
of planned maneuver steps. In contrast the nonlinear controller, incorporating
desired state information, demonstrates significantly superior tracking, with a
mean error 105 times smaller than the linear controller. Specific performance
metrics are provided in Table 4.2. The position error fluctuations observed
for the nonlinear controller during the initial 100 minutes reflect the limit of
numerical precision for the simulation, approximately 10?8 meters. The translation
tracking performance for both controller types degrades during the combined
translation/attitude maneuver. This effect is attributed to the discrete nature of the
simulation. Translation commands assume the spacecraft maintains a fixed attitude
during each propagation interval. Discrete modeling of the simultaneous attitude
maneuver has the effect of generating a thruster misalignment which degrades the
tracking performance. A similar effect is observed in Figure 4.7 for attitude tracking.
Table 4.2: Scenario 1 ? Controller Tracking Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation < 10?8 ? 101 3.4?10?1 meters
Nonlinear Translation < 10?8 ? 10?5 2.4?10?6 meters
Linear Attitude 10?11 101 1.9?100 arcsec
Nonlinear Attitude < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
98
Figure 4.6: Range Error and Translation Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms
99
Figure 4.7: Attitude Error and Torque Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms
Figure 4.8 and Table 4.3 compare the fuel requirement for each controller.
The nonlinear controller performs with higher fuel efficiency. In fact the nonlinear
controller consumes slightly less fuel than required for perfect tracking. This benefit
is derived from the inability of the controller to perfectly track instantaneous changes
in the velocity/attitude rate profile. The instantaneous thrust profile of the linear
controller indicates the transient response is the principal factor in the degradation
in fuel efficiency.
100
Figure 4.8: Instantaneous Thrust Level and Fuel Consumption for Linear (solid)
and Nonlinear(dashed) Control Algorithms
Table 4.3: Scenario 1 ? Controller Fuel Performance Summary
Deviation from Ideal
Controller Type Fuel Consumption (Perfect Tracking)
(meters/second) (Percent)
Ideal Translation/Attitude 23.25535 ?
Linear Translation/Attitude 23.63285 +1.6%
Nonlinear Translation/Attitude 23.25498 -0.002%
101
4.4 Scenario 2 ? Close Formation, 50 Meter Range
For the close formation the Follower is initially positioned from the Leader
with a 75 meter range along the inertial X-axis. The maneuver scheme is similar
to Scenario 1. The attitude maneuver, depicted in Figure 4.5, is the same for both
scenarios. The translation maneuver follows the same time sequence with shorter
range changes, as shown in Figure 4.9. The sequence of maneuvers for the close
formation are tabulated as:
Simulation Time Commanded Maneuver
0?5 minutes: Maintain Steady State
5?65 minutes: Decrease Range to 50 meters
65?75 minutes: Maintain Steady State
75?95 minutes: 90 Degree Attitude Slew
95?105 minutes: Maintain Steady State
105?165 minutes: Increase Range to 100 meters
-90 Degree Attitude Slew
165?175 minutes: Maintain Steady State
102
Figure 4.9: Desired Leader/Follower Range and Ideal (Perfect Tracking) Control
Profile
The controller performance is shown in Figures 4.10, 4.11 and 4.12. Tables
4.4 and 4.5 provide numerical performance metrics. As expected, both controllers
demonstrate improved translation trajectory tracking compared to the results in
Section 4.3. Linear controller translation tracking improved by two orders of
magnitude. Nonlinear translation tracking improved by one order of magnitude.
The result is attributed to the significant reduction in the range change during
the maneuver, 10?s of meters versus kilometers. Similar degradation in translation
tracking performance is observed for both controllers during the combined
translation/attitude maneuver. Unaffected by relative spacecraft range, attitude
tracking performance is similar for both scenarios. While the overall fuel requirement
is reduced for scenario 2, both scenarios exhibit similar fuel efficiency relative to the
ideal (perfect tracking), Table 4.5.
103
Figure 4.10: Range Error and Translation Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms
104
Figure 4.11: Attitude Error and Torque Control Profile for Linear (solid) and
Nonlinear(dashed) Control Algorithms
Table 4.4: Scenario 2 ? Controller Tracking Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation < 10?8 ? 10?2 1.7?10?3 meters
Nonlinear Translation < 10?8 ? 10?6 2.4?10?7 meters
Linear Attitude < 10?11 101 1.9?100 arcsec
Nonlinear Attitude < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
105
Figure 4.12: Instantaneous Thrust Level and Fuel Consumption for Linear (solid)
and Nonlinear(dashed) Control Algorithms
Table 4.5: Scenario 2 ? Controller Fuel Performance Summary
Deviation from Ideal
Controller Type Fuel Consumption (Perfect Tracking)
Ideal Translation/Attitude 0.1182103 ?
Linear Translation/Attitude 0.1201096 +1.6%
Nonlinear Translation/Attitude 0.1182084 -0.002%
106
4.5 Perturbations
In an ideal environment both linear and nonadaptive nonlinear control
strategies provide good performance. This section examines the performance impact
due to unmodeled dynamics for Scenario 1, presented in Section 4.3. Perturbation
sources include: solar pressure, mass uncertainty and thruster placement/alignment
uncertainty. Additionally, the impact of assumed time invariant dynamics is assessed
by modeling maneuver performance 28 days after the scenario epoch. Chapter 6
details the modeling of each perturbation.
4.5.1 Compensation for Solar Pressure
Solar pressure generates an unmodeled differential acceleration between the
Leader and Follower spacecraft. The position tracking performance comparison for
the linear controller is presented in Figure 4.13 and Table 4.6. Compared to the
results in Section 4.3, there is no perceptible difference in the performance of the
linear controller with the addition of solar pressure effects. This result is attributed
to the inclusion of the integrator in the linear controller. The integrator provides
a generic adaptive mechanism that compensates for the differential acceleration
between the two spacecraft, a combined effect of a gravitational gradient and
differential solar pressure.
107
Figure 4.13: Range Error and Translation Control Profile for Linear Controller,
Perturbed with Solar Pressure (solid) and Unperturbed (dashed)
In contrast there is a significant impact of the position tracking performance
of the nonlinear controller as shown in Figure 4.14 and Table 4.6. An order
of magnitude increase is observed in the mean translation tracking error. The
nonadaptive nonlinear controller depends on a truth model of the environmental
forces. The controller lacks a mechanism to compensate for unmodeled differential
solar pressure effects. The nonlinear controller still outperforms the linear controller
by several orders of magnitude.
Despite the performance degradation the control effort follows a similar profile
108
for both the perturbed and unperturbed cases. This is expected since the control
effort is principally governed by the desired trajectory. Thus, with reasonable
tracking performance, the actual control profile must correlate with the ideal (perfect
tracking) control, Figures 4.4 and 4.5.
Figure 4.14: Range Error and Translation Control Profile for Nonlinear Controller,
Perturbed with Solar Pressure (solid) and Unperturbed (dashed)
As modeled, there is no torque generated by solar pressure, thus the attitude
tracking performance, Figure 4.14 and Table 4.6, for both the linear and nonlinear
controllers is unchanged from the results presented in Section 4.3.
109
Table 4.6: Scenario 1 with Solar Perturbation ? Controller Tracking
Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation < 10?8 ? 101 3.4?10?1 meters
Nonlinear Translation 10?5 10?5 1.6?10?5 meters
Linear Attitude < 10?11 101 1.9?100 arcsec
Nonlinear Attitude < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
4.5.2 Compensation for Mass Property Uncertainty
Uncertainty in the spacecraft mass properties is introduced as an error in the
overall spacecraft mass estimate, as well as the inertial properties. Modeling details
for mass uncertainty are discussed in Section 6.2.2. Uncertainty in the spacecraft
mass affects translation control performance, shown in Figure 4.15 and Table 4.7.
Uncertainty in the inertial mass properties affects attitude tracking
performance, shown in Figure 4.16 and Table 4.7.
As with solar pressure, the linear controller exhibits the ability to compensate
for mass uncertainty for both translation and attitude tracking, as evidenced by
the tracking performance metrics in Table 4.6. In contrast the performance of
the nonlinear controller is significantly degraded for both position and attitude
tracking. Nonlinear translation mean tracking error is increased by 5 orders of
110
magnitude, attitude tracking increases by 3 orders of magnitude. However, the
nonlinear controller still outperforms the linear controller based on mean tracking
error. The linear controller provides superior steady state tracking error, evidenced
by the minimum tracking error.
Figure 4.15: Range Error Profile Comparison with Mass Uncertainty Effects (solid)
and without (dashed) for Linear (top) and Nonlinear (bottom) Control Algorithms
111
Figure 4.16: Attitude Error Profile Comparison with Mass Uncertainty Effects
(solid) and without (dashed) for Linear (top) and Nonlinear (bottom) Control
Algorithms
112
Table 4.7: Scenario 1 with Mass Property Uncertainty ? Controller
Tracking Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation < 10?8 ? 101 3.4?10?1 meters
Nonlinear Translation 10?6 10?1 1.9?10?1 meters
Linear Attitude 10?11 101 1.9?100 arcsec
Nonlinear Attitude < 10?11 ? 100 2.4?10?1 arcsec
? Reflects limit of numerical precision.
4.5.3 Compensation for Thruster Misalignment/Misplacement
Errors in thruster alignment and placement specifications affect the mapping
of acceleration and torque control commands to thruster firing commands. As a
result the actual net acceleration and torque applied to the spacecraft deviates from
the desired values. Perturbed and unperturbed performance profiles for the linear
and nonlinear controllers are shown in Figure 4.17. Modeling details for thruster
misalignment/misplacement are discussed in Section 6.2.3.
113
Figure 4.17: Range Error Profile Comparison with Thruster
Misalignment/Performance Effects (solid) and without (dashed) for Linear
(top) and Nonlinear (bottom) Control Algorithms
The linear controller effectively compensates for thruster misalignment and
misplacement during the pure translation maneuver. Performance degradation
is manifested during sequences involving attitude maneuvers, indicating a
coupling action between translation and attitude control. The nonlinear
controller performance significantly degrades during the entire sequence. Thruster
misalignment/misplacement significantly impacts attitude tracking performance of
both the linear and nonlinear controllers, shown in Figure 4.18. Table 4.8 provides
114
controller performance metrics. Based on the mean error, both controllers have
comparable performance for both translation and attitude tracking. The linear
system provides an order of magnitude better steady state tracking based on the
minimum error. The maximum error for the nonlinear controller is approximately
an order of magnitude better than the linear controller.
Figure 4.18: Attitude Error Profile Comparison with Thruster
Misalignment/Performance Effects (solid) and without (dashed) for Linear
(top) and Nonlinear (bottom) Control Algorithms
115
Table 4.8: Scenario 1 with Thruster Alignment Uncertainty ? Controller
Tracking Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation 10?7 101 3.4?10?1 meters
Nonlinear Translation 10?6 100 2.3?10?1 meters
Linear Attitude 10?2 104 5.2?103 arcsec
Nonlinear Attitude 10?1 104 5.1?103 arcsec
? Reflects limit of numerical precision.
4.5.4 Compensation for Solar Pressure, Mass Properties, Thruster
Misalignment/Misplacement
For completeness, the performance of both the linear and nonlinear controllers
in the presence of all modeled disturbances is shown in Figures 4.19 and 4.20,
and Table 4.9. As expected from previous results, the minimum tracking error
indicates the linear controller possesses greater robustness to model uncertainty for
steady state tracking. Based on the mean error, the linear and nonlinear controllers
have equivalent performance. Comparing results presented in Tables 4.8 and 4.9,
thruster alignment/placement uncertainty is the dominant source of performance
degradation.
116
Figure 4.19: Range Error Profile Comparison with All Perturbation Effects (solid)
and without (dashed) for Linear (top) and Nonlinear (bottom) Control Algorithms
117
Figure 4.20: Attitude Error Profile Comparison with All Perturbation Effects (solid)
and without (dashed) for Linear (top) and Nonlinear (bottom) Control Algorithms
Table 4.9: Scenario 1 with Solar Pressure, Mass Properties, Thruster
Alignment/Position Uncertainty ? Controller Tracking Performance
Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation 10?8 101 3.4?10?1 meters
Nonlinear Translation 10?6 100 2.6?10?1 meters
Linear Attitude 10?2 104 5.2?103 arcsec
Nonlinear Attitude 10?2 104 5.1?103 arcsec
118
4.5.5 Performance Impact of Time Invariant Dynamics Model
As discussed in Section 3.2.2.3, the dynamics model for the linear control
design is assumed time invariant. The robustness analysis ensured stability provided
scheduled gain updates with a frequency of 14 days, or better. However, as a
practical matter, linear controller performance is not expected to degrade beyond
14 days, even without gain update. Due to the benign dynamic environment, the
natural dynamics do not influence the LQR gains. The LQR design yields diagonal
gain matrices for translation control. The gains are equal to those derived with
assumed double integrator dynamics. To demonstrate the point, Scenario 1 (with
perturbations) is initiated 28 days after the epoch for controller gain computation.
Tracking performance, Table 4.10, is unchanged. There is a slight fuel penalty, shown
in Table 4.11. Graphically the performance is equivalent to the results presented in
Figures 4.6, 4.7, and 4.8.
119
Table 4.10: Scenario 1, 28 days after Epoch ? Linear Controller Tracking
Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation (delay) < 10?8 ? 101 3.4?10?1 meters
Linear Translation (no delay) < 10?8 ? 101 3.4?10?1 meters
Linear Attitude (delay) < 10?11 ? 101 1.9?100 arcsec
Linear Attitude (no delay) 10?11 101 1.9?100 arcsec
? Reflects limit of numerical precision.
Table 4.11: Scenario 1 ? Linear Controller Fuel Performance Summary
Deviation from Ideal
Controller Type Fuel Consumption (Perfect Tracking)
Ideal Translation/Attitude 23.25535 ?
Linear Translation/Attitude (no delay) 23.63285 +1.6233%
Linear Translation/Attitude 23.63308 +1.6243%
120
4.5.6 Summary
The results in Sections 4.3 and 4.4 demonstrate the superior performance
of the nonlinear control in the absence of unmodeled dynamics. In contrast, the
performance characteristics for the linear and nonlinear controllers are mixed in
the presence of unmodeled effects due to the presence of an integrator. The
linear controller is more robust to the unmodeled dynamics. While less robust,
the nonlinear controller performance degrades, but maintains superior tracking
performance, except for steady state conditions. Minimum tracking error, occuring
at steady state, is lower for the linear controller. The robustness of the linear
controller suggests adding an integrator feature to the nonlinear controller would
provide the best controller performance. This motivates the discussion of adaptive
nonlinear control in the next chapter.
As a closing comment, simulating Scenario 2 with the same perturbations
generates similar performance characteristics. The results are omitted for the sake
of brevity.
121
Chapter 5
Adaptive Control System Design
Implementation of control designs developed in Chapter 3 assumes availability
of spacecraft mass properties and absolute state data. As an example, although
the linear control is computed with relative state measurements, computation of
the dynamics matrix requires the absolute state of the Leader. Also, real systems
challenge the control designer with the need to compensate for uncertainty in system
properties and dynamics, as well as measurement noise. This chapter discusses
nonlinear adaptive control strategies to meet these challenges. The goal is to
design a 6DOF adaptive control algorithm to compensate for dynamic uncertainty,
uncertainty in spacecraft mass properties, and uncertainty in thruster performance.
5.1 Model Uncertainty and Unmodeled Dynamics
This section provides a general overview of expected disturbances associated
with spacecraft formation flying due to model uncertainty and dynamic
disturbances. Further characterization of uncertainties and disturbances appear
in section 5.3 along with specific adaptive control designs.
122
5.1.1 Spacecraft Parameter Uncertainty
While spacecraft properties are not completely unknown, uncertainty limits
knowledge of the true values. Two principal contributors to control error are
mass property uncertainty and thruster performance/alignment uncertainty. Mass
properties are measured prior to launch, but are expected to change during the life
of a mission. Mass property changes typically link to fuel consumption. Errors
in estimated mass properties and thruster performance are not extreme. However,
uncertainty impacts the performance of the control strategies developed in Chapter
3, as shown in Section 4.5.4. Adaptive control provides a mechanism to compensate
for uncertainty in the spacecraft parameters.
5.1.2 Unmodeled Dynamics
Gravity and solar pressure are the principal drivers for the formation dynamics.
The spacecraft dynamics, developed in Chapter 2, neglect the perturbing effect of
solar pressure and unmodeled gravitational sources. Since formation control focuses
on tracking desired relative states, the principal concern is the differential effects of
these perturbations. Ideally, spacecraft design could eliminate, or at least minimize,
differential solar pressure.
There are three principal sources of gravitational perturbations. First, while
the Earth/Moon system is reasonably modeled as a single mass located at the
barycenter, the true field exhibits variation in magnitude and direction matched
to the orbital period of the moon. Next, planets contribute to gravitational
123
perturbations. The net magnitude and direction of the gravity field associated
with the other planets varies as their relative orbital positions evolve. The final
source is indirect. The gravity model depends on our knowledge of the formation
relative to the Sun and Earth/Moon barycenter. Uncertainty in time estimates
affect knowledge of the relative positions of the Sun, Earth and Moon. The position
of the spacecraft is determined by Earth-based tracking data. Position tracking
error is an additional source of uncertainty in the gravity field model.
While mission managers require reasonable knowledge of the absolute state
of the formation, the relative states between formation elements principally drive
the control system. In addition to compensating for uncertainty in the spacecraft
parameters, adaptive control provides a mechanism to compensate for the gravity
field model without absolute spacecraft state knowledge.
The scheme for attitude control expects each spacecraft to maintain a desired
orientation referenced to inertial space. Relative attitudes are determined by
differencing individual spacecraft attitudes. The basic assumption is that each
spacecraft will be equipped with sensors for determining its inertial referenced
attitude.
5.2 Nonlinear Adaptive Control Theory
Nonlinear adaptive control is one method employed to compensate for
modeling errors. Several resources on this topic are references: [28, 34, 58].
124
5.2.1 Parameter/Model Uncertainty
Adaptive control provides a mechanism to adjust the system model parameters
to improve the tracking response. A system lends itself to adaptive control
provided the unknown dynamics are structured and can be expressed as a linear
parametrization. The dynamics are assumed to have a modified form of Equation
3.12, expressed as:
H(?) ?? +C(?,?) ? +E(?,?)+?(?,?,?r,?r)? = u (5.1)
?(?,?,?r,?r) is a matrix of known functions. ? is a vector of uncertain
constants.
Assume ??= ?, and express the dynamics as:
H(?) ?? +C(?, ??) ?? +E(?, ??)+?(?, ??,?r, ??r)? = u (5.2)
In this case the control law, Equation 5.3, and adaptive rule, Equation 5.4,
are proposed:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)?KDs+?(?, ??,?r, ??r) ?? (5.3)
Where the design parameter KD is specified with: KD = KDT > 0.
??? = ??? = ???(?, ??,?
r, ??r)T s (5.4)
The adaptive gain matrix, ?, is specified with: ? = ?T > 0
Note: ?? = ????, ? constant ? ??? = ???
Equate the control law and dynamics:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)+?(?, ??,?r, ??r)???KDs
125
u = H(?) ?? +C(?, ??) ?? +E(?, ??)+?(?, ??,?r, ??r)?
? H(?)[??? ??r] +C(?, ??)[ ??? ??r]??(?, ??,?r, ??r)[????] +KDs = 0
? H(?) ?s+C(?, ??)s??(?, ??,?r, ??r) ?? +KDs = 0
? H(?) ?s = ?[C(?, ??)+KD]s+?(?, ??,?r, ??r) ?? (5.5)
For the stability proof choose the candidate Lyapunov function:
V(s,t) = 12sTH(?)s+ 12??T??1 ?? (5.6)
? ?V(s,t) = sTH(?)?s+ 12sT ?H(?)s+ ??T??1 ???
Substitute Equation 5.5: H(?) ?s = ?[C(?, ??)+KD]s+?(?,?,?r,?r) ??,
and the adaptive rule: ??? = ???(?, ??,?r, ??r)T s
? ?V(s,t) = ?sT[C(?, ??)+KD]s+ 12sT ?H(?)s+sT ?(?, ??,?r, ??r) ??
? ??T ?(?,?,?r,?r)T s
? ?V(s,t) = ?sTKDs+ 12sT[ ?H(?)?2 C(?, ??)]s
[ ?H(?)?2 C(?, ??)] is skew ? 12sT[ ?H(?)?2 C(?, ??)]s = 0
? ?V(s,t) = ?sTKDs < 0 ? snegationslash= 0 ?s? 0 (5.7)
The remainder of the proof of asymptotic stability of ?? follows the same line
of reasoning presented in the section 3.1.2. This method does not guarantee
identification of the unknown parameter, ?.
126
5.2.2 Actuation Uncertainty
The above algorithm adaptively determines the desired control. However,
implementation requires mapping the desired control to a set of actuator commands.
As discussed in Sections 2.3 and 3.4, actuator commands are determined by a reverse
mapping, given by:
Fthrust = BthrustT(Bthrust BthrustT)?1 u (5.8)
The approach assumes knowledge of the Bthrust matrix, which reflects the
placement, orientation and output of the individual thrusters. As with other
parameters, Bthrust represents an estimate of the true value. Adaptive methods
can be applied to compensate for the uncertainty. The control law is implemented
as discussed in Section 5.2.1 with the addition of an adaptive rule for modifying
Bthrust. With u computed from Equation 5.3, the thrust command is:
Fthrust = ?BthrustT( ?Bthrust ?BthrustT)?1 u (5.9)
Compute BthrustFthrust as:
BthrustFthrust = ( ?Bthrust ? ?Bthrust)Fthrust = ?BthrustFthrust ? ?BthrustFthrust
= ?Bthrust[ ?BthrustT( ?Bthrust ?BthrustT)?1 u]? ?BthrustFthrust
= u? ?BthrustFthrust
The dynamics are:
BthrustFthrust = H(?) ?? +C(?, ??) ?? +E(?, ??)+?(?, ??,?r, ??r)?
The control is:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)+?(?, ??,?r, ??r)???KDs
127
Then the closed loop dynamics are formed as:
u = H(?) ??r +C(?, ??) ??r +E(?, ??)+?(?, ??,?r, ??r)???KDs
=
braceleftBig
H(?) ?? +C(?, ??) ?? +E(?, ??)+?(?, ??,?r, ??r)?
bracerightBig
+ ?BthrustFthrust
? H(?) ?s+C(?, ??)s??(?, ??,?r, ??r) ?? +KDs+ ?BthrustFthrust = 0
For the stability proof modify the candidate Lyapunov function from Equation 5.6:
V(s,t) = 12sTH(?)s+ 12??T??1 ?? + 12?[ ?Bthrust]ij[ ?Bthrust]ij (5.10)
The term [ ?Bthrust]ij represents the individual elements of ?Bthrust. The
repeated indices in Equation 5.10 imply summation over all values of i and j. The
derivative of the Lyapunov function is:
?V(s,t) = sTH(?)?s+ 12sT ?H(?)s+ ??T??1 ??? + 1?[ ?B
thrust]ij[
??B
thrust]ij
Substitute H(?) ?s = ?[C(?, ??) + KD] s + ?(?,?,?r,?r) ?? ? ?BthrustFthrust,
and the adaptive rule: ??? = ? ? ?(?, ??,?r, ??r)T s. Also, define and substitute
??B
thrust =
??B
thrust = ?s [Fthrust]
T ? [ ??B
thrust]ij = si[Fthrust]j
? ?V(s,t) = ?sT[C(?, ??)+KD]s+ 12sT ?H(?)s+sT ?(?, ??,?r, ??r) ??
?sT ?BthrustFthrust ? ??T ?(?,?,?r,?r)T s+si[ ?Bthrust]ij[Fthrust]j
Note:
sT?(?, ??,?r, ??r)?? = ??T?(?,?,?r,?r)Ts
sT ?BthrustFthrust = si[ ?Bthrust]ij[Fthrust]j
[ ?H(?)?2 C(?, ??)] is skew ? 12sT[ ?H(?)?2 C(?, ??)]s = 0
Therefore: ?V(s,t) = ?sTKDs < 0 ? snegationslash= 0 ?s? 0
128
The remainder of the proof for asymptotic stability of ?? follows the same
line of reasoning presented in the section 3.1.2. This method simultaneously
estimates Bthrustand ?. As before, while the tracking error is asymptotically stable,
identification of Bthrust and ? is not guaranteed.
5.3 Parametrization for Nonlinear Adaptive Formation Control
5.3.1 Linear Parametrization for Translation Control
The development of a linear parametric dynamic model is based on the relative
orbital dynamics, recall Equation 2.26. Assuming the Leader follows a ballistic
trajectory, uthrust,L = 0, the dynamics are expressed as:
uthrust,F = ?x+{ ?em||r
EF ||3
+ ?s||r
SF ||3
}x+?em{ 1||r
EF ||3
? 1||r
EL||3
}rEL
+?s{ 1||r
SF ||3
? 1||r
SL||3
}rSL +?fsolar +?fpert
(5.11)
The goal is to parameterize dynamic uncertainty in the form ?T?T. ?T is
known, ?T is a constant with uncertain elements. The gravitational parameters
represent the only true constants in Equation 5.11. However, for the typical
formation flying mission profile, range vectors linking each spacecraft to the
Earth and Sun evolve at a slow rate compared to the control cycle. While the
range vector direction sweeps inertial space, the magnitude exhibits much less
variation. Therefore, an adaptive mechanism will track the dynamics associated
129
with the variation in the range vector magnitude as if it were constant. The range
vector direction is measured. Recall, the adaptive mechanism does not guarantee
identification of ?T.
The spacecraft are considered equipped with a suite of sensors capable of
resolving the relative range, x, in inertial coordinates. Also, sensors measure the
inertial direction to the Earth and Sun. While the dynamics model references
the position of the Earth/Moon barycenter, the position of the Earth provides a
reasonable measure of the barycenter position. The barycenter lies below the surface
of the Earth. Equation 5.11 is restructured as the measured vectors with constant
coefficients.
uthrust,F = ?x+d1 x+d2 eEL +d3 eSL +?fsolar +?fpert (5.12)
Where:
d1 = { ?em||r
EF ||3
+ ?s||r
SF ||3
}
d2 = ?em{ 1||r
EF ||3
? 1||r
EL||3
}||rEL||
d3 = ?s{ 1||r
SF ||3
? 1||r
SL||3
}||rSL||
eEL = rEL||r
EL||
, Unit vector pointing from Earth toward Leader (measured)
eSL = rSL||r
SL||
, Unit vector pointing from Sun toward Leader (measured)
Expressing the gravity and solar pressure perturbation terms as a linear
parametrization requires engineering judgement. The force generated by solar
pressure is proportional to its coefficient of reflectivity and the {cross-sectional area
130
of incidence/mass} ratio. Balancing the acceleration due to solar pressure serves
to minimize fuel requirements for formation maintenance. Assume the spacecraft
configurations are designed to minimize differential solar pressure, ||?fsolar|| is
considered constant, possibly with a trackable, slow variation. Also, with the
sun as the dominant radiation source, ?fsolar will maintain alignment with the
Sun-to-spacecraft vector. Therefore, differential solar pressure is modeled as:
?fsolar = d4eSL ? d4eSF. Note: eSF ? eSL since the spacecraft range to the
sun 1.5?108 kilometers, is much larger than the relative spacecraft ranges within
a formation.
Figure 5.1: Variation in True Earth Moon Gravity Field Experienced at L2
Perturbations in the gravity field are small, but deserve consideration. The
most significant variation is associated with the treatment of the Earth/Moon system
as a single point mass, versus two distinct masses. As noted in Figure 5.1, the
131
difference between the true field and the modeled field at L2 is on the order of
10?10 kilometers/sec2. In contrast, the gravity field due to the Earth/Moon is
approximately 2 ? 10?7 kilometers/sec2, and the gravity field due to the Sun is
approximately 6?10?6 kilometers/sec2. These values represent the total gravity field
experienced by the spacecraft, rather than the differential field across the formation,
yet reflect the relative magnitude of the contribution of each source. Additional
perturbations are generated by the gravity field of the other planets. At the point
of closest approach, the gravity fields of Venus and Jupiter have the same order of
magnitude as the variation in the Earth/Moon field. The gravity field perturbation
is modeled as: ?fpert = d5eEL ? d5eEF, recognizing eEF ? eEL. The model
structure recognizes the variation in the Earth/Moon field generally aligns with the
direction to the Earth. Also, the field variations due to Venus and Jupiter are
much slower than the 28 day cycle of the Earth/Moon field variation. Therefore,
an adaptive mechanism that tracks the Earth/Moon field variations will be able to
approximately track the variations due to Jupiter, Venus and the other planets.
Equation 5.12 is rewritten to reflect the linear parametrization of the
perturbations, as discussed.
uthrust,F = ?x+d1 x+d2 eEL +d3 eSL +d4eSL +d5eEL
= ?x+d1 x+ (d2 +d5) eEL + (d3 +d4) eSL
(5.13)
Expressed in matrix form:
uthrust,F = ?x+ [x|eEL|eSL]
?
??
??
d1
(d2 +d4)
(d3 +d5)
?
??
?? = ?T ?T (5.14)
132
As discussed, eEF ? eEL and eSF ? eSL, allowing an equivalent
implementation with ?T = [x|eEF|eSF]. The reference spacecraft selection in the
controller is driven by the metrology architecture.
Based on Equation 5.3, the control law is expressed as:
uthrust,F = ?xr +?T ??T ?KDs, With: ???T = ??T ?T T s (5.15)
Knowledge of the spacecraft mass is necessary to determine thrust required
to generate the desired control, uthrust,F. Mass properties can be incorporated
in the control law provided in Equation 5.15. Alternatively, the mass properties
can be included in the actuator mapping matrix, Bthrust. As discussed in section
5.2.2, compensation for actuator mapping uncertainty is achieved through a separate
adaptive rule. The specific formation flying discussion on this topic is provided in
section 5.3.3, addressing both translation and attitude control.
Including estimated mass properties in Equation 5.15, the modified control
law is:
[ ?mFuthrust,F] = ?mF ?xr + ?mF?T ??T ? ?mFKDs
= mF ?xr + [?xr|x|eEL|eSL]
?
??
??
??
?
( ?mF ?mF)
d1
(d2 +d4)
(d3 +d5)
?
??
??
??
?
? ?mFKDs (5.16)
133
Implemented as:
[ ?mFuthrust,F] = [?xr|x|eEL|eSL]
?
??
??
??
?
?mF
d1
(d2 +d4)
(d3 +d5)
?
??
??
??
?
? ?mFKDs
= ?T ,m ??T ,m ? ?mFKDs, With: ???T ,m = ??T [?T ,m]T s
(5.17)
The dynamics model, Equation 5.14, is modified to include the true spacecraft
mass. Equating the dynamics with the control, Equation 5.17, provides the closed
loop dynamics. Asymptotic stability of the error dynamics follows from the analysis
presented in Section 5.2.1.
5.3.2 Linear Parametrization for Attitude Control
Reference [14] presents an adaptive control design for the case where the
inertia matrix, HR, is unknown. The method requires an expression of HR ??r ?
[HR ??]?r as a linear parameterization of the form:
HR ??r ?[HR ??]?r = ?R(?,?r, ??r) ?R(HR) (5.18)
The term?R(HR) is a constant vector formed from the unknown components of
HR. The control law and adaptive rule that guarantee global stability are expressed
as:
Control Law: ? = ?R(?,?r, ??r) ??R ?KR sR (5.19)
Adaptive Rule: ???R = ??R ?R(?,?r, ??r)T sR (5.20)
Where:
?R ? Design parameter, ?R = ?RT > 0
134
The parameterization for Equation 5.18 is not unique. The specific definition
of ?R(?,?r, ??r) and ?R applied in this work follows:
For HR =
?
??
??
?
hR11 hR12 hR13
hR12 hR22 hR23
hR13 hR23 hR33
?
??
??
?
, define ?R ?
?
??
??
??
??
??
??
??
?
hR11
hR22
hR33
hR12
hR13
hR23
?
??
??
??
??
??
??
??
?
, and
?R(?,?r, ??r) ?
?
??
??
?
??r1 ??2?r3 ?3?r2 ??r2 ??1?r3 ??r3 +?1?r2 ?2?r2 ??3?r3
?1?r3 ??r2 ??3?r1 ??r1 +?2?r3 ?3?r3 ??1?r1 ??r3 ??2?r1
??1?r2 ?2?r1 ??r3 ?1?r1 ??2?r2 ??r1 ??3?r2 ??r2 +?3?r1
?
??
??
?
Design modification allows compensation for a constant disturbance torque.
Adding a disturbance torque to Equation 5.18 gives:
HR ??r ?[HR ??]?r = ?R(?,?r, ??r) ?R(HR) +?dist
Redefine the adaptive parametrization as:
HR ??r ?[HR ??]?r = ?primeR(?,?r, ??r) ?primeR(HR) (5.21)
Where:
?primeR(?,?r, ??r) = [I3|?R(?,?r, ??r)], ?primeR = [?Tdist|?RT]T (5.22)
135
5.3.3 Linear Parametrization for Thruster Performance and
Misalignment
The control sensitivity matrix, Bthrust, expresses the mapping of the
thruster outputs to the net translational acceleration and torque experienced
by the spacecraft. The mathematical structure of Bthrust is defined as a
set of column vectors representing the translational acceleration and torque
generated by each individual thruster. The construction appears as: Bthrust =
[b1|b2|b3|???|b(n?2)|b(n?1)|bn|], with n representing the total number of thrusters.
Each bi is comprised of six elements. By design, the first three elements form a
vector in the direction of the thrust vector with a magnitude equal to the reciprocal
spacecraft mass. The second set are formed as the cross product of the torque arm
and the unit vector in the direction of thrust. The inertial mass properties are not
included in Bthrust, as they appear in the attitude control law.
The sources of error in Bthrust link to uncertainty in the spacecraft mass,
thruster output, thrust direction, and thruster location (torque arm). Based on the
discussion in section 5.2.2, the adaptive rule to compensate for the control sensitivity
matrix uncertainty is defined as:
??B
thrust = ?s[Fthrust]
T (5.23)
As previously noted, the adaptive rule does not guarantee identification of
Bthrust. It neither serves to identify individual components of the uncertainty,
such as the spacecraft mass. The design assumes the composition of Bthrust is
136
constant. In reality the mass of the spacecraft decreases at the rate the thrusters
expel mass. However, the change is negligible for the thrust levels associated
with formation maintenance. For a formation stationed in the vicinity of the
Earth/Moon-Sun L2 point, the thrust levels required to maintain a 100 kilometer
separation are on the order of 10?7 meters/sec2. For a spacecraft with a mass
of 1,000 kilograms and a thruster Isp = 1000 seconds, the mass flow rate is on
the order of 10?8 kilograms/second. More significant mass changes will result
from maneuvers required to stabilize the global trajectory. The interval between
these impulsive maneuvers is on the order of weeks. Therefore, the adaptive
mechanism is expected to effectively compensate for mass property changes during
the intermediate intervals.
5.4 Parametrization for Adaptive Translation without Absolute
Navigation
The translation control strategies developed in Chapter 3 require absolute
spacecraft states for implementation. Section 5.3.1 presents an adaptive strategy
that reduces the requirement for absolute navigation, but still depends on knowledge
(measurement) of the range vector directions from the spacecraft to the Sun and
Earth. This section provides a modified adaptive algorithm based on relative
navigation only. Recall the parameterized dynamics from Equation 5.13:
uthrust,F = ?x+d1 x+ (d2 +d5) eEL + (d3 +d4) eSL
137
Consider the unit vectors eEL and eSL in the rotating frame of the RTBP. Based
on the benchmark problem discussed in sections 1.2.1 and 4.1.5, the formation
follows a halo orbit about L2 with a nominal radius of 300,000 kilometers. The
orbit period about L2 is approximately 6 months. Assuming a perfectly circular
trajectory, the velocity of the formation within the rotating frame is approximately
0.1 kilometers/second. The distance between the Earth and L2 is approximately
1.6?106 kilometers. Thus, the unit vector eEL has a negligible rotation rate on the
order of 10?6 degrees/second. The rotation rate of eSL is two orders of magnitude
smaller. Therefore, it is reasonable to treat eEL and eSL as constant vectors within
the rotating RTBP frame.
Recall from the discussion of the RTBP in Chapter 2, DRI(t) transforms
vectors from the inertial frame to the RTBP frame. Following the definition of
the RTBP frame, computation of DRI(t) is based on the position/velocity of the
Earth/Moon barycenter relative to the Sun. An accurate measure of time with
corresponding planetary ephemeris data is sufficient to determine DRI(t). The
parameterized translational dynamics are expressed as:
uthrust,F = ?x+d1 x+ (d2 +d5) DRI(t)T ReEL + (d3 +d4) DRI(t)T ReSL
= ?x+d1 x+DRI(t)T [(d2 +d5) ReEL + (d3 +d4) ReSL]
= ?x+d1 x+DRI(t)T ?T1
(5.24)
with ?T1 ? [(d2 +d5) ReEL + (d3 +d4) ReSL], constant.
Equation 5.24 parameterizes the uncertain quantities in terms of a constant
138
vector in the RTBP frame. Considering some unmodeled perturbations may appear
as constant disturbances within the inertial frame, an additional term, ?T2, is added
to the parametrization for completeness and increased adaptive flexibility. The
linear parametrization of the translation dynamics is expressed as:
uthrust,F = ?x+d1 x+DRI(t)T ?T1 +?T2
= ?x+ [x|DRI(t)T|I3] [d1|?T1T|?T2T]T
= ?x+?T ?T
?T ? [x|DRI(t)T|I3]
?T ? [d1|?T1T|?T2T]T
(5.25)
With this form of the dynamics the adaptive control law and adaptive rule are
expressed as:
uthrust,F = ?xr +?T ??T ?KT sT
???
T = ?T?T
T s
T
(5.26)
The elegance of this parametrization is the absence of any dependence on global
navigation factors. Implementation of the control law only depends on relative
navigation data. The rotation matrix, DRI(t), is the coordinate transformation
from the inertial to the rotating RTBP frame. Computation of DRI(t) is based on
planetary ephemeris data, and therefore easily determined based on knowledge of
the ephemeris time.
Further simplification is possible if the alignment of x is known to be constant
within either the RTBP or inertial frame. In this case d1 x can be absorbed into
?T1 or ?T2, respectively. The control is implemented as in Equation 5.26 with ?T ?
[DRI(t)T|I3] and ?T ? [?T1T|?T2T]T. The strategy is reasonable considering the
139
science for most formation flying astronomy missions requires maintaining x fixed
in inertial space for long observing periods.
5.5 Combined Translation/Attitude Adaptive Control
As discussed in Sections 2.3 and 3.4 the net force and torque generated by the
spacecraft thrusters is expressed as:
bU =
?
??
?
bu
thrust,F
?
?
??
? = Bthrust Fthrust (5.27)
Based on the parametrization of the translation and attitude dynamics
provided in Sections 5.3 and 5.4, the 6DOF dynamics are expressed as:
bU = B
thrust Fthrust =
?
??
?
?x
0
?
??
?+
?
??
?
?T 0
0 ?R(?,?r, ??r)
?
??
?
?
??
?
?T
?R
?
??
?
= ?? + ??
(5.28)
The control law is implemented as:
bU = ??
r + ? ???KD s
Fthrust = [ ?Bthrust]T ([ ?Bthrust] [ ?Bthrust]T)?1 bU
(5.29)
Where:
??r =
?
??
?
?xr
0
?
??
?; s =
?
??
?
sT
sR
?
??
?; ?? =
?
??
?
??
T
??
R
?
??
?
140
The adaptive rules are expressed as:
??? = ???T s
??B
thrust = ?s [Fthrust]
T
(5.30)
141
Chapter 6
Model Uncertainty Effects: Simulation Results and Analysis
This chapter characterizes the performance of the nonlinear adaptive control
method discussed in Chapter 5. The nonlinear control design developed in Chapter
3 serves as the baseline for comparison. The simulation architecture is the same
as presented in Section 4.1. As both mission scenarios generate similar results,
discussion in this chapter is limited to Scenario 1, defined in Section 4.3. Section
6.1 discusses the details of the control design. Simulation results are presented in
Section 6.2. Section 6.3 provides a summary analysis of the simulation results.
6.1 Control Design
Recall from Chapter 5, the adaptive control designs are implemented as follows:
Translation control from Equation 5.17:
[ ?mFuthrust,F] = ?T ,m ??T ,m ? ?mFKDsT, With: ???T ,m = ??T [?T ,m]T sT (6.1)
Attitude control from Equations 5.19 and 5.20:
Control Law: ? = ?R(?,?r, ??r) ??R ?KR sR (6.2)
Adaptive Rule: ???R = ??R ?R(?,?r, ??r)T sR (6.3)
142
Adaptation for thruster misalignment and performance uncertainty from Equations
5.8 and 5.23:
Control: Fthrust = ?BthrustT( ?Bthrust ?BthrustT)?1 u
Adaptive Rule: ??Bthrust = ?s[Fthrust]T
The control design is based on the linear control gains presented in Section
4.2. The design goal is to implement the nonlinear adaptive control with comparable
gains, removing gain selection as a factor in the performance comparison. As
presented in Section 4.2, the equivalent proportional and derivative gains are
computed as:
KD,trans = Kd,trans; ?T = [Kd,trans]?1Kp,trans
KD,att = Kd,att; ?R = [Kd,att]?1Kp,att
The adaptive mechanism for the nonlinear controller is a structured integrator
built around the system dynamics. In contrast the linear control design incorporates
adaptation through a simple integrator which accumulates position error, limiting
gain correlation between the two designs. In fact direct correlation exists only
between the gain for the linear integrator and the portion of the adaptive translation
control law designed to compensate for differential gravitational and solar pressure
effects.
The design goal is to specify ?T from Equation 6.1 such that the nonlinear and
linear control outputs have similar sensitivity to variations in the integrated position
error. Excluding the gain associated with mass uncertainty, ?T is specified as a
143
diagonal matrix with elements specified as:
?T(i,i) = ?SingularValue(Ki,trans [?T]?1) (6.4)
The remaining control gains are specified based on engineering judgement and
testing. The adaptive gains for mass uncertainty and thruster misalignment and
performance have no direct counterpart in the linear control design. The gains
specified to compensate for mass uncertainty are:
Translation Control, Mass Uncertainty Adaptive Gain: 106 kg/m
Attitude Control, Mass Uncertainty Adaptive Gain: 1011 kg-m2/rad
For thruster misalignment and performance uncertainty the adaptive gain, ?,
from Equation 5.23 is specified as 0.15 (N-m)?1.
6.2 Simulation Results
This section compares the performance of the adaptive and nonadaptive
nonlinear control designs under a varied set of model uncertainties. Model
uncertainty is introduced as solar pressure effects, mass property uncertainty, and
thruster misalignment/performance uncertainty. Adaptive controller performance
is also simulated with a 28-day delay from the scenario epoch. Performance
is independently characterized for each source of model uncertainty. The final
simulation includes all sources of uncertainty. As a baseline, the simulation is run
without perturbations. Results are presented in Figure 6.1 and Table 6.1. Fuel
consumption for both cases is: 23.25498 meter/second. As expected the results are
144
nearly identical. The nonadaptive controller performance without perturbations is
the benchmark for characterizing the adaptive controller tracking performance with
unmodeled dynamics. Fuel consumption is compared to the ideal/perfect tracking
model, discussed in Section 4.3.
Figure 6.1: Translation/Attitude Error Profile Comparison with No Perturbation
Effects for Nonlinear Nonadaptive (solid) and Adaptive (dashed) Control Algorithms
145
Table 6.1: Scenario 1 with No Perturbations ? Nonlinear Controller
Tracking Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Nonadaptive < 10?8 ? 1.5?10?5 2.4?10?6 meters
Translation/Adaptive < 10?8 ? 2.2?10?5 2.0?10?6 meters
Attitude/Nonadaptive < 10?11 ? 6.8?10?2 3.8?10?4 arcsec
Attitude/Adaptive < 10?11 ? 6.8?10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
6.2.1 Compensation for Solar Pressure
As discussed in Section 4.1, solar pressure generates unmodeled differential
acceleration between the two spacecraft. For the simulation, the solar pressure area
is treated as attitude independent. The net force due to solar pressure is assumed
to act at the spacecraft center of mass, resulting in zero net torque. Results are
presented in Figures 6.2 and 6.3, and Tables 6.2 and 6.3. As presented in Section
5.5, the nonadaptive nonlinear controller shows approximately an order of magnitude
performance degradation in the presence of solar pressure effects. In contrast, the
adaptive controller exhibits equivalent performance to the baseline results. As noted
in Figure 6.2 and Table 6.2, the nonlinear adaptive controller recovers the baseline
nonlinear control performance presented in Section 5.3. Since the solar pressure
146
effects are modeled as torque free, the attitude performance is not affected, as shown
in Figure 6.3 and Table 6.2. From Table 6.3 fuel consumption is similar for both
the nonadaptive and adaptive controller.
Figure 6.2: Range Error Profile, Scenario 1 with Solar Pressure Effects, Comparison
with Baseline (solid), and Nonlinear Nonadaptive (top/dashed) and Adaptive
(bottom/dashed) Control Algorithms
147
Figure 6.3: Attitude Error Profiles (Identical), Scenario 1 with Solar Pressure
Effects, Comparison with Baseline (solid), and Nonlinear Nonadaptive (top/dashed)
and Adaptive (bottom/dashed) Control Algorithms
148
Table 6.2: Scenario 1 with Solar Pressure ? Nonlinear Controller Tracking
Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Nonadaptive 10?5 10?5 1.6?10?5 meters
Translation/Adaptive < 10?8 ? 10?5 1.9?10?6 meters
Translation/Baseline < 10?8 ? 10?5 2.4?10?6 meters
Attitude/Nonadaptive < 10?11 ? 10?2 3.8?10?4 arcsec
Attitude/Adaptive < 10?11 ? 10?2 3.8?10?4 arcsec
Attitude/Baseline < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
Table 6.3: Scenario 1 with Solar Pressure ? Controller Fuel Performance
Summary
Nonlinear Controller Type Fuel Consumption Deviation from Ideal
(meters/second) (Percent)
Ideal Translation/Attitude 23.25535 ?
Nonadaptive Translation/Attitude 23.25498 -0.0015933%
Adaptive Translation/Attitude 23.25492 -0.0018465%
149
6.2.2 Compensation for Mass Property Uncertainty
Spacecraft mass properties are normally determined prior to launch. However,
the true mass properties remain uncertain due to initial measurement errors and
changes over the course of the mission, principally due to fuel consumption. Mass
uncertainty is introduced by initializing the mass properties with ?estimated?
values. The estimated mass properties are used to compute the control. True mass
properties are employed in the propagation. For the spacecraft mass the ?estimated?
value is set at 90% of the true value with the specific values:
Follower Mass: 1,980 kg, versus true value: 2,200 kg
Inertial properties are varied in several ways. First, the off-diagonal elements
are ignored, and set to zero. The diagonal elements are varied by 5% of their true
values and entered in the incorrect order. Specific values are:
Estimated Inertia Mass Properties:
HR =
?
??
??
??
?
190 0 0
0 210 0
0 0 315
?
??
??
??
?
kg m2 (6.5)
True Inertia Mass Properties:
HR =
?
??
??
??
?
200 10 5
10 300 15
5 15 200
?
??
??
??
?
kg m2 (6.6)
150
Figures 6.4 and 6.5, and Tables 6.4 and 6.5, present simulation results. Figure
6.4 and Table 6.4 describe the tracking performance for translation control. Figure
6.5 and Table 6.4 depict attitude tracking performance. Compared to solar pressure
effects, mass uncertainty generates greater degradation in the nonlinear controller
performance. The nonadaptive controller performance degrades by 5 orders of
magnitude for translation tracking. In contrast the adaptive controller performance
degrades only two to three orders of magnitude. Attitude tracking performance
is also degraded, approximately three orders of magnitude for the nonadaptive
controller and two orders of magnitude for the adaptive controller.
151
Figure 6.4: Range Error Profile Comparison, Scenario 1 with Mass Uncertainty
Effects, Comparison with Baseline (solid), and Nonlinear Nonadaptive (top/dashed)
and Adaptive (bottom/dashed) Control Algorithms
152
Figure 6.5: Attitude Error Profile Comparison, Scenario 1 with Mass Uncertainty
Effects, Comparison with Baseline (solid), and Nonlinear Nonadaptive (top/dashed)
and Adaptive (bottom/dashed) Control Algorithms
For the nonlinear adaptive controller the initial performance is perceptibly
degraded, yet performance remains superior to the nonadaptive design. As the
simulation progresses the performance improves to the unperturbed baseline model.
To further demonstrate the advantage of the adaptive strategy the simulation was
restarted using the mass parameter estimates from the first run. The values were:
Follower Mass: 2,198 kg (estimated after first run)
153
Inertia Mass Properties (estimated after first run):
HR =
?
??
??
??
?
186.9 48.3 ?16.1
48.3 251.6 34.5
?16.1 34.5 194.1
?
??
??
??
?
kg m2 (6.7)
Although the adaptive theory does not guarantee convergence of the estimated
parameters to their true values, the Follower mass and the diagonal elements of the
inertia matrix are closer to their true values than originally estimated. It is noted
that after the second run there is no change is the estimated values for the inertia
matrix. However, the estimated mass actually converges to the true value of 2,200
kg. The performance improvement during the second run is presented in Figure
6.6 and Table 6.4, comparing the results from the first and second run. The mean
tracking error improves almost two orders of magnitude for both translation and
attitude control. Both translation and attitude control performance approach the
performance of the unperturbed system.
As shown in Table 6.5, the fuel consumption is close to the perfect tracking
baseline.
154
Figure 6.6: Nonlinear Adaptive Controller Performance Improvement for Second
Run with Updated Mass Property Estimates. Position Tracking (top), Attitude
Tracking (bottom) with Updated Parameters (solid), Original Simulation (dashed)
155
Table 6.4: Scenario 1 with Mass Uncertainty ? Nonlinear Controller
Tracking Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Nonadaptive 10?6 10?1 1.9?10?1 meters
Translation/Adaptive < 10?8 ? 10?1 9.1?10?4 meters
Translation/Adaptive (2nd Run) < 10?8 ? 10?4 1.0?10?5 meters
Translation/Baseline < 10?8 ? 10?5 2.4?10?6 meters
Attitude/Nonadaptive < 10?11 ? 100 2.4?10?1 arcsec
Attitude/Adaptive < 10?11 ? 100 4.3?10?2 arcsec
Attitude/Adaptive (2nd Run) < 10?11 ? 10?2 1.2?10?4 arcsec
Attitude/Baseline < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
Table 6.5: Scenario 1 with Mass Uncertainty ? Controller Fuel
Performance Summary
Nonlinear Controller Type Fuel Consumption Deviation from
(meters/second) Ideal (Percent)
Ideal Translation/Attitude 23.25535 ?
Nonadaptive Translation/Attitude 23.27020 0.0638510%
Adaptive Translation/Attitude 23.25583 0.0020318%
Adaptive Translation/Attitude (2nd Run) 23.25499 -0.0015807%
156
6.2.3 Compensation for Thruster Misalignment/Misplacement
Thruster placement and pointing are determined during the pre-launch phase
of a mission. As discussed in Section 2.4, the mapping of the thruster output to
the net force and torque on the spacecraft is expressed in the form of the matrix
Bthrust. The physical location and general pointing of the thrusters is not expected
to change during the life of a mission. However, the spacecraft center of mass, the
origin of the thruster mapping coordinate frame, will drift principally due to fuel
consumption. Also, thruster output levels will vary during the life of the mission.
Consequently, the true mapping matrix Bthrust exhibits variations during the life of
a mission. The changes are sufficiently slow to track the changes using the adaptive
mechanism discussed in Section 5.5.
Uncertainty is introduced in Bthrust by offsetting the position and direction
for each thruster. Each thruster position is offset by 0.1 meters. A random set of
unit vectors determines the direction of the offset. In a similar fashion the pointing
direction of each thruster is subjected to a 30 degree Euler axis rotation. The Euler
axis for each thruster is chosen from a random set of vectors. While each thruster
is significantly displaced and repointed, the net effect is much smaller due to the
random nature of the applied uncertainties. Recall the commanded thrust/torque on
the spacecraft is transformed to the actual thrust torque according to the relation:
Uactual = Bthrust [ ?BthrustT( ?Bthrust ?BthrustT)?1]Ucommanded (6.8)
157
The matrix Bthrust [ ?BthrustT( ?Bthrust ?BthrustT)?1] represents the combined
effect of thruster misplacement and misalignment. To facilitate analysis, the matrix
is partitioned as:
Bthrust [ ?BthrustT( ?Bthrust ?BthrustT)?1] =
?
??
?
B1,1 B1,2
B2,1 B2,2
?
??
? (6.9)
The elements B1,1 and B2,2 reflect the net repointing of the thrust and torque
vector, respectively. The elements B1,2 and B2,1 represent the coupling action
introduced between the commanded thrust and torque. Based on the thruster
placement/alignment uncertainty applied in the simulation, the net pointing error
for the thrust vector is approximately 2.3 degrees, and 3.5 degrees for the net torque.
Based on the singular values of B1,1 and B2,2, the magnitude of the net thrust vector
varies between 72% and 116% of the commanded value. Variation of the net torque
ranges between 87% and 125% of the commanded values.
158
Figure 6.7: Range Error Profile Comparison, Scenario 1 with Thruster
Misalignment/Performance Effects, Comparison with Baseline (solid), and
Nonlinear Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms
Simulation results are shown in Figures 6.7 and 6.8, and Tables 6.6 and 6.7.
Similar to the previous cases the adaptive translation controller demonstrates the
ability to compensate for uncertainty in model parameters. Translation tracking
performance for the adaptive controller is significantly degraded during attitude
maneuvers due to the uncertainty induced coupling between the commanded thrust
and torque. The attitude error for the nonadaptive controller exhibits significant
degradation during the entire control sequence.
159
Figure 6.8: Attitude Error Profile Comparison, Scenario 1 with Thruster
Misalignment/Performance Effects, Comparison with Baseline (solid), and
Nonlinear Nonadaptive (top/dashed) and Adaptive (bottom/dashed) Control
Algorithms
The nonlinear adaptive control performance experiences similar degradation
for both translation and attitude control, as shown in Table 6.6. However,
the adaptive mechanism provides performance improvement as the simulation
progresses. To test the benefit of adaptation a second simulation is run using
Bthrust, as estimated from the first run. The results show improved tracking for
the second run, Figure 6.9 and Table 6.6. The second run exhibits an order of
160
magnitude improvement in translation tracking, and almost an order of magnitude
improvement in attitude tracking.
Fuel data, Table 6.7, shows a half percent fuel penalty for the adaptive control
strategy. The penalty is reduced for the second run.
Figure 6.9: Nonlinear Adaptive Controller Performance Improvement for Second
Run with Parameter Estimate Update for Thruster Misalignment/Performance
Effects. Position Tracking (top), Attitude Tracking (bottom) with Updated
Parameters (solid), Original Simulation (dashed)
161
Table 6.6: Scenario 1 with Thruster Misalignment/Performance Effects
? Nonlinear Controller Tracking Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Nonadaptive 10?6 10?1 2.3?10?1 meters
Translation/Adaptive < 10?8 ? 10?1 2.5?10?2 meters
Translation/Adaptive (2nd Run) < 10?8 ? 10?4 1.3?10?3 meters
Translation/Baseline < 10?8 ? 10?5 2.4?10?6 meters
Attitude/Nonadaptive 10?1 100 5.1?103 arcsec
Attitude/Adaptive 10?5 100 7.1?101 arcsec
Attitude/Adaptive (2nd Run) 10?6 10?2 1.1?101 arcsec
Attitude/Baseline < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
Table 6.7: Scenario 1 with Thruster Misalignment/Performance Effects
? Controller Fuel Performance Summary
Nonlinear Controller Type Fuel Consumption Deviation from
(meters/second) Ideal (Percent)
Ideal Translation/Attitude 23.25535 ?
Nonadaptive Translation/Attitude 23.27592 0.088434%
Adaptive Translation/Attitude 23.36557 0.473950%
Adaptive Translation/Attitude (2nd Run) 23.36178 0.457650%
162
6.2.4 Compensation for Solar Pressure, Mass Properties, Thruster
Alignment/Position Uncertainty
As a final step in characterizing the performance of the adaptive control
design, performance is simulated with all sources of uncertainty applied. Perhaps
expected, the results shown in Figures 6.10 and 6.11, and Tables 6.8 and
6.9, exhibit performance characteristics similar to those generated for thruster
alignment/placement uncertainty, the worst case individual perturbation. Although
similar, results show a slight improvement in performance with all adaptive
mechanisms activated despite the presence of the perturbations associated with solar
pressure and mass property uncertainty. Comparing Figures 6.7 and 6.10, the most
prominent feature of improved performance is the position tracking error at the
t = 110 minute mark in the simulation. This represents the start of the combined
translation/attitude maneuver.
163
Figure 6.10: Range Error Profile Comparison, Scenario 1 with All Perturbation
Effects, Comparison with Baseline (solid), and Nonlinear Nonadaptive (top/dashed)
and Adaptive (bottom/dashed) Control Algorithms
164
Figure 6.11: Attitude Error Profile Comparison, Scenario 1 with All Perturbation
Effects, Comparison with Baseline (solid), and Nonlinear Nonadaptive (top/dashed)
and Adaptive (bottom/dashed) Control Algorithms
As with the previous cases, the simulation is rerun with updated parameter
estimates. As shown in Figure 6.12 and Table 6.8, tracking performance for both
position and attitude continue to improve. From Table 6.9 the adaptive controller
does experience a minor half percent increase in fuel usage.
165
Figure 6.12: Nonlinear Adaptive Controller Performance Improvement for Second
Run with Parameter Estimate Update for All Perturbation Effects. Position
Tracking (top), Attitude Tracking (bottom) with Updated Parameters (solid),
Original Simulation (dashed)
166
Table 6.8: Scenario 1 with All Perturbations ? Nonlinear Controller
Tracking Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Nonadaptive 10?6 100 2.6?10?1 meters
Translation/Adaptive < 10?8 ? 10?1 7.3?10?3 meters
Translation/Adaptive (2nd Run) < 10?8 ? 10?1 4.1?10?3 meters
Translation/Baseline < 10?8 ? 10?5 2.4?10?6 meters
Attitude/Nonadaptive 10?2 104 5.1?103 arcsec
Attitude/Adaptive 10?5 104 7.4?101 arcsec
Attitude/Adaptive (2nd Run) 10?6 103 1.2?101 arcsec
Attitude/Baseline < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
Table 6.9: Scenario 1 with All Perturbations ? Controller Fuel
Performance Summary
Nonlinear Controller Type Fuel Consumption Deviation from
(meters/second) Ideal (Percent)
Ideal Translation/Attitude 23.25535 ?
Nonadaptive Translation/Attitude 23.26750 0.05222%
Adaptive Translation/Attitude 23.36459 0.46973%
Adaptive Translation/Attitude (2nd Run) 23.36178 0.46148%
167
6.2.5 Performance Impact of Time Invariant Dynamics Model
As discussed in Section 4.5.5, the linear controller sustained equivalent
performance through a 28 day period without a gain update. In comparison, the
nonlinear adaptive controller design does not required gain updates. To demonstrate
this point, Scenario 1 (without perturbations) is initiated with a start time 28
days after the epoch for gain computation. Tracking performance, Table 6.10, is
unchanged. There is a slight improvement in fuel cost, as shown in Table 6.11.
Graphically the performance is equivalent to the results presented in Figures 4.6,
4.7, and 4.8
Table 6.10: Scenario 1, 28 days after Epoch ? Nonlinear Adaptive
Controller Tracking Performance Summary
Error
Tracking Metric Minimum Maximum Mean Units
Translation (delay) < 10?8 ? 10?5 1.9?10?6 meters
Translation (no delay) < 10?8 ? 10?5 2.2?10?6 meters
Attitude (delay) 10?11 10?3 3.8?10?3 arcsec
Attitude (no delay) < 10?11 ? 10?3 3.8?10?3 arcsec
? Reflects limit of numerical precision.
168
Table 6.11: Scenario 1 ? Nonlinear Adaptive Controller Fuel Performance
Summary
Controller Type Fuel Consumption Deviation from
(meters/second) (Ideal Percent)
Ideal Translation/Attitude 23.25535 ?
Adaptive Translation/Attitude (no delay) 23.25498 -0.0015930%
Adaptive Translation/Attitude 23.25494 -0.0017685%
6.3 Summary Analysis
The results presented in the previous section sufficiently demonstrate that
the nonlinear adaptive control design outperforms a similar nonadaptive controller.
Based on the results in Chapter 4, the nonadaptive controller outperformed a similar
linear controller. Therefore, the nonlinear adaptive controller provides the best
overall performance. The superiority of the nonlinear adaptive control design lies
within the adaptive integrator and inclusion of the desired acceleration term in
the control law. The desired acceleration allows the controller to anticipate thrust
commands, eliminating the observed characteristic transient response of the linear
controller to a step input.
The adaptive mechanism of the nonlinear controller provides an enhanced
integrator based on the system dynamics and a structured uncertainty model. The
169
nonlinear controller has the key advantage of adapting to a set of constant values.
Exercising the nonlinear controller provides a learning mechanism which enhances
future performance. In contrast the integrator in the linear controller chases a
generally time varying control offset, and is unable to capitalize on performance
history.
The chosen simulation control sequence is fairly simplistic. Further
improvement in the performance of the nonlinear controller is expected with
additional varied maneuvers.
These results support a general conclusion that the nonlinear adaptive
controller is a superior design. Detailed analysis of control design options is required
for specific missions to determine the best course of action.
Table 6.12: Scenario 1 with No Perturbations ? Controller Tracking
Performance Summary
Error
Controller Type Minimum Maximum Mean Units
Linear Translation < 10?8 ? 101 3.4?10?1 meters
NL Nonadaptive Translation < 10?8 ? 10?5 2.4?10?1 meters
NL Adaptive Translation < 10?8 ? 10?5 2.0?10?1 meters
Linear Attitude 10?11 101 1.9?100 arcsec
NL Nonadaptive Attitude < 10?11 ? 10?2 3.8?100 arcsec
NL Adaptive Attitude 10?11 10?2 3.8?100 arcsec
? Reflects limit of numerical precision.
170
Table 6.13: Scenario 1 ? Linear Controller Fuel Performance Summary
Controller Type Fuel Consumption Deviation from
(meters/second) Ideal (Percent)
Ideal Translation/Attitude 23.25535 ?
Linear Translation/Attitude (no delay) 23.63285 +1.6233%
Linear Translation/Attitude 23.63308 +1.6243%
171
Chapter 7
Summary and Conclusions
The main summary and conclusions for this research are provided in Section
1.5 at the end of Chapter 1. This chapter provides a synopsis of the contributions,
topics for future research, and concluding remarks.
7.1 Contributions
This research characterizes the relative performance of linear and nonlinear
control strategies for precision formation flying. The comparative analysis employs
?equivalent? controller gains and incorporates an integrator in the linear control
design to ?level the playing field? between controller designs.
Controller performance is characterized through various simulations. The
simulation design reflects a realistic space environment, including a full gravitational
model and solar pressure effects. Spacecraft model properties are based on realistic
mission design parameters. A fixed set of thrusters serve as control actuators for
both translation and attitude control. Therefore, the control law design is based
on a 6DOF control architecture, characteristic of precision formation flying control.
Analysis includes the impact on controller performance due to omitted dynamics
(gravitational sources and solar pressure) and model uncertainty (mass properties,
thruster placement and thruster alignment). In general the prior work, cited in
172
Section 1.4, examines translation control only with ideal actuation and models.
Linearized equations of relative motion were derived for spacecraft operating in
the context of the Restricted Three Body Problem. Prior work employed linearized
dynamics based on numerical analysis, or neglected the gravity field. Although time
varying, analysis demonstrated controller stability with gain scheduling. Gains are
determined with a Linear Quadratic Regulator design assuming a constant linear
model. For the studied case, gain scheduling within a 14-day period is recommended.
Gain scheduling requirements are mission specific, and must be determined through
analysis.
7.2 Results Summary
Linear and nonlinear control designs are considered to address the problem
of precision formation flying. The linear control design includes development of an
analytic model for the linearized equations of relative motion within the context of
the restricted three body problem (RTBP). Restated results, presented in Tables
4.9 and 6.8, demonstrate the ability of the nonlinear controller to compensate for
unmodeled dynamics and uncertainties. Also, the adaptive controller performance
shows improvement with updated adaptation parameters.
173
Table 7.1: Scenario 1 with All Perturbations ? Controller Tracking
Performance Comparison
Nonlinear Controller Error
Mode/Type Minimum Maximum Mean Units
Translation/Linear 10?8 101 3.4?10?1 meters
Translation/Nonadaptive 10?6 100 2.6?10?1 meters
Translation/Adaptive < 10?8 ? 10?1 7.3?10?3 meters
Translation/Adaptive (2nd Run) < 10?8 ? 10?1 4.1?10?3 meters
Translation/Baseline < 10?8 ? 10?5 2.4?10?6 meters
Attitude/Linear 10?2 104 5.2?103 arcsec
Attitude/Nonadaptive 10?2 104 5.1?103 arcsec
Attitude/Adaptive 10?5 104 7.4?101 arcsec
Attitude/Adaptive (2nd Run) 10?6 103 1.2?101 arcsec
Attitude/Baseline < 10?11 ? 10?2 3.8?10?4 arcsec
? Reflects limit of numerical precision.
The results do not guarantee nonlinear adaptive control is the best design
strategy. Rather they demonstrate a potential for improving controller performance
to meet the extreme control requirements associated with formation flying missions
like MAXIM and Stellar Imager. Mission specific analysis from a systems perspective
is required to determine the best controller design. A systems level analysis considers
all aspects of the control system, including sensors, actuators, and testing with high
fidelity dynamic models.
174
7.3 Future Research
This research represents one step toward improving our understanding of
control technology for precision formation flying. Further research is required
to characterize controller performance with realistic sensor models, estimation
algorithms and enhanced dynamic models. The following highlights some of the
issues to be addressed.
The controller design and analysis presented in this research assumes perfect
knowledge of the spacecraft states. In practice sensor measurements are employed
to determine the states. A complete analysis of a controller design must incorporate
appropriate sensor models. A typical sensor suite for a spacecraft formation flying
application enables state measurement for relative translation and absolute attitude.
Star trackers are currently the best sensor technology for fine attitude measurement
in deep space. Relative attitude can be determined by differencing the absolute
attitude of spacecraft pairs within the formation. Relative position/velocity sensors
are an emerging technology. Sensor models are required to support controller
performance analysis.
Sensor models include noise and quantization characteristics. Also, sensor
performance may be state dependent. For example the design of a relative
position/velocity sensor may restrict the relative spacecraft attitude for optimal
performance.
Introducing noise in the measurement model motivates the requirement to
consider estimation algorithms. This is particularly important with regard to
175
the nonlinear controller designs. A stable linear estimation algorithm can be
independently designed for a linear controller application. The Separation Principal
guarantees stability of a closed-loop system incorporating a stable linear controller
and a stable linear estimator. However, if the controller and/or the estimator
are based on a nonlinear design, combined controller/estimator stability is not
guaranteed. The closed-loop stability of the combined estimator/controller must
be independently verified.
An ideal sensor suite measures the full state required for controller
implementation. Sensor limitations or failure can limit the measurements to a
reduced state. This requires the development of strategies to extract the required
full state information from the available measurements, or design controllers with
reduced state requirements.
Shifting attention to enhanced spacecraft dynamic models, a number of issues
arise. Higher fidelity modeling of thruster performance requires inclusion of realistic
output characteristics, such as delay, saturation, range limitation and quantization.
The potential for significant degradation in thruster performance (or complete
failure) requires development of controller strategies with a reduced actuator set.
The controller designs presented in this research limit the actuator set to
thrusters. An alternative approach incorporates momentum wheels to support
attitude control. Momentum wheels add a degree of complexity to the controller
design, and introduces additional challenges. Momentum wheels require inclusion
of strategies to limit momentum build-up. Also, momentum wheels can introduce
176
spacecraft jitter due to wheel imbalance and digital wheel control.
Another important research area relates to spacecraft flexible dynamics.
Spacecraft designs typically include flexible structures, i.e. solar arrays, booms
and sun shades. Controller designs must incorporate provide flexible mode damping
and avoid mode stimulation.
Solar torque is a related issue to spacecraft appendages and asymmetric design.
Solar torque is generated by solar pressure acting on the surface of the spacecraft
with a center of pressure offset from the spacecraft center of mass. The controller
must incorporate compensation for attitude dependent solar torque.
These issues represent important areas for future research. In addition
formation flying poses many other technology challenges related to initialization,
maintenance of the global formation trajectory, communication delays, active optics,
etc.
7.4 Concluding Remark ? The End and a Beginning
Looking back to the opening remarks of Chapter 1, envision a small child
standing in an open field beneath a clear night sky, staring up with eyes full of
wonder, pondering the secrets of the universe. This quest for knowledge, this desire
to explore motivates development of space-based astronomical platforms. Precision
formation flying is one of many enabling technologies required to accomplish these
missions.
Our quest to explore the universe is viewed as a journey with an intricate
177
network of paths leading to a boundary of discovery. This work represents a segment
of that network, enhancing the knowledge base for spacecraft control with specific
application to precision formation flying.
So this segment of the journey ends, having pushed the knowledge limit of
spacecraft control a little further. Many new paths lie ahead, requiring the efforts of
future researchers to pave the way for ultimate development of new space-based
observatories, enabling the science community to unveil new knowledge of the
universe.
178
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