ABSTRACT
Title of dissertation: Electrolocation-Based Obstacle Avoidance
and Autonomous Navigation in
Underwater Environments
Kedar D. Dimble, Doctor of Philosophy, 2013
Dissertation directed by: Professor J. Sean Humbert
Department of Aerospace Engineering
Weakly electric fish are capable of performing obstacle avoidance in dark and
complex aquatic environments efficiently using a navigation technique known as
electrolocation. That is, electric fish infer relevant information about surrounding
obstacles from the perturbations that these obstacles impart to their self-generated
electric field. This dissertation draws inspiration from electrolocation to demon-
strate unmapped reflexive obstacle avoidance in underwater environments. The
perturbation signal, called the electric image, contains the spatial information of
the perturbing objects regarding their location, size, conductivity etc. Electrostatic
equations elucidate the concept of electrolocation and the mechanism of obstacle
detection using electric field perturbations. Spatial decomposition of an electric im-
age using Wide-Field Integration processing extracts relative proximity information
about the obstacles. The electric field source is changed to an oscillatory one and a
quasistatic approach is taken. Simulations were performed in straight tunnel, clut-
tered corridor and an obstacle field. Experimental validation was conducted with
a setup comprising a tank, a computer-controlled gantry system and an electro-
sensor. Consistency between the simulations and the experiments was maintained
by recreating similar environments.
Simulations using both the electrostatic and the quasistatic approach demon-
strate that the algorithm is capable of performing various maneuvers like tunnel
centering, wall following and clutter navigation. The experimental results agree
with the simulation results and validate the efficacy of the approach in perform-
ing obstacle avoidance. The presented approach is computationally lightweight and
readily implementable, making underwater autonomous navigation in real-time fea-
sible.
ELECTROLOCATION-BASED OBSTACLE AVOIDANCE
AND AUTONOMOUS NAVIGATION IN UNDERWATER
ENVIRONMENTS
by
Kedar D. Dimble
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
2013
Advisory Commmittee:
Associate Professor J. Sean Humbert, Chair/Advisor
Associate Professor Derek A. Paley
Associate Professor Raymond Sedwick
Associate Professor Dave Akin
Professor Amr Baz, Dean?s Representative
c? Copyright by
Kedar D. Dimble
2013
To my beloved, Ovee and Dhrupud, for filling my life outside the lab with loads of
joy and happiness.
ii
ACKNOWLEDGEMENTS
A stream will carry a boat with it in the right direction. My stream has
brought me to this juncture where I get to defend my doctoral thesis. I am very
thankful to my current advisor, Dr. J. Sean Humbert, for awarding this research
topic to me. Moreover, he has been extremely patient with me throughout, when
things were working great or even when results were hard to come through. His
advice of understanding things deeply to explain them clearly is something that I
may remember for the entirety of my life. The support he provided, both intellectual
and financial, played a major role in seeing me through this program. My advisor
during my MS stint at the University of Kansas, Dr. Trevor C. Sorensen, was very
instrumental in motivating me to pursue my doctoral studies at the University of
Maryland. I am extremely thankful to him too. I would also like to thank the rest
of my committee comprising Dr. Paley, Dr. Sedwick, Dr. Akin and Dr. Baz for
their time and help in making this thesis possible.
I am glad that I had my labmates on whose shoulders I could stand. Badri,
you are extremely good at what you do, maybe one of the best, but I know your
humility won?t let you admit that. Without you this research may have not gone
as well as it did. Greg and Imraan, in my opinion you two are extremely good with
working out complex problems in your head by visualizing them correctly and I am
happy I could make use of that quality of yours. It has always been great to talk
to you Joe as I learn something new every time. Hector, Jishnu and Mac have been
great friends that I could always count on to discuss my research problems. Lina,
Andrew K., Julia and Alex have been unknowingly making my days in the lab less
boring. My undergrad associates, Shaun and Mihir, have played very important
parts in this thesis? completion as well. It was a great pleasure to cross paths with
the lab alumni, Andrew H., Scott, Bryan, Nick, Doug, Derek, Renee, Steve and
Jon. Sachit and Praveen who were fellow doctoral students and good family friends
iii
showered me with great advice on various topics from time to time. Amar played
the role of my go-to stress reliever and financial advisor.
I am really blessed to have a family as amazing as mine. My mother and
father, Aai and Baba, strived very hard to put me and my brother through college
and to make us good human beings. I am, in all my capacity, still trying to be one.
My father being an engineer himself and my mother being from an industrialist
family were able to maintain an educational and engineering oriented environment
at home when I was growing up. That has in some way worked, I suppose. They
finally get to see me attain this doctoral degree, which they were long waiting for.
I think I can never thank them enough for all they have done for me. My elder
brother, Dada, has served as a big inspiration for me throughout my life due to
his hard working and scientific approach towards engineering. I thank him and my
sister-in-law for being supportive and encouraging. My lovely niece Ovee and my
u?ber energetic son Dhrupud are two utterly entertaining kids who kept me going
by helping me switch off the stress from work. Finally, my wife, Vibha, has been
a great support while I was being a grad student. Somehow she figured out a way
to maintain a family on a grad student salary. She doesn?t say but I must have
bored her at times by my lab tales and PhD humor. Not to mention her sitting
through my presentation rehearsals. In addition there are so many others who have
contributed in one way or the other to my work, like my in-laws, old roommates,
professors from other schools working on similar projects etc., that it is impossible
to thank everyone here.
Finally, I would like to quote a Sanskrit prayer in praise of Saraswati, the
Hindu Goddess of knowledge and wisdom.
(Gist: O Goddess Saraswati, I pray to you to always protect me from evil and
remove all my apathy, laziness and ignorance.)
iv
Table of Contents
List of Tables viii
List of Figures ix
Nomenclature xiii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Basic Physiology of the Electrolocation System . . . . . . . . 2
1.1.2 Concept of an Electric Image . . . . . . . . . . . . . . . . . . 3
1.1.3 Motivation for an Artificial Electro-sensor . . . . . . . . . . . 5
1.2 Electrolocation in Robotics . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Approach/Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Electrostatic Expressions for Electric Images 13
2.1 Electro-Sensor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Electric Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Electric Image in a Straight Tunnel . . . . . . . . . . . . . . . 22
2.3.2 Electric Image in a Circular Arena . . . . . . . . . . . . . . . 27
3 Wide-Field Integration and Autonomous Control 30
3.1 Wide-Field Processing of Electric Images . . . . . . . . . . . . . . . . 30
3.2 Relative State Estimation . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Static Output Feedback Control . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Control Strategy for Centering Behaviors . . . . . . . . . . . . 39
3.3.2 Robust Stability Analysis of the Controller . . . . . . . . . . . 41
3.4 Control Strategy for Wall Following Behaviors . . . . . . . . . . . . . 46
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Centering in a Straight Tunnel . . . . . . . . . . . . . . . . . . 47
3.5.2 Wall Following in a Circular Arena . . . . . . . . . . . . . . . 49
3.6 Modifications for Non-conductive Objects . . . . . . . . . . . . . . . . 51
3.6.1 Non-conductive Obstacles . . . . . . . . . . . . . . . . . . . . 52
3.6.2 Mix of Conductive and Non-conductive Objects . . . . . . . . 52
4 Quasistatic Analysis and Simulations 56
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Finite Elements Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Modeling of geometry . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Meshing the environment . . . . . . . . . . . . . . . . . . . . . 63
v
4.2.3 Application of Loads and Boundary Conditions . . . . . . . . 64
4.2.4 Solution and Postprocessing . . . . . . . . . . . . . . . . . . . 65
4.2.5 Linking of ANSYS with MATLAB . . . . . . . . . . . . . . . 66
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Straight Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1.1 Conductive Walls . . . . . . . . . . . . . . . . . . . . 68
4.3.1.2 Non-conductive Walls . . . . . . . . . . . . . . . . . 70
4.3.1.3 One Conductive and One Non-conductive Wall . . . 71
4.3.2 Cluttered Corridor . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2.1 Conductive Objects . . . . . . . . . . . . . . . . . . 74
4.3.2.2 Non-conductive Objects . . . . . . . . . . . . . . . . 78
4.3.2.3 Conductive and Non-conductive Objects . . . . . . . 81
4.3.3 Obstacle Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.3.1 Conductive Cylinders . . . . . . . . . . . . . . . . . 85
4.3.3.2 Non-conductive Cylinders . . . . . . . . . . . . . . . 87
4.3.3.3 Conductive and Non-conductive Cylinders . . . . . . 88
5 Experimental Validation 90
5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 The Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 The X-Y Gantry Robot . . . . . . . . . . . . . . . . . . . . . 92
5.1.2.1 Electromagnetic Noise from Stepper Motors . . . . . 94
5.2 Electro-sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 Signal Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.2 Quasistatic-like Approximation . . . . . . . . . . . . . . . . . 99
5.4 Operation Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5.1 Straight Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5.1.1 Both Walls Conductive . . . . . . . . . . . . . . . . . 105
5.5.1.2 Both Walls Non-conductive . . . . . . . . . . . . . . 107
5.5.1.3 One Conductive and One Non-conductive Wall . . . 109
5.5.2 Cluttered Corridor . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.2.1 Conductive Cylinders . . . . . . . . . . . . . . . . . 113
5.5.2.2 Non-conductive Cylinders . . . . . . . . . . . . . . . 115
5.5.2.3 Conductive and Non-conductive Cylinders . . . . . . 118
5.5.3 Obstacle Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.3.1 Conductive Cylinders . . . . . . . . . . . . . . . . . 121
5.5.3.2 Non-conductive Cylinders . . . . . . . . . . . . . . . 124
5.5.3.3 Conductive and Non-conductive Cylinders . . . . . . 124
6 Conclusions and Future Work 129
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1.1 Sensor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
vi
6.1.3 Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.4 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography 139
vii
List of Tables
4.1 Comparison of Maxwell?s general and quasistatic electric field equations 58
4.2 Materials used in simulations . . . . . . . . . . . . . . . . . . . . . . 62
viii
List of Figures
1.1 Effect of an object on fish?s electric field, (A) Unperturbed field, (B)
Field lines are dispersed by a non-conductive object, e.g., a rock, (C)
Field lines are concentrated by a conductive object, e.g., sea grass. . . 4
1.2 Electric image of an object as perceived by a fish. . . . . . . . . . . . 4
2.1 Model of electro-sensor used for electrostatics analysis . . . . . . . . . 14
2.2 Method of images applied to the straight-wall problem. Red color
stands for positive charge(s) and green color stands for negative charge(s). 16
2.3 A multipole placed within a conducting circular arena. . . . . . . . . 17
2.4 Electro-sensor multipole field perturbed by a straight wall. (A) The
setup, (B) Electrostatic image formed . . . . . . . . . . . . . . . . . . 19
2.5 Variation of electric image due to a straight wall. (A) Effect of change
in lateral offset, (B) Effect of change in angular offset. . . . . . . . . . 21
2.6 Electro-sensor introduced in a straight tunnel forming an infinite se-
ries of images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Variation of electric image in a 12 inch wide tunnel. (A) Effect of
change in lateral offset, (B) Effect of change in angular offset. . . . . 26
2.8 Electro-sensor in a circular arena. . . . . . . . . . . . . . . . . . . . . 27
3.1 Optimal state extraction functions. (A) y-extraction function, (B)
?-extraction function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Feedback control system for performing obstacle avoidance . . . . . . 39
3.3 Variation of structured singular value, ? with respect to frequency. . . 45
3.4 Simulations demonstrating centering behavior in a straight tunnel.
(A) Trajectories, (B) Actual and estimated lateral offsets, y and y?,(C)
Actual and estimated angular offsets, ? and ??. . . . . . . . . . . . . . 48
3.5 Simulations demonstrating wall following behavior in a circular arena.
(A) Trajectories, (B) Angular rates, (C) Actual and estimated lateral
states, y and y?, (D) Actual and estimated angular states, ? and ??. . . 50
3.6 Absolute values of electric images. (A) Conducting objects, (B) Non-
Conducting objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Optimal state extraction functions for mixed conductivity environ-
ments. (A) y-extraction function, (B) ?-extraction function. . . . . . 54
4.1 Different environments used for simulations. (A) Straight tunnel, (B)
Cluttered corridor, (C) Obstacle field. . . . . . . . . . . . . . . . . . . 61
4.2 Mesh quality choices for simulations. (A) Coarse mesh, (B) Fine
mesh, (C) Locally refined mesh. . . . . . . . . . . . . . . . . . . . . . 64
4.3 Simulation results for a straight tunnel with conductive walls. (A)
Trajectories, (B) Actual and estimated lateral offsets, y and y?,(C)
Actual and estimated angular offsets, ? and ?? . . . . . . . . . . . . . 69
ix
4.4 Simulation results for a straight tunnel with non-conductive walls.
(A) Trajectories, (B) Actual and estimated lateral offsets, y and y?,(C)
Actual and estimated angular offsets, ? and ??. . . . . . . . . . . . . . 70
4.5 Simulation results for a straight tunnel with one wall each of conduc-
tive and one non-conductive properties. (A) Trajectories, (B) Actual
and estimated lateral offsets, y and y?,(C) Actual and estimated an-
gular offsets, ? and ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Steady-state trajectory for the cluttered corridor consisting of con-
ducting objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.7 Simulation results for a corridor with all conducting objects. (A)
Trajectories, (B) Time traces of actual and estimated lateral offset, y
and y?, (C) Time traces of actual and estimated angular orientations,
? and ??, (D)Time traces of the velocity components of the vehicle in
the inertial frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Steady-state trajectory for the cluttered corridor consisting of non-
conducting objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.9 Simulation results for a corridor with all non-conducting objects. (A)
Trajectories, (B) Time traces of actual and estimated lateral offset, y
and y?, (C) Time traces of actual and estimated angular orientations,
? and ??, (D)Time traces of the velocity components of the vehicle in
the inertial frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.10 Steady-state trajectory for the cluttered corridor consisting of con-
ducting and non-conducting objects. . . . . . . . . . . . . . . . . . . 82
4.11 Simulation results for a corridor with conducting and non-conducting
objects. (A) Trajectories, (B) Time traces of actual and estimated
lateral offset, y and y?, (C) Time traces of actual and estimated angu-
lar orientations, ? and ??, (D)Time traces of the velocity components
of the vehicle in the inertial frame. . . . . . . . . . . . . . . . . . . . 83
4.12 Simulation results for an obstacle field with all cylinders being con-
ductive. (A) Trajectories, (B) Time traces of estimated lateral offset
y?, (C) Time traces of estimated angular orientation ??, (D)Time traces
of the velocity components of the vehicle in the inertial frame. . . . . 86
4.13 Simulation results for an obstacle field with all cylinders being non-
conductive. (A) Trajectories, (B) Time traces of estimated lateral
offset y?, (C) Time traces of estimated angular orientation ??, (D)Time
traces of the velocity components of the vehicle in the inertial frame. 87
4.14 Simulation results for an obstacle field with conductive and non-
conductive. (A) Trajectories, (B) Time traces of estimated lateral
offset y?, (C) Time traces of estimated angular orientation ??, (D)Time
traces of the velocity components of the vehicle in the inertial frame. 88
5.1 Experimental Setup consisting of glass tank and gantry. . . . . . . . . 91
5.2 Electro-sensor in experiments. (A) CAD model, (B) Electro-sensor
assembled on the setup. . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Filtering of the sensory signal. . . . . . . . . . . . . . . . . . . . . . . 99
x
5.4 Fitting a sine wave to the filtered signal. . . . . . . . . . . . . . . . . 100
5.5 Environments for experimentation (A) Tunnel, (B) Clutter, (C) Ob-
stacle field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Experiment demonstrating centering in a straight tunnel having con-
ductive walls (A) Trajectory, (B) Experimental setup, (C) Time traces
of actual and estimates lateral offsets, y and y?, (D) Time traces of
actual and estimated angular orientations, ? and ?? . . . . . . . . . . . 105
5.7 Effect of variation in initial conditions on trajectories in a tunnel with
conductive walls, (A) Variation in initial lateral offset, (B) Variation
in initial angular offset . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 Experiment demonstrating centering in a straight tunnel having non-
conductive walls (A) Trajectory, (B) Experimental setup, (C) Time
traces of actual and estimates lateral offsets, y and y?, (D) Time traces
of actual and estimated angular orientations, ? and ?? . . . . . . . . . 108
5.9 Effect of variation in initial conditions on trajectories in a tunnel
with non-conductive walls, (A) Variation in initial lateral offset, (B)
Variation in initial angular offset . . . . . . . . . . . . . . . . . . . . 109
5.10 Experiment demonstrating centering in a straight tunnel having one
conductive and one non-conductive wall (A) Trajectory, (B) Experi-
mental setup, (C) Time traces of actual and estimates lateral offsets,
y and y?, (D) Time traces of actual and estimated angular orientations,
? and ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.11 Effect of variation in initial conditions on trajectories in a tunnel with
one conductive and one non-conductive wall, (A) Variation in initial
lateral offset, (B) Variation in initial angular offset . . . . . . . . . . 111
5.12 Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with conducting cylinders, (A) Trajectory, (B) Experi-
mental setup, (C) Time trace of relative state, y?, (D) Time trace of
relative states, ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.13 Effect of variation in initial conditions on trajectories in a cluttered
corridor comprising of conductive cylinders, (A) Variation in initial
lateral offset, (B) Variation in initial angular offset . . . . . . . . . . 115
5.14 Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with non-conducting cylinders, (A) Trajectory, (B) Ex-
perimental setup, (C) Time trace of relative state, y?, (D) Time trace
of relative states, ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.15 Effect of variation in initial conditions on trajectories in a cluttered
corridor comprising of non-conductive cylinders, (A) Variation in ini-
tial lateral offset, (B) Variation in initial angular offset . . . . . . . . 117
5.16 Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with conductive cylinders on one side and non-conductive
cylinders on the other, (A) Trajectory, (B) Experimental setup, (C)
Time trace of relative state, y?, (D) Time trace of relative states, ?? . . 119
xi
5.17 Effect of variation in initial conditions on trajectories in a cluttered
corridor comprising of conductive and non-conductive cylinders, (A)
Variation in initial lateral offset, (B) Variation in initial angular offset 120
5.18 Experiment demonstrating obstacle avoidance in an obstacle field
comprising of conductive cylinders, (A) Trajectory, (B) Experimental
setup, (C) Time trace of relative state, y?, (D) Time trace of relative
states, ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.19 Effect of variation in initial conditions on trajectories in an obsta-
cle field comprising of conductive cylinders, (A) Variation in initial
lateral offset, (B) Variation in initial angular offset . . . . . . . . . . 123
5.20 Experiment demonstrating obstacle avoidance in an obstacle field
comprising of non-conductive cylinders, (A) Trajectory, (B) Experi-
mental setup, (C) Time trace of relative state, y?, (D) Time trace of
relative states, ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.21 Effect of variation in initial conditions on trajectories in an obstacle
field comprising of non-conductive cylinders, (A) Variation in initial
lateral offset, (B) Variation in initial angular offset . . . . . . . . . . 126
5.22 Experiment demonstrating obstacle avoidance in an obstacle field
comprising of conducting and non-conductive cylinders, (A) Trajec-
tory, (B) Experimental setup, (C) Time trace of relative state, y?, (D)
Time trace of relative states, ?? . . . . . . . . . . . . . . . . . . . . . . 127
5.23 Effect of variation in initial conditions on trajectories in an obsta-
cle field comprising of conductive and non-conductive cylinders, (A)
Variation in initial lateral offset, (B) Variation in initial angular offset 128
xii
Nomenclature
WFI Wide-Field Integration
FEM Finite Element Methods
CNC Computerized Numerical Control
RMS Root Mean Square
R Radius of the electro-sensor ring
a Tunnel width
am Nominal tunnel width
?a Allowed Perturbation in tunnel width
? Uncertainty in tunnel width
? Locations of electro-receptors
V Total potential due to the source
V ? Total potential due to image multipole
? Electric image
Q Charge carried by the source on the sensor
? Permittivity of water
P (r, ?) Radial position of point P in space
r0 Distance of P from the origin
r Distance of P from center of the sensor
re Distance of P from each electro-receptor
y Lateral offset of the sensor
? Angular offset of the sensor
q State vector [y, ?]
vi ith WFI output
Fi ith basis function
C? State extraction matrix
m Number of receptors on the sensor
n Total number of basis functions
Ky, K? Static feedback gains
xiii
Chapter 1
Introduction
Autonomous underwater vehicles have been in existence for more than half a century
now, the first being developed at the University of Washington from the late 1950?s
until early 1960?s [1]. Their objective was to measure temperature and sound veloc-
ity at varying depths in oceans. In time, there have been significant improvements in
vehicles, sensors, algorithms and controllers in order to achieve even complex tasks.
Contemporary vehicles are required to be designed to perform tasks like reconnais-
sance [2, 3], aquatic structures, flora and fauna examination [4], ocean bed imaging
[5] etc. Obstacle avoidance forms the basic behavior required by an autonomous
vehicle for completing many of these tasks. The obstacle avoidance problem has
been handled extensively for robotic vehicles operating on ground [6, 7], in air [8, 9]
as well as under water [10, 11, 12], using visual or sonar sensing modalities. Vision
and sonar are the two most widely used sensing modalities in underwater robotics.
While these work almost flawlessly for airborne and ground vehicles, there are chal-
lenges in using them on underwater vehicles. Vision deteriorates at a rapid rate
with depth under water. Additionally, dissolved matter in water poses difficulties
in clearly detecting objects from visual images. Sonar on the other hand does not
1
exhibit this problem, however it has a directional nature where only the objects
within the sound cone can be realized correctly. Thus several successive scans and
added processing of the acquired data is necessary. Sonar reflection and echo from
unwanted sources in the environment is another difficulty that is required to be
resolved by the processing algorithms. Hence, the success of an obstacle avoidance
scheme depends on factors other than the underlying signal processing algorithm.
This is evident from the above limitations of underwater vision and sonar based
sensing.
This project aims towards addressing these issues with the conventional sens-
ing modalities. Nature presents us with a simple yet effective underwater sensing
technique called electrolocation, which does not depend on visibility in water and is
omnidirectional in nature [13]. By providing a similar capability to an underwater
vehicle, a principal, adjunct or redundant method of obstacle sensing and avoidance
can be developed.
1.1 Background
1.1.1 Basic Physiology of the Electrolocation System
The two key elements in the electroreception system are the electric organ, which
produces the field, and the electro-receptors, which senses the electric field after
it has been perturbed by the environment. The electric organ is usually situated
in the tail region of the fish [14]. It produces electric field by releasing varying
concentrations of positively charged sodium and potassium ions [15]. This process
2
of ion exchange is carried out by arrays of cells called electrocytes possessing a
complex neural network to control them [16]. The electro-receptors are electrolytic
gel-filled narrow-mouthed cups submerged in the skin [17]. They act as biological
voltmeters and transmit the sensed voltage to the brain via afferent neurons [16].
Thousands of electro-receptors are distributed over the fish?s body, more of them
being at preferred locations like the head and trunk [18]. To ensure that no current
is drawn by the skin of the fish, the skin has a very high resistance [19].
1.1.2 Concept of an Electric Image
The field lines of the electric field of the fish originate at the electric organ in the
tail region and terminate at the electroreceptors dispersed all over the body. When
a conductive object is introduced in the fish?s field, a local increase in the electric
flux and thus a local positive perturbation is observed. By conductive, it is implied
that the object is more conductive than the surrounding water, e.g., sea grass, metal
pieces, etc. Similarly, a non-conductive object or an object that is less conductive
than the water, like rocks, glass pieces, other fish, etc. weaken the electric field
locally, thus producing a negative perturbation [20]. This can also be seen in Figure
1.1.
The perturbations are perceived by the fish in the form of voltage spikes at
the locations of electro-receptors. The profile of all the spikes together produces
a surface with certain ?hot-spots? representing the perturbations. Such profiles
are called electric images [21] and constitute the sensory signal of electric fish.
3
A B C
Figure 1.1: Effect of an object on fish?s electric field, (A) Unperturbed field, (B)
Field lines are dispersed by a non-conductive object, e.g., a rock, (C) Field lines are
concentrated by a conductive object, e.g., sea grass.
Perturbations to the field of magnitudes as small as 1?V can also be registered by
electric fish in order to interpret the location information of the perturbing object
[22]. When one horizontal slice of the fish is considered, the electric image from all
the electro-receptors is in the form of a 2D curve instead of a surface, as shown in
Figure 1.2. This work considers only planar geometries, because the concept of a
planar electric image is important.
Figure 1.2: Electric image of an object as perceived by a fish.
4
1.1.3 Motivation for an Artificial Electro-sensor
Weakly electric fish in the Gymnotiformes and the Mormyridae groups are cham-
pions at electrolocation, although each group is known for a different mechanism
to perform it. Gymnotiformes, found in South America, produce an electric field
that temporally varies as a cyclic wave with a few constituent frequencies super-
imposed on each other [18], hence referred to as a wave-type fish. Mormyridae,
found in Africa, generate pulses to create their electric field [18] and are fittingly
called pulse-type fish. The frequency response of the electric field of wave-type fish
is understandably narrow as only a few frequencies are contained in it. On the
other hand, the frequency response of a pulse-type fish is wide due to presence of
instantaneous pulses. Due to this reason, pulse-type fish are capable of detecting
objects with a variety of complex impedance values. However extra effort is required
for maintaining a pulse-type electric field as compared to a wave-type field. In this
project, while designing the electro-sensor, many of the features found in wave-type
fish are adopted directly or after simplification to be compatible with contemporary
hardware available.
1.2 Electrolocation in Robotics
Lissmann and Machin [13, 23] performed experiments on electric fish wherein rods
of materials with different conductivity were placed in two porous pots, and the fish
were rewarded on approaching the pot containing higher conductivity rod. These
experiments qualitatively established for the first time that these fish are able to
5
differentiate objects based on the object?s conductivity. It has been known since
the work of Lissmann [24] and later Knudsen [25] that the electric field observed
sufficiently far from the fish resembles that of an oscillating electric dipole. This
finding made possible mathematical analysis of behaviors of electric fish based on
electric images. Rasnow [26] derived a formula using electrostatics for analytically
computing the electric image of a sphere, with its electrical properties, diameter,
and distance from the electric field dipole as variables. Assad et al. [14] further
reconstructed the electric fields of different species of electric fish using measure-
ments at multiple sites in the fields. He suggested through behavioral experiments
that relative width [14] could be used to model the process by which an electric
fish extracts the proximity information of an object from the corresponding electric
image. In an independent study by Von der Emde [20], slope-to-amplitude ratio
was believed to be a better metric to model the proximity information extraction
behavior of a fish. Irrespective of the study, it is clear that electric images possess
artifacts that reflect proximity information of the perturbing object.
The response of resistive objects and objects with complex impedance to a
fish?s electric field were modeled in [27] and [19]. In the case of a resistive ob-
ject results showed modulation of the magnitude of the peak of the electric image
whereas in an object with complex impedance a distortion of the image occurs. Fi-
nite Element Method (FEM) was used to model the effect of objects with complex
impedance on an electric field similar to that of a fish; similar distortion of electric
images were observed [28]. Significant amount of experimental work has also been
carried out in this field. In a prototypical setting, it was shown that the exact loca-
6
tion and the approximate shape of an object can be reconstructed from the electric
images obtained by sensing it from multiple distinct locations in the environment
[29]. In another study, spherical objects of different materials were moved past a
biomimetic underwater electro-sensor array to study the effect of object distance on
the corresponding electric images [22]. It was verified that the amplitude of electric
images decreased with increasing distance and vice versa. In [30], the location of an
object was computed in simulation and experiments by solving an inverse problem
using a map of electric images previously acquired with the same object at different
locations in the environment. Solving the inverse problem involved an interpolation
routine based on the predetermined electric images from the map and the electric
image due to the object at the required position.
The advent of advanced model estimation methods has enabled researchers
to accurately estimate the locations of obstacles in an artificially generated electric
field. Solberg et al. [31] used a particle filter to estimate the location of a spher-
ical object in an electric dipole field using two sensing electrodes. Starting with
a large number of uniformly dispersed particles, their algorithm was able to iter-
atively converge on the actual position of the object, either stationary or moving.
After extending their two electrode sensor to take a form of a cylindrical array of
sensors, the same signal processing technique proved helpful in performing obstacle
avoidance by a robot in underwater scenes [32]. They also showed, in a separate
work, that Electrical Impedance Tomography can be used to compute the location
of objects based on their impedance values computed as a reverse problem from
the corresponding electric images [33]. An unscented Kalman filter was shown to
7
be effective in electrolocating spherical and cubical objects and in performing wall-
following maneuvers using a multipole sensor [34, 35]. Using the same experimental
setup, in [36] a sensor was demonstrated to travel along electric field lines towards
or away from an object depending on whether the object was conductive or non-
conductive. The sensor was propelled between non-conductive obstacles towards a
conductive object by the resultant field.
For learning a way to extract proximity information from electric images due
to global stimuli we used insects as motivation, because insects perceive their visual
fields as patterns of relative motion between themselves and the environment, more
commonly known as optic flow [37, 38]. Specialized neurons called tangential cells
are used to spatially decompose the optic flow patterns to generate visual cues like
visual odometry, useful for navigation [39, 40]. In doing so, they extract the relative
information contained in optic flow for control and do not compute the absolute
quantities [41]. The approach found in naturally evolved systems not only provides
the relevant control parameters to the insects, but makes the process less intensive
for their brain, which has limited computational capabilities. The transition of these
efficient signal processing methods from the physiology and neuroethology domains
to the robotics domain is made possible by translating the spatial decomposition
routine into mathematical terms using a method termed as Wide-Field Integration
(WFI) [42, 43, 44, 45]. Due to inherent simplicity of the algorithm it has been proven
that a control system developed around the WFI technique can completely eliminate
the need for dynamic estimation of absolute quantities by utilizing static output
feedback of optic flow [46]. The algorithm has been successfully demonstrated on
8
ground and air borne vehicles to perform unmapped reflexive obstacle avoidance [47,
45, 8]. In this project, it has been demonstrated that WFI processing of an electric
image yields relative proximity information of the perturbing objects which can be
confined in a control theoretic framework to perform reflexive obstacle avoidance
maneuvers.
1.3 Approach/Methods
This thesis focuses on extracting control-relevant information from electric images
with low computational effort. For doing so, the approach proposed and taken is
encapsulated below.
? Derive qualitative equations for computing electric images for a straight tunnel
with conductive walls. These equations will be based on electrostatic principles
due to their simplicity.
? Constitute a control strategy by deriving feedback laws for a straight tunnel
with conductive walls.
? Assume that geometries that are more complex than a straight tunnel locally
look like a straight tunnel.
? Perform robust analysis to get an idea of the stability of the controller with
respect to variation of tunnel width.
? Evaluate the control strategy in simulations and experiments by applying the
control strategy designed for a straight, conductive tunnel.
9
The direct implication of this approach will be the use of a common control strategy
for variety of environments. Various combinations of object geometries and materials
can then be used to form to test and evaluate the controller.
1.4 Contribution
The contributions of this research to the contemporary state-of-the-art are primarily
a simple representation of electric images and an easier procedure to extract relevant
proximity information from them. The detailed contributions are listed below.
? Formulation of the Physics of Electrolocation: Implemented the principles
from electrostatics to compute electric images for our version of a bio-inspired
electro-sensor.
? Adaptation and Evaluation of WFI-based Algorithms : Adapted the existing
WFI technique into a numerical version to extract proximity information from
distributed electric field potential patterns. Evaluated obstacle avoidance al-
gorithms in simulation by processing electric images computed for simpler
environment geometries.
? FEM based simulations for complicated geometries : Confirmed the working of
the WFI-based sensing and obstacle avoidance routines in environments with
complicated geometries using FEM generated electric images.
? Creation of a sensor and signal processing scheme: For hardware implemen-
tation, an electro-sensor and a signal acquisition and processing scheme were
10
designed to mimic an electric fish.
? Experimental validation: Validated the physical model of electrolocation and
the control algorithm using a gantry system to autonomously navigate the
electro-sensor between obstacles under water.
The following chapters categorically detail each of the above contributions.
1.5 Thesis Organization
The thesis is organized as follows. Chapter 2, derives electrostatics equations for
electric images due to straight and circular walls using a technique called the method
of images. An important set of equations for a straight tunnel is also derived in this
Chapter.
Chapter 3, describes in detail the procedure to extract control-relevant infor-
mation from electric images using the WFI method. We propose the use of two state
extraction functions to compute the relative states of the electro-sensor with respect
to an obstacle. A static feedback controller is designed to use these extracted rela-
tive states effectively to manipulate the motion of a robotic vehicle and to perform
obstacle avoidance. The WFI algorithm was adapted to process the electric images
by performing numerical integration of generally non-integrable equations that are
encountered in this problem. The state-extraction functions and thus the entire
analysis is derived for a straight tunnel with conductive walls. Simulation results
are presented in order to demonstrate the use of the relative states in performing
obstacle avoidance in environments with conductive obstacles of relatively simple
11
geometries.
Chapter 4, presents simulations in environments containing non-conductive
obstacles. For this purpose, electric images are generated using the quasistatic
approach found under finite element methods. This approach also allows the use
of an oscillatory source to power the electric field of the electro-sensor, as opposed
to a static source in the electrostatics case. Additionally, obstacle geometries can
be made more complicated, thus enabling simulations of more realistic environment
scenes.
Chapter 5, describes the presented analysis and simulations are validated ex-
perimentally. Effort was made to replicate environments similar to their simulation
counterparts. The sensor design is maintained to be consistent with the design used
for the electrostatic analysis. The experimental setup consists of a large glass water
tank and an X-Y positioning gantry and an electro-sensor.
Chapter 6, contains concluding remarks and directions for future research for
this project.
12
Chapter 2
Electrostatic Expressions for Electric Images
This chapter presents the electrostatic equations for electric images formed due
to the perturbation of an electric field from various environments like a straight
wall, a tunnel and a circular arena. An electro-sensor model motivated from the
electroreception system in fishes is used to formulate these equations. (For greater
insight into electrolocation in fishes, the reader is referred to [19, 48, 49, 50, 51, 52,
53].) Further, we take assistance of a well-known tool from electrostatics used to
simplify the electric field differential equations, called the method of images. There
are two main assumptions behind this analysis: all obstacles are conductive in nature
and the electric field source is static in nature. Although a non-oscillatory source
is rarely observed in nature, this treatment is considered here due the its inherent
accessibility and the convenience it presents in introducing the concepts of electric
image processing and control methodology.
Electric images of a fish need a complete electromagnetic analysis to capture
the effect of all the variables involved, which typically is a cumbersome task. How-
ever, for visualizing the nature and variation of the electric image due to perturbing
objects, it is easier to analyze them using electrostatics. Electrostatic analysis is not
13
a substitute for full electromagnetic study, but it is a faster way to understand the
concept of electrolocation and to design controllers for robotic vehicles as performed
in Chapter 3.
2.1 Electro-Sensor Model
Source
(Positive pole)
X
Y
?
Electro-receptor
P(r,?)
r
?
(Negative poles)
Figure 2.1: Model of electro-sensor used for electrostatics analysis
The sensor model assumed is shown in Figure 2.1. It consists of an electric
multipole system with the source pole located centrally to a circular ring of ra-
dius R carrying m electro-receptors. The electro-receptors are modeled as small
spherical electrodes and are distributed evenly along the ring. The centrally located
source can be considered as the positive pole carrying a charge Q and the peripheral
electro-receptors as negative poles, each possessing a charge ?Qm . The variables in
an electrostatic potential equation at a point in a field of a known charge are the
value of the charge itself and the distance between the point and the charge [54, 55].
14
On similar lines, the equation to evaluate the potential at a general point P , having
a radial position (r0, ?), as shown in Figure 2.1, due to the multipole system of the
sensor is thus the sum of the potentials due to all of the individual charges as given
by Eq. (2.1) below, according to the superposition theorem [54, 55]:
V (r, ?) = 14??
(
1
r ?
m
?
k=1
Q
mre,k
)
(2.1)
Here, r is the distance of point P from the source and re,k is the distance of point P
from each of the m electro-receptors. Subscript k iterates the equation for each of
the m receptors. The presence of the factor m in the denominator in the second half
of the right hand side of the equation comes directly from the value of the charge
on each electro-receptor.
2.2 Method of Images
For a point charge in free space, electric field intensity and voltage potential at any
point can be computed using Coulomb?s Law and the work done against the field,
respectively. But when an object is inserted in the field, both of the above computa-
tions become exceedingly difficult. The method of images is a tool in electrostatics
that is useful in solving Coulomb?s equation and the voltage differential (or inte-
gral) equation. It states that charges and surfaces are equivalent to a system of
the charges and their corresponding images due to the surfaces only. The boundary
conditions due to the surfaces are replaced by image charges and the effect of the
surfaces is accounted for [55].
A popular example for explaining this concept is that of a point source placed
15
near a conducting wall. Consider a positive pole with charge Q at a position y from
a straight conducting wall. The effect of the wall on the electric field of the pole
can be reproduced by introducing another pole at a position ?y from the wall with
opposite polarity. This secondary pole is called as image pole and is illustrated in
Figure 2.2. The total potential at a point in space is expressed as the sum of those
due to the source and the image poles as given by Eq. (2.2) below. The point of
interest P (r0, ?) is the location in space where the potential is computed.
P(r0, ?)
y
y
y
Q
Q?
O
r
r?
Conducting Surface
Q
Y
X
A B
?
r0
Figure 2.2: Method of images applied to the straight-wall problem. Red color stands
for positive charge(s) and green color stands for negative charge(s).
Vwall =
Q
4??
(1
r ?
1
r?
)
(2.2)
Here, r and r? are the lengths P?Q and ?PQ?, respectively; they are equal to the
2-norm of the following vectors.
r =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
?
?
?
?
?
0
y
?
?
?
?
r? =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
+
?
?
?
?
0
y
?
?
?
?
(2.3)
16
This analysis holds for a general charged body of any shape provided the
image has the same shape and geometry as that of the original body. Moreover,
for conducting surfaces that are not planar, the analysis is still valid if certain
modifications are made to the image, as demanded by the shape of the surface.
Thus for a circular arena, a similar expression for potential can be derived. For a
charge Q inside the arena, the image can be assumed to have formed outside the
arena. This image will lie on the same radius vector of the arena on which the
original charge is located, as illustrated in Figure 2.3.
P(r0 , ?)
r
r?
B D
Q Q?
a
b
d
O
Y
X
Figure 2.3: A multipole placed within a conducting circular arena.
Following analysis is obtained from [54, 55]. The image charge and its distance
from the center of the arena are,
Q? = ?adQ d =
a2
b (2.4)
This means that the total voltage potential at point P (r0, ?) is,
Vsphere =
Q
4?? r +
Q?
4?? r? (2.5)
17
The variables r and r? are the distances of the pole from the point of interest as
mentioned before and are the 2-norms of the following vector expressions.
r =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
?
?
?
?
?
b
0
?
?
?
?
r? =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
?
?
?
?
?
d
0
?
?
?
?
(2.6)
The reader is referred to [54, 55] for detailed derivations. The two examples provided
in this Section prove to be the stepping stones for further analysis. The next section
introduces the our sensor model.
2.3 Electric Image
An electric image is the modulation caused to the nominal spatial and temporal
patterns of transdermal potentials of an electric fish due to the presence of an object
near it. To introduce the mathematical expressions of electric image, consider the
case of a single wall as shown in Figure 2.4A.
Assume a straight conducting wall at a position y relative to the center of the
electro-sensor. Its effect on the electric field of the multipole can be reproduced by
introducing another multipole at a position ?y and having opposite polarity. The
secondary multipole is called an image multipole. The total potential at a point in
space is the sum of those due to the source and the image multipoles as given by
Eq. (2.7). The point of interest P (r0, ?) is the location in space where the potential
18
?,Degrees
Pe
rtu
rb
at
io
n
Po
te
nt
ial
0
0.1
0.3
 0.2
0.4
0.5
0 90 180 270 360
Electric Image
0.6
A B
Electro-sensor
?
b
b
X
Y
?
Wall
x
y y
P(r0 , ?)
X
Y
Image
multipole
r r?
Figure 2.4: Electro-sensor multipole field perturbed by a straight wall. (A) The
setup, (B) Electrostatic image formed
is computed. We have,
Vwall(q, r, ?) =
Q
4??
[
1
r ?
1
r? ?
1
m
m
?
k=1
(
1
re,k
? 1r?e,k
)]
(2.7)
Here, r and re,k are the distances of P (r0, ?) from the source and the m electro-
receptors on the sensor, respectively. The distances r? and r?e,k pertain to the image
multipole and are the counterparts of r and re,k. The subscript k above is stands
for the image of the kth, k = 1, ..., m electro-receptor. The variable m in the
denominator of the summation term on the right hand side of the equation comes
from the charge on each electro-receptor which is assumed to be ?Q/m.
In the presence of a wall, two additional independent variables are included in
the equation for electric image: the lateral offset, y, and orientation, ?, with respect
to the wall. These two variables can be grouped as q = (y, ?) giving the pose of the
sensor with respect to the wall. It is assumed that the coordinate system is located
on the wall with the positive X-axis directed forward along the wall and the positive
Y-axis perpendicular to the wall so that the Z-axis is directed out of the plane of the
19
paper. The sensor is located at (x, y) with respect to the origin of the coordinate
system. The distances, r, re,k, r? and r?e,k can be derived by computing the 2-norms
of the following vector expressions.
r =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
?
?
?
?
?
x
y
?
?
?
?
re,k = r?
?
?
?
?
R cos
(
2pi(k?1)
m + ?
)
R sin
(
2pi(k?1)
m + ?
)
?
?
?
?
r? = r?
?
?
?
?
0
?2y
?
?
?
?
r?e,k = r? ?
?
?
?
?
R cos
(
?2pi(k?1)m ? ?
)
R sin
(
?2pi(k?1)m ? ?
)
?
?
?
?
(2.8)
In the above equations, ?k = 2pi(k?1)m gives the locations of the electro-receptor
k = 1, ..., m. The field of the source multipole is perturbed by the wall, the effect of
which is replicated by introducing the image multipole, i.e., the effect on the source
field is the field perturbation. Hence the potential due to the image multipole
computed at the locations of the electro-receptors on the sensor gives the following
expression for the electric image of a straight wall:
?wall(q, ?) =
Q
4??
[
1
m
m
?
k=1
(
1
r?e,k
)
? 1r?
]
(2.9)
The variable Q stands for the magnitude of the total charge on the electro-receptors,
thus giving the magnitude of the charge on each of the m electro-receptors as Q/m.
Q is also the charge on the source pole of the multipole. This expression gives
the value of the perturbation potential ? at each of the m electro-receptors on the
20
electro-sensor located at a azimuthal spacing of ? between each other. The depen-
dence of the electric image on ? is explicit from the expressions for the distances
r?e,k and r? which are computed as given by Eq. (2.8) with the values for r0 and ?
computed at the electro-receptors using the vector expression:
r0 =
?
?
?
?
R cos(? + ?)
R sin(? + ?)
?
?
?
?
+
?
?
?
?
x
?y
?
?
?
?
(2.10)
Figure 2.4B above shows the electric image for a particular pose where the ve-
hicle is directed towards the wall. With an increase in the distance y, the magnitude
of the electric image decreases. On varying the orientation of the sensor with re-
spect to the wall, the peak of the electric image is shifted along the electro-receptors
linearly. The variation in electric image with the lateral and angular offsets are illus-
trated in Figure 2.5A&B respectively. The offsets mentioned here are with respect
to the wall or the coordinate system mentioned above.
0
0.005
0.01
0.015
0.02
0.025
Pe
rtu
rb
at
io
n 
Po
te
nt
ial
, ?
, v
ol
ts
Electric Images for variable lateral o!set
0 90 180 270 360
Electro-receptor position, ?, degrees
Electric Images for variable angular o!set
0 90 180 270 360
Electro-receptor position, ?, degrees
Closest to the wall
Farther away
Even farther away
0
0.005
0.01
0.015
0.02
0.025
Pe
rtu
rb
at
io
n 
Po
te
nt
ial
, ?
, v
ol
ts
? = 0?
? = 20?
? = 40?? = 60?
Figure 2.5: Variation of electric image due to a straight wall. (A) Effect of change
in lateral offset, (B) Effect of change in angular offset.
21
2.3.1 Electric Image in a Straight Tunnel
The straight-tunnel case is a direct extension of the straight-wall problem. Hence,
to study the electrostatics of a straight tunnel we introduce a second wall to the
straight-wall case. Due to the inherent symmetry of the environment, the origin of
the coordinate frame is shifted to the centerline of the tunnel. Proceeding with the
straight-wall analysis, the boundary conditions imposed by the two walls are satisfied
by one image multipole associated with each wall. But each image multipole will
introduce one new boundary condition due the wall farther from it. So another set
of two images arises due to each image in the first set. The process of formation of
new image multipoles for satisfying the boundary conditions is repeated infinitely
many times, which results in the formation of subsequent images. (It is analogous
to placing an object in between two parallel mirrors to obtain an infinite number
of mirror images.) Thus, when the electro-sensor is introduced between a set of
parallel conducting walls, the method of images analysis leads to the formation of
an infinite number of image multipoles as shown in Figure 2.6.
a+y a+y2(a+y) a-y a-y 2(a-y)
ay
X
Y
Straight
TunnelSuccessive
Image Multipoles
Sensor
?
b
b
X
Y
?
Figure 2.6: Electro-sensor introduced in a straight tunnel forming an infinite series
of images.
22
It is assumed that the X-axis is aligned along the centerline of the tunnel and
directed forward, while the Y-axis is perpendicular to the centerline. The sensor is
located at coordinates (x, y). The images can be divided into two groups so that
all the images in a group have the same orientation. One of the groups contains all
image multipoles with the same heading as that of the source multipole; this group
is assigned the subscript a and the other group is assigned the subscript b. The
total potential at point P (r0, ?) due to all of the image multipoles and the source
multipole is,
Vtunnel(q, r, ?) = V + V ? (2.11)
As is the case with the single wall formulation, non-primed variables denote quanti-
ties due to the source multipole, whereas primed variables give the terms due to the
image multipoles. Thus V ? is the contribution due to the image multipoles. Both
terms on the right-hand side of the above equation are defined as:
V = Q4??
[
1
r ?
1
m
m
?
k=1
1
re,k
]
(2.12)
V ? = Q4??
?
?
l=1
[
1
m
m
?
k=1
(
1
r?b,e,k,l+
+ 1r?b,e,k,l?
? 1r?a,e,k,l+
? 1r?a,e,k,l?
)
?
(
1
r?b,l+
+ 1r?b,l?
? 1r?a,l+
? 1r?a,l?
)]
(2.13)
The summation index k in Eq. (2.12) and Eq. (2.13) iterates over all of the m
electro-receptors on source multipole and the image multipoles. The index l in Eq.
(2.13) iterates all all of the infinite image multipoles. Only the first few of the infinite
image multipoles are needed to provide satisfactory accuracy in the computed value
of the potentials. As before, the terms r, re,k, r?a,k,l+ etc. are the distances of point
23
P from multipole centres and receptors. The subscripts contain extra information
about the particular multipole considered. Subscript e, k stands for electro-receptors
and denote that the particular distance is measured between point P and the kth
electro-receptor on that multipole. Subscripts a and b stand for the set of image
multipoles as described earlier. All the image multipoles are considered to have
formed on a line parallel to the Y-axis. The additional subscripts l+ and l? stand
for the lth image formed on the positive or negative side of the Y-axis. Thus, r?a,e,k,l+
stands for the distance between point P and the kth electro-receptor on the lth image
multipole that belongs to the set a of images and is formed on the positive Y-axis.
Similarly, r?b,l? is the distance of point P from the center of the lth image multipole
that belongs to the set b and is formed on the negative Y-axis. The distances in Eq.
(2.12) and Eq. (2.13) are the 2-norms of the vector expressions given below:
r =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
?
?
?
?
?
x
y
?
?
?
?
re,k = r?
?
?
?
?
R cos
(
2pi(k?1)
m + ?
)
R sin
(
2pi(k?1)
m + ?
)
?
?
?
?
r?a,l+ = r?
?
?
?
?
0
4al
?
?
?
?
r?a,l? = r?
?
?
?
?
0
?4al
?
?
?
?
24
r?b,l+ = r?
?
?
?
?
0
2(a? y) + 4a(l ? 1)
?
?
?
?
r?b,l? = r?
?
?
?
?
0
?2(a + y)? 4a(l ? 1)
?
?
?
?
r?a,e,k,l+ = r?a+ ?
?
?
?
?
R cos
(
2pi(k?1)
m + ?
)
R sin
(
2pi(k?1)
m + ?
)
?
?
?
?
r?a,e,k,l? = r?a? ?
?
?
?
?
R cos
(
2pi(k?1)
m + ?
)
R sin
(
2pi(k?1)
m + ?
)
?
?
?
?
r?b,e,k,l+ = r?b+ ?
?
?
?
?
R cos
(
?2pi(k?1)m ? ?
)
R sin
(
?2pi(k?1)m ? ?
)
?
?
?
?
r?b,e,k,l? = r?b? ?
?
?
?
?
R cos
(
?2pi(k?1)m ? ?
)
R sin
(
?2pi(k?1)m ? ?
)
?
?
?
?
. (2.14)
The perturbation potential recorded at the electro-receptors on the sensor
constitutes the electric image. In this case, the perturbation due to the tunnel walls
has been replaced by the image multipoles. Hence the electric image is the potential
due to the image multipoles computed at the electro-receptors on the sensor and is
given as,
?tunnel(q, ?) = V ? (2.15)
For computing the electric image, V ? is computed at the electro-receptor locations
25
?. The values for r0 and ? in V ? in Eq. (2.13) can be computed using:
r0 =
?
?
?
?
R cos(? + ?)
R sin(? + ?)
?
?
?
?
+
?
?
?
?
x
y
?
?
?
?
(2.16)
The tunnel electric image is similar in shape to that obtained in the straight-
wall case. It tends to decrease in magnitude when the sensor is moved towards
the centerline and the peak tends to shift laterally with the change in inclination
?. Additionally, the peak is formed on the region of the sensor that is closest to
either of the walls. Similar to a straight wall, the electric image for a tunnel requires
the knowledge of the pose of the sensor q = (y, ?) with respect to the centerline of
the tunnel. The variation in electric image with the lateral and angular offsets are
illustrated in Figures 2.7A&B, respectively.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 90 180 270 360
Electro-receptor position, ?, degrees
Pe
rtu
rb
at
io
n 
Po
te
nt
ial
, ?
, v
ol
ts
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Pe
rtu
rb
at
io
n 
Po
te
nt
ial
, ?
, v
ol
ts
0 90 180 270 360
Electro-receptor position, ?, degrees
Near the left wall Near the right wall
? = - 60?
? = - 40?
? = - 20?? = 0?
A B
y = 4? y = -4?
y = 3?
y = -3?
y = 2?
y = -2?
Figure 2.7: Variation of electric image in a 12 inch wide tunnel. (A) Effect of change
in lateral offset, (B) Effect of change in angular offset.
26
2.3.2 Electric Image in a Circular Arena
The expression for the electric image in a circular arena is presented here to provide
a complete procedure of formation of electric images using the method of images.
The curvature of the arena shapes the equations in this problem. These equations
are derived only for simulating wall-following behavior in such an environment.
Experimental validation or FEM based simulations for this environment have not
been performed.
Consider a circular arena of radius a within which a multipole is placed at a
radial distance b from the center. As before, the source of the multipole carries a
charge Q and each electro-receptor carries a charge equal to ?Q. The coordinate
system is assumed to have its origin at the center of the arena. Without loss of
generality, assume the multipole is located on the radius vector of the arena along
the X-axis as shown in Figure 2.8.
?
y
y' ?
y
a
x b
y b
?
Dipole
Image
Circular
Arena
Figure 2.8: Electro-sensor in a circular arena.
Recall, an image of a pole with a charge Q is formed at a distance d = a2b
from the center of the arena and has a charge Q? = ?adQ. The image of the central
27
pole of the multipole has a charge Q?. It is formed on the same azimuth as the
original multipole at a distance d from the center of the arena. Each of the images
of the m electro-receptors will have a charge ?Q?/m. These images will be formed
at distances proportional to the distances of the corresponding electro-receptors of
the original multipole according to the relation between b and d above. The image
multipole does not have a pole located exactly at the center of the ring of electro-
receptor images, due to the curvature of the arena. Following Eq. (2.17) gives the
expressions for the position vectors of the central poles and the electro-receptors of
both the original and the image multipole:
? =
?
?
?
?
b
0
?
?
?
?
?? =
?
?
?
?
a2
b
0
?
?
?
?
?e,k = ?+
?
?
?
?
R cos
(
2pi(k?1)
m + ?
)
R sin
(
2pi(k?1)
m + ?
)
?
?
?
?
??e,k =
a2
||?e,k||2
?e,k (2.17)
The actual distances of all the poles are the 2-norms of these position vectors.
The expressions for distances of the point P (r0, ?) from the source as well as
28
image multipole can be derived as in Eq. (2.18). We have
r =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
? ?
re,k = r? ?e,k
r? =
?
?
?
?
r0 cos(?)
r0 sin(?)
?
?
?
?
? ??
r?e,k = r? ? ??e,k (2.18)
The expression for total potential is given by the sum of the contributions due to
the source and the image multipoles, i.e., Vtunnel(q, r, ?) = V + V ?, where,
V = Q4??
[
1
r ?
1
m
m
?
k=1
1
re,k
]
V ? = Q4??
[
1
r? ?
1
m
m
?
k=1
1
r?e,k
]
(2.19)
Finally, the electric image for a circular arena is,
?arena(q, ?) = V ?, (2.20)
when computed at the locations of the electro-receptors on the source multipoles.
Chapter 3 delineates the procedure to extract relative states from the electric
image equations using Wide-Field Integration processing. The relative states are
then used to design the closed-loop control system that carries out autonomous
navigation tasks.
29
Chapter 3
Wide-Field Integration and Autonomous Control
This chapter discusses the entire procedure to extract relevant information from
electric images in the form of relative states using the electro-sensor model and the
equation for electric image in a tunnel from Chapter 2. Wide-Field Integration
(WFI) processing decomposes electric images into useful quantities contain the de-
sired state information of the vehicle carrying the sensor. Optimal values of relative
states are computed in order to form the feedback input to the vehicle actuation
system. Simulations showing centering in a tunnel and wall-following behavior in a
circular arena demonstrate the working of the signal processing routine and the con-
troller. Finally, the robustness of the controller when the tunnel width is different
from the nominal value is proven by providing a ?-analysis based argument.
3.1 Wide-Field Processing of Electric Images
Insects perceive their visual fields in the form of the relative speed of the environment
with themselves instead of absolute distances of objects, a phenomenon called optic
flow. Farther objects tend to move slower in comparison to closer objects, thus
setting up a speed/distance relationship. For avoiding collisions with closer objects,
30
it is imperative to move away from fast-moving visual fields towards slow-moving
fields. With a limited brain-size, it is necessary for these organisms to process optic
flow efficiently. WFI is a mathematical analogue of the processing of optic flow by
the tangential cells in insects [56].
For a planar tunnel geometry, an insect aligns with the centerline of the tunnel
by equalizing the optic flow in their visual fields. The nominal optic flow pattern
is generated during translation along the centerline of the tunnel. This pattern is
roughly sinusoidal [45], with the peak and the trough sections indicating the presence
of a wall on either side of the insect. The equal magnitude and zero spatial phase
shift of the peak and the trough indicate that the walls are at an equal distance
from the insect and parallel to its heading, respectively. A lateral offset from the
centerline of the tunnel increases one half of the optic flow pattern while the other
half diminishes. An angular offset produces a phase angle shift. Inner products
of such perturbed patterns with appropriate basis functions increase or decrease in
value accordingly. Thus, WFI can extract relevant information from the optic flow
perturbations in the form of lateral or angular offsets.
There is no evidence that electric fish use this approach to extract information
about their environments from electric images. However, electric images with the
multipole model considered in Section 2.3 exhibit amplitude modulation and phase
shift properties that can similarly provide lateral and angular offsets in a planar
tunnel. Let Fi represent the ith basis function used to decompose the electric image
to yield the ith WFI output vi. The WFI equation for extracting similar information
31
from electric images is,
vi(q) = ??(q, ?), Fi(?)? (3.1)
Recall from Figure 2.6, in a tunnel, small angular and lateral offsets from
the centerline produce low frequency spatial changes in electric images. So low
frequency harmonics of an electric image can capture all the information of the
offsets that produce such changes in the electric image. Fourier basis functions are
chosen for all the subsequent analysis due to their simplicity of implementation.
Hence the functions Fi are selected to be F1 = cos(?), F2 = sin(?), F3 = cos(2?),
F4 = sin(2?) and so on. The inner product for a continuous function, f(x, ?) with
g(?) for L2[0, 2?] (space of a square-integrable function, integrable over the bounded
interval 0, 2?]) is,
h(x) =
? 2pi
0
f(x, ?) g(?) d?
The corresponding inner product relevant to this problem is obtained by sub-
stituting the electric image ? for function f and various basis functions for function
g. The basis functions can follow any orthonormal basis for L2[0, 2?]. The above
inner products are the projections of the electric image on the basis functions. These
projections are responsible for decomposing the electric image in the desired direc-
tion of the basis function. To numerically integrate the electric image ?, the integral
is replaced with a summation, yielding the expression for WFI outputs,
vi(q) =
2pi
?
?=0
?(q, ?) Fi(?) ??
The above equation is supplied with a set of electric images computed for discrete
poses on either side of the centerline. For instance, a possible set of lateral displace-
32
ments is Y = {y| ? a/2 ? y ? a/2} and ? = {?| ? ?/4 ? ? ? ?/4}.
The objective is to implement a linear or a computationally simple estimator
to derive the vehicle states from electric images. Values on both the sides of the
centerline are necessary for the linearization of the WFI outputs, vi(q). The WFI
outputs were linearized in terms of the states q = [y, ?]T about the centreline of the
tunnel, the desired steady state condition q0, using the multivariable Taylor series,
vi(q) ? vi(q0) +?q|q0 ? (q? q0) (3.2)
The gradient operator operates on a matrix of values of v(q); it is numerically
implemented using the five-point central finite difference formula for each variable.
The constant terms arising from the 1st term in Eq. (3.2) are small enough to be
neglected. In matrix form, Eq. (3.2) resemble the linear equation, v = C q, which
has the form of the output of a linear system:
?
?
?
?
?
?
?
?
?
?
?
?
v1
v2
...
vn
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
C1y C1?
C2y C2?
... ...
Cny Cn?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
y
?
?
?
?
?
(3.3)
The matrix C contains the linearization coefficients of the Fourier harmonics with
the sensor in its desired steady-state pose. For centering in a tunnel, the desired pose
is when the sensor is perfectly aligned with the centerline and directed forward along
it. Cy and C? provide the linearized Fourier coefficients to form linear combinations
of the states q = [y, ?]T .
33
3.2 Relative State Estimation
In this Section, the procedure to extract control relevant information from WFI
outputs is described. The states computed using the formulation presented here are
static in nature as opposed to dynamic. The approach of relative state estimation
has the advantage of low computational effort and fast processing, as will be evident
from the following formulation. The relative states of the system can be estimated
by inverting v = C q as in Eq. (3.3). The residual errors in computation of the
WFI equation from Eq. (3.3) with these state estimates is given as e = v ? Cq,
where v is the set of measured WFI outputs whereas Cq is the set of estimated WFI
outputs. The optimal solution that minimizes the summed squares of the residual
errors given by the cost function J = 12eT e is obtained from the principle of least
squares [57],
q = (CT C)?1 CT v (3.4)
The matrix C? = (CT C)?1 CT is the pseudo-inverse of the matrix of linearization
coefficients C. The states estimated by Eq. (3.4) are not absolute states; they
represent the sensor pose relative to the obstacles causing the perturbations to the
electric field.
For n WFI harmonics, m electroreceptors and j states, the dimensions of C
are n? j, C? is j ? n and ? is m? 1. The set of Fourier harmonics F consists of n
harmonics denoted by Fi, i = 1, ..., n, each having the dimensions 1 ? m. Further
examination of Eqs. (3.1?3.4)reveals that the pseudo inverse C? can be pushed
inside the inner product to form a compound term C? Fi as shown below:
34
qj = C?ijvi = C?ji??, Fi? i = 1, ..., n
qj =
?
?,
n
?
i=1
C?jiFi
?
(3.5)
The state-estimation procedure is summarized by Algorithm 1.
0 90 180 270 360
?0.15
?0.075
0
0.075
0.15
 
 
State Extraction Fcn.
sin(? ) Approx.
0 90 180 270 360
?0.3
?0.15
0
0.15
0.3
State Extraction Fcn.
sin(2?) Approx.
Electroreceptor position, ?, degrees Electroreceptor position, ?, degrees
A B
y-State Estimation Function ?-State Estimation Function
Figure 3.1: Optimal state extraction functions. (A) y-extraction function, (B) ?-
extraction function.
The advantage of Eq. (3.5) is that the relative states q = [y, ?]T are esti-
mated directly from the electric image ? [46]. The functions
?n
i=1 C
?
jiFi used to
estimate the states are locally optimal state extraction functions, henceforth called
state extraction functions. For use with electric images, the state extraction func-
tions resemble sin(?) for y-estimation and sin(2?) for ?-estimation. The y? and
?? state extraction functions overlaid with their respective sinusoidal approxima-
tions are shown in Figure 3.1A and B respectively. These functions are pertinent
to a conducting tunnel for which the inner products in Eq. (3.5) were numerically
computed.
35
Algorithm 1: Relative State Estimation
Input: Electric images for different poses in a straight conductive tunnel,
? = V ?
. Output: Optimal state extraction matrix, C?
. begin
Compute a set of complex potentials for a tunnel, for various combinations
of lateral offset y and orientation ? with respect to the tunnel centreline.
for Lateral offset y ? Y do
for Orientation ? ? ? do
q = (y, ?), ? = [0, 2?].
Numerically integrate to compute the inner products
vi(q) = ??(q, ?), Fi(?)? to obtain Fourier harmonics a0(q), a1(q),
b1(q) etc.
Numerically linearise the harmonics about the desired steady state
condition.
Assemble the linearised WFI output equation v = Cq.
Find the pseudo-inverse of the matrix C using linear least squares
principle as C? = (CTC)?1CT .
Estimate the states of the sensor at any location in a tunnel, relative to
the steady state condition, by performing the operation
qj =
?
?,?ni=1 C
?
jiFi
?
.
36
3.2.1 Discussion
In the absence of any object, the electric image is equal to zero at all electro-receptor
positions. When the sensor is introduced in a straight tunnel, a peak is developed
in the electric image. The inner product of such an electric image with sin(?) will
be positive if the peak is in the ? = (0, ?) range and will be negative if the peak is
in the ? = (?, 2?) range. A peak is formed in the ? = (0, ?) range if the sensor is
closer to the left wall or, in other words, a positive lateral offset is provided with
respect to the centreline. Similarly, a peak in the ? = (?, 2?) range indicates that
the sensor is closer to the right wall as a negative lateral displacement is provided.
Thus the sign of the inner product is indicative of the nature of the lateral offset
provided. Additionally, the magnitude of the inner product is useful in judging the
amount of lateral offset: a smaller magnitude corresponds to smaller offset and vice-
versa. Along similar lines, the angular offset can be estimated by examining the
inner product between an electric image and sin(2?). When the sensor is traveling
parallel to either of the walls, the peak of the electric image is formed symmetrically
about the ?/2 or 3?/2 sensor depending on the side to which the closest wall is.
Thus the inner product yields a zero output and we get the estimate for angle ? as
zero. In the case when the sensor is directed into a wall closer to it, the peak is
formed in one of the four quadrants. By computing the inner product, it is possible
to find out the quadrant in which the peak of the electric image occurs. The value
of the inner product is positive in the region where sin(2?) has a positive polarity
and is negative where the sin(2?) function has negative polarity. The relative states
37
are employed in forming a feedback input to the system, as explained in the next
section.
3.3 Static Output Feedback Control
In this section, a controller is designed to perform centering behavior in a straight
tunnel and its robustness properties are checked for application in other similar en-
vironments. We assume a linearized unicycle-style kinematics model for the vehicle
similar to [45]. The controller is required to be simple and easy to implement in
order to consume limited computational capability. Due to the availability of the
relative state estimates from the electric image, a natural choice of a controller is
output feedback. The idea is to form the input from linear combination of the esti-
mates using the desired gain values. On closing the feedback loop by utilizing the
estimated relative states in Eq. (3.5), suitable motion commands can be supplied
to the vehicle to perform obstacle avoidance. These commands will be equal to zero
when the vehicle aligns with the desired steady state condition. Also, the character-
istics of the trajectory like settling time, overshoot etc. can be modified by noting
that the kinematic model of the vehicle is a second order system. Thus the two
gains that are used to form the feedback input are in fact similar to stiffness and
damping in a spring?mass?damper system.
38
3.3.1 Control Strategy for Centering Behaviors
The goal of WFI-based control is to navigate a sensing vehicle to the desired steady
state condition. In a tunnel, the steady state condition is the centerline, heading
straight ahead. For other corridor-like environments, the steady state condition is
the line equidistant from the boundaries formed by the objects on the either side of
the corridor.
The schematic for the control loop is shown in Figure 3.2. The optimal state
Plant 
Kinemacs
Electro-
sensor
Wide-field
Integraon
Linear Least 
Square 
Esmator
Environment
Opmal State Extracon
v
v
v
v
v
Figure 3.2: Feedback control system for performing obstacle avoidance
extraction operation in Eq. (3.5) is depicted by combining the WFI and the Linear
Least Squares Estimator (LLSE) blocks together.
A planar linearized unicycle model is assumed for the vehicle kinematics, based
on the behavioral observations of fish in general. The simplest maneuvers exhibited
by fish in response to an obstacle consist of continued straight-line motion and a
yawing action similar to a unicycle [58]. Coupled kinematics from other out of
plane degrees of freedom are small enough to be neglected [58]. In addition, the
non-holonomic constraint x? sin ? ? y? cos ? = 0 is enforced so that the body-frame
39
velocities become y?b = 0 and x?b = U . This constraint prescribes a motion to the
model such that there is no motion perpendicular to the vehicle Xb axis, thereby
preventing side-slip motion. The unicycle kinematics in the inertial frame are
x? = U cos ?
y? = U sin ?
U? = uy
?? = u? (3.6)
Here, uy and u? are the control inputs provided to the plant.
For a vehicle having a constant forward velocity U , we have U? = uy = 0 and
the linearized open-loop kinematics are
x? = U
?
?
?
?
y?
??
?
?
?
?
=
?
?
?
?
0 U
0 0
?
?
?
?
?
?
?
?
y
?
?
?
?
?
+
?
?
?
?
0
1
?
?
?
?
u (3.7)
Here, the matrices can be put in a standard state space form to yield the matrices
A and B as under.
A =
?
?
?
?
0 U
0 0
?
?
?
?
B =
?
?
?
?
0
1
?
?
?
?
. (3.8)
Output feedback is provided to the system using the control law u? = K [y?, ??]T ,
i.e.,
u = Ky y? +K? ??. (3.9)
40
In the case of zero estimate error, i.e., y? = y and ?? = ?, the closed-loop system is
?
?
?
?
y?
??
?
?
?
?
=
?
?
?
?
0 U
Ky K?
?
?
?
?
?
?
?
?
y
?
?
?
?
?
(3.10)
As mentioned earlier, this is a second order system similar to a spring-mass-damper
system. The characteristic equation for this closed loop system is s2 ? sK? ?
U Ky = 0 and eigenvalues are given as s1,2 =
K? ?
?
K2?+4U Ky
2 . It is clear from
the characteristic equation and the eigenvalues that Ky corresponds to the stiffness
term while K? to the damping term. For the roots s to lie in the left half plane, we
select K? < 0 and Ky < 0. On further analysis it is revealed that K2? < |4UKy|
yields complex roots whereas K2? > |4UKy| yields real roots. For K2? = |4UKy|, the
meaningless solution of two unique and real roots is obtained. We selected the gains
using Ky = ?K
2
?
4U (second condition above) so that real roots are obtained. Stability
as well as the desired performance can be achieved by carefully selecting the value
for Ky. Simulations with electrostatic images in some environments having simple
geometries can be found later in this chapter as well as in previous work by the
authors [59].
It is worth mentioning that the controller designed for a straight conductive
tunnel with a half width a = 6 inches ( 15 cm) will be shown via simulationto work
for environments made up of objects of different conductivities and geometries.
3.3.2 Robust Stability Analysis of the Controller
In this Section, we analyze the robustness properties of the designed controller.
The controller is intended to perform satisfactorily in various environments. In a
41
straight tunnel, the controller is expected to work flawlessly as it is designed for such
an environment. For environments with complex geometries, we assume that the
obstacles look like local tunnels and hence the same controller can be implemented.
This assumption is verified in the following study.
The C matrix is parametrized in terms of the tunnel width parameter a. Thus
C matrices are computed for various tunnel widths. A curve can then be fit to all the
values of each of the elements in the C matrix by assuming a suitable trendline. We
use a power fit Y = AXB for this purpose. The resulting parametrized C matrix,
which we call C matrix itself for convenience, is given below.
C =
?
?
?
?
?
?
?
?
?
?
?
?
3.66? 10?3a?5.14 0
0 3.49? 10?5a?5.16
?4.33? 10?6a?6.96 0
0 ?2.83? 10?8a?7.22
?
?
?
?
?
?
?
?
?
?
?
?
(3.11)
The elements in the above matrix originate from the bn Fourier harmonics of electric
images. The an harmonics are negligible are hence not included. High frequency
harmonics are not included as the values of successive harmonics decrease signifi-
cantly, as can be seen from the coefficients of the elements in the C matrix above.
As a result, only the first four bn harmonics are found in the above parametrization.
For performing the parametrization itself, the Curve Fitting Toolbox in MATLAB is
used. The power fit is selected owing to the quality of fit it provides for the available
data and the inherent simplicity of representing these fits as closed form equations.
The options selected in the toolbox settings are NonlinearLeastSquares, Bisquare
(for robust fits) and Levenberg-Marquart (curve fitting algorithm). This combina-
42
tion is found to give lowest residuals for the fits.
The Curve Fitting Toolbox also yields the 99% confidence bounds of the fits.
Of all the data points 99% will always lie within these bounds. These bounds are
useful in computing the allowed perturbations ?n to each of the element in the C
matrix given above. The ?n bounds found for above fits are as shown below.
C1,1 ? ?1 = ?0.5075
C2,2 ? ?2 = ?0.0041
C3,1 ? ?3 = ?0.0161
C4,2 ? ?4 = ?0.0002 (3.12)
If we assume that a 20% uncertainty exists in estimating each of the curve fit
coefficients, then the corresponding element in the C matrix can be multiplied by
a factor (1 + 0.2?n). The sign of ?n is irrelevant in this formulation. The C matrix
therefore changes to,
C =
?
?
?
?
?
?
?
?
?
?
?
?
4.03? 10?3a?5.14 0
0 3.493? 10?5a?5.16
?4.34? 10?6a?6.96 0
0 ?2.83? 10?8a?7.22
?
?
?
?
?
?
?
?
?
?
?
?
. (3.13)
As expected, the C matrix does not change significantly due to the better quality of
fits obtained using the settings mentioned above. Following this, we dissociate the
tunnel width parameter a as am + ?a, where am is the nominal tunnel width for
which the controller is intended for and ?a is the uncertainty in the tunnel width.
The uncertainty in a is defined as the ratio ?aam or the change in tunnel width ?a with
43
respect to the nominal tunnel width, am. Similarly, the C matrix can be written as
Cm +?C, due to am and ?a above. The theoretical proofs for above procedure can
be found in [60, 61]. Cm is given as,
Cm =
?
?
?
?
?
?
?
?
?
?
?
?
4.03? 10?3a?5.14m 0
0 3.493? 10?5a?5.16m
?4.34? 10?6a?6.96m 0
0 ?2.83? 10?8a?7.22m
?
?
?
?
?
?
?
?
?
?
?
?
. (3.14)
?C is obtained by differentiating C with respect to am as follows.
?C =
?
?
?
?
?
?
?
?
?
?
?
?
?5.14? 4.03? 10?3a?5.14 ?aam 0
0 ?5.16? 3.493? 10?5a?5.16 ?aam
6.96? 4.34? 10?6a?6.96m ?aam 0
0 7.22? 2.83? 10?8a?7.22m ?aam
?
?
?
?
?
?
?
?
?
?
?
?
.(3.15)
Rearranging Eq. (3.15) we have,
?C =
?
?
?
?
?
?
?
?
?
?
?
?
?5.14 0 0 0
0 ?5.16 0 0
0 0 6.96 0
0 0 0 7.22
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?a 0 0 0
0 ?a 0 0
0 0 ?a 0
0 0 0 ?a
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
4.03? 10?3a?5.14 0
0 3.493? 10?5a?5.16
4.34? 10?6a?6.96m 0
0 2.83? 10?8a?7.22m
?
?
?
?
?
?
?
?
?
?
?
?
(3.16)
The first matrix in Eq. (3.16) is a matrix of weights, W , and the second matrix is
the uncertainty matrix ?. The remaining third matrix is the nominal matrix Cm
44
from Eq. (3.14). Substituting in C = Cm + ?C we have,
C = Cm + ?C = Cm +W?Cm
? C = Cm (I +W?) (3.17)
For this C matrix an M ? ? structure can be formulated as prescribed by [60].
The robust stability of such a plant is guaranteed if and if the following condition
is satisfied.
RS ? ? (M(j?)) < 1, ?? (3.18)
In the above equation ? is the structured singular value for the M ?? form. It is
found that this condition is satisfied an uncertainty in the tunnel width a,
RS ? ? ? ?aam
= 17.3393 = 13.63%. (3.19)
The following plot shows the variation of ? with respect to frequency. Analysis was
10?3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
M
ag
ni
tu
de
 (a
bs
)
Frequency (rad/sec) 
10?2 10?1 100 101 102 103
Figure 3.3: Variation of structured singular value, ? with respect to frequency.
45
performed to obtain similar plots for other uncertainty values for ?1, ?2, ?3 and ?4.
In all the cases the allowed uncertainty was found to be almost the same. This is
an indicator of the high accuracy of the curve fits.
Thus, the robust stability of the controller is guaranteed if the variation in tunnel
width is less than 13.63% with respect to the nominal tunnel width am. This implies
that the assumption of a local tunnel in place of objects with complicated geometries
should hols as long as the lateral separation between them is less than 13.63% from
the designed tunnel width. The simulations in the next section confirm the validity
of this assumption.
3.4 Control Strategy for Wall Following Behaviors
The next objective is to manipulate a sensing vehicle in order to traverse a path
along a wall (a circular arena or a straight wall) while maintaining a fixed distance
from it. Here, we consider the desired steady state of the vehicle to be at the required
offset from the wall and heading parallel to it.
For a circular arena, the steady state is achieved when the vehicle moves along
a circle of radius equal to 80% of the arena radius and concentric to it. Thus the offset
clause is met by traveling at a fixed distance of 20% of the radius from the wall and
the parallel trajectory clause is met by traversing along a concentric circle. Consider
the electric image equation for a circular arena given by Eq. (2.20). The procedure
to find the state extraction functions is similar to those for the centering case in
Section 3.3.1. As before, the state extraction functions are sin(?) for estimating the
46
y-state and sin(2?) for estimating the ?-state.
3.5 Simulation Results
This Section presents simulation results demonstrating obstacle avoidance in a
straight tunnel and a circular arena using WFI processing and controller strategy
explained in Sections 3.1?3.3. The basic assumption in these simulations is that the
electric image is exactly given by the analytical equations given by Eq. (2.15) and
Eq. (2.20). Thus, the estimated relative states are absolute and exactly correspond
to the electric image. Another implicit assumption is the absence of sensor and
actuator noise. The presence of such noise will corrupt the signal to some degree
which will require additional filtering and signal processing, which is not performed
here. We demonstrate centering response in a straight tunnel and wall-following
behavior in a circular arena from multiple initial conditions.
3.5.1 Centering in a Straight Tunnel
In a straight tunnel, it is expected that the vehicle will eventually move away from
walls and align itself with the centerline of the tunnel. We chose a 1 foot (30 cm)
wide tunnel with conductive walls. Several starting locations and orientations were
simulated in order to illustrate the response of the algorithm to variation in initial
conditions. The state extraction functions for estimating the y and ? states were
sin(?) and sin(2?) as prescribed in Section 3.2. The velocity U of the vehicle was
assumed to be equal to 6 inches/sec (15 cm/sec). From Eq. (3.9) it can be seen
47
1 0.5 0 ?0.5 ?1
0
5
10
15
Y?direction, feet
X?
di
re
ct
io
n,
 fe
et
0 5 10 15 20
0
?0.1
?0.2
?0.3
?0.4
?0.5
Time, sec
Di
st
an
ce
, fe
et
A
y y
B
0 5 10 15 20
0
0.1
0.2
0.3
0.4
Time, sec
An
gl
e, 
ra
d
??
C
Figure 3.4: Simulations demonstrating centering behavior in a straight tunnel. (A)
Trajectories, (B) Actual and estimated lateral offsets, y and y?,(C) Actual and esti-
mated angular offsets, ? and ??.
that the feedback input is large when the vehicle has a larger offset with respect to
the centerline of the tunnel (either a large lateral offset or a large angular offset).
Thus, when the steady-state condition is achieved i.e., the vehicle is aligned with
the centerline of the tunnel, the feedback input is equal to zero. Figure 3.4A shows
the environment and the trajectories generated in the simulations. The arrows on
each trajectory indicate the respective initial condition. Figure 3.4B plots the time
traces of the relative y?estimates and the actual lateral offset y. Similar plots for
the relative ??estimate and the actual angular estimate ? are plotted in Figure
3.4C. These traces correspond to the solid line trajectory in Figure 3.4A.
Observe that irrespective of the initial conditions the desired steady state is
achieved eventually. The response to the obstacles depends on the gain values se-
lected, thus deciding the values for overshoot, settling time and oscillations of the
trajectory about the steady state. The selected gains provide a consistent perfor-
mance with negligible overshoot and less setting time. The gains for forming the
48
feedback input u? were selected using a trial and error procedure to be Ky = ?20
and K? = ?40. Convergence of trajectories is observed along the steady-state con-
dition within the first few seconds. This behavior suggests the usefulness of the
WFI-based control strategy in corridor-like environments.
3.5.2 Wall Following in a Circular Arena
Similar to the straight tunnel case, simulations were performed to demonstrate wall
following behavior in a circular arena of diameter 3.3 feet ( 1 m) using the control
strategy described in Section 3.4. The vehicle was expected to align with a circular
trajectory of 2.6 feet ( 0.8 m) diameter, concentric to the arena. Consistent with the
straight-tunnel case, multiple initial conditions were considered. The linear offsets in
this case were defined with respect to the steady state circular trajectory, whereas
the angular offsets were defined with respect to the perpendicular to the radius
vector on which the vehicle is located at the particular instance. The controller
tries to drive these offsets (states) to zero. This condition set by the controller was
automatically satisfied when the vehicle traveled along the steady state trajectory.
Figure 3.5A shows the trajectories originating from different initial conditions. The
arrowheads on each trajectory indicate the initial conditions. In all the cases, it
was observed that the controller successfully drove the vehicle to the desired steady-
state condition. The gains Ky = ?20 and K? = ?40 were selected by trial and
error to limit the trajectory overshoot and settling time. The forward velocity of
the vehicle was 6 inches/sec (15 cm/sec). If allowed to continue, all the trajectories
49
?2 0 2
?3
?2
?1
0
1
2
3
X?direction, feet
Y?
di
re
ct
io
n,
 fe
et
??
A B
0 5 10 15 20
?2
?1
0
1
Time (sec)
An
gu
lar
 ra
te
, r
ad
/s
0 5 10 15 20
?0.5
0
0.5
1
1.5
Time (sec)
La
te
ra
l O
!s
et
, fe
et
 
 
0 5 10 15 20
?0.5
0
0.5
1
1.5
Time (sec)
An
gu
lar
 O
!s
et
, ra
d
 
 
y y
C D
Figure 3.5: Simulations demonstrating wall following behavior in a circular arena.
(A) Trajectories, (B) Angular rates, (C) Actual and estimated lateral states, y and
y?, (D) Actual and estimated angular states, ? and ??.
50
continue along the circular trajectory. Figure 3.5B shows the time trace of the
angular velocity of the vehicle. The vehicle shows high angular rate in the beginning
which corresponds to the window when the vehicle approaches the desired steady
state offset from the wall rapidly. Once the vehicle has aligned itself with this offset
trajectory, the angular velocity is held constant to continue along the circular path.
The steady state offset observed here is due to this reason. Figures 3.5C shows the
time traces of actual and estimated lateral offset of the vehicle. Similarly, the time
traces for the actual and the estimated angular offsets are shown in Figure 3.5D.
In both the plots showing the offsets, the vehicle states and the state estimates
converged when the steady state condition was attained.
3.6 Modifications for Non-conductive Objects
The entire algorithm discussed thus far is based on the premise that the perturb-
ing objects are conductive in nature. Electric images due to such objects exhibit
positive peaks. According to [26] the electric images due to similarly sized non-
conductive objects is negative in polarity as compared to that due to a conductive
objects. In addition, these electric images are smaller in magnitude (nearly half)
than their conductive counterparts. There is also a phase shift in time involved in
non-conductive electric images due to their capacitive qualities [19]. However the
phase-shift property is not considered in this work.
51
3.6.1 Non-conductive Obstacles
To produce control inputs from electric images of negative polarity and reduced
magnitude, it is required to negate the gains Ky and K? in Eq. (3.9) and increase
their magnitudes. This negation does not however destabilize the system. The state
extraction functions used are the ones that were derived for metallic corridors. These
functions hence yield negative state estimates from the negative electric images of
non-conductive objects. Thus by using negative gains the sign of the control input
is guaranteed to have the right sign as in the metal corridor case. This approach is
however only valid for environments constructed entirely out of non-metallic objects.
When there is mix of conductive and non-conductive obstacles, a different strategy
is used as shown next.
3.6.2 Mix of Conductive and Non-conductive Objects
This analysis is relevant for FEM simulations and experiments. The electric im-
age has a definite positive peak for conductive objects and negative peak for non-
conductive objects. The controller, with prescribed gains, can be used as defined
by Eq. (3.9) when all the objects are positive, whereas negated gains as explained
above can be used when all the objects are non-conductive. For environments with
a mix of these two types of objects, the same controller cannot be used directly.
This is because the controller needs to decide from the electric image, whether the
obstacle was conductive or non-conductive and apply appropriately signed gains.
It can be possible to do this by looking at the signs of the peaks of the electric
52
images. In simulations, where noise is assumed to be absent, the electric images of
small or distant objects possess the right polarities for the peaks. So it is possible
to use this method to tell the two types of materials apart, in that case. However,
in experiments, with noise corrupting low magnitude signals due to small or distant
objects, this method may or may not work. So a more robust method is sought.
The absolute value of an electric image due to a conductive object will look
similar to that of a non-conductive object as is depicted in Figure 3.6. The procedure
in Algorithm 1 to determine the state extraction function from the absolute-valued
electric images can still be followed. In this procedure, absolute values of electric im-
ages of a conductive tunnel are used. State extraction functions so obtained extract
states from electric images due to either conductive or non-conductive objects. It is
known from [26], that electric images due to non-conductive objects are about half
the magnitude of those due to conductive objects. Thus the above state extraction
functions will yield lower state estimates for non-conductive objects. Note the pro-
0 90 180 270 360
0
0.01
0.02
0.03
?0.02
?0.01
0
0.01
0.02
?
|?|
?
|?|
0 90 180 270 360
Electroreceptor position, ?, degrees Electroreceptor position, ?, degrees
El
ec
tri
c P
ot
en
tia
l, ?
, v
ol
ts
El
ec
tri
c P
ot
en
tia
l, ?
, v
ol
ts
A B
Figure 3.6: Absolute values of electric images. (A) Conducting objects, (B) Non-
Conducting objects
53
cedure is expected to yield state extraction functions with higher frequency content
in them due to the presence of the discontinuity at the zero crossing. The state
extraction functions for this mixed environment case are plotted in Figure 3.7. The
0 90 180 270 360
?0.15
?0.075
0
0.075
0.15
 
 
State Extraction Fcn.
sin2(?
sin(? ) Approx.
) Approx.?
0 90 180 270 360
?0.3
?0.15
0
0.15
0.3
State Extraction Fcn.
sin(4?) Approx.
Electroreceptor position, ?, degrees Electroreceptor position, ?, degrees
A B
Figure 3.7: Optimal state extraction functions for mixed conductivity environments.
(A) y-extraction function, (B) ?-extraction function.
state extraction function for the y state, shown in Figure 3.7A can be approximated
by a sin2(?) for 0 ? ? < ? and ? sin2(?) for ? ? ? < 2? or by the sum of sines
of higher frequencies. We use an approximation for this function given by sin(?).
The sin(?) approximation has the advantage that it provides a stronger response
near small angles to accelerate the vehicle to the steady state faster. The ? state
estimation function involves higher frequencies and is given by sin(4?) as shown in
Figure 3.7B.
Consider the linearized unicycle kinematics in Eq. (3.10). As before, for the
mixed tunnel case the characteristic equation for the closed loop system is s2 ?
sK? ? U Ky = 0 and eigenvalues are given as s1,2 =
K? ?
?
K2?+4U Ky
2 . The signs of
54
the relative states estimated in this case are the same as their exact counterparts. So
to the same argument that was true for selecting the gains in the conductive tunnel
case holds for this problem as well, i.e., K? < 0 and Ky < 0 for stability. Further,
the gains are selected using Ky = ?K
2
?
4U so that real valued roots are obtained and
desired performance is achieved.
The advantage of using this approach is that, it is not required to distinguish
between conductive and non-conductive materials. Direct application of these state
extraction functions to absolute-valued electric images yields the required states
irrespective of the nature of the perturbing object. In the following chapters, more
complex simulation problems and experiments are presented in order to demonstrate
the usefulness of the control strategy. These problems will test the algorithm against
change in material properties, shape of objects and complexity of the environment.
In all of these problems, the control strategy developed in this chapter is used.
55
Chapter 4
Quasistatic Analysis and Simulations
This chapter describes simulation problems for demonstrating reflexive obstacle
avoidance in unmapped environments of complicated geometries. Such geometries
are comprised of multiple, staggered obstacles that are either conductive or non-
conductive. In addition, the source of the electro-sensor is made oscillatory for
achieving a higher degree of bio-mimicry. For solving the obstacle-avoidance prob-
lem in such environments, the control method developed in Chapter 3 is used.
4.1 Motivation
The control strategy in Chapter 3 navigates a sensing vehicle in a straight tunnel
with conductive walls. In more realistic situations, the environment can be lined
with complex-shaped objects of different conductivities. Electrostatics falls short on
two counts in analyzing these situations. Firstly, electrostatics cannot handle non-
conductive objects very efficiently. Secondly, method of images type analysis for a
complex shaped object is extremely difficult. In order to compute electric images
in such a case, a full electromagnetic analysis by solving the general Maxwell?s
equations in Table 4.1 can be performed. But doing so is almost always a very
56
tedious assignment. There are methods to solve the problems involving dielectric
walls and slabs. So potentially solving such a problem could possibly yield an
analysis similar to the method of images analysis in Chapter 2. However, there are
quite a few issues involved in this approach.
The most common method involves solving an integral called Sommerfeld?s
formula [62, 63]. Such integrals are generally impossible to solve either analyti-
cally or numerically due to their oscillatory nature. Some solutions that have been
proposed by making some crucial asymptotic assumptions are still very oscillatory
and slow converging. This is not a favorable situation if such computations are
to be performed during simulations involving multiple iterations. Another popular
method to solve electromagnetic problems is by using Green?s functions. This has
been demonstrated in many works like [64, 65, 66]. However, with the addition of
boundary conditions and singularities, the solution of a Green?s function gets com-
plicated and computationally heavy. All such methods involve the electric field as
the solution. To arrive at the electric potential, which is the variable of interest,
the gradient equation E = ??? is to be solved. Here, E is the electric field and
? is the potential. Integrating a 3D gradient equation is generally a difficult task.
Moreover, such an intensive computation where the solution is computed for the
entire spectrum of wavelengths is not required in our case. This work attempts to
mimic the electrolocation observed in wave-type electric fish, which produce low
frequency quasi-sinusoidal electric fields. Hence, the full electromagnetic approach
is not pursued.
So to simulate obstacle avoidance in such environments, a different approach
57
is required. Finite Elements Methods (FEM) for quasistatic problems tends to
address all of the shortcomings of electrostatics mentioned previously while avoiding
the cumbersome approach of a full electromagnetic study. FEM tools provide the
ability to model objects of complex shapes and assign to them different electrical
properties. The prime objective of this project is to mimic electrolocation in electric
fish. Electrostatics, with a static electric field source, is not the absolute way to
explain electrolocation. With FEM through quasistatic analysis it is possible to
consider a sinusoidally varying source, which is a very close approximation of the
electric field source in electric fish. Hence, the quasistatic approach is consider here.
If the velocity of propagation of the source electromagnetic wave is negligible
with respect to the speed of light, then the problem can be treated as a quasistatic
problem and the time variance of the current density in Maxwell?s equations can
be neglected [55]. Table 4.1 shows the comparison between the general Maxwell?s
electromagnetic equations and quasistatic versions of them. The notation used
Table 4.1: Comparison of Maxwell?s general and quasistatic electric field equations
General Maxwell?s equations Quasistatic field equations
Faraday?s Law ?? E(r, t) = ??B(r,t)?t ?? E(r, ?) = ?j?B(r, ?)
Ampere?s Law ??H(r, t) = J(r, t) + ?D(r,t)?t ??H(r, ?) = J(r, ?) + j?D(r, ?)
Gauss?s Law ? ?D(r, t) = ?v(r, t) ? ?D(r, ?) = ?v(r, ?)
No monopoles ? ?B(r, t) = 0 ? ?B(r, ?) = 0
in the general Maxwell?s equations is the standard one, where E is the electric
field intensity, B is the magnetic flux density, H is the magnetic field intensity,
58
J is the current density, D is the electric flux density and ?v is the (volumetric)
charge density. In general, all the quantities in the general Maxwell?s equations
are arbitrary functions of position r and time t, and hence time-dependent. For
quasistatic equations, we assume that the quantities evolve sinusoidally in time and
are functions of r and the frequency of the sinusoid ?. The relation between B and
H and D and E are given by the constitutive relations,
B = ?H
D = ?E (4.1)
More explanation can be found in [55]. Additionally, in quasistatic field equations
j =
?
?1 and ? is the frequency of the sinusoidally varying source electric field.
It can be seen that the general Maxwell?s equations are grossly simplified due
to the assumption that the electric field is produced by a source varying sinusoidally
at low frequencies. Due to these simplifications, the quasistatic equations behave
similar to their static counterparts by satisfying Laplace?s and Poisson?s equations.
Further, many methods and ideas that are valid for electrostatics problems, which
are otherwise void for electromagnetic problems defined by the general Maxwell?s
equations, can be extended to the quasistatic equations. Thus in our case, the
quasistatic approximation not only simplifies the problem, but is a good way to
mimic electrolocation in fishes by including a slowly varying electric field.
For implemention in simulations, the process of solving the quasistatic Maxwell?s
equations is required to be quick and fairly accurate. Hence an FEM-based approxi-
mation using ANSYS was used to solve the problem and generate the electric images.
59
The quasistatic electric images resemble electrostatic images with the additional
ability to resolve objects of different materials into conductive or non-conductive.
For in-depth study of electromagnetic principles, the reader is referred to [55, 54].
4.2 Finite Elements Analysis
Finite Elements Analysis is a tool to find approximate solutions to boundary value
problems for differential equations. The solution is found using numerical methods
by solving for small subdomains making up the entire domain of the problem. The
equations solved for the subdomains are simpler than the differential equation that
applies to the entire domain as a whole. The final result is obtained by systematically
combining, or performing a summation of the results of all the subdomains. We used
the ANSYS program to perform these simulations. The FEM simulation process
involves the following steps.
? Modeling the geometry
? Meshing
? Application of loads and boundary conditions
? Solution and postprocessing
Here, all these steps are explained in the context of the quasistatic simulations
performed to demonstrate electrolocation.
60
4.2.1 Modeling of geometry
The environments in which the obstacle avoidance is performed either consist of
straight tunnels or a corridor constructed from complex-shaped objects or a struc-
tured obstacle field. All commercially available FEM tools possess a capability to
draw shapes either by defining the geometry or by providing coordinates of points
on the boundary of the objects. Thus geometrically simpler obstacles like straight
walls and circles were drawn accordingly. But when complex shaped objects were
involved, a different approach was used. A bitmap image file was drawn for such
environments. A MATLAB script was used to extract the coordinates of the points
on the boundaries of the objects. These coordinates were then input to the FEM
program to form the environment. Three different environments, shown in Figures
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5
X?Distance, m
Y?
Di
st
an
ce
, m
A
?0.5
0
0.5
0 0.5 1 1.5 2 2.5 3
X?Distance, m
Y?
Di
st
an
ce
, m
0
1
2
1 2 3
Y?
Di
st
an
ce
, m
X?Distance, m
B C
Straight Tunnel
Cluttered Corridor Obstacle Field
Figure 4.1: Different environments used for simulations. (A) Straight tunnel, (B)
Cluttered corridor, (C) Obstacle field.
4.1A?C were modeled in ANSYS for performing the simulations. The first environ-
ment was a basic straight tunnel (Figure 4.1A), the second one contained a cluttered
corridor (Figure 4.1B) and the third one had an obstacle field of cylinders (Figure
4.1C). For each type of environment, three cases were considered wherein the con-
61
ductive properties of the objects were changed. The first case was designed to have
all the obstacles to be conductive, the second case with all non-conductive objects
and the third with a mix of conductive and non-conductive objects.
After drawing the geometries of the environment and obstacles, different re-
gions are assigned material properties. The three materials that have been consid-
ered in these simulations are detailed in Table 4.2 below. Relative permittivity of a
Table 4.2: Materials used in simulations
Material
name
Relative
Permittivity, ?r
Loss Tangent,
tan ?
Application
Water 80 5.542? 10?8 The environment
Aluminum 1.3? 1014 3? 1012 Conductive obstacles
Wood 5 0.04 Non-conductive obstacles
material defines its degree of conducting an electric field in comparison to vacuum.
Loss tangent, on the other hand, is the property that defines a material?s rate of
dissipation of electromagnetic energy. As can be seen from the table, metals pos-
sess very high permittivity and high loss tangent values as compared to non-metals.
Loss tangent is convenient in that it relates the conductivity and the permittivity
of a material with the wave frequency [55]. Both, relative permittivity and loss
tangent depend on the frequency of the wave. We have considered the values for
these variables at 100 Hz, which is the desired frequency of our electric field source
in simulations. These two material properties are sufficient for ANSYS to perform
quasistatic analysis.
62
4.2.2 Meshing the environment
Meshing the environment splits it into discrete elements which obey the same dif-
ferential equations as the entire environment. However, due to some simplifications
and assumptions made during splitting the environment, it is possible to apply a
simplified set of equations to each element in lieu of the original more complicated
equations. On connecting the solutions obtained for all the elements, an approxi-
mate solution to the complete domain is obtained. The accuracy of the approximate
solution depends on the average size of the segments or the quality of the meshing:
The smaller the size, the more accurate the solution, in general.
Thus, meshing forms a very important part in the FEM procedure. Ideally, it is
desirable to make the mesh size or the element size as small as possible to arrive at an
accurate solution. This kind of a mesh is called a fine mesh. However, this approach
has some limitations. Smaller element size implies more elements. Consequently,
more computations are to be performed, which makes the entire process slow and
computationally very demanding. However, making the mesh coarse sacrifices the
accuracy of the solution. Therefore, there should be a good compromise between
the number of elements and the desired accuracy of the solution. There are various
mesh-size optimization algorithms that are useful in computing the right degree of
meshing [67, 68, 69]. In contrast, a brute force approach of local refinement of a
coarse mesh is also very popular [70, 71]. (The popularity of this approach is also
due to the availability of this capability built into most commercially available FEM
software like ANSYS.) The three scenarios mentioned above are depicted in Figure
63
4.2A?C.
A B C
Figure 4.2: Mesh quality choices for simulations. (A) Coarse mesh, (B) Fine mesh,
(C) Locally refined mesh.
The approach of local mesh refinement is used here where the mesh is refined
in the prime region of interest, i.e. in the vicinity of the sensor. The electric image
is computed at the sensor and, with a refined mesh, the accuracy of the solution is
increased. The remaining regions with a coarse mesh yield a less accurate solution.
However, these regions contribute little towards the electric image. In fact, the less
computational effort required for obtaining the solutions in these regions is a very
desirable quality.
4.2.3 Application of Loads and Boundary Conditions
An FEM problem consists of the environment and the load set applied for specified
boundary conditions. These load sets in electromagnetic problems typically include
voltages, charges, currents, magnetic fields etc. Whereas, boundary conditions in-
clude the materials and geometry of objects in the environments and impedance of
64
the contact surfaces between different objects in the environment.
In most commercially available FEM software, the boundary conditions can be
set through built-in tools. The assignment of materials and geometry to the objects
in the environments was explained in the previous two steps. In addition to these,
contacts are created between water and all the objects, as required by ANSYS.
ANSYS provides the required tool to do so. The contacts were configured to have
high conductivity or very low impedance for both conductive and non-conductive
objects. Doing so ensures computing the correct effect of the electric field on both
type of objects.
In the simulations in this work, the load applied was of the voltage type. The
source of the multipole is defined as a 3 volt peak-to-peak, 100 Hz, sinusoidally
varying voltage waveform. In addition, the outermost boundary of the environment
is grounded by the program by default. This boundary condition does not affect
the simulation since the boundaries of the environment are maintained sufficiently
far from the objects to avoid their effect on the electric field of the sensor.
The three steps mentioned so far are grouped under a process called prepro-
cessing.
4.2.4 Solution and Postprocessing
Once the preprocessing steps are carried out, the simulation is ready to be solved.
On issuing the solve command, ANSYS solves the quasistatic problem as per the
variables defined during preprocessing, like mesh size, loads, contacts etc. Depending
65
on the mesh size the solution time varies. As is true with any FEM program, ANSYS
performs successive iterations to arrive at a solution by minimizing the relative error.
It does so by combining solutions for all the elements starting from the outermost
elements to the innermost elements and vice versa in alternate iterations. The theory
governing this procedure can be found in any standard FEM textbook [72, 73].
Voltages at all the element nodes are obtained as the solution to the simulated
problem. The values at the nodes in the vicinity of the source are of particular
interest. The fine mesh size in this region is important for converging to accurate
solutions. The values at the locations of the electro-receptors are computed by inter-
polating within the set of the nodal values in their vicinity. By concatenating voltage
values at all the electro-receptors, the required electric image is obtained. On imple-
menting the WFI processing and controller subroutines using the simulated electric
images, the next position of the sensor is computed. This new position is computed
with the objective of moving the sensor to a position where the perturbations on
both the sides of the sensor are equalized.
4.2.5 Linking of ANSYS with MATLAB
To automate the process of FE simulation discussed above, it was felt necessary to
bypass the ANSYS GUI and perform the tasks. In addition, this was meant to be
done iteratively to perform autonomous navigation in such environments. The signal
processing and control loop were implemented in MATLAB for the electrostatics
study and it was decided to port the same code to the quasistatic simulations.
66
Owing to these reasons, ANSYS and MATLAB were operated jointly.
ANSYS is capable of running batch files that are written in the ANSYS pro-
gramming language. A Macro was written in MATLAB to perform the tasks listed
below.
? Write an ANSYS batch file.
? Send the file to ANSYS server.
? Send an instruction to the ANSYS server to run the batch file
? Interpolate the nodal solutions at the positions of the electro-receptors to
obtain the electric image.
? Relay the electric image back to MATLAB for further processing.
The simulations are carried out in the background thus saving a plenty of compu-
tational resources.
4.3 Simulation Results
Four environments, namely straight tunnel, variable width corridor and an obstacle
field, are used in simulations. The objective in all the simulations is to navigate
within the obstacle while avoiding all the collisions. The following results for differ-
ent test cases show composite figures containing the trajectories, the time traces of
actual states and the relative state estimates. For some problems, the time traces
of body velocities in the inertial frame are plotted as well. Trajectories due to dif-
ferent initial conditions are included. The time traces are plotted for the case for
67
which the trajectory is plotted using a continuous line. The arrowheads at the be-
ginning of each trajectory indicate the initial conditions, position and orientation,
for that particular trial. The control strategy developed in Chapter 3 was used.
In all the figures that follow, conductive objects are shown in grey color, whereas
non-conductive objects are shown in brown.
4.3.1 Straight Tunnel
A straight tunnel consisting of conductive walls was considered in Chapter 3. Here,
two additional cases are considered, one with both the walls being non-conductive
and the other with one conductive and one non-conductive wall. In each of the three
cases, four trajectories corresponding to four initial conditions have been plotted.
The objective was to center between the two walls thus demonstrating the centering
behavior.
4.3.1.1 Conductive Walls
As expected, the results obtained here were similar to those obtained using electro-
statics. A 1 meter (3.3 feet) was used with both the walls being conductive. The
sensor disc radius was taken to be 75 mm (3 inch). The electric field source on the
sensor was set to generate a sinusoidal waveform of 3 volt peak-to-peak at 100 Hz
frequency. ANSYS solved the quasistatic problem and an iterative control scheme
was implemented via MATLAB. It can be seen from Figure 4.3A that, irrespective
of the initial conditions, the vehicle carrying the electro-sensor was able to converge
68
to the desired steady-state condition by aligning with the centerline and heading
straight down the tunnel. The feedback gains Ky = ?20 and K? = ?40 were used.
The magnitude of the forward velocity of the vehicle was maintained to be 0.5 m/sec
(3.3 ft/sec). Figures 4.3B&C show the time traces of the actual and relative states
?0.5
0
0.5
0 0.5 1 1.5 2 2.5 3
X?Distance, m
Y?
Di
st
an
ce
, m
Both Walls Conductive
A
0 2 4
0
0.3
0.6
time, t, sec
La
te
ra
l o
!s
et
, y
, m
et
er
s
 
 
y
y?
?0.6
?0.3
B
0 2 4
?0.6
?0.3
0
0.3
0.6
time, t, sec
An
gu
lar
 o
!
se
t, 
?,
 r
ad
 
 
??
?
C
Figure 4.3: Simulation results for a straight tunnel with conductive walls. (A) Tra-
jectories, (B) Actual and estimated lateral offsets, y and y?,(C) Actual and estimated
angular offsets, ? and ??
for the trajectory drawn using a continuous line in Figure 4.3A. A general agreement
in terms of nature and shape is observed between the actual states, y and ?, and
relative state estimates, y? and ??, as expected for the control strategy.
69
4.3.1.2 Non-conductive Walls
This case is analogous to the previous one with both the walls being non-conductive
in nature. The tunnel was 1 meter (3.3 feet) wide. The controller gains were changed
to Ky = 30 and K? = 60 in this case. All the trajectories showed convergence with
the steady state condition. The time traces of the actual and relative states exhibited
A
0 2 4
0
0.3
0.6
time, t, sec
La
te
ra
l o
!s
et
, y
, m
et
er
s
 
 
y
y?
?0.6
?0.3
B
0 2 4
?0.6
?0.3
0
0.3
0.6
time, t, sec
An
gu
lar
 o
!
se
t, 
?,
 r
ad
 
 
??
?
C
?0.5
0
0.5
0 0.5 1 1.5 2 2.5 3
X?Distance, m
Y?
Di
st
an
ce
, m
Both Walls Non-conductive
Figure 4.4: Simulation results for a straight tunnel with non-conductive walls. (A)
Trajectories, (B) Actual and estimated lateral offsets, y and y?,(C) Actual and esti-
mated angular offsets, ? and ??.
behavior similar to the conductive wall case above.
70
4.3.1.3 One Conductive and One Non-conductive Wall
We implement the modified controller to achieve obstacle avoidance in the case of
one conducting and one non-conducting wall. Accordingly, sin(?) and sin(4?) were
used as the optimal state extraction functions, with controller gains Ky = ?20 and
K? = ?40. All of the variables, including tunnel width, sensor radius, forward
velocity etc., the same as the other tunnel simulations described above. The results
are shown in Figure 4.5. The trajectories originating from different initial conditions
exhibited convergence but not to the centerline of the tunnel, because of the steady
state error in the y state. This behavior can be explained as follows.
Originally, the control strategy to navigate through such a hybrid tunnel was
designed by taking absolute values of electrostatic electric images for a conductive
tunnel because electrostatic equations to compute the electric image due to the non-
conductive wall were not available. It was assumed that the electric image due to
a conductive wall is comparable to that due to a non-conductive wall. However,
this assumption is not accurate according to [26]. In this problem, the modified
electric image is obtained by taking the absolute value of the FEM generated electric
image. Such an image due to a non-conductive wall is smaller in magnitude than its
conductive counterpart. In other words, the non-conductive wall produced smaller
perturbations to the electric field of the sensor. This made the algorithm believe
the wall to be farther away than it actually is. The objective of WFI based control
strategies i.e., to equalize the perturbations on both the sides of the sensor. In this
case, this condition is satisfied when the sensor moves closer to the non-conductive
71
A0 2 4
0
0.3
0.6
time, t, sec
La
te
ra
l o
!s
et
, y
, m
et
er
s
 
 
?0.6
?0.3 y
y?
B
0 2 4
?0.6
?0.3
0
0.3
0.6
time, t, sec
An
gu
lar
 o
!
se
t, 
?,
 r
ad
 
 
??
?
C
?0.5
0
0.5
0 0.5 1 1.5 2 2.5 3
X?Distance, m
Y?
Di
st
an
ce
, m
One Conductive and One Non-conductive Wall
Figure 4.5: Simulation results for a straight tunnel with one wall each of conductive
and one non-conductive properties. (A) Trajectories, (B) Actual and estimated
lateral offsets, y and y?,(C) Actual and estimated angular offsets, ? and ??.
72
wall to balance the perturbations. Hence the steady-state offset is encountered.
In general, in the presence of two walls or objects of different electrical prop-
erties on either side of the sensor will produce a steady state offset towards the one
with lower conductivity.
4.3.2 Cluttered Corridor
A cluttered environment is constructed by filling the sides of a curved corridor with
objects of complex geometries. Care was taken while constructing these environ-
ments to not to create gaps between objects that were wide enough for the vehicle
to take an alternative path and escape. To the vehicle, the randomly shaped obsta-
cles look like walls forming a local tunnel at every instance and the straight tunnel
algorithm is followed. The vehicle is repelled away strongly by objects of large size.
Corrections are made to the trajectory after this behaviour and the vehicle is sub-
sequently brought near the center of the corridor. This continues until the vehicle
has traversed until the end of the corridor. Three cases have been considered as
before. The first one had all conductive objects, whereas the second one had all
non-conductive objects. The third case is the one where all the objects on one side
of the corridor were non-conductive and those on the other were conductive. Four
trajectories originating due to four initial conditions have been plotted for each of
the three cases.
73
4.3.2.1 Conductive Objects
Here all the objects were defined to be conductive in the FEM meshing step. The
width of the tunnel varied between 0.7 m to 1.1 m (2.8 ft to 3.6 ft). The steady-
state trajectory for this corridor was found by simulating at different locations in
the tunnel and gathering those which yielded zero relative state estimates using the
state extraction functions mentioned in Chapter 3. The process is summed up in
Algorithm 2.
The resulting steady-state trajectory is shown in Figure 4.6. The steady states are
not unique as the vehicle can find steady-state paths on the both sides of the small
objects in the corridor. The paths B and (C) in the figure show these alternate steady
state conditions. For computing the actual states of the vehicle, the appropriate
patches of the steady-state trajectories shown above is used.
In the simulations to generate trajectories originating at various initial con-
ditions the forward speed of the vehicle was maintained to be constant at 0.5 m/s
(1.65 ft/s). Four trajectories due to as many initial conditions are shown in figure
4.7A. The gains of Ky = ?5 and K? = ?10 were used in all the simulations. It
was observed that the vehicle was repelled away strongly by objects of large size,
e.g. the objects labeled L1 and L2. Following these large perturbations, the vehicle
adjusted to the steady state condition by generating large control inputs formed by
amalgamating large relative state estimates. It is interesting to note that the two
small obstacles in the corridor present a situation where the vehicle can proceed from
either side of the object, thus causing the variability in the resultant trajectories.
74
Algorithm 2: Steady-state estimation for cluttered corridors.
Input: Cluttered Corridor
. Output: Steady-state location in the corridor
. begin
for Inertial frame x-coordinate xn ? X (X is a linear space of
x-coordinates) do
Select a y0-coordinate and orientation ?0 (in the inertial frame)
Compute electric image at (xn, y0, ?0).
Compute relative-state estimates, y?0 and ??0.
if |y?0| > 1? 10?4 and |??0| > 1? 10?4 (convergence limits) then
y1 = y0 + y?0
?1 = ?0 + ??0
Compute electric image at (xn, y1, ?1).
Compute relative-state estimates, y?1 and ??1.
k = 1
while |y?k| > 1? 10?4 and |??k| > 1? 10?4 do
yk = ?y?k?1 yk?1?yk?2y?k?1?y?k?2 + yk?1
?k = ???k?1 ?k?1??k?2??k?1???k?2 + ?k?1
(Linear extrapolations assuming (??k) = 0)
Compute electric image at (xn, yk, ?k).
Compute relative-state estimates, y?k and ??k.
k = k + 1
Assign steady state location xssn = xn, yssn = yk, ?ssn = ?k
75
0 0.5 1 1.5 2 2.5
0
0.4
0.8
1.2
1.6
2
X?Distance, m
Y?
Di
st
an
ce
, m
A B C
Figure 4.6: Steady-state trajectory for the cluttered corridor consisting of conduct-
ing objects.
76
0 1 2 3 4 5 6
?0.5
0
0.5
time, t, sec
Ve
lo
cit
y, 
m
et
er
/s
Body velocities
 
 
vx
vy
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5
X?Distance, m
Y?
Di
st
an
ce
, m
Conductive Obstacles
L1
L2
A
D
0 2 4 6
?0.5
0
0.5
1
time, t, sec
Lateral O!sets
 
 
0 2 4 6
?2
0
4
time, t, sec
An
gu
lar
 o
"s
et
, ra
d
Angular O!sets
 
La
te
ra
l o
!s
et
, m
et
er
s
B
C
y y?
? ??
2
Figure 4.7: Simulation results for a corridor with all conducting objects. (A) Tra-
jectories, (B) Time traces of actual and estimated lateral offset, y and y?, (C) Time
traces of actual and estimated angular orientations, ? and ??, (D)Time traces of the
velocity components of the vehicle in the inertial frame.
77
The actual states of the vehicle are the difference between the inertial frame co-
ordinates of the trajectories traversed by a vehicle and the steady-state trajectory.
Plots for actual and estimated states are shown in figure 4.7B and C. The large
peak in the relative states plots towards the end is due to the large adjustment the
controller makes to bring the vehicle back towards the steady-state. The evolution
of velocity components of the vehicle in the inertial frame is shown in figure 4.7D.
The resultant velocity components of the vehicle (or the body frame velocity) at all
times was 0.5 m/s (1.65 ft/s) as specified in the beginning.
4.3.2.2 Non-conductive Objects
The steady-state trajectory for this case is exactly the same as that for the conduc-
tive obstacles case shown in Figure 4.8. the procedure mentioned in Algorithm 2 is
followed here.
The behavior of the vehicle in a corridor with all the objects being non-
conductive, was similar to the previous case. By modifying the gains as Ky = 5
and K? = 10, the vehicle showed similar responses to the objects in the corridor
as illustrated in figure 4.9A. It was provided a large perturbation signal by each of
the large objects L1 and L2 from which it recovered by making appropriate changes
to the the control input. The relative states showed convergence with the steady
state around 1 second after the start of the simulation as can be seen from figure
4.9B and C. The actual states of the vehicle are the difference between the inertial
frame coordinates of the trajectories traversed by a vehicle and the steady-state
78
0 0.5 1 1.5 2 2.5
0
0.4
0.8
1.2
1.6
2
X?Distance, m
Y?
Di
st
an
ce
, m
A B C
Figure 4.8: Steady-state trajectory for the cluttered corridor consisting of non-
conducting objects.
79
Non-Conductive Obstacles
0 2 4 6
?0.5
0
0.5
1
time, t, sec
Lateral O"sets
 
 
0 2 4 6
?2
0
4
time, t, sec
An
gu
lar
 o
"s
et
, ra
d
Angular O"sets
 
0 2 4 6
?0.5
0
0.5
time, t, sec
Ve
lo
cit
y, 
m
et
er
/s
Body velocities
 
 
La
te
ra
l o
"s
et
, m
et
er
s
vx
vy
A B
C
D
0.5
1
1.5
2
0.5 1 1.5 2 2.5
X?Distance, m
Y?
Di
st
an
ce
, m
L1
L2
0
y y?
? ??
2
Figure 4.9: Simulation results for a corridor with all non-conducting objects. (A)
Trajectories, (B) Time traces of actual and estimated lateral offset, y and y?, (C)
Time traces of actual and estimated angular orientations, ? and ??, (D)Time traces
of the velocity components of the vehicle in the inertial frame.
80
trajectory. The sudden peaks in the relative states between 4 and 5 seconds into
the simulation are due to the adjustment the controller makes to the trajectory in
order to converge with the steady-state. The velocity profiles in figure 4.9D are the
velocity components of the inertial frame velocity of the vehicle. The total velocity
of the vehicle is specified and maintained to be constant at 0.5 m/s (1.65 ft/s).
4.3.2.3 Conductive and Non-conductive Objects
Among all the cases considered under the cluttered corridor problem, the case involv-
ing conducting and non-conducting objects simultaneously is the most challenging.
It was shown in [52] that the composite electric image of two or more objects in jux-
taposition is considerably different from a simple addition or superposition of their
individual electric images. This deviation depends on a lot of factors like, distance
between the objects, material and the geometry of the objects.
A corridor with objects on one side being all conductive and the other side
being all non-conductive was considered here. The steady-state trajecory for this
kind of an environment is computed using the procedure mentioned in Algorithm 2
and is shown in Figure 4.10.
In simulations demonstrating obstacle avoidance, the objects were assumed by
the controller to be a local tunnel at any given instance. The results are presented
in figure 4.11A?D.
The state extraction functions sin(?) and sin(4?) were used to generate the
y and ? relative estimates. Ky = ?20 and K? = ?40 were used as control gains.
81
0.4
1.2
1.6
2
0.5 1 1.5 2 2.5
X?Distance, m
Y?
Di
st
an
ce
, m
0
A
B
C
0.8
Figure 4.10: Steady-state trajectory for the cluttered corridor consisting of conduct-
ing and non-conducting objects.
82
Mixed Conductivity Obstacles
0 2 4 6
?0.8
?0.4
0
0.4
0.8
time, t, sec
 
 
0 2 4 6
?2
0
2
time, t, sec
An
gu
lar
 o
"s
et
, ra
d
 
0 2 4 6
?0.5
0
0.5
time, t, sec
Ve
lo
cit
y, 
m
et
er
/s
Body velocities
 
 
La
te
ra
l o
"s
et
, m
et
er
s
vx
vy
A B
C
D
0.5
1
1.5
2
0.5 1 1.5 2 2.5
X?Distance, m
Y?
Di
st
an
ce
, m
L1
L2
0
y y?
? ??
Lateral O"sets
Angular O"sets
Figure 4.11: Simulation results for a corridor with conducting and non-conducting
objects. (A) Trajectories, (B) Time traces of actual and estimated lateral offset,
y and y?, (C) Time traces of actual and estimated angular orientations, ? and ??,
(D)Time traces of the velocity components of the vehicle in the inertial frame.
83
The trajectories are shifted towards the non-conductive (top) side of the corridor,
consistent with the mixed tunnel case. The time traces of the actual states, relative
state estimates and body velocities have the same explanation as before. There was
a steady state error in the value of the relative y estimate as the vehicle stabilizes
away from the pseudo-center of the tunnel. The bumps at 2 sec and 4.5 sec instant
in figures 4.11B?D occurred when the vehicle passed close to the smaller objects in
the corridor.
4.3.3 Obstacle Field
An obstacle field is an environment with multiple obstacles that do not form a
corridor. An obstacle field can give rise to multiple possible paths that the vehicle
can take in order to avoid collisions. Here, an obstacle field was chosen to have a
structured pattern of cylinders of same size. Each of the cylinders was placed roughly
at an equal distance from its closest neighbors. Two walls were placed on the farthest
sides of the environment to form barricades so that the vehicle does not escape from
the sides. Due to the small sensing range of the sensor, these walls do not perturb the
field of the sensor to induce a modulation in the electric image due to the cylinders.
These walls were also selected to be of the same material as the cylinders near it.
Depending on the initial conditions the trajectory which provides least obstruction
was taken, out of the presented choices. Cases involving all conductive, all non-
conductive and a mix of conductive and non-conductive cylinders were considered.
Four trajectories originating due to four initial conditions have been plotted for
84
each of the three cases. Additionally, the time traces for actual states, relative state
estimates and body velocities have also been plotted.
4.3.3.1 Conductive Cylinders
In this case, all the cylinders in the environment were conductive. They were ar-
ranged as shown in figure 4.12A. Conductive walls were placed as barricades at
outer edges of the environment. The trajectories originating from 4 different initial
conditions are plotted above. For the trajectory plotted in continuous line, the time
traces of the actual states, relative state estimates and body velocities are plotted
alongside. Here, the gains were selected to be Ky = ?20 and K? = ?40. The state
extraction functions for a 1 meter (3.3 ft). All the trajectories converged with the
steady state condition after the initial adjustment phase. For this case where all the
cylinders were conductive, the steady state condition loci were equidistant from the
cylinders on either of its sides. The vehicle passed from either sides of the central
cylinders in different iterations to demonstrate obstacle avoidance. Due to absence
of a unique steady-state trajectory, actual states were not computed in these sim-
ulations. The relative state estimates are shown in Figure 4.12B and C. The plots
in Figure 4.12D show the evolution of the inertial frame velocity components with
time. The body velocity at all times was maintained constant at 0.5 m/sec (1.65
ft/sec). The trajectories arriving from either side of the central object towards the
end did not converge afterwards and continued their course since there were no more
objects present to induce a turn in the trajectories to initiate convergence.
85
01
2
0 1 2 3
Y?
Di
st
an
ce
, m
X?Distance, m
0
0.1
0.2
 
 
?1
?0.5
0
0.5
 
 
0 2 4 6
?0.4
?0.2
0
0.2
0.4
time, t, sec
Ve
lo
cit
y, 
m
/se
c
 
 
0 2 4 6
time, t, sec
0 2 4 6
time, t, sec
A
B C D
vx
vy
St
at
e e
sti
m
at
e, 
?,
 r
ad
St
at
e e
sti
m
at
e, 
y, 
m
^
^
Figure 4.12: Simulation results for an obstacle field with all cylinders being conduc-
tive. (A) Trajectories, (B) Time traces of estimated lateral offset y?, (C) Time traces
of estimated angular orientation ??, (D)Time traces of the velocity components of
the vehicle in the inertial frame.
86
4.3.3.2 Non-conductive Cylinders
This case is similar to the conductive cylinders case above except that the value of
the gains selected here were Ky = 30 and K? = 60. The trajectories generated from
4 different initial conditions are shown in Figure 4.13A. Irrespective of the initial
position, all the trajectories converged together and were completed without any
collisions with the cylinders. The estimates states y? and ?? are plotted in Figure 4.13B
0
1
2
0 1 2 3
Y?
Di
st
an
ce
, m
X?Distance, m
St
at
e e
sti
m
at
e, 
?,
 r
ad
time, t, sec
Ve
lo
cit
y, 
m
/se
c
time, t, sectime, t, sec
St
at
e e
sti
m
at
e, 
y, 
m
A
B C D
0 2 4 6 8
?0.5
0
0.5
 
 
0 2 4 6 8
?1
?0.5
0
0.5
1
 
 
0 2 4 6 8
?0.4
?0.2
0
0.2
0.4
 
 
vx
vy
^
^
Figure 4.13: Simulation results for an obstacle field with all cylinders being non-
conductive. (A) Trajectories, (B) Time traces of estimated lateral offset y?, (C) Time
traces of estimated angular orientation ??, (D)Time traces of the velocity components
of the vehicle in the inertial frame.
and C. Figure 4.13D shows the evolution of the inertial frame velocity components
of the vehicle with time.
87
4.3.3.3 Conductive and Non-conductive Cylinders
The arrangement of the cylinders to form the obstacle field is shown in Figure 4.14A.
The controller gains of Ky = ?20 and K? = ?40 were used. It can be observed that
all of the 4 trajectories were biased towards the non-conductive cylinders. This can
be attributed to the perturbations of smaller magnitude due to the non-conductive
cylinders in comparison to the conductive cylinders.
0
1
2
0 1 2 3
Y?
Di
st
an
ce
, m
X?Distance, m
time, t, sec
Ve
lo
cit
y, 
m
/se
c
time, t, sectime, t, sec
A
B C D
0 2 4 6 8
1
0
0.5
 
 
0 2 4 6 8
?1
0
1
 
 
0 2 4 6 8
?0.4
?0.2
0
0.2
0.4
 
 
vx
vy
St
at
e e
sti
m
at
e, 
?,
 r
ad
St
at
e e
sti
m
at
e, 
y, 
m
^
^
Figure 4.14: Simulation results for an obstacle field with conductive and non-
conductive. (A) Trajectories, (B) Time traces of estimated lateral offset y?, (C)
Time traces of estimated angular orientation ??, (D)Time traces of the velocity com-
ponents of the vehicle in the inertial frame.
The relative states in 4.14B and C show peaks that were a result of the con-
troller making adjustments to the trajectory in order to avoid collisions with any of
88
the cylinders. The inertial velocity components of the vehicle are plotted in 4.14D.
The total velocity was maintained at a constant value of 0.5 m/sec (1.65 ft/sec).
From the simulations, it was realized that electrolocation can be used to per-
form reflexive unmapped obstacle avoidance with WFI based electric image process-
ing. Assumptions like absence of noise and infinite sensor resolution are unrealis-
tic in real world implementation. But for simulation purposes, these assumptions
are useful since the viability of the algorithm could be tested. It is interesting to
note that the control strategy derived using electrostatics equations worked well for
these simulations. This proves that the WFI processing is efficient at computing
navigational parameters from perturbation signals, electric images in this case. The
quasistatic simulations provided a basis for designing the hardware for performing
experimental autonomous navigation using electrolocation. In the next chapter, the
details of the experimental setup and the electro-sensor hardware are provided. Ad-
ditionally, the signal processing routine to form electric image signals resembling
quasistatic electric images is also mentioned. The experimental results obtained
using the electro-sensor and the signal processing routine are similar to the ones
obtained here.
89
Chapter 5
Experimental Validation
In this Chapter, the experimental implementation of electric image acquisition and
processing, Wide-Field Integration (WFI) based control and reflexive unmapped ob-
stacle avoidance is described in detail. This includes an overview of the construction
and operation details of the experimental setup. The electro-sensor design is mo-
tivated from the sensor model in Chapter 2. From the acquired signal, an electric
image resembling a quasistatic image is generated. For this purpose, various digital
signal conditioning routines were used, which have also been described. Finally, ex-
periments were performed to demonstrate centering, clutter navigation and general
obstacle avoidance. Results to these experiments are also reproduced here.
The experimental setup consists of an X-Y positioning gantry robot con-
structed above a glass water tank. For consistency, the environments used for per-
forming experiments were a straight tunnel, a cluttered environment and an obstacle
field, similar to the simulations in Chapter 4. For each environment three cases were
considered, first with all objects being conductive, second with all non-conductive
objects and third with a mix of conductive and non-conductive objects.
90
5.1 Experimental Setup
The experimental setup consists of two components. The first component is a large
water tank that holds water for building different underwater environments. The
second component is a gantry system to move the electro-sensor to positions com-
puted by the controller. While the tank was obtained from a commercial vendor,
the gantry and rest of the structure was custom-built in-house. Figure 5.1 shows
the overall setup.
Figure 5.1: Experimental Setup consisting of glass tank and gantry.
5.1.1 The Tank
The tank measures 4 feet ? 4 feet ? 1.5 feet (1.2 m ? 1.2 m ? 0.45 m) and is made
up of 0.75 inch (2 cm) glass. A sturdy wooden frame for the tank was constructed
for additional structural strength and with an intent of installing wheels for ease
91
of motion. The tank was fitted with a 3 inch (7.5 cm) diameter drainage in one
of its corners. A grid of resolution 2 inches ? 2 inches (5 cm ? 5cm) was drawn
on the floor of the tank and sprayed on with clear acrylic spray. A 1/16 inch (0.16
cm) thick plexiglass sheet was laid on the floor for protecting the grid from being
scraped off. The edges of the plexiglass were sealed using a silicone sealant.
5.1.2 The X-Y Gantry Robot
The electro-sensor is desired to be navigated between obstacles while the controller
computes its motion. Here, the demonstration of the effectiveness of the WFI based
electric image processing for extraction of proximity information of obstacles was
the goal. This information was used to perform obstacle avoidance in different
environments.
The X-Y gantry robot moves the electro-sensor from one location to the next
as commanded by the controller. The design of the gantry was based on the XY
robot design available on the website http://www.solsylva.com/. It is constructed
on the water tank and is an integral part of the experimental setup. It consists of a
horizontal beam that can move along the X-axis of the tank on leadscrews. Another
carriage which is capable of moving along the Y-axis is mounted on a leadscrew on
the beam. For Z-axis motion, another leadscrew driven carriage is mounted on the
Y-axis carriage. All the leadscrews are driven by stepper motors for accurate motion.
In addition to these motions, yaw rotational motion is added to the setup by adding
a stepper motor directly on the Z-axis carriage. The electro-sensor is mounted on
92
a long rod which is coupled with the spindle of the yaw stepper. Thus a motion
in X,Y,Z and yaw can be prescribed to the sensor. Since this work concentrates
on a planar electrolocation problem, the Z-axis motion is not used. However, it is
available for any future experiments which may possibly include more DOF motion
of the sensor.
All the leadscrews have 4-starts, with a maximum diameter of 3/8 inch (?10
mm), a pitch of 0.375 inch (?10mm) and are made up of stainless steel. A multistart
leadscrew translates the parts coupled to it farther in fewer rotations. Thus, the
stepper motors are made to rotate fewer number of times thus increasing their over-
all life. Most of the components required to build the gantry robot were machined
in the machine shop facilities available. These parts included, brackets, carriage
structures, bearing flanges and other additional parts. Remainder of the parts in-
cluding, fasteners, bearings, and other supplementary hardware were procured form
commercial sources.
Generic High-torque Nema 23 380 oz-in stepper motors were used to drive the
leadscrews through toothed belts and pulleys. The stepper motors are controlled
using the Geckodrive G540 4-axis motor controller and powered by a 48 volt, 12.5
ampere switching power supply. The G540 receives CNC commands directed to all
the steppers in parallel from a Linux based system running on a 3 GHz Core 2 Duo
processor and 4 GB RAM. The gantry has a positioning accuracy of 0.001 inches
(0.025 mm) and a workspace that covers an area of 42 inches ? 42 inches (?1 m
? 1 m) in the X-Y plane. Additional 15 inches (38 cm) motion along the Z-axis
plane can be added if necessary in future. The gantry is programmed to move at a
93
nominal forward speed of 2 inches/s (0.05 m/s). and maximum rotational speed of
90?/s. However it can be configured to operate at other values.
5.1.2.1 Electromagnetic Noise from Stepper Motors
It was realized after beginning experimentation, that the stepper motors from the
gantry robot induce high frequency electromagnetic noise to the electric field of the
sensor. The consequence of this is that the electric image generated were extremely
noisy and unworkable at times. Various analog and digital filtering techniques were
used to mitigate this noise, but results did not improve. Multiple layers of high order
digital filters could clean up the signal to a satisfactory level. We used two layers of a
9th order digital filter for this purpose. However, such high order filtering introduced
considerable time delay to the filtered output. Thus, first few wavelengths of the
captured signal were to be neglected in order to compute RMS value of the filtered
signal. This called for capturing more data at every iteration and for performing such
heavy filtering. This slowed the process significantly and a continuous operation of
the gantry by updating the controller at high rates was not viable any more.
This prompted us to use a different approach at capturing electric images. The
algorithm is now modified to momentarily shutdown the motors completely while the
sensor readings are acquired. Following this the motors are started again to perform
the motion as dictated by the controller. This algorithm sacrifices the ability to
perform a continuous motion in an environment due to the starting and stopping
of motors. But it is believed that this approach is better than acquiring noisy
94
readings, spending significant amount of resources at filtering them and updating
the actuators at a slow rate.
5.2 Electro-sensor
The electro-sensor used in the experiments is directly adapted from the electro-
sensor model used in the simulations as shown in Chapter 2. It is constructed in a
manner similar to that explained by MacIver and Nelson [22] using EKG sensors with
the difference being the shape of the electro-receptor array being circular instead
of linear and the location of the source being at the center of the array. A total
of 16 electro-receptors were used. The CAD model for this construction and the
electro-sensor hardware are shown in figure 5.2. The conical projections from the
A B
Figure 5.2: Electro-sensor in experiments. (A) CAD model, (B) Electro-sensor
assembled on the setup.
sensor disk are the fixtures to hold the EKG electrodes and the source electrode is
placed at the center underneath the disc of the sensor. The sensor was designed with
a maximum diameter of 3 inches (75 mm) in which the actual solid disc occupied
95
a diameter of 258 inches (?65 mm) and the conical projections were 3/8 inches (?10
mm) long. Each EKG sensor used as electro-receptors had a diameter of 3/8 inches
(?10 mm). The source is supplied with a 3 V peak-to-peak, 100 Hz sinusoidal
voltage. This source was selected in order to mimic wave-type weakly-electric fish,
which generate electric fields similar to the one used here. The sensor body is built
using a rapid prototyping machine and is held on the yaw axis motor on a shank
and kept immersed at least 8 inches (0.2 m) below the surface of water in the tank
to avoid any effects due to the ripples generated at the water surface. Each of
the sixteen electro-receptors on the periphery of the sensor is connected to ground
through a 10 k? resistor. These resistors mimic the fish?s skin resistance. For a
fish the voltage drop across the skin is the electric image. In this electro-sensor
the potentials measured at the electro-receptors are the voltage drops across the 10
k? resistors which constitute the electric image. The electro-sensor is capable of
measuring the effect of the resistance of an object (amplitude modulation of signal)
on the electric field and not the complex impedance (phase shift) as observed in
electric fish [19]. Conductive as well as non-conductive objects can be detected
using this electro-sensor.
The electric images similar to the ones described in Section 2.3 are obtained
by unwrapping the electric potential profile formed at the electro-receptor locations
starting at the electro-receptor located at ? = 0. This interpretation of the elec-
tric image is relevant to the controller design explained in Chapter 3. Thus, the
state extraction functions described there, can also be applied to the electric im-
ages so obtained. However, it should be noted that the electric potential sensed
96
at the electro-receptors is not static but sinusoidal in nature due to a sinusoidal
source voltage. Hence, to acquire electric images similar to the quasistatic images
in Chapter 4, a signal processing routine is employed as mentioned below.
5.3 Signal Processing
The signal processing step is made up of two parts, signal acquisition and the pro-
cessing of the acquired data to form a quasistatic image. Two prerequisites of the
signal processing routine are that the signal should be acquired at high sampling
rates and that the metric used to compute the quasistatic image from the dynamic
data should be easy to implement in realtime. With high sampling rates, various
operations like selecting a viable sample for analysis, filtering the sample etc. can be
computed in realtime. Hence, a high sampling rate Data Acquisition (DAQ) system
is used. Due to a simple metric to convert the data into quasistatic electric image,
extra computational effort is not wasted in carrying out the computation. For this
purpose, the root-mean-square (RMS) of the acquired sample is considered as the
potential value. It is not only a straightforward computation, but it has a practical
similarity to the quasistatic potential.
5.3.1 Signal Acquisition
With a sinusoidally varying source, the signal sensed at each electro-receptor is a
sinusoid. This sensed signal is significantly lower in magnitude as compared to the
source potential. The reason behind this is that the electric field of the source
97
attenuates with increasing distance from it. The equipotential lines at the distance
of the electro-receptors are considerably lower in magnitude due to low conductivity
of water. In addition, the sensed signal is the voltage drop across the 10 k? resistors
in series with the electro-receptors symbolizing the skin voltage. The total effect of
these two factors decrease the magnitude of the sensed potentials. For this reason a
data acquisition system which has a digitization resolution high enough to resolve
potentials of this order is required.
The signals from all of the 16 electro-receptors were acquired using National
Instruments PCI-6224 Data Acquisition board. This board has a 16 bit analog to
digital conversion (ADC) resolution. With this sort of a resolution and a 5 volt
basis used for the ADC, an accuracy of 5216 = 76?V . Such an accuracy is extremely
high even for our purposes, but is a very desirable quality of the DAQ system. The
particular DAQ board has a maximum sampling rate of 250 kS/s. This provides a
sensing bandwidth of 15.625 kS/sec for each receptor, which imparts a very high
sampling rate to the acquired signal. Thus the DAQ board is an extremely good
choice to acquire the level of accuracy desired for this setup.
We acquire 5 cycles every iteration at each receptor. The frequency of the
sensed signal at each receptor is the same as that of the source frequency of 100 Hz.
With a sampling rate of 15.625 kS/sec, there will be 5100?15.625?103 ? 781 samples
in the recorded 5 seconds of data. This large number of data points recorded in 0.05
seconds is helpful in implementing a lowpass filter to attenuate the sensor noise.
An example of filtering is shown in figure 5.3. A digital lowpass filter of 1st order
with the cutoff frequency of 500 Hz was used. Higher order filters could yield better
98
results, but they impart time delays due to which the first few cycles of the filtered
data are not useful for computation of RMS. So 1st order filter was considered to be
sufficient for this problem. This filtered signal is then further processed to compute
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
?1.2
?0.8
0
0.4
0.8
1.2
Time, sec
El
ec
tri
c p
ot
en
tia
l, v
ol
ts
Filtering of sensory signal
?0.4
Figure 5.3: Filtering of the sensory signal.
the RMS value. This is repeated for each of the 16 electro-receptors. The code to
acquire data and process it digitally employs multithreading to perform this task
faster in parallel for all the receptors.
5.3.2 Quasistatic-like Approximation
It was mentioned earlier that a property of the acquired signal at each receptor
similar to a quasistatic value was sought. This quasistatic approximation enables in
application of the previously developed control strategy. a quasistatic approximation
takes into account the dynamic nature of the field by performing the computations
in frequency domain and hence treating the field as static in frequency. However,
this is an artificial method of simplifying the computations. However nature does
99
not follow this method even if the source is oscillating at a single frequency, in
these experiments for instance. The electric potential values that are read at each
electro-receptor are arrived at by performing the full electromagnetic computations
by nature. Thus the signal available for measurement is oscillatory and in time do-
main. However, if the RMS value of the sensed signal over a few cycles is computed,
it can act as a representative for entire sensed data. The RMS takes into account the
magnitude and the frequency of oscillation of the signal. Thus it is analogous to a
quasistatic measurement. Continuing with the same example as before, the filtered
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
?1.2
?0.8
0
0.4
0.8
1.2
Time, sec
El
ec
tri
c p
ot
en
tia
l, v
ol
ts
Sine wave "t to "ltered signal
?0.4
?0.0082 ? 0.19 cos(690.20 x) + 0.84 sin(690.20 x)
Figure 5.4: Fitting a sine wave to the filtered signal.
data can be seen to have a sinusoidal shape. It is known [22] that this filtered data
should in theory be perfectly sinusoidal since the source signal is sinusoidal. The
distortions in the sensed signal are caused due to various noise sources. To stay
consistent with the theory, a sine wave could be fitted to the sensed signal as shown
in figure 5.4 above. Once we have a sine wave fitted to the data, we can easily
compute the RMS voltage of the wave. Two points are to be noted here. We fit
100
the sine wave to the filtered data and not the raw data since the noise present in
the sensor data tends to make the fitted sine wave larger in magnitude and thus it
does not represent the real data. Secondly, we do not compute the RMS off of the
filtered data as we would need exactly full cycles. Thus fractional wavelengths of
data will not give the correct RMS value unless the number of recorded cycles is
very large. More cycles will however increase the time required for data acquisition
and processing. So computing RMS off of filtered data is not considered and a sine
wave it fitted to the data for this purpose.
We implement a least-squares four-parameter sine wave fitting [74] routine
for this purpose. This algorithm yields an equation of the form c + a1 cos(bx) +
a2 sin(bx). In the above example, the fit to the filtered data is given as ?0.0082 ?
0.19 cos(690.20x) + 0.84 sin(690.20x) where x is the time after start at any given
instance. Three of the parameters yielded by this algorithm are useful in construct-
ing a zero-mean sine function fit to the dataset. The fourth parameter provides the
value of the DC shift of the entire dataset. It was observed that DC shift varies and
drifts over time. To reject this drift, we ignore the fourth parameter of the sine-
fit routine to get a zero-mean sine wave every time. Thus, in the above example
c = ?0.0082 can be ignored. The RMS value of the fitted wave is then easy find
using the discrete values of the rest of the equation.
Performing this sine wave fitting for each sensor in parallel yields the quasistatic-
type potentials at the electro-receptors, Vpert.. A precomputed set of such potentials,
Vnom. for the nominal case where no perturbing obstacles are present is available a
priori. The electric image is then given as the difference between the potentials in
101
the presence of the obstacles and in the nominal case.
? = Vpert. ? Vnom.
The control strategy derived in Chapter 3 can be applied to an electric image so
obtained.
5.4 Operation Sequence
The entire sequence of operations is summarized below:
1. Stop the motors.
2. Acquire readings.
3. Process readings and Start motors.
4. Move according to received commands.
Out of these, steps 1?3 are completed within 0.1?0.15 sec. The actual motion of the
gantry adds another 0.25?0.3 sec. since the algorithm advances the gantry by about
half an inch (1.25 cm) at a configured forward speed of 2 in/sec (5 cm/sec). Thus
the total time per iteration is 0.35?0.45 sec.
All the above operations are performed on the same computer that controls the
gantry motion. For starting and stopping the motors, instructions are sent by the
computer over an XBee module attached to it to another of such module attached
to gantry. These instructions are read by an Arduino board which operates a relay
circuit to start and stop the motors all at once. Following the stopping of motors, the
102
computer instructs the DAQ board to read all the 16 sensors at once. The motor are
instructed to start up as soon as all the readings are acquired to minimize the down-
time. The acquired readings are processed in parallel to perform WFI from which
relative states are obtained. The parallel processing is achieved by implementing
16 threads simultaneously. The controller then computes the control input to the
actuators which is further processed to compute the next location of the sensor. A
CNC command is then issued via a parallel port to the motor controller facilitating
the motion of the gantry. This entire chain of events is executed by a single code
written in Python.
In the following Section, the results to experiments in different environments
are presented.
5.5 Experiments
Experiments were conducted on the lines similar to the quasistatic simulations. Ex-
periments demonstrating tunnel centering, clutter response and obstacle field navi-
gation were performed in environments having all conductive, all non-conductive or
a mix of conductive and non-conductive objects. Conductive objects had aluminum
plates to form a straight tunnel and steel pipes to form cluttered and obstacle field
environments. Among non-conductive objects were wooden boards for the tunnel
and PVC pipes for the remaining two types of environments. Example setup of
each of the three types of environments can be seen in figure 5.5. A total of nine
sets of experiments were performed. Three for conductive environments, three for
103
A CB
Figure 5.5: Environments for experimentation (A) Tunnel, (B) Clutter, (C) Obstacle
field
non-conductive and another three for mix conductivity type environments. Each set
was further divided into two subsets, each arising due to changing either the initial
lateral offsets or the initial angular orientations, keeping the other initial condition
constant. Also included is a selected trial among these for which actual and relative
states along with body velocities are plotted. In the plots that follow, conductive
objects are shown in grey color while non-conductive objects are shown in brown.
The arrowheads on each trajectory indicate the initial position for that particular
trial.
5.5.1 Straight Tunnel
The expected behavior of the vehicle carrying the sensor in a straight tunnel was
to center itself along the centerline of the tunnel and directed forward. Through
experiments, this particular behavior has been demonstrated in tunnels made out of
conductive and non-conductive walls. We have used the control strategy developed
in Chapter 3 for this purpose.
104
5.5.1.1 Both Walls Conductive
In this setup, two aluminum plates were used to form a 9 inch (23 cm) tunnel. The
plates were 1/16 inch (1.6 cm) thick, 36 inches (91 cm) long and 12 inches (30 cm)
tall. Figure 5.6B shows the experimental setup.
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive Walls
?1
0
1
2
3
0 50 100 150
?0.3
?0.2
0
0.1
Time step number
0 50 100 150
Time step number
A
C D
B
?0.2
An
gu
lar
 o"
se
ts,
 ra
d
La
te
ral
 o"
se
ts,
 in
ch
es
??
?
y
y?
Figure 5.6: Experiment demonstrating centering in a straight tunnel having con-
ductive walls (A) Trajectory, (B) Experimental setup, (C) Time traces of actual and
estimates lateral offsets, y and y?, (D) Time traces of actual and estimated angular
orientations, ? and ??
The trajectory in figure 5.6A indicates that the vehicle was centered in the
tunnel soon after start and was maintained in that condition afterwards. Gains of
Ky = ?20 and K? = ?40 were used in this experiment. The controller designed in
Chapter 3 was used. As was also seen in simulations, the objective of the controller
105
is to drive the relative states, y? and ?? to zero. The relative states plots in Figure
5.6C and D confirmed this behavior of the controller. In contrast to the simulations
for the tunnel problem, the actual states did not resemble the relative states exactly.
A slight overshoot and a small steady state error in the lateral offset was observed as
illustrated in Figure 5.6C and D. The overshoot was due to the gains not being good
enough to damp out the oscillations in the trajectory. The steady state error can be
attributed to misalignment in the tunnel and possible presence of exogenous electric
field perturbations. The controller tries to balance the electric image perturbations
in the right half with those in the left half of the electric image at all times. This
balancing is interpreted by the controller as the achievement of the steady state
condition and results in the relative states going to zero. However, in the presence of
misalignment in tunnel geometry and exogenous perturbations, a bias is introduced
which misguides the controller in thinking of a non-steady state condition as being
a steady state one. This results in a steady state error. Since the error in the
above trial was constant, the exogenous perturbation if present must have existed
constantly throughout the length of the tunnel.
The controller was tested for variations in either one of the two initial con-
ditions y0 and ?0 to test it for repeatablity and adaptability as shown in Figure
5.7. When the initial lateral offset was varied within the tunnel, the initial effort
required to approach steady state was varied in proportion of the offset. After this
initial stage, all the trajectories converge with the centerline of the tunnel. Similar
behavior is observed when the initial angular offset is varied and the initial lateral
offset is held constant. All the trajectories make initial turn maneuvers based on
106
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?axis, inches
Y?
ax
is,
 in
ch
es
Initial lateral o"setsA
0 5 10 15 20 25 30 35?9
?4.5
0
4.5
9
X?axis, inches
Y?
ax
is,
 in
ch
es
Initial angular o"setsB
Figure 5.7: Effect of variation in initial conditions on trajectories in a tunnel with
conductive walls, (A) Variation in initial lateral offset, (B) Variation in initial an-
gular offset
the initial angular offset and merge together to center in the tunnel. It is interesting
to note that the overshoot was consistently observed in all the cases.
5.5.1.2 Both Walls Non-conductive
In this problem the tunnel was constructed out of two wooden walls that were
0.75 inch (1.9 cm) thick, 8 inches (20 cm) tall and 36 inches (91 cm) long. The
experimental setup can be seen in Figure 5.8B.
The controller used in this trial is the same as that used for the conductive
walls case. The only difference is the gains that need to be negated for the reasons
explained in Section 3.6. In addition, electric images due to non-conducting walls are
smaller in magnitude than due to their conductive counterparts. So the estimated
states are of lower magnitude as well, requiring higher gains. Accordingly, the gains
that were used in the experiment were Ky = 30 and K? = 60. These gains do not
cause instability as they multiply relative states that opposite in sign as that for
107
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Non-conductive Walls
?3
?2
?1
0
1
0 50 100 150
?0.1
0
0.1
0.2
0.3
Time step number
An
gu
lar
 o"
se
ts,
 ra
d
 
0 50 100 150
Time step number
La
te
ral
 o"
se
ts,
 in
ch
es
A
C D
??
?
y
y?
B
Figure 5.8: Experiment demonstrating centering in a straight tunnel having non-
conductive walls (A) Trajectory, (B) Experimental setup, (C) Time traces of actual
and estimates lateral offsets, y and y?, (D) Time traces of actual and estimated
angular orientations, ? and ??
.
108
the conduucting tunnel case. Centering was achieved in the tunnel with negligible
steady state error as can be seen in Figure 5.8A. Overshoot was observed in the
trajectory and the relative state y? due to the particular choice of the gains.
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?axis, inches
Initial lateral o"sets
0 5 10 15 20 25 30 35?9
?4.5
0
4.5
9
X?axis, inches
Initial angular o"sets
Y?
ax
is,
 in
ch
es
Y?
ax
is,
 in
ch
es
Figure 5.9: Effect of variation in initial conditions on trajectories in a tunnel with
non-conductive walls, (A) Variation in initial lateral offset, (B) Variation in initial
angular offset
On changing exactly one initial condition out of two, the control signal to the
vehicle is varied accordingly to center itself in the tunnel. Convergence of trajectories
was observed when initial lateral as well angular conditions were varied as seen in
Figure 5.9.
5.5.1.3 One Conductive and One Non-conductive Wall
A tunnel was constructed with one aluminum wall and another with a wooden wall.
At each iteration, absolute value of electric image was computed. State extraction
functions of sin(?) and sin(4?) were used as per the discussion in Section 3.6. These
functions were computed for a conductive walled tunnel but were used in this case
nonetheless. The experimental setup is shown in Figure 5.10B.
109
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive and Non-conductive Walls
Time step numberTime step number
A
C D
0 40 80 120 160
?4
?2
0
2
4
0 40 80 120 160
?0.04
?0.02
0
0.02
0.04
?0.06
B
An
gu
lar
 o"
se
ts,
 ra
d
La
te
ral
 o"
se
ts,
 in
ch
es
??
?
y
y?
Figure 5.10: Experiment demonstrating centering in a straight tunnel having one
conductive and one non-conductive wall (A) Trajectory, (B) Experimental setup,
(C) Time traces of actual and estimates lateral offsets, y and y?, (D) Time traces of
actual and estimated angular orientations, ? and ??
.
110
Interestingly, the vehicle did not align with the centerline but with a line 2
inches (5 cm) offset from it, as shown in the trajectory plot in Figure 5.10A. This
behavior is in fact a very good result and informative result. With the above men-
tioned state extraction functions, the controller would center the vehicle perfectly if
both the walls were made up of aluminum. However, a wooden wall generates per-
turbations that are smaller in magnitude than that of an aluminum wall. In a tunnel
with one aluminum wall and one wooden wall, the perturbation due to one wall was
lower than other. The controller tries to balance the electric field perturbations on
right and left sides recorded by the electro-sensor. This balance is achieved at a
position closer to the wooden wall and not at the center of the tunnel. The offset is
also apparent in the plots for actual states in Figure 5.10C where the lateral offset
y settled at a steady state value of -2. The relative states however settled at zero as
shown in Figure 5.10D. This confirms that the controller worked as expected.
0 5 10 15 20 25 30 35
?9
?4.5
0
4.5
9
X?axis, inches
Initial lateral o"sets
A
0 5 10 15 20 25 30 35?9
?4.5
0
4.5
9
X?axis, inches
Initial angular o"setsB
Y?
ax
is,
 in
ch
es
Y?
ax
is,
 in
ch
es
Figure 5.11: Effect of variation in initial conditions on trajectories in a tunnel with
one conductive and one non-conductive wall, (A) Variation in initial lateral offset,
(B) Variation in initial angular offset
When the experiment was repeated for multiple initial conditions where only
111
one variable out of two was varied at a time, consistent results were observed.
Trajectories originating from different locations but with zero initial orientation
values all converged to align with the line offset from the centerline. Same is also
observed when the initial lateral offset was maintained constant and the the initial
angular offset was varied. In both the cases, the steady state offset was found out
to be the same. The results are plotted in Figure 5.11.
5.5.2 Cluttered Corridor
A cluttered corridor is formed when objects of complex geometries are placed in
water to form a corridor, not necessarily straight. However, objects of such random
shapes are not easily found commercially. Therefore, steel and PVC cylinders of
different diameters were used to form this kind of environments. Cylinders of 3,4
and 6 inches (7.5, 10 and 15 cm) diameters were used. There was no particular
template that was used to form these environments. The only care that was taken
while forming these environments was to not to leave gaps of more than 5 inches
(13 cm) between cylinders to prevent the vehicle from escaping the corridor. A
small cylinder was placed midway in the corridor near the center to act as a local
perturbation. This cylinder was small enough to not to be sensed by the sensor from
far away. This cylinder was intended to induce a sudden reaction and trajectory
correction by the vehicle. Three different cases were studied as before, first when
all the cylinders were made up of steel, second when only PVC cylinders were used
and third when a mix of the two types of cylinders were used. The steady state
112
trajectory similar to the one in the quasistatic simulations was not computed here.
As a result, the actual states are not available.
5.5.2.1 Conductive Cylinders
In this case only steel cylinders were used to form the corridor. The steel cylinders
were each 8 inches (20 cm) tall and made up of 1/32 inch (0.8 mm) thick steel sheet.
Three cylinder sizes (diameters) were used, 3,4 and 6 inches (7.5, 10 and 15 cm).
The arrangement if the cylinders is shown in Figure 5.12B.
As is seen in Figure 5.12A, the vehicle navigated between the obstacles without
any collisions. The corridor had a varying width. Gains of Ky = ?20 and K? = ?40
were used. On encountering the small object placed between the two sides of the
corridor towards the end of the corridor, the vehicle made a sharp turn away from
it in order to avoid collision.
This turn maneuver is seen in the relative states plot in Figure 5.12C and D
when a sudden jump in the estimates occurs. For all other times, the estimates are
seen to have values close to zero after the initial adjustment. On varying the initial
conditions, one variable at a time, convergence of trajectories was observed. Both
the cases when linear and angular offsets were varied separately, are shown in Figure
5.13.
113
0 10 20 30 40
?20
?10
0
10
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive Objects
?0.4
?0.2
0
0.2
0.4
0 50 100 150 200
Time step number
Re
lat
ive
 st
at
e, 
?,
 ra
d
A B
C D
?0.4
?0.2
0
0.2
0.4
0 50 100 150 200
Time step number
Re
lat
ive
 st
at
e, 
y, 
in
ch
es
Figure 5.12: Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with conducting cylinders, (A) Trajectory, (B) Experimental setup, (C)
Time trace of relative state, y?, (D) Time trace of relative states, ??
114
0 10 20 30 40
?20
?10
0
10
X?axis, inches
Y?
ax
is,
 in
ch
es
Initial lateral o"setsA
0 10 20 30 40
?20
?10
0
10
X?axis, inches
Y?
ax
is,
 in
ch
es
Initial angular o"setsB
Figure 5.13: Effect of variation in initial conditions on trajectories in a cluttered
corridor comprising of conductive cylinders, (A) Variation in initial lateral offset,
(B) Variation in initial angular offset
5.5.2.2 Non-conductive Cylinders
In this set of experiments, the corridor is created using cylinders made out of PVC.
Each cylinder was 1 foot (30 cm) in height. The arrangement of cylinders was not
repeated from the one with conductive cylinders but a completely new one was
used. However, a cylinder was placed in the corridor towards the end to produce a
rapid correction in the trajectory by the vehicle similar to the conductive cylinders
problem. The arrangement is shown in Figure 5.14B.
The gains used here were Ky = 30 and K? = 60. The change in sign and
higher magnitude in comparison to the previous case can be noticed. These changes
were made in order to work with non-conductive cylinders due to the electric images
resulting from them as explained in Section 3.6. The vehicle navigated between the
115
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Non-conductive Objects
?1
?0.5
0
0.5
1
0 50 100 150 200
Time step number
Re
lat
ive
 st
at
e, 
y, 
in
ch
es
Re
lat
ive
 st
at
e, 
?,
 ra
d
A B
C D
0 10 20 30 40
?20
?10
0
10
0 50 100 150 200
Time step number
?1
?0.5
0
0.5
1
Figure 5.14: Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with non-conducting cylinders, (A) Trajectory, (B) Experimental setup,
(C) Time trace of relative state, y?, (D) Time trace of relative states, ??.
116
cylinders without any collisions as is evident from Figure 5.14A. Near the small
cylinder in the corridor acting as a small obstacle, the vehicle made a sharp turn
away in order to avoid colliding with it. The relative states plotted in Figure 5.14C
and D stay close to zero after the initial adjustment except for the time when the
vehicle encounters the small cylinder in its course. At this point, a large correction
is made by the controller due to the occurrence of the peak in the state estimates.
0 10 20 30 40
?20
?10
0
10
X?axis, inches
Initial lateral o"setsA
0 10 20 30 40
?20
?10
0
10
X?axis, inches
Initial angular o"setsB
Y?
ax
is,
 in
ch
es
Y?
ax
is,
 in
ch
es
Figure 5.15: Effect of variation in initial conditions on trajectories in a cluttered cor-
ridor comprising of non-conductive cylinders, (A) Variation in initial lateral offset,
(B) Variation in initial angular offset
The consistency and repeatability of the controller was tested by varying the
initial conditions, one variable at a time. In all the cases, convergence of trajectories
was observed irrespective of the initial conditions which can be seen in Figure 5.15.
117
5.5.2.3 Conductive and Non-conductive Cylinders
In this case, both types of cylinders, steel and PVC, were used. It was realized
that any random arrangement of the cylinders would not work. If a conductive
and a non-conductive cylinder were placed next to each other, the resulting electric
image was observed to be smaller in magnitude and wider. These images would fool
the controller in thinking that the vehicle was away from the objects and it would
move the vehicle close to them. In many cases, the vehicle would simply collide
with the objects due this misinterpretation by the controller. Hence, after a lot of
failed attempts to use a random arrangement of cylinders, a simpler arrangement
was finalized. Here, the corridor was formed with all the conductive cylinders on its
one side and non-conductive cylinders on the other. This arrangement is analogous
to the tunnel case with one conductive wall and one non-conductive wall. The
arrangement is shown in Figure 5.16B.
The trajectory is shown in Figure 5.16A. It can be seen to approach the non-
conductive cylinders in the initial stage of its journey. This behavior is consistent
with the results from the mixed conductivity tunnel case. However, on getting very
close to such a cylinder, the controller made a big correction to avoid colliding.
This correction made the vehicle traverse a path which goes close to the conductive
cylinders. However, the vehicle can be seen to move away from the conductive
cylinders by the end of its journey. The gains of Ky = 20 and K? = 40 were used.
The big correction made by the controller is a result of higher values of relative
estimate y? near the start of the trial as seen in Figure 5.16C. This relative state
118
0 10 20 30 40
?20
?10
0
10
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive Objects
0 60 120 180 240
?1
?0.5
0
0.5
Time step number
0.4
?0.4
0
0.2
0 60 120 180 240
Time step number
Re
lat
ive
 st
at
e, 
y, 
in
ch
es
Re
lat
ive
 st
at
e, 
?,
 ra
d
A B
C D
?0.2
1
Figure 5.16: Experiment demonstrating obstacle avoidance in a cluttered corridor
constructed with conductive cylinders on one side and non-conductive cylinders on
the other, (A) Trajectory, (B) Experimental setup, (C) Time trace of relative state,
y?, (D) Time trace of relative states, ??
119
stays positive for the rest of the experiment which indicates locations closer to the
conductive cylinders. Towards the end of the experiment, it is seen to decrease
to zero indicating the attempt made by the controller to move closer to the non-
conductive cylinders.
0 10 20 30 40
?20
?10
0
10
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
0 10 20 30 40
?20
?10
0
10
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Initial lateral osets
A
Initial angular osets
B
Figure 5.17: Effect of variation in initial conditions on trajectories in a cluttered
corridor comprising of conductive and non-conductive cylinders, (A) Variation in
initial lateral offset, (B) Variation in initial angular offset
Additional set of trials were performed with change in either the initial lateral
offset or initial angular offset. It was observed that the vehicle stayed closer to the
non-conductive cylinders in most of the cases as can be seen in Figure 5.17. The
exceptions were the cases when the vehicle got very close to the non-conducting
cylinders prompting the controller to make a big correction to avoid correction.
Satisfactory level of convergence of trajectories was observed in these trials.
120
5.5.3 Obstacle Field
In these environments cylinders were arranged in a structured pattern providing as
staggered obstacles for the vehicle. Walls were placed outside this arrangement of
cylinders in order to prevent the vehicle escaping from the sides. Steel and PVC
cylinders of 6 inches (15 cm) diameter and aluminum and wood plates were used
to construct the environments. As before, three cases were considered, first with
all conductive cylinders and conducting walls, second with non-conductive cylinders
and walls and the third with a mix of conductive and non-conductive walls and
walls. Due to the placement of the cylinders, the trajectory which provides least
obstruction was taken, out of the presented choices. This is a property of reflexive
obstacle avoidance. In this problem, there exist multiple steady state conditions
unlike the previous problems where unique steady state conditions existed. Here
the steady state condition can be defined as the locus of all the points between the
cylinders where the electric perturbations in the left and right halves of the electric
image are equal. This is a generic definition and in this problem could result into
many such co-existing loci.
5.5.3.1 Conductive Cylinders
The obstacle field was constructed by placing the cylinders at an equal distance
from all its neighbors. The aluminum plates were used to prevent the vehicle from
escaping from the sides of the environment. The arrangement can be seen in Figure
5.18B.
121
0 10 20 30 40
?20
?10
0
10
20
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive Cylinders 
0 50 100 150 200
0
Time step number
Re
lat
ive
 s
ta
te
,
 
?,
 
ra
d
A B
C D
0 50 100 150 200
?0.1
0
0.1
0.2
Time step number
Re
lat
ive
 St
at
e, 
y, 
in
ch
es
?0.1
0.1
0.2
Figure 5.18: Experiment demonstrating obstacle avoidance in an obstacle field com-
prising of conductive cylinders, (A) Trajectory, (B) Experimental setup, (C) Time
trace of relative state, y?, (D) Time trace of relative states, ??
122
The relative states are plotted in Figure 5.18C and D staying closer to zero.
The gains of Ky = ?20 and K? = ?40 were used.
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Y?
ax
is,
 in
ch
es
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Y?
ax
is,
 
in
ch
es
Initial lateral o"sets Initial angular o"setsA B
Figure 5.19: Effect of variation in initial conditions on trajectories in an obstacle
field comprising of conductive cylinders, (A) Variation in initial lateral offset, (B)
Variation in initial angular offset
Initial conditions were varied, one out of two variables at a time and trials
were performed. It was observed that the vehicle traversed through the environment
without any collisions and achieved satisfactory level of convergence of trajectories as
shown in Figure 5.19. More interestingly, it took a path from either side of the central
cylinders. The trajectories arriving from either side of the central cylinders towards
the end did not converge afterwards and continued their course since there were no
more objects present to induce a turn in the trajectories to initiate convergence.
123
5.5.3.2 Non-conductive Cylinders
In this case, all the cylinders that were used to build the environment were made
out of PVC and the barricade walls were made out of wood. This case is similar to
the conductive cylinders case above except that the value of the gains selected here
were Ky = 30 and K? = 60. The environment is shown in Figure 5.20B.
No collisions occurred while the vehicle traversed through the environment as
seen from the trajectory plot in Figure 5.20A. The relative state estimates in Figure
5.20C and D converge and stay close to zero indicating the attainment of the steady
state.
On varying the initial conditions, the vehicle was found to take the path of
least resistance around the central cylinders. Satisfactory level of convergence was
also observed as seen in Figure 5.21.
5.5.3.3 Conductive and Non-conductive Cylinders
In this case both, steel and PVC cylinders were used to build the environment. The
issue of juxtaposed objects encountered in the cluttered corridor problem consisting
of both types of objects was not experienced here since all the cylinders maintained
a certain a distance from all their neighbors. So the cylinders were placed randomly
as shown in Figure 5.22B.
As seen from the trajectory plot in Figure 5.16A, the vehicle made some turn
maneuvers where it was not expected to. The vehicle made a turn just before
the final central cylinder but it eventually corrected for this turn and completed
124
0 10 20 30 40
?20
?10
0
10
20
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Non-conductive Cylinders 
0 50 100 150 200
?0.1
0
0.1
0.2
Time step number
Re
lat
ive
 s
ta
te
,
 
?,
 
ra
d
A B
C D
?0.2
0 50 100 150 200
?0.1
0
0.1
0.2
Time step number
Re
la
tiv
e S
ta
te
, y
, 
in
ch
es
?0.2
Figure 5.20: Experiment demonstrating obstacle avoidance in an obstacle field com-
prising of non-conductive cylinders, (A) Trajectory, (B) Experimental setup, (C)
Time trace of relative state, y?, (D) Time trace of relative states, ??
125
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Initial lateral o"sets
A
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Initial angular o"sets
B
Y?
ax
is,
 in
ch
es
Y?
ax
is,
 in
ch
es
Figure 5.21: Effect of variation in initial conditions on trajectories in an obstacle
field comprising of non-conductive cylinders, (A) Variation in initial lateral offset,
(B) Variation in initial angular offset
its journey. This random turning behavior could be attributed to the difference in
magnitudes of the electric image due to the two types of objects. The vehicle tried to
move away from the central conductive object at the right side of the environment
and towards the first central non-conductive object. So a turn was induced to
complete this operation. However the vehicle sensed the second conductive object
in the top row and corrected it path soon afterwards. This explanation was inferred
from the relative states in Figure 5.22C where a dip in the state estimate y? occurs
due to the presence of a non-conductive object nearby and signals the controller to
make the turn mentioned above. Similarly a positive peak indicates a conductive
object nearby triggering the corrective turn.
Similar random turn maneuvers were also experienced when the initial condi-
126
0 10 20 30 40
?20
?10
0
10
20
X?Distance, inches
Y?
Di
st
an
ce
, in
ch
es
Conductive Cylinders 
0 50 100 150 200
?0.05
0
0.05
0.1
Time step number
Re
lat
ive
 St
at
e, 
y, 
in
ch
es
Re
lat
ive
 s
ta
te
,
 
?,
 
ra
d
A B
C D
?0.1
0 50 100 150 200
?0.05
0
0.05
0.1
Time step number
Re
lat
ive
 St
at
e, 
y, 
in
ch
es
Re
lat
ive
 st
at
e,
 
?,
 
ra
d
?0.1
Figure 5.22: Experiment demonstrating obstacle avoidance in an obstacle field com-
prising of conducting and non-conductive cylinders, (A) Trajectory, (B) Experimen-
tal setup, (C) Time trace of relative state, y?, (D) Time trace of relative states,
??
127
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Initial lateral o"sets
A
0 10 20 30 40
?20
?10
0
10
20
X?axis, inches
Initial angular o"sets
B
Y?
ax
is,
 in
ch
es
Y?
ax
is,
 in
ch
es
Figure 5.23: Effect of variation in initial conditions on trajectories in an obstacle
field comprising of conductive and non-conductive cylinders, (A) Variation in initial
lateral offset, (B) Variation in initial angular offset
tions were varied. These turn maneuvers make the trajectories jagged, but collisions
were avoided in all the cases. The corresponding plots are shown in Figure 5.23.
There is a clear agreement between the simulation and experimental results.
The inferences drawn from the observations in all the previous work have been
summed up in the following Chapter. In addition the scope for future work is also
discussed.
128
Chapter 6
Conclusions and Future Work
This study was set out to explore the possibility of use of electric images in perform-
ing reflexive obstacle avoidance in different environments in an attempt to mimic
the navigational technique called electrolocation found in some electric fishes. In
that respect, an electrostatics based model was developed to represent the electric
images formed due to objects of simpler geometries. Wide-Field Integration (WFI)
methodology was applied to extract proximity information from the above electric
images in order to develop a general control strategy to perform autonomous nav-
igation. Simulations demonstrating obstacle avoidance behaviors were generated
using this control strategy. For hardware implementation, an electro-sensor and
an experimental setup were designed and constructed. The same control strategy
also proved useful in performing autonomous navigation and obstacle avoidance be-
haviors. While previous attempts at performing such tasks using electrolocation
have been made, this study sought to achieve this goal by designing a controller
that is computationally efficient and easy to implement on hardware with limited
computational capabilities.
Following is the list of key findings and their technical implications.
129
Chapter 2: Electrostatic Equations
1. In Chapter 2, the equations for electric images due to objects of simple ge-
ometries were derived using principles from electrostatics. Of particular in-
terest was the formulation for a straight tunnel. For deriving this equation,
a technique called method of images was used. This subsequently allowed us
to apply mathematical techniques to design weighting patterns that decode
relevent information from electric image measurements.
Chapter 3: Control Strategy
2. In Chapter 3, the procedure to map an electric image directly to actuator
commands was described. This consisted of two tasks, WFI processing of the
electric image to obtain optimal estimates of relative states and linear feedback
control.
3. WFI processing calls for computing inner products of the electric image with
various sensitivity functions followed by linearization of the outputs of the
inner products. The linearized equations directly relate the sensory signal
i.e. the electric image with proximity information such as relative position
and orientation. Since the electric image equations are difficult to integrate,
a numerical scheme to perform the inner product operation was developed.
Another routine was developed to linearize these inner products by performing
numerical differentiation.
4. The linearized equations revealed that the number of WFI harmonics that
130
contained information about the states was more than the number of states. In
other words, different sensitivity functions brought out different information
from the electric image about the states of the vehicle. This resulted in a
skinny C matrix on arranging the harmonics in the standard output form
v = Cx. Thus the system of WFI equations became overdetermined due to
the presence of more WFI outputs than the number of states involved. This
is particularly helpful in optimal state estimation since the solution to the
system of equations exists only if the above criteria is satisfied.
5. By inverting the set of linearized WFI outputs, the relative states of the system
were computed. To minimize the error between the actual and relative states,
a linear least squares estimation scheme was used. The end result of this
process was a set of state extraction functions. Inner products between an
electric image and the state extraction functions yielded the relative states in
one step. This is a very desirable quality for hardware implementation.
6. The state extraction functions for y and ? states were computed to be sin(?)
and sin(2?) respectively. The inner product between an electric image and
sin(?) yielded a positive value if the peak of the electric image existed between
0??180? of the sensor. Such an electric image was formed if the sensor was
close to an object on its right. A negative value indicated an object on the
left side and a peak in the 180??360? of the sensor. Thus this state extraction
function was helpful in estimating the relative lateral offset from the centerline
of a tunnel. On the other hand a zero value for the inner product between the
131
electric image and sin(2?) function indicated a motion parallel to the tunnel
walls. Any deviation from zero indicated a direction of travel along a direction
in one of the four quadrants. Thus this function computes the relative angular
offset of the vehicle with respect to the centerline of the tunnel.
7. It was shown using elementary simulations that different behaviors like cen-
tering (in a tunnel) and wall following (in a circular arena) could be achieved
by selecting the appropriate steady state while designing the state extraction
functions.
8. It can also be noticed that this entire procedure did not require computing
the actual kinematic states or the distances of the objects. Nor did it need
any prior knowledge of the environment. These qualities made the algorithm
computationally efficient and easily integrable with hardware. It was possible
to pre-compile and hard-code the state extraction functions in the controller
due to their static nature, thus saving a lot of on-board computational effort.
Chapter 4: Quasistatic Simulations
9. FEM based quasistatic simulations were used to simulate environments con-
sisting non-conductive as well as complex shaped objects. In addition the
electric field was made oscillatory. Quasistatic approximation for an oscilla-
tory electric field enabled the use of the methods and state extraction functions
derived for electrostatic electric images.
10. The state extraction functions were designed for a straight tunnel but they
132
were found to work for other environments as well. The reasoning behind
this was that the algorithm viewed every environment as a local tunnel and
computed relative states accordingly. Thus as long as there are objects on
either side of the sensor, this control strategy should work.
11. The scope of the state extraction functions is limited in that they apply to
environments comprised of conductive objects. Hence modifications were sug-
gested to make the same state extraction functions work for non-conductive
objects as well. The obvious implication of this was that the control strategy
could now be used for a larger variety of environments.
12. Normally FEM simulations are computationally heavy and time consuming.
However, the background operation and localized refining of meshes adopted
in this study made the process less arduous and quick. It was assumed that
there were no noise sources present in the environments. This assumption
although unrealistic, helped develop a better understanding of the process. In
addition, this criteria also helped in tuning the controller gains. Controller
gains designed this way provided a starting value of gains for all other simu-
lations and even for experiments. In theory, fine adjustments to these gains
should give the actual gains necessary in a particular situation.
13. The simulations suggested that the algorithm is quite robust to initial condi-
tions and nature of objects perturbing the electric field of the sensor.
14. The controller was also found to be robust to slight changes in the corridor
133
width.
Chapter 5: Experimentation
15. The electro-sensor was designed to measure the electric field and perturbations
to it in the plane of its disc. Rapid prototyped body of the sensor ensured
perfect alignment of the electro-receptors in this plane. The source electrode
was located on the bottom of the disc at its center. This location was favorable
since a symmetric electric field could be generated in all direction. The gantry
was designed and constructed to move the sensor at rapid rates and through
large distances while providing a positioning accuracy of 0.001 inch (0.025
mm).
16. For mitigating noise in the readings, digitally filtered data was further fitted
with a sine wave in a least squares fashion. Performing these operations was
necessary to minimize the effect of noise on the RMS value computed off of
the captured data.
17. EMI noise induced due to the stepper motors was considered to be an un-
desirable influence on electric images and was extremely difficult to get rid
of using traditional filtering. Hence, the stepper motors were stopped before
recording the electric images and started back again before the issuing of the
command to move the gantry. This introduced a start and stop motion to
the experiments however. Multithreading was used to perform some of the
tasks in parallel. This was helpful in keeping the idle time of the gantry to a
134
minimum during experiments.
18. The experiments showed that the controller was robust to initial conditions,
noise and environment structure.
19. Finally, the entire WFI routine and the control strategy was confirmed to be
computationally very efficient and suitable for implementing on a real robot.
6.1 Future Work
The study has offered an evaluative perspective on autonomous navigation using
electrolocation. In future, certain improvements can be made to the current state
of this project. Additionally, the study encountered a few limitations, which also
need to be considered in the future work.
The scope of the project was limited to planar electric fields and computations
thereof. With 3D enhancements, to the sensor and the algorithm, it might become
feasible to mount the sensor on a 6 DOf platform. The sensor itself can be modified in
order to make compact and modular. With improvement in the sensor, the algorithm
and the controller would require improvements as well. Another limitation of this
study was that a random arrangement of objects was not possible. Finally, the
gantry robot performed exceptionally in experiments and furnished some very good
results. However, due to EMI issues, it was required to start and stop the motors
while recording electric images, thus sacrificing continuous motion of the vehicle.
This issue can be addressed in future as well.
Accordingly, the future work is divided into four categories. These are sensor
135
design, algorithm, environments and hardware.
6.1.1 Sensor Design
The sensor can be modified to measure electric images due to objects in entire 3D
space and not just in the plane of the sensor?s disc. For this purpose, a sensor
consisting of three orthogonal discs can be constructed. However, a better option
would be to construct a sphere with electro-receptors spread all over. Full spher-
ical measurements of electric images provide redundancy and hence better quality
electric images.
Another improvement in the sensor can be in the form of smaller and custom
made electro-receptors. In the current version, EKG sensors were used as electro-
receptors. However these receptors are large and bulky. Instead, tiny MEMS type
sensors can be custom-made and used. This can facilitate construction of small
and compact sensors. Further in future, the spherical sensor can be modified into a
fish shaped sensor by performing conformal mapping. Such a sensor will promote a
higher degree of bio-mimicking.
6.1.2 Algorithm
With improvements in sensor, the algorithm will be required to process 3D electric
images of larger array of receptors. Another implication of this upgrade is that
2D sensitivity functions could no longer be used to perform WFI processing of the
3D electric images. Fourier harmonics were used in this study, whereas spherical
136
harmonics can be one of the candidate sets of the sensitivity functions for processing
3D electric images. For a fish shaped sensor, the algorithm will need to include
fish kinematics instead of unicycle kinematics. As another enhancement to the
algorithm, a dedicated robust controller, like a H? loop shaping controller can be
used to mitigate noise and disturbance introduced in the system through various
sources.
6.1.3 Environments
It was observed in the experiments in cluttered corridors that a conductive and a
non-conductive object could not be placed next to one another as this arrangement
becomes almost invisible to the algorithm. So these type of environments were never
used. The algorithm can be developed to test in such environments which were not
considered in this study. The complete knowledge of the reasons for corruption of
electric images in such cases can be useful in reverse processing such images to arrive
at the original images.
6.1.4 Hardware
The current experimental setup was shown to perform autonomous navigation in
different environments. However, it was required to be stopped while recording
electric images and started back again for motion afterwards. This was done to
prevent EMI induced noise entering the captured readings. One way to solve this
issue is by designing better magnetic field shields around the motors. Another
137
approach that is often suggested against such issues is to completely redesign the
drive system.
With the development of a fish shaped sensor, the gantry would eventually
become unnecessary as a propulsion system could be mounted with the vehicle
carrying the sensor. In that case, shielding will be required for the motors driving
this propulsion system.
The demonstration of obstacle avoidance in simulation and experiments proves
that spatial sensitivity function based methodology can be extended for extracting
relative states from electric images. In fact the major contribution of this work is
the development of the procedure to extract relative proximity from electric im-
ages. Bio-inspired systems are a relatively new area in the robotics field. This
work is significant in that bio-mimicking was achieved while utilizing WFI which
is a signal processing and control algorithm that was modeled by studying insects.
Hence, this algorithms are highly efficient with regards to implementation. WFI
processing has a marked advantage that a static feedback controller is used instead
of dynamic controllers [31, 35]. The inner product operation is comparatively simple
and can be used to compute the navigational parameters from an electric image in
a single step. The property of conservation of computational resources makes this
WFI based algorithm highly efficient and greatly simplifies hardware and software
implementation.
138
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