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Control Configured Design for Smooth,
Highly-Maneuverable, Underwater Vehicles
F
JUL 0 6 2015
LIBRARIES
by
Anirban Mazumdar
B.S., Mechanical Engineering, Massachusetts Institute of Technology, 2007
M.S. Mechanical Engineering, Massachusetts Institute of Technology, 2009
Submitted to the Department of Mechanical Engineering in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2013
@ Massachusetts Institute of Technology 2013. All rights reserved.
Author
.................................
Signature redacted
Department of Mechalical Engineering
May 22, 2013
Certified by....................S
redacted
Signature
igr
rd..........
H. Harry Asada
Ford Professor of Mechanical Engineering
Thesis Supervisor
Signature redacted
A ccepted by ........................................ .,,...................................
David E. Hardt
Chairman, Department Committee on Graduate Students
Department of Mechanical Engineering
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Control Configured Design for Smooth,
Highly-Maneuverable, Underwater Vehicles
by
Anirban Mazumdar
Submitted to the Department of Mechanical Engineering
on May 22, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
We describe the development of a new type of robotic underwater vehicle designed
specifically for the inspection of critical infrastructures such as boiling water reactor
nuclear power plants. These applications require vehicles that can access confined
areas, maneuver precisely, and move easily in several directions. In addition, external
appendages such as fins or propellers should be avoided in order to reduce the risk of
damage through collisions.
We propose a smooth, spheroid shaped vehicle that uses an internal propulsion
system to generate and direct water-jets for propulsion and maneuvering. Drawing inspiration from aeronautics and ocean engineering, we treat this system as a
control-configured vehicle (CCV) and design the system specifically for superior control performance. Like many modern CCV aircraft, our robot is designed to be open
loop unstable in order to avoid bulky external stabilizers. We refer to this new type
vehicle as a Control Configured Spheroidal Vehicle (CCSV).
An integrated pump-valve maneuvering system is developed by combining powerful centrifugal pumps with compact Coanda-effect valves. This system is used to
design and construct a compact, multi-degree-of-freedom (DOF) prototype vehicle.
To achieve precision orientation control, high speed valve switching is exploited using
a unique Pulse Width Modulation (PWM) control scheme. Dead zones and other
complex nonlinear dynamics of traditional propeller thrusters and water jet pumps
are avoided with use of integrated pump-valve control. Three simple control algorithms for coordinating valve switching and pump output are presented and are
verified through experiments.
Planar control is complicated by the presence of hydrodynamic instability. A dynamic control system that augments stability and achieves high maneuverability is
outlined and implemented. A nonlinear hydrodynamic model is formulated, and its
linearized dynamics are analyzed to attain insights into how physical design parameters, such as jet direction and body shape, influence controllability and stability. The
integrated design method is implemented and shown to achieve high maneuverability
and stability.
Finally, this thesis concludes with a discussion on broader CCSV design ap-
3
proaches. The vehicle open loop dynamics are studied and a plant zero is shown
to significantly influence closed loop performance. Jet angle and vehicle shape are
explored through the lens of optimizing this plant zero location, and design recommendations are presented for both ideal and practical situations. These lessons can
be used to design new CCSV systems for a variety of scales and applications.
Thesis Supervisor: H. Harry Asada
Title: Ford Professor of Mechanical Engineering
4
Acknowledgments
I would like to start by thanking my adviser, Professor Harry Asada, for all his
guidance over the past six years. I still remember how excited I was to join this lab
as a fresh graduate student, and I look back on those days and marvel at how much
I have learned and grown as a researcher and as a person. Professor Asada pushed
me to take on new and exciting challenges, and in doing so allowed me to learn both
creativity and confidence. To have an adviser who helps you learn and develop every
day is truly a privilege.
I also must thank my thesis committee members. Professor Triantafyllou's classes
introduced me to Ocean Engineering and underwater vehicle design, and I have been
hooked ever since. His immense knowledge on vehicle design and hydrodynamics was
an invaluable resource for me throughout this process, and his feedback was critical to
several key achievements on this project. Professor Youcef-Toumi also deserves special
thanks for introducing me to control system design. He gave me my first research
opportunity at MIT, and my work on robotic fish for my Senior Thesis helped me
learn many valuable skills that I used for this project. Professor Youcef-Toumi was
never too busy to help me with my work, and his knowledge of design and controls
was very helpful to both this project and to my growth as an engineer.
I gratefully acknowledge the Electric Power Research Institute (EPRI) as well
as the National Science Foundation Graduate Research Program for their generous
support of this work.
John Lindberg and Greg Selby at EPRI were valuable and
knowledgeable partners, and I appreciate their practical insights and expertise on
nuclear power plant inspection in particular.
Similarly, this project as a whole was a collaborative effort. Meagan Roth was
my first senior thesis student, and her work on initial prototypes, circuit design and
software set the framework for much of this project.
Martin Lozano was my star
2.12 student who I managed to convince to join our laboratory.
His work on the
electronics and "brain board" as he calls it was pivotal to miniaturizing the robot
and achieving wireless control.
We still use the circuit boards he designed and I
5
always enjoy showing them to envious graduate students. Wyatt Ubellacker recently
joined our group, but has already made real contributions to communications, and
robust design. Finally, I must thank my good friend Aaron Fittery for his immense
contributions over the past 2 years. Aaron, on his own initiative, learned CFD and
used it to miniaturize and design new valves. It has been a pleasure to work alongside
Aaron and watch him develop into a confident researcher.
I owe thanks to my my fellow lab members both past, and present.
Ian Rust
was my collaborator on this project and deserves special thanks. Similarly, I must
specifically thank lab alumni Levi Wood, Manas Menon, Tom Secord, Geoff Karasic,
Patrick Barragan, and Shinichiro Tsukahara for their advice and assistance over the
past years.
One of the best things about MIT is the people.
My basketball friends Kimi
Shirasaki, Daniel Kraemer, Yasu Shirasaki, Victor Wang, Kevin Sung, Cody Fleming,
Conrad Miller, and Miguel Saez helped keep me humble and provided something to
look forward to at the end of the day. Dyan Melvin was perhaps my closest friend
during graduate school. While I appreciated his visits to lab and encouragement
during quals, I am most grateful for him letting me beat him at one on one every
once in awhile on the Rockwell Courts. Lastly I must thank Ellen Chen for her kind
support and encouragement for the past couple years. She is my confidante and my
number one supporter, and I truly cherish the time we have spent together.
I should also like to thank my "extended family" back in Irvine. Michelle and Irv
Walot have always been there for me, and Dan and Jeanne Stokols are a constant
source of encouragement. Christine King has been my ally since my days at University
High School. Without her tough love I would not have made it to MIT.
Finally I will attempt to convey my deep gratitude to my parents. I cannot thank
them enough for all their love and for teaching me to truly value learning and higher
education. Going to school thousands of miles away from my parents has been one of
the hardest things I have ever done. But whether through late night phone calls or
impromptu cooking lessons, I feel like they have been part of this journey for every
step of the way. I certainly know none of this would have been possible without them.
6
Contents
1
2
3
Introduction
23
1.1
Underwater Robots for Cluttered Environments
. . . . . . . . . . . .
23
1.2
Nuclear Power Case Study . . . . . . . . . . . . . . . . . . . . . . . .
24
1.3
Functional Requirements . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.3.1
Motions in Multiple Directions . . . . . . . . . . . . . . . . . .
26
1.3.2
Bidirectional Motions . . . . . . . . . . . . . . . . . . . . . . .
27
1.3.3
Maneuverability at a Range of Speeds
. . . . . . . . . . . . .
27
1.3.4
Robust to Collisions
. . . . . . . . . . . . . . . . . . . . . . .
27
1.4
Design Concept: Control Configured Spheroidal Vehicle . . . . . . . .
28
1.5
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Appendage Free Propulsion and Maneuvering
33
2.1
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.2
Current Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
Pump-Valve Concept . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4
Modeling Pump-Valve Static Performance
. . . . . . . . . . . . . . .
39
2.4.1
Static Force Performance . . . . . . . . . . . . . . . . . . . . .
39
2.4.2
Switching Length . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.4.3
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
. . . . . . . . . . . . . . . . . . .
46
2.5
Pump Model
2.6
Pump-Valve Dynamic Performance
Implementation and Experimental Characterization of Pump-Valve
49
Systems
7
3.1
Pump Selection ........
50
3.2
Valve Design . . . . . . . . .
51
3.3
Implementation . . . . . . .
52
3.3.1
Valve Construction .
52
3.3.2
Switching Design . .
52
Experimental Evaluation . .
53
3.4.1
Static Performance .
54
3.4.2
Dynamic Performance
54
.
.
.
.
.
Multi-DOF Propulsion . . . . . . . .
. . . . . . . . .
57
4.2
Pump Valve Design . . . . . . . . . .
. . . . . . . . .
58
4.2.1
Pump Design . . . . . . . . .
. . . . . . . . .
59
4.2.2
Reduced Actuation Design . .
. . . . . . . . .
61
. . . . . . . . . . .
. . . . . . . . .
64
4.3.1
Mechanical Design . . . . . .
. . . . . . . . .
64
4.3.2
Electrical Design
. . . . . . . . .
65
4.3.3
Waterproofing Components
.
. . . . . . . . .
68
Full Robot Summary . . . . . . . . .
. . . . . . . . .
69
.
.
.
.
.
.
.
.
.
.
.
CCSV Prototype
.
.
.
. . . . . . .
.
.
4.4
.
4.1
.
Control Configured Underwater Vehicle Design and Implementation 57
4.3
Dual Pump-Valve Control
73
Nonlinear Pump and Valve Dynamics . . . .
73
5.2
Combined Pump-Valve Control
. . . . . . .
74
5.2.1
Summary of Pulse Width Modulation
74
5.2.2
Pump Speed Control . . . . . . . . .
76
5.2.3
Valve PWM Control . . . . . . . . .
76
5.2.4
Hybrid Control . . . . . . . . . . . .
77
Heading Control Case Study . . . . . . . . .
77
5.3.1
Heading Control Dynamics . . . . . .
77
5.3.2
Response to PWM Signals . . . . . .
79
.
.
.
.
.
.
.
5.3
.
5.1
.
5
.
3.4
Sizing Exit Nozzle
.
.
3.1.1
4
49
8
5.4
5.5
5.6
5.7
6
5.3.3
Modeling PWM Induced Vehicle Oscillations .8
80
5.3.4
Approximate Closed Form Expression . . . . . . . . . . . . . .
81
Controller Designs
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.4.1
Selecting PWM Amplitude . . . . . . . . . . . . . . . . . . . .
85
5.4.2
Selecting PWM Frquency
. . . . . . . . . . . . . . . . . . . .
85
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.5.1
Pump Speed Control . . . . . . . . . . . . . . . . . . . . . . .
88
5.5.2
Valve PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.5.3
Hybrid Control . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.6.1
Pump Speed Control . . . . . . . . . . . . . . . . . . . . . . .
92
5.6.2
Valve PWM Control
. . . . . . . . . . . . . . . . . . . . . . .
92
5.6.3
Hybrid Control . . . . . . . . . . . . . . . . . . . . . . . . . .
94
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
CCSV Directional Stability
97
6.1
Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.1.1
Munk Moment
. . . . . . . . . . . . . . . . . . . . . . . . . .
99
6.1.2
Countering the Munk Moment . . . . . . . . . . . . . . . . . .
100
6.1.3
Closed Loop Stabilization of Yaw . . . . . . . . . . . . . . . .
102
Linearized Planar Dynamics . . . . . . . . . . . . . . . . . . . . . . .
103
6.2.1
Linearized Equations of Motion . . . . . . . . . . . . . . . . .
103
6.2.2
State Space Representation
. . . . . . . . . . . . . . . . . . .
105
6.2.3
Linearized System Characteristics . . . . . . . . . . . . . . . .
106
6.2
6.3
Feedback Controller Design
. . . . . . . . . . . . . . . . . . . . . . .
107
6.3.1
Open Loop Performance
. . . . . . . . . . . . . . . . . . . . .
108
6.3.2
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . .
109
6.4
Comparison with Conventional Approaches . . . . . . . . . . . . . . .
110
6.5
Implementation
110
6.5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . .111
9
6.6
7
Vehicle Simulations . . . . . . . . . . . . . . . . . . . . . . . .
111
6.5.3
Vehicle Experiments
. . . . . . . . . . . . . . . . . . . . . . .
114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
Sum m ary
CCSV-RAD Prototype Performance
117
7.1
117
Vehicle Performance .......
...........................
7.1.1
Surge Performance
........................
118
7.1.2
Sway Performance
........................
118
7.1.3
Stationary Turning . . . . . . . . . . . . . . . . . . . . . . . .
120
7.1.4
Turning at Speed . . . . . . . . . . . . . . . . . . . . . . . . .
122
7.1.5
Heave Translations . . . . . . . . . . . . . . . . . . . . . . . .
123
7.1.6
Pitching Motions . . . . . . . . . . . . . . . . . . . . . . . . .
125
7.2
Switching Between Motion Families . . . . . . . . . . . . . . . . . . .
125
7.3
Exploring Passive Yaw Stability . . . . . . . . . . . . . . . . . . . . .
125
7.3.1
Hydrofoil Modeling . . . . . . . . . . . . . . . . . . . . . . . .
126
7.3.2
Sample Tail Fin . . . . . . . . . . . . . . . . . . . . . . . . . .
127
7.3.3
Experimental Evaluation . . . . . . . . . . . . . . . . . . . . .
129
7.4
8
6.5.2
Sum mary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
General Design Approaches
133
8.1
General Properties of the CCSV Design . . . . . . . . . . . . . . . . .
133
8.2
Controllability of the CCSV Design . . . . . . . . . . . . . . . . . . .
134
8.3
Designing for Control Performance
136
8.4
. . . . . . . . . . . . . . . . . . .
8.3.1
Determinant of Controllability Matrix
. . . . . . . . . . . . .
136
8.3.2
Vehicle Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
Using Zero Location to Inform Vehicle Design
. . . . . . . . . . . . .
138
8.4.1
Jet Angle
8.4.2
Vehicle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.5
Examining Vehicle Performance . . . . . . . . . . . . . . . . . . . . .
141
8.6
Additional Performance Considerations . . . . . . . . . . . . . . . . .
142
8.6.1
Low Speed Turning . . . . . . . . . . . . . . . . . . . . . . . . 143
10
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
147
Fundamental Control Performance Limitations
149
. . . . . . . . . . . .
149
.
. . . . . . . . . . . .
150
. . . . . . . . . . . .
151
. . . . . . . . . . . .
153
Bode's Integrals
9.2
Sensitivity Analysis
.
9.3
CCSV Performance
.
9.4
A General Tool . . . . .
.
.
.
.
9.1
.
9
Summ ary
144
.
8.7
Practical Considerations . . . . . .1
.
8.6.2
155
10 Additional Vehicle Prototypes
155
10.1.1 Jet Design . . . . . . . .
155
10.1.2 Vehicle Prototype .....
157
.
.
. . . . . . . . .
10.1 4-Pump CCSV
159
10.2.1 Vehicle Design . . . . . .
159
. .
159
. . . . . . . . . . . .
161
10.3 Summary
.
10.2.2 Vehicle Performance
.
.
.
10.2 Propeller Based Design . . . . .
163
11.1 Summary of Thesis Contributions
. . . . . . . . .
163
11.2 Future Work . . . . . . . . . . .
.
. . . . . . . . .
165
.
11 Conclusions
. . . . . . . . .
165
11.2.2 Potential Research Directions
11
. . . . . . . . . . . . . . . . .
.
11.2.1 Practical Considerations
166
12
List of Figures
1-1
A diagram illustrating a GE Boiling Water Reactor system . . . . . .
25
1-2
A simple diagram for illustrating desired vehicle motions. . . . . . . .
26
1-3
A rendering of the CCSV concept. . . . . . . . . . . . . . . . . . . . .
28
2-1
An illustration of the coordinate frame convention.
. . . . . . . . . .
34
2-2
An illustration of the bistable fluidic amplifier concept. . . . . . . . .
36
2-3
A CFD illustration of the bistable fluid amplifier fluid dynamics. . . .
37
2-4
An illustration showing the full pump-valve concept.
38
2-5
A figure outlining the relevant geometric parameters for pump-valve
modeling.
2-6
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A simulation plot illustrating the dependence of output force on the
exit area for the combined pump-valve system. . . . . . . . . . . . . .
2-7
39
41
CFD simulations showing a "good" valve design that avoids spillover
(a) and a poor design that suffers from spillover (b). . . . . . . . . . .
42
2-8
A plot comparing the analytical model predictions of L, with CFD. .
43
2-9
A plot illustrating how the switching length, L, does not vary with a
range of values for the input flow rate,
Q. . . . . . . . . . . . . . . .
2-10 A visual illustration of pump voltage control.
44
The voltage control
model is shown (a) as well as the use of voltage level to modulate
output force (b).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2-11 A visual illustration of pump speed control. Using speed control to
modulate force is shown in (a) as well as the speed-force input output
relationship (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
45
2-12 CFD simulation illustrating valve switching dynamics for various force
levels.
3-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A photograph of the TCS M400 micropump that we will use throughout this thesis.
3-2
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
A figure illustrating how to use the exit nozzle area to optimize the
output force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3-3
Diagrams showing the dimensions of the final valve design.
52
3-4
Photograph of a full valve prototype with the switching mechanism (a)
. . . . . .
and a plot showing the high-low switching scheme. . . . . . . . . . . .
53
3-5
Photograph of a fully assembled pump-valve system.
54
3-6
Pump voltage control over pump-valve system. Both directions of the
. . . . . . . . .
valve are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7
Experimental data illustrating the switching dynamics of the valve.
The pump is running at full voltage.
3-8
55
. . . . . . . . . . . . . . . . . .
55
Experimental data illustrating the switching dynamics of the valve for
several pump voltages.
. . . . . . . . . . . . . . . . . . . . . . . . . .
56
4-1
A diagram illustrating the jet arrangement for the CCSV design. . . .
58
4-2
A diagram from The Encyclopedia Britannica illustrating a common
type of centrifugal pump design. . . . . . . . . . . . . . . . . . . . . .
59
4-3
A close up photograph showing the symmetric impeller. . . . . . . . .
60
4-4
A CFD simulation illustrating an in-line dual-output pump design.
61
4-5
A CFD simulation illustrating the 90 degree pump. Note how when the
.
impeller direction is reversed, the jet switches exits completely without
any back-flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4-6
Experimental data comparing 3 pump design methodologies. . . . . .
62
4-7
Photograph of a BAU prototype. Two fluidic valves are combined with
an orthogonal dual output port pump to generate forces in 4 directions. 62
4-8
A rendering showing the pump-valve system for the CCSV-RAD prototype.........
....................................
14
64
4-9
Photographs of the CCSV-RAD prototype. A fully assembled prototype is shown (a) along with a closer view of the maneuvering system
is shown (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4-10 Photographs of the outside of the CCSV-RAD prototype. . . . . . . .
66
.
.
67
4-12 Eagle software schematic layout for the robot brain board PCB. .
.
68
4-13 Images showing the brain board layout and its fully assembled form.
69
4-11 An overview of the electrical components for the CCSV system.
4-14 A photograph showing the inner components of the robot. Note the
metal cap, this is used for sealing the watertight electronics chamber.
70
4-15 A photograph showing the CCSV-RAD prototype performing a diving
maneuver.
5-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
A diagram illustrating the principal of combined pump-valve control
with a single input from the controller and and a single force output.
74
5-2
A diagram illustrating the components of a PWM Signal. . . . . . . .
75
5-3
Dual pump-valve control algorithms: (a) pump speed control, (b) valve
PWM control, and (c) hybrid control.
Each sub-figure includes i)
PWM Duty Cycle, ii) pump output, and iii) resultant force, all in
relation to the controller output, u. . . . . . . . . . . . . . . . . . . .
5-4
Illustration of vehicle heading angle (a) and a diagram showing the
capability of the vehicle to turn in place using dual jets (b).
5-5
A plot of simulated yaw oscillations.
labeled.
5-6
75
. . . . .
78
The 4 solution cases are each
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
A comparison of the full nonlinear solution with the closed form nonlinear approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5-7
A block diagram of the heading controller. . . . . . . . . . . . . . . .
85
5-8
A plot illustrating how a design curve can be used to tune the PWM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
Illustration of the simplified pump-valve static model. . . . . . . . . .
87
5-10 Simulated response with and without a dead zone. . . . . . . . . . . .
88
frequency.
5-9
15
5-11 Simulated response with PID pump speed control with deadzone. The
integral action slowly eliminates steady state error (a). Note the presence of chattering in the valve signal (b). . . . . . . . . . . . . . . . .
89
5-12 Simulated response with PD valve PWM control. The improvement in
the overshoot and settling time is shown in the angle trajectory (a) as
well as the force output of the pump-valve system(b). . . . . . . . . .
90
5-13 Valve PWM using 2 different PWM frequencies. Note how the oscillations for fPWM = 1.5 are significantly higher than those for fPwM
=
2.5H z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5-14 Simulated response comparing valve PWM with hybrid control. Note
the improvement in the rise time (a) as well as the amplitude adjustment in the force output (b) . . . . . . . . . . . . . . . . . . . . . . .
92
5-15 A set of frames from the hybrid control experiment. The dashed black
lines are a reference to illustrate the angle tracking. The dot is used to
indicate the front of the robot, and the arrows indicate the direction
of the vehicle angular velocity. . . . . . . . . . . . . . . . . . . . . . .
93
5-16 Experimental data from the CCSV-RAD prototype showing the performance improvements from using valve PWM control. . . . . . . . .
93
5-17 Zoomed in view of the valve PWM control experimental results. . . .
94
5-18 Experimental data from the CCSV-RAD prototype showing how hybrid control can be used to improve the transient response. . . . . . .
95
6-1
An illustration of the body fixed coordinate frame.
98
6-2
An illustration of the Munk moment and how the stagnation points
create a turning motion on streamlined shapes.
6-3
. . . . . . . . . .
. . . . . . . . . . . .
100
A diagram illustrating how fins can provide both passive stabilization
or destabilization based on the direction of the vehicle.
. . . . . . . .
101
6-4
An illustration of the key parameters for the planar vehicle model. . .
104
6-5
A pole-zero plot (a) and root locus plot (b) for the SISO vehicle maneuvering system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
108
6-6
A schematic diagram showing the role of jet angle in vehicle control
perform ance.
6-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A pole zero plot for the closed loop SISO system.
The closed loop
poles (pcL,1, PcL,2, PcL,3) are shown as are the zeros (zc, zi) and the
open loop poles (pi, P2, P3). Note how the system is stabilized using
PD control.
6-8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
Simulated response for the open loop performance of the robot vehicle.
The trajectory in fixed global coordinates,(XI, Y1 ), is provided in (a)
and the yaw angle is provided in (b).
6-9
. . . . . . . . . . . . . . . . . .
113
Simulated results illustrating the poor control performance when the
jet angle, -y, is set to zero.
. . . . . . . . . . . . . . . . . . . . . . . .
6-10 Simulations of the closed loop system response.
113
The angle (a) and
velocity (b) results of both the linearized model and the full nonlinear
model are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
6-11 Experimental video data showing the vehicle trajectories for both open
loop control (a) and PD closed loop stabilization (b).
. . . . . . . . .
115
6-12 Experimental data showing the vehicle angle trajectory for both a
straight test (a) and a disturbance rejection test (b).
7-1
. . . . . . . . .
Video (a) and numerical (b) data illustrating the forward and back
performance of the CCSV-RAD prototype. . . . . . . . . . . . . . . .
7-2
. . . . . . . . . . . . . . . . .
121
Video (a) and angle tracking (b) data illustrating the turn-at-speed
capability of the CCSV-RAD prototype.
7-6
121
Video (a) and numerical (b) data illustrating the turn-in-place capa-
bility of the CCSV-RAD prototype. . . . . . . . . . . . . . . . . . . .
7-5
120
Angle trajectory data for pure sway translation (a) as well as angle
adjustments while moving sideways (b). . . . . . . . . . . . . . . . . .
7-4
119
Video (a) and numerical (b) data illustrating the sway translation capability of the CCSV-RAD prototype.
7-3
116
. . . . . . . . . . . . . . . .
122
Video (a) and velocity (b) data illustrating the turn-at-speed capability
of the CCSV-RAD prototype.
. . . . . . . . . . . . . . . . . . . . . .
17
123
7-7
Video (a) and velocity (b) data illustrating the diving capability of the
CCSV-RAD prototype. . . . . . . . . . . . . . . . . . . . . . . . . . .
124
7-8
Pump reversal dynamics. . . . . . . . . . . . . . . . . . . . . . . . . .
126
7-9
A simple diagram illustrating the hydrodynamic forces on a hydrofoil.
127
7-10 Lift and drag coefficients for a NACA 0015 airfoil. . . . . . . . . . . .
128
7-11 Photographs comparing the OL stable prototype (a) and the OL unstable prototype (b).
. . . . . . . . . . . . . . . . . . . . . . . . . . .
129
7-12 Video data illustrating the stabilizing effect of the tail fin. The vehicle
moves relatively straight without any feedback control.
. . . . . . . .
130
7-13 Video (a) and yaw rate data (b) data comparing the stationary turning
performance of the two vehicle designs. . . . . . . . . . . . . . . . . .
131
7-14 Video (a) and angle tracking (b) data comparing the turning at speed
performance of the two vehicle designs. . . . . . . . . . . . . . . . . .
132
8-1
A diagram illustrating the jet arrangement for the CCSV design. . . .
134
8-2
Simulations illustrating the poor control performance of a spherical
shape. ........
...................................
136
8-3
A diagram illustrating the key design parameters, a, b, and yj. . . . .
8-4
A plot illustrating the dependence of z, on the aspect ratio for m
=
0.9kg. Note that for these aspect ratios, z1 is actually negative.
.
8-5
138
140
Simulation data illustrating the control performance for various aspect
ratios vary from 1.01 to 4. . . . . . . . . . . . . . . . . . . . . . . . .
142
8-6
Drag coefficients and forces for various aspect ratios.
143
8-7
A figure showing all 3 performance metrics and the role of aspect ratio. 144
8-8
A figure showing all 3 performance metrics. In this case a minimum
. . . . . . . . .
radius, b, is im posed. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-9
145
A figure showing all 3 performance metrics. In this case a minimum
radius, b, is imposed, and aspect rations above 1.3 are penalized. . . .
146
8-10 Simulated data showing the XY trajectory (a) and the sway response
(b) for various aspect ratios when a minimum radius is imposed. . . .
18
147
9-1
A block diagram of the SISO yaw control system.
9-2
A prototype sensitivity function that should provide good control perform ance.
9-3
9-4
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
150
151
A plot showing how the Bode Integral analysis can be used to find the
vehicle speed limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
A plot showing the values for P for various aspect ratios. . . . . . . .
154
10-1 A rendering illustrating the maneuvering system for the 4-Pump CCSV. 156
10-2 A photograph of the maneuvering system for the 4-Pump CCSV prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-3 A photograph of the outside of the 4-Pump CCSV prototype.
157
The
lights on the camera are clearly visible. . . . . . . . . . . . . . . . . .
158
10-4 An image taken from the onboard recording camera during a vertical
weld inspection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
10-5 A rendering illustrating the maneuvering and propulsion system for
the propeller based design. . . . . . . . . . . . . . . . . . . . . . . . .
160
10-6 Photographs illustrating the maneuvering system as well as the outer
shape of the propeller based robot prototype.
. . . . . . . . . . . . .
160
10-7 Heading angle data for both the straight test (a) and the disturbance
rejection test (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
10-8 Heading angle data for both turning at speed (a) and stationary turning
(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
162
20
List of Tables
. . . . . . . . . . . . . . . . . . . . . .
50
3.1
Summary of the M400 Pump.
4.1
Summary of Maneuvering Primitives. .......................
58
4.2
Summary of Pump and Valve Combinations for Vehicle Motions. . . .
64
5.1
Summary of Solution Cases
. . . . . . . . . . . . . . . . . . . . . . .
80
5.2
Summary of Region Boundary Conditions
5.3
Summary of CCSV-RAD prototype yaw dynamics properties. ....
6.1
Summary of Physical Properties for CCSV-RAD Prototype.
. . . . .
111
7.1
Summary of Vehicle Performance. . . . . . . . . . . . . . . . . . . . .
118
7.2
Summary of Physical for Sample Tail Fin.
. . . . . . . . . . . . . . .
129
21
. . . . . . . . . . . . . . .
81
85
22
Chapter 1
Introduction
1.1
Underwater Robots for Cluttered Environments
Modern societies have become increasingly dependent on water based infrastructures whether they are power systems, ports, piping systems or water treatment
plants. As these systems age, they require repairs and inspections with increasing
frequency.
Since many of these systems are essential to public safety, there exist
strict protocols for inspection. For example, for Boiling Water Reactor (BWR) nuclear powerplants, there exist very strict and specific visual inspection protocols. For
many of these applications, it is difficult, costly, dangerous, and sometimes even impossible to send humans to inspect and assess. In addition, for many systems, inspections cannot be performed with the system running and must instead be performed
during a shutdown. These shutdowns are not only inconvenient and disruptive, they
can be extremely costly economically. As a result, the inspection of cluttered aquatic
environments is a rapidly growing area of research and technical innovation.
Already underwater robots are being developed and deployed for the inspection
of ports [1], [2], dams [3], water piping systems [4], shipwrecks [5], and nuclear power
plants [6], [7].
Some key challenges for these types of robots relate to accessing
and inspecting complex environments where small size and high maneuverability are
required. In addition, robustness is an important characteristic for vehicles operating
in constrained environments where collisions are inevitable.
For example, nuclear
power plants are subject to foreign material exclusion (FME) rules that stipulate that
23
no outside materials can be left within the plant after an inspection. This means that
the robot must be able to survive collisions without breaking and without external
components falling off.
1.2
Nuclear Power Case Study
The challenging nature of underwater infrastructure inspection is perhaps best
illustrated with boiling water nuclear plants. Boiling water reactors serve as a good
case study example because they are complex water-filled systems, highly regulated,
and must satisfy strict inspection protocols. In addition, nuclear power plants are
clearly areas where direct inspection using human workers is something that must be
avoided.
There exist 35 boiling water reactors in the USA [8], and 84 world wide [9]. Nuclear
power is an extremely highly regulated industry, and the plants must satisfy regulations developed by the Nuclear Regulatory Commission (NRC), ASME (American
Society of Mechanical Engineers) and EPRI (Electric Power Research Institute). Visual inspections using cameras are the most common inspection methodology. Sometimes this will be combined with techniques such as ultrasound in order to provide
more detailed measurements.
As Fig. 1-1 illustrates, the inspection of the reactor environment is a very challenging problem. The system can be treated roughly as a 15m diameter, 40m deep
pool of water filled with nozzles, guides, pipes and tubes that must all be navigated
and inspected.
An illustration of the inside components of a BWR plant can be
found in Fig. 1-1, which provides a popular diagram of a General Electric (GE) reactor assembly [10]. As Fig. 1-1 shows, the environment is very complex with small
areas such as the top guide (item 12 in the figure) placing restrictions on size and access. Deploying and then finely maneuvering tools within this environment is clearly
challenging and requires robots that are relatively small and nimble as well as robust.
As mentioned previously, nuclear powerplants are shut down during inspection
(usually during the refuel cycle). This means that there is a clear economic incentive
for rapid inspections. Inspections that are slowed by sluggish or unreliable equipment
can cost power companies and inconvenience thousands of people and businesses.
24
BWR /6
REACTOR ASSEMBLY
1. VENT AND HEAD SPRAY
2.
IIFTING
STEAM DRYER
.UO
3, STEAM DRYER ASSEMBLY
4.
STEAM OUTLET
G. CORE SPRAV INLET
6. STEAM SEPARATOR ASSEMBLY
?, F.FDWATER INLET
8. FEMDWATER SPAROER
9.
LOW PRES"IRE COOLANT
INJECTION INLET
10.
CORE SPRAY
INE
11, CORE SPRAY SPARGER
12.
-
TOP GUIDE
13. JET PUMP ASSEMBLY
14. CORE
SHROUD
15.
FUELt
16.
CONTROL 1LADE
ASSEMBLIES
17. CORE PLATE
11. JET PUMPI RECIRCULATION
WATER INLET
I9. RECIRCULATION WATER
2D.
OUTLET
VESSEL SUPPORT SKIRT
21. SHIELD WALL
22. CONTROL ROD DRIVES
23. CONTROL ROD DRLVE
HYDRAULIC LIKES
24. IN-CORE FLUX MONITOR
DENEIALO
ELECTRIC
Figure 1-1: A diagram illustrating a GE Boiling Water Reactor system.
25
Therefore, it is not surprising that developing robotic systems which can enter the reactor environment, maneuver precisely, and obtain visual data are an area of growing
innovation and research.
1.3
Functional Requirements
We can use the nuclear powerplant inspection case study to help us generate
functional requirements for inspection robots. These functional requirements will be
used to examine prior work, generate design concepts, and evaluate our designs. As
illustrated below, these functional requirements are very general and therefore also
summarize the needs for a wide class of underwater inspection robots. Therefore,
robots systems that meet these requirements will likely be generally applicable to a
wide class of emerging inspection applications.
1.3.1
Motions in Multiple Directions
In order to traverse a complex environment and aim a camera or other sensors,
inspection robots should have the ability to move in 5 directions. To illustrate these
motions we will use the coordinate frame fixed to the center of mass of the vehicle
in Fig. 6-1 The motions that we require are include surge (translation alog x), sway
(translation along y), and heave (translation along z), as well as yaw (rotations about
z) and pitch (rotations about y). will also be required. The only degree of freedom
that we will not need is roll (rotation about the x axis). Roll is generally not useful
for visual inspection tasks, as turning sideways or upside down serves no obvious
purpose. This is especially true for symmetric vehicles where rolling would not help
fit into confined areas.
rx
Figure 1-2: A simple diagram for illustrating desired vehicle motions.
26
1.3.2
Bidirectional Motions
Precise motions within a constrained space require the ability to move forwards
and backwards with equal ease. While this may seem like an obvious requirement,
many underwater vehicles due to their design and propulsion system cannot simply
reverse direction. The inspection robots we aim to develop must be able to move in
both the positive and negative direction for each degree of freedom. This means that
they will able to move quickly, stop, reverse, and carefully adjust their position and
orientation in order to navigate obstacles and aim their sensors precisely and quickly.
Bidirectional capability will greatly simplify vehicle control and path planning.
1.3.3
Maneuverability at a Range of Speeds
The ability to maneuver well at many operating speeds is essential for efficient
inspections. A robot that can only maneuver at higher speeds makes precision inspection very challenging. This is especially true for tasks such as visual inspection where
the sensors sample slowly and are susceptible to blurring.
Similarly, a robot that
can only maneuver at slow speeds will not be able to move quickly between inspection locations and will not be able to react dynamically to changing conditions. The
inability to maneuver at higher speeds therefore reduces efficiency, prolongs mission
times, and leaves the robot vulnerable in uncertain environments.
1.3.4
Robust to Collisions
As discussed previously, nuclear power plant inspection requires extremely robust
robot systems. In fact, the fear of robot damage and violation of FME rules has meant
that the industry has been cautious in their adaptation of robotic technologies. Robot
systems must be extremely robust to collisions by virtue of their design. This means
that components that can potentially break and fall off during a collision must be
avoided. Conventional underwater systems have many such systems such as spinning
propellers, fins, and rudders. These types of systems that protrude from the vehicle
must be re-evaluated due to their high risk of damage.
27
1.4
Design Concept: Control Configured Spheroidal
Vehicle
Based on these functional requirements we developed a conceptual design which is
a novel and innovative to approach the challenge of infrastructure inspection. Specifically, we propose a vehicle that is completely smooth and spheroidal in shape. The
vehicle propels itself and maneuvers using water jets that can be modulated and
switched between various exit ports. The symmetric and smooth nature of the shape
allow for bi-directional motions, high maneuverability, and robustness to collisions.
The use of jets for propulsion rather than propellers means that the risk of a spinning
propeller breaking during a collision or becoming tangled is removed.
In addition,
the absence of stabilizers such as fins means that there are no components that can
snag on obstacles. We describe this approach as the "Control Configured Spheroidal
Vehicle (CCSV)" We use this title because we emulate the Control Configured Vehicle
ideas from aeronautical engineering [11]. Specifically, the vehicle is designed specifically to achieve multi-degree-of-freedom (DOF) motions and high fidelity feedback
control performance.
We are faithful to the CCV concept by thoroughly analyzing
projected control performance and using it to educate vehicle design. Finally, just like
many modern CCV type vehicles, our design is open loop unstable and uses feedback
control instead of passive stabilizers to achieve superior performance.
Figure 1-3: A rendering of the CCSV concept.
28
While this type of approach appears simple and intuitive, examples of such smooth,
multi-DOF and bidirectional robots are extremely rare in the literature. The Eyeball
ROV proposed by Rust and Asada [12], uses a creative internal moving mass and
gimbal system to achieve rotational motions, but still relies on external propellers
to generate its forward thrust. Similarly, the University of Hawaii's ODIN robot is
spherical in shape but uses several external propeller thruster to propel and maneuver
[13]. Work by Lin describes a smooth sphere propelled by jets, but only achieves 3
DOF by using modified bow thrusters [14].
The vehicle design that comes closest
to matching the performance and overall shape we desire is the BFFAUV design developed by Licht and
T
riantafyllou. This design is capable of bidirectional motions,
multiple DOF, and high maneuverability through the use of biologically inspired flapping foils [15]. However the large external flapping foils means that the vehicle does
not have a completely smooth external shape.
In our view there are two main technical challenges that make designs like this
CCSV concept difficult to realize. The first is the design of a propulsion and maneuvering system that can fit within a small, streamlined shell. The most popular
approach for underwater vehicles is to use propeller thrusters, but these have to sit
in the ambient fluid to operate properly. In addition, combining several external propellers to achieve multi-DOF propulsion results in several propellers on the outside
of the vehicle, making it less hydrodynamically efficient and more difficult to position precisely. The second major technical challenge is associated with the presence
of hydrodynamic stability. This instability is generally dealt with by adding fins at
the tail of the vehicle. However, these are not only large appendages, but they will
destabilize the vehicle if the vehicle direct is reversed, making bidirectional motions
challenging.
1.5
Thesis Overview
This work will focus on the analysis, design and evaluation of this new CCSV
concept. We discuss the development of a novel pump plus fluidic valve propulsion
system that can be built into a smooth shell and used to achieve very precise control.
We will also discuss how this system can be used as an enabling technology to design
29
small and smooth robots. We will also analyze in depth the nature of the hydrodynamic instability and propose a unique and realizable stabilizing controller. Finally
we will perform general analysis on how to design specific vehicle components and
how to determine the best vehicle shape. Lastly, a full robot prototype is constructed
and used as a test-bed for these new concepts and ideas.
Chapter 2 of this thesis focuses on the development of the unique pump plus
fluidic valve system that serves as the building block for this work.
We describe
the current state of the art in jet propulsion and examine the role our system plays
in advancing the field. We then illustrate the functionality of the device with both
analytical models and computational fluid dynamics (CFD) simulations.
Chapter 3 describes the construction of a functional pump-valve system. Experiments are used to evaluate the pump and valve performance and the results are shown
to correspond well to the models in Chapter 2.
Chapter 4 illustrates how the pump-valve systems can be used to design a robot
that is completely smooth and capable of motions in 5 directions.
A unique or-
thogonal, dual-output port pump is outlined and used to generate reduced actuation
designs. The Control Configured Underwater Vehicle (CCSV) concept is introduced,
and a fully functional prototype which uses reduced actuation design is presented.
This prototype is referred to as CCSV-RAD. Each critical component is described in
detail.
Chapter 5 discusses the use of combined pump-valve control to achieve precision
orientation control.
The pump-valve nonlinearities are addressed through the use
of two novel pump-valve control algorithms that exploit Pulse Width Modulation
(PWM) of the fluidic valves. Rigorous mathematical analysis is performed to predict vehicle oscillations, and design methods are provided. The pump-valve control
systems are evaluated using both simulations and experiments.
Chapter 6 focuses on planar vehicle control in the face of hydrodynamic instability. The hydrodynamics of the smooth vehicle are examined, and the Munk moment
is shown to result in instability. The coupled nonlinear equations are linearized about
a trim state and used to design a stabilizing control system. This linear analysis pro-
30
vides several unique insights into vehicle control and performance. Finally, a unique
stabilizing controller which uses only angle and angle rate measurements is outlined
and used to achieve impressive planar performance. The controller is implemented
on the CCSV-RAD prototype robot and is shown to stabilize the vehicle even in the
face of substantial disturbances.
Chapter 7 uses the stabilizing control system from Chapter 6 to illustrate the
unique performance capability of the CCSV prototype. Multi-DOF motions are shown
including forward and backwards translations, sideways motions, diving, and high
speed turning. This vehicle performance is also compared with a prototype that uses
a fixed tail fin to achieve stability. The actively stabilized CCSV-RAD prototype is
shown to provide substantially improved performance, especially with regard to low
and high speed turning.
Chapter 8 explores CCSV design from a more general perspective by examining
the role of jet angle and vehicle shape. Linear analysis is used to explore uncontrollable
shapes and behaviors, and control system analysis is used to explore optimal aspect
ratios. Simulations are used to validate these concepts, and ideal ranges for both jet
angle and vehicle aspect ratio are provided.
Chapter 9 analyzes the fundamental limitations of using closed loop control to
stabilize unstable underwater vehicles. Bode's Integrals and results from a seminal
paper in the field are used to explore "speed limits" for our control configured design.
These concepts are then used to examine the broader ramifications of closed loop
stabilization.
Chapter 10 develops some additional designs that combine the pump-valve systems with additional pumps or propellers in order to improve control and efficiency.
Photographs of two prototypes along with preliminary experimental results are pro-
vided.
The thesis concludes with Chapter 11 which provides final conclusions and a
summary of the key contributions.
31
32
Chapter 2
Appendage Pree Propulsion and
Maneuvering
In this chapter we describe our unique pump-valve propulsion and maneuvering system for smooth underwater vehicles. We first introduce basic terminology and nomenclature, then we provide an overview of existing technologies. We then propose a novel
system based on powerful centrifugal pumps and fluidic valves. Models for predicting
force output are outlined and a design procedure for maximizing force output while
ensuring proper valve performance is described. Finally, this chapter discusses pump
models and pump-valve dynamic modeling.
2.1
Nomenclature
Throughout this thesis we will be using terminology and nomenclature derived
from the field of Ocean Engineering.
Specifically we will use kinematics and and
dynamics based on a body centered coordinate system shown in Fig.
6-1.
This
system was developed by the Society of Naval Architects and Marine Engineers in
1952 and is prevalent in the underwater vehicle literature [16], [17].
As the figure illustrates, u, v, w represent translational velocities about the x, y,
and z axes respectively.
These motions are also described as "surge," "sway," and
"heave," respectively. Similarly, rotational velocities about the x, y, z axes are referred
to as p, q, r respectively. These motions are also known as "roll," "pitch," and "yaw."
33
p
Y
X
z
Figure 2-1: An illustration of the coordinate frame convention.
2.2
Current Approaches
Traditionally, propeller thrusters and lifting surfaces (fins or rudders) have been
the most common forms of propulsion and steering.
Lifting surfaces in particular
function extremely well for vehicles moving at high speeds because they provide forces
that are proportional to the square of the fluid velocity. However, at low speeds, these
forces become very small and many lifting surfaces such as rudders lose their ability
to provide turning forces and moments (this does not necessarily apply to flapping
or heaving foils). As a result, a common approach is to use a number of propeller
thrusters to achieve multi-DOF motions.
In order to function properly, propellers must have access to a relatively unobstructed flow. This means that many systems must be placed so that they protrude
off the vehicle body.
As a result, having many thrusters make the vehicle body
bulkier and less maneuverable. One emerging solution to this issue is the use of tunnel thrusters [181. These systems use a concentric stator with the propeller and rotor
sitting in the middle. As a result, these types of thrusters can be placed in an opening
in the vehicle hull. These devices hold promise because they can be incorporated into
a streamlined vehicle shape.
However, since the fluid must flow through the hull,
tunnel thrusters still take up substantial space. In addition, the prototypes presented
in [18] are quite large (70mm diameter).
34
For precision maneuvering propeller thrusters present additional challenges. Propeller thrusters tend to have substantial nonlinearities such as varying time response,
dead zones, and asymmetric performance [19], [20], [21]. Precision maneuvering tasks
frequently involve switching the thrust direction back and forth rapidly. Reversing
the propeller direction causes large current spikes when the motor is at zero velocity. In addition, many screw type propellers are designed to turn in one direction.
Reversing the direction changes the flow profile substantially. Similarly, propellerthruster nonlinearities become more pronounced at lower operating speeds [18], [19].
The presence of dead zones can further degrade control performance. Dead zones are
common in propeller thrusters and are present in examples from the literature such
as [20] and [21].
As a result of these issues with propeller thrusters, developing alternatives is a
growing research field. Much of this work has focused on the development of biologically inspired robots with flexible bodies such as [22, 23, 24]. A field that has recently
emerged is using alternative actuation technologies to propel flexible, biologically inspired fish. The authors in [25] and [26] use Ionic Polymer-Metal Composite (IPMC)
to develop very small fish-like robots. Another novel approach is to use jet action and
fluttering fluid to create oscillatory tail motions [27]. These vehicles emulate fishlike
swimming to achieve unique efficiency and robustness. However, in this case, fish-like
behavior is not necessarily appropriate for our applications of multi-DOF motions in
confined spaces. Caudal fin propulsion meas that turning and longitudinal motions
are coupled and turning in place is challenging. In addition, these types of fishlike
robots cannot achieve pure sway rotations.
Another area of particular relevance is the development of biologically inspired
synthetic jet thrusters. These systems can be incorporated into streamlined shapes
and used to achieve impressive multi-DOF performance. Examples from the literature
include [28], [29], and [30].
Thus far these systems have been paired in order to
achieve bidirectional performance. This adds to the size requirements for multi-DOF,
bidirectional robot systems.
35
2.3
Pump-Valve Concept
In this thesis, we propose a different approach. We construct a propulsion system
with powerful DC motor based centrifugal pumps. The pumps are used to generate
high velocity water jets that can be used to propel the robot. We choose centrifugal
pumps due to their mechanical and electrical simplicity, small size, and commercial
availability. One issue with centrifugal pumps is that they often have a preferred direction. This means that achieving equal bidirectional forces is challenging. The most
obvious approach would be to combine two centrifugal pumps in a back back configuration in order to achieve forces 1800 apart. However, such an approach increases
the size and weight of the robot substantially.
In order to achieve bidirectional forces we draw inspiration from the field of fluidics.
Fluidic technology emerged in the 1960s and 1970s as a way of making fluid logic
circuits and computers.
One such component that is particularly relevant is the
"bistable fluidic amplifier". This device which emerged as early as the 1960's was
used to switch the direction of a powerful input jet by modulating two small control
ports [31]. The system exploits the Coanda Effect (discovered by Henri Coanda in
1932), or the tendency of fluid jets to attach themselves to curved surfaces [32, 33, 341.
Exit E2
Exit E,
rEntrained:
fluid
Entrained
flud
Control
PortPor
open to
ambient
Control
PortC 1
closed to
ambient
D
Control
tort C1
open to
Control
Port C2
closed to
ambient
ambient
Input flow Q
Input flow Q
Figure 2-2: An illustration of the bistable fluidic amplifier concept.
As Fig. 2-2 illustrates, the device can be used to switch a high velocity fluid jet
between two output ports. The device sits in the ambient fluid, and an input flow
36
Q
is injected at the input. The control ports, C1 and C2 are used to switch the direction
of the jet. If control port C1 is closed while control port C2 is open to the ambient
fluid, a small amount of fluid will be entrained through port C2 and the jet will bend
and exit through exit E1 . If the control port C1 is closed and control port C2 is closed
the jet will then switch and exit through exit E 2 . Depending on the dimensions and
jet parameters, very high switching speeds can be achieved using this type of system
[35]. If the dimensions of the valve are designed improperly, the the jet will not bend
completely (or not at all) and flow will exit through both exit ports. This is problem
is called "spillover" and will be discussed further in this chapter.
Figure 2-3: A CFD illustration of the bistable fluid amplifier fluid dynamics.
A CFD plot in Fig. 2-3 provides a visual illustration of the fluid dynamics of
bistable switching amplifiers. The figure shows clearly how if the system is designed
properly it can be used to achieve bidirectional jet switching. The switching mechanism itself is quite simple, all that is required is to open and close the two small
control ports. Since the control ports are never open or shut simultaneously, a simple
switching system can be developed using a small solenoid or DC motor. Since we will
use the fluidic amplifiers to switch a jet direction, we will refer to our fluid amplifier
systems as fluidic valves for the remainder of this thesis.
We develop a pump-plus-fluidic-valve system by attaching one of these fluidic
valves to the output of a centrifugal pump. A visual illustration of this system is
provided in Fig. 2-4. The pump draws fluid inward radially (blue arrows) and then
37
Fluid Force Fon Control Volume
Output Jet
C, Open
ILX
1NC2 Closed
Inlet Flow
lnlet
~
-
-
Inlet Flow
Inlet Flow
Reaction Moment MR Exerted On
Control Volume
Figure 2-4: An illustration showing the full pump-valve concept.
injects the jet into the fluidic valve. If C1 is open and C2 is closed, the output water
jet will exit in the positive X direction (jet labeled in red). Analyzing the output
force, F, will be discussed later in this chapter.
This mechanical simplicity as well as the high switching speed has meant that such
devices have become popular for a variety of diverse applications.
A few examples
from the literature include amplifier architectures have been used for mechanical and
biomedical system identification [35], [36]. Similar systems have also been explored for
aerospace applications [37], [38] and fuel injection systems [39]. There exist numerous
other examples for aeronautical and gas-jet flow control, but very few cases exist for
use in underwater vehicles. One such example is [40], where the author explored using
a jet diverter to explore hovering control. A patent describes using fluidic amplifier
architectures to maneuver ships but experimental validation and published works are
not provided [41]. In addition, no previous work describes the design for a multi-DOF
underwater robot system using pumps and fluidic valves.
38
2.4
2.4.1
Modeling Pump-Valve Static Performance
Static Force Performance
To analyze and properly design the pump-valve system, we need simple models
that can be used to predict performance.
In Fig.
2-5, we illustrate the relevant
physical dimensions. An input volumetric flow rate,
Q, is injected through a nozzle
of area A, and then exits through either exit of area Ae. We assume a fixed ratio
between A, and Ae (in practice A, = 2A,). We assume the exits are square shapes.
Throughout this thesis we treat the fluid as an incompressible and inviscid fluid with
density p.
We assume that the pump has a no load flow of
Qm
and a no flow
pressure of Pma.
Exit El
--
Exit E2
-
Ae
C1
LsC2
A
Inlet I
Figure 2-5: A figure outlining the relevant geometric parameters for pump-valve
modeling.
We use a simple relationship to determine the flow,
Q, as a function of the pressure.
The equation is provided in eq.2.1.
Q(P) =
x P +Qmax
PM=
39
(2.1)
0=-P(
2
An
)
'+ '
Qina
Q-Pmax
(2.2)
We can use Bernoulli's equation to model the flow exiting the pump. Note that
we neglect minor pipe losses due to elbows and the contraction at the nozzle. The
result from eq. 2.1 can be substituted resulting in the following relationship for the
flow rate based on the nozzle area.
Finally, we can use conversation of linear momentum to determine the output
force, F. Since the pump and valve are fixed to a floating vehicle, all the resulting
reaction forces and moments will affect vehicle motion. To analyze the reaction forces
and moments we create a control volume that encloses both the pump and valve. This
control volume is illustrated in Fig. 2-4. Note that the fluid enters the pump radially
and therefore creates no planar force. The output jet bends 900 before exiting. We
assume that the jet exits through one port completely (no spillover). This is treated
as an internal force based on our choice of control volume. This modeling convention
is consistent with the literature [35]. The reaction force on the control volume, F
is in the -x direction. The rotation of the pump impeller generates a pure reaction
moment, MR on the control volume. If this moment is large enough it can cause
the robot to spin. In practice, for the systems analyzed in this thesis, the reaction
moment, MR, is negligible.
Using this control volume we can determine the reaction force, F. The expression
for F as a function of the flow rate,
Q, is provided
in eq. 2.3.
F = p-Q(2.3)
Ae
We assume that Ae = 2An and then perform a simulation using parameters and
dimensions from real devices. The output force, F, is plotted in Fig. 2-6 as a function
of the exit area, Ae. The figure corresponds with intuition for the extreme cases. For
example, a very small exit leads to nearly zero flow rate due to large pressure buildup.
40
Similarly, very large exit area leads to small force due to very small exit velocities.
Most importantly, there exists an optimal exit area at which F is maximized. This
result can be used to size the fluidic valve in order to match the pump properties.
Making the ratio between Ae and An smaller can improve the output force but will
greatly affect the switching performance.
Most analysis assumes that An acts as
a nozzle with respect to the other dimensions.
In fact, for other systems in the
literature, such as [35] and [36] use much larger ratios. From our experience through
CFD and experiments, reducing the ratio of Ae/An below 2 leads to poor switching
performance.
0.25
0.2
Z 0.15-
0.1-
20
40
60
AAn,pmp [min]
[M
80
100
Figure 2-6: A simulation plot illustrating the dependence of output force on the exit
area for the combined pump-valve system.
2.4.2
Switching Length
As noted in the previous section, the static force analysis assumes that the jet
bends completely and exits through only one exit port. If this does not occur and flow
instead exits through both exit ports, we refer to this as "spillover." An illustration of
an improperly designed valve is provided in Fig. 2-7. The control ports are configured
so that the flow exits through the right hand exit, but some flow does not switch and
instead exits through the left hand exit. This substantially degrades the output force
41
performance of the system and therefore must be avoided.
Spillover
2.242
1 993
2.19
1790
1.44
1.495
1.702
1.450
1.246
I-0.996
1.216
a 973
030
0.747
0.498
0.486
0249
0.243
VlciyJms
Velocit [mf8
(a) Good Design: No spillover
(b) Poor Design: Spillover through left exit.
Figure 2-7: CFD simulations showing a "good" valve design that avoids spillover (a)
and a poor design that suffers from spillover (b).
One of the most critical geometric parameters is the switching length, L,. As
Fig. 2-5 illustrates, the switching length represents the distance between the nozzle
exit and the splitter. If the switching length is too short, spillover will occur. If the
switching length is too long, the valve is excessively large and takes up unnecessary
space.
The previous work by Xu on gas jet actuators [35] provides a valuable model for
determining L,. In this work they approximate the jet using Gortler's equation and
analyze the bending of the streamlines [31]. This model is elegant and can be easily
solved using a
MATLAB
script. We primarily use this model to choose the minimal
switching length, L, based on the choice of A,.
As the plot in Fig. 2-8 reveals, the switching length scales approximately linearly
with the nozzle exit dimension.
Since this model is based on planar analysis we
4
use the nozzle width, b,. For our geometries, b, = vx/ A/v.
The analytical model
corresponds very well with the the CFD simulations. It should be noted that the CFD
simulations do not explicitly calculate L, but rather evaluate whether a design has
spillover or not. Therefore the CFD simulations have limited resolution. Nonetheless,
the results in Fig.
2-8 show that the analytical model from [35] can serve as an
42
effective tool for sizing the fluidic valve.
7
Similar studies were performed to gauge
I
K
I
I
6.5,
6- C
5.51
E
5
E
JO
4.54-
-Moe
3.5-
MDoe
1.8
2
2.2
2.4
2.8
2.6
bn [mm]
3
3.2
3.4
3.6
Figure 2-8: A plot comparing the analytical model predictions of L, with CFD.
the dependence on the input flow rate,
a range of input flow rates,
Q.
Q.
As Fig. 2-9 shows, L., is not affect by
This is a significant result for two reasons. First,
from the perspective of control it makes sense to use the flow rate to modulate the
output force. If the switching length changes with flow rate, valve performance could
degrade as the flow rate is changed. In addition, this result shows that the fluidic
valve concept is scalable. As more powerful pumps emerge, the valve size can remain
relatively constant.
2.4.3
Summary
This static analysis is very valuable from the perspective of design. We can outline
a set of steps for developing the optimal valve system based on a certain pump.
" First a pump is selected based on size constraints and availability.
" Second, the pump parameters, P,,. and
Q,,.
are used to determine the optimal
nozzle area, A,. This nozzle area also determines the exit area (Ae = 2A,).
" Third, the length of the valve can be determined using the cited model.
This analysis is also valuable because it can be used to educate pump design. The most
appropriate pump geometries for combination with fluidic valves achieve maximum
43
8.4-
E
E
8.3-
8.28.1-
1
8.5
1.5
2
2.5
3
Q [m 3/s]
3.5
4
4.5
5
x 10 3
Figure 2-9: A plot illustrating how the switching length, L,, does not vary with a
range of values for the input flow rate, Q.
force at relatively small nozzle areas. These smaller nozzle areas require shorter fluidic
valves.
2.5
Pump Model
The pump can be used to modulate the output force levels, while the valve can
be used to modulate the force direction. For the scope of this work we assume the
centrifugal pumps are driven by DC motors. The simplest form of controlling the
pump is to adjust the input voltage, or pump voltage control. We assume that the
pump parameters,
Q
,
and Pma, scale linearly with the pump voltage level, VP.
Figure 2-10 provides a visual illustration of pump voltage control.
A more sophisticated approach is to use speed control of the pump and use an
input voltage to control the pump impeller speed, w. This technique is less prone
to disturbances and unforseen effects because the circuit will increase the current in
order to achieve the desired impeller speed. Figure 2-11 provides a visual illustration
of pump speed control. Notice how in output force scales linearly with W 2 . This type of
relationship also exists for propeller thrusters [191. This result is reasonable because
both centrifugal pumps and propeller thrusters use a spinning device (impeller or
propeller) to generate fluid forces.
44
0.2
0.15
Qmax
0.10.05
7
Increasing VP
0
-0.05
-0.1-
Exit
xit 2
-
-0.15
max
'
0.2
0.4
0.6
P
0.8
1
max
(b)
(a)
Figure 2-10: A visual illustration of pump voltage control. The voltage control model
is shown (a) as well as the use of voltage level to modulate output force (b).
0.15
0.12
-Exit 1
0.1 ---Exit 2
0.1
0.05
LL
-0.05
z 0.06
Exit 1-
Exit 2
0.2
.
.
.
0.08
0
0.04
-0.1
0.021
-
-0.15
-0%
0.2
0.4
W
0.6
0.8
0.2
1
0.4 20.6
2
0.8
I
W
Wmax
(a)
(b)
Figure 2-11: A visual illustration of pump speed control. Using speed control to
modulate force is shown in (a) as well as the speed-force input output relationship
(b).
45
2.6
Pump-Valve Dynamic Performance
Developing simple analytical models for the pump and valve dynamics is challenging. This is because the unsteady fluid dynamics now must also be considered.
The valve dynamics are especially challenging and complex. Nonlinear and intricate models can be found in [31, 42, 43, 44, 45]. These models are quite accurate but
require numerically solving numerous nonlinear differential equations. This process
is challenging and sacrifices physical intuition. Since the design choices are mainly
made during the design phase, we seek to identify the dynamic performance mainly
for feedback controller design. Therefore we will rely on simple CFD simulations. A
more interested reader can see the aforementioned references. Additionally, [39] provides a simpler modeling approach but uses an amplifier design that is significantly
different from the one we have outlined.
An example of these simulations is provided in Fig. 2-12. In these simulations
we used a valve of dimensions similar to those in the static CFD simulations. The
pump-valve system was used to generate different force levels and at t = 0.3s the
valve was commanded to switch the flow direction. Three key results emerge from
this study. First, the switching dynamics are nonlinear; the switching time varies with
the force level. The switching dynamics slow down as the force level (based on flow
rate) decreases.
This result is actually common in underwater propulsion and is a
key nonlinearity in propeller thruster performance [19]. The second key result is that
the switching dynamics are quite fast. The switching times range from 0.s
to 0.2s,
and even at the slower levels the switching dynamics are substantially faster than the
vehicle dynamics. The vehicle dynamics we explore in this work have dynamics of
approximately 0.5Hz which are about an order of magnitude slower than the valve
dynamics. We will exploit this feature in the following chapters.
The third and final observation is that the switching dynamics have a slight undershoot which is a characteristic of a nonminimum phase system. This relates to
the unsteady fluid dynamics associated with the switching. When the control ports
are switched fluid is entrained and temporarily acts to increase the exit flow. This
46
0.15
(=0.50)
0.1...-.--------------
0.max
-- o= 0.7%=
= 0.87o,
...---....-.--...--
-Valve Command
-
0.05
0
r
-0.05--
.10
0.1
0.2
0.3
lime [s]
0.4
0.5
0.6
Figure 2-12: CFD simulation illustrating valve switching dynamics for various force
levels.
behavior, which can be approximated using the Pade approximation for a time delay
(right half plane zero, left half plane pole). In order to match the CFD results, the
pole-zero pair had dynamics that correspond to a time delay of 20ms. This places an
upper bound on the system dynamics. Attempting to achieve closed loop bandwidth
of about 75 rad/s would lead to a reduction in the phase margin of about 90*. The
vehicle dynamics are on the order of 2 rad/s, and the closed loop bandwidths discussed in this work will be on the order of 5 rad/s. This would lead to relatively small
effects (phase margin reduction of 5*. For this reason we will neglect the undershoot
dynamics for the remainder of this thesis.
47
48
Chapter 3
Implementation and Experimental
Characterization of Pump-Valve
Systems
In this chapter we describe the implementation of the pump-valve system outlined
in Chapter 2. While we make use of commercially available centrifugal pumps, we
outline a novel valve design that combines small size, mechanical simplicity, and high
force output. We follow the design procedure outlined in Chapter 2 and construct
and verify both the static and dynamic performance of this new pump-valve system.
3.1
Pump Selection
For this study we desired a pump system that was small (cm scale), and mechanically and electrically simple. Since pump design is not the core of this research we
sought systems that are readily commercially available. The small size plays a critical role because we desire robot designs that are on the order of 1-3 kg so that they
can fit within the tight constraints imposed by the BWR environment.
We found
that TCS MICROPUMPS provides a wide series of small and powerful micropumps
(http://www.micropumps.co.uk/). The TCS micropumps are submersible, small (1070g), and come with speed control circuits. Specifically we use the M400Sub model
which is pictured in Fig. 3-1. The M400 is driven by a powerful brushless motor and
is driven by an internal speed control circuit. A summary of the pump parameters
49
Table 3.1: Summary of the M400 Pump.
Parameter
Value
m
0.33[kg]
Pmax
Qmax
35[kPa]
Vmax
i0
6.37 x 10- 5 [m 3 /s]
12[Vj
0.5[A]
such as the no flow pressure, Pmax, the no load flow,
Qmax,
the maximum voltage,
Vax, and the nominal current draw, io can be found in Table 3.1.
Figure 3-1: A photograph of the TCS M400 micropump that we will use throughout
this thesis.
3.1.1
Sizing Exit Nozzle
The first step of our design procedure is to use the simple models outlined in
Chapter 2 to determine the best nozzle area, Anpump. We choose to verify the role
of the nozzle area separately from the full valve because the nozzle area must be
identified before designing the entire valve.
To verify the model, we used a force
testing system designed by Ian Rust to measure the output force performance based on
various nozzle exit areas. These measurements are provided in Fig. 3-2. As the figure
shows, the model predictions match the experimental data well. The experimental
data illustrates the presence of an "optimal" area where the output force is maximized.
50
0.25
-Model
0 Experimental
Data
0
0.20
Z.0.15-
0.1-
0.0%
20
40
6080
AAn,pump [mini
[m
100
Figure 3-2: A figure illustrating how to use the exit nozzle area to optimize the output
force.
3.2
Valve Design
Once the best nozzle areas are identified, the valve can be designed using the
nozzle area. In this case we chose A, = 22mm 2 which is a bit larger than the optimal
size. This is because the pump fittings are designed for this size, allowing simpler
designs and retrofitting (discussed in a subsequent section). This decision means that
we operate at about 20 percent less than the optimal force.
Once the nozzle area was verified, we selected the switching length, L, and then
design the remaining components. The switching length model from [35] provides an
estimate of 6.1mm. In order to ensure switching in the face of manufacturing errors
and incomplete control port closings, we introduced a safety factor of about 1.6, and
selected L, = 10mm. For the designs outlined in this thesis, multiplying the modeled
switching length by 1.5 to 2 provides reliable valve switching without adding to size
dramatically. Schematic diagrams of this finalized valve configuration are provided
in Fig. 3-3. This figure provides both a cross sectional view of the valve design as
well as a three dimensional model showing the external shape of the device.
51
3.3
3.3.1
Implementation
Valve Construction
Due to the intricate and unique geometry of the valve, using traditional manufacturing techniques is challenging. However, the growth of 3D printing technology
means that these valves can be made as one piece out of strong plastic. In this case we
use the d'Arbeloff Laboratory's Dimension 1200es SST 3D printer. The printer uses
ABS plastic which is lightweight and durable. Valves such as the one photographed
in Fig. 3-4 take about 45 minutes to print.
A, = 43.6mm 2
00
L, =10mm
Figure 3-3: Diagrams showing the dimensions of the final valve design.
3.3.2
Switching Design
The final step before full implementation is the development of a small, powerefficient, and simple device for controlling the valve. A previous work used a a spring
loaded solenoid to cover and open the two control ports [36]. However, in order to hold
the solenoid against the spring force, a substantial steady state current was required.
This created a level of power consumption that would be troublesome for a mobile
robot. For this work we developed a new switching system that uses a micro DC
motor to move a "flapper" over the two control ports. The design exploits geometry
to open and close the two control ports simply by rotating the motor clockwise or
counterclockwise.
Mechanical stops prevent the flapper from rotating continuously
52
A photograph of the fluidic valve
and mean that feedback control is not required.
with its switching mechanism is shown in Fig. 3-4-a. The switcher can also be printed
using the 3D printer, allowing it to be very light and flexible. This reduces the loads
on the switching motor.
A high-low switching scheme is used where a small burst of power is used to switch
the valve and a much smaller level is used to hold it in place. The fluid pressures act to
keep a closed port shut, so very little power is required to maintain a configuration.
An illustration of this switching scheme is provided in Fig. 3-4-b.
Typically, the
valve is sent a voltage of ~ 5V for a burst of 100ms and then reduced to a a lower
value of
-
0.75V. These values were determined empirically and can be changed to
accommodate different hardware.
Valve Direction
-Switching Motor Voltage
Time [s]
(a) Fluidic valve with switching mechanism
(b) High-low valve switching scheme
Figure 3-4: Photograph of a full valve prototype with the switching mechanism (a)
and a plot showing the high-low switching scheme.
3.4
Experimental Evaluation
A full pump valve prototype was created by combining the elements discussed
throughout this chapter. The pump and valve were combined into a single component
and 3D printed as one piece. Figure 3-5 shows a photograph of the full prototype. The
pump motor and the smaller switching motor are clearly illustrated. Not only is the
switching motor substantially smaller and lighter than the pump motor, but a simple
brushed DC motor can be used without speed control or commutation circuitry.
53
Figure 3-5: Photograph of a fully assembled pump-valve system.
3.4.1
Static Performance
The first test was to study the pump-valve static performance under pump-voltage
control. The pump valve system was attached to the force test setup described previously and used to measure the force at various pump voltages, V. The valve was
used to switch the direction of the jet. The results from this experiment are provided
in Fig. 3-6. As the figure shows, the experimental data matches the theoretical model
of a
from Chapter 2 quite well. In addition, it is important to note the the presence
dead zone around V = 3V. This is the result of both fluid, mechanical, and electrical
considerations. The electrical circuits need a minimum voltage to operate, and at low
voltages friction can affect the impeller performance.
Finally, if the impeller spins
a
too slowly the fluid dynamics may not be fully developed. The net result is there is
region where a nonzero voltage results in a nearly zero force. Dealing with this issue
will be a key area of focus in Chapter 5.
3.4.2
Dynamic Performance
The second test was to evaluate the dynamic performance of the valve system.
Figure 3-7 shows the response of the system when the valve is operating at full
speed (V
= 12V) and the valve is commanded to switch direction at time t
=
0. As the figure shows, the valve switches very quickly, reaching the steady state
value in about 0.11s. This shows that the physical system matches the high speed
switching performance predicted by the simulations in Chapter 2. We also examined
54
Exit 1 (Exp.)
0.15 0$Valve
Valve Exit 2 (Exp.)
-Valve
I
I
6
8
Ex 2 (Theory)
0.1 -- Valve ExA 1 (Theory)
0.05
-xt1L
LL
x
-0.1
-0.15
C
4
2
Pump Voltage V
12
10
Figure 3-6: Pump voltage control over pump-valve system. Both directions of the
valve are shown.
the switching dynamics at various pump speeds. These results are provided in Fig.
3-8. Due to the low force levels and the presence of noise from waves and vibrations
from the pump spinning it was difficult to estimate the exact rise times for each level.
However, the figure does illustrate that even at the lower levels, the valve is able to
switch direction fully within ~ 150ms. As discussed previously, these dynamics are
substantially faster than the vehicle dynamics which are about 0.5Hz.
0.
0.1
----
5-- -------------------
-----
0.
Sc0.11
SS
0
-
0.01
-0.01
-0.
-0.1 5:
-0.
0.2
0.4
'~~~~
0.6
1
0.8
Time [s]
1.2
-Switch Signal
-[NJ
'
1.8
1.6
1.4
Figure 3-7: Experimental data illustrating the switching dynamics of the valve. The
pump is running at full voltage.
55
U. I Z)
0.10.05,0
U-0.05
-V =12V
V =10
-0.1
P
-V P= 8V
-V = 6V
-0.15
-0.
0.05
0.1
0.15
Time [s)
0.2
0.25
0.3
0.35
Figure 3-8: Experimental data illustrating the switching dynamics of the valve for
several pump voltages.
Less emphasis was placed on the pump dynamics. Simple experimental results
have shown that the pump-voltage dynamics have have similar dynamics to the valve
(switching times of about 100ms). As a result we will treat both the pump and valve
dynamics as a single first order system. We treat both the pump and valve as having
a roll off frequency of 5Hz, which corresponds to a time constant of about 0.033s.
Lumping the pump and valve dynamics together greatly simplifies control system
design.
56
Chapter 4
Control Configured Underwater
Vehicle Design and Implementation
In this chapter we explore using the novel pump valve system discussed in previous
sections. We start by exploring a unique way to achieve dual forces from a centrifugal
pump by specially orienting the exit nozzles.
We use these orthogonal dual out-
put ports to reduce the number of pumps required for multi-DOF actuation. These
concepts are used to create a novel Control Configured Spheroidal Vehicle (CCSV)
capable of motions in 5 directions. This vehicle is completely smooth and designed
specifically for high feedback control performance.
This chapter concludes with a
detailed description of the implementation of this new design.
4.1
Multi-DOF Propulsion
As we described in the introductory chapter, the vehicles we seek to design require
motions in 5 directions (surge, sway, heave, yaw, pitch). In Fig. 8-1 we provide an
illustration of the coordinate system as well as our proposed jet propulsion design.
Figure 8-1 and Table 4.1 show an arrangement of water jets that allow us to achieve
these motions. We number the jets in order to simplify implementation, discussion,
and to provide physical intuition. While Fig. 8-1 provides a visual illustration of
the jet configurations, Table 4.1 provides an in depth summary of how various jet
combinations can be used to achieve the desired motions.
A significant contribution of this thesis is the diamond configuration of Jets 1
57
and 2 which are angled inward at an angle -y.
This feature allows the robot to
achieve translations in the sway direction (ty) without adding additional jets. The
diamond configuration also increases the moments about the center of mass that is
created by the jets. Finally, the jet angle has very significant ramifications for control
performance which will be explored in great detail in later chapters.
-jet 1
-Jet 2
+Jet 4
+Jet 3
,4
A
* ,
L
%
II
ZX
+Jt
%%
.
0+e
I
z
* -Jet 3
4 -Jet 4
Figure 4-1: A diagram illustrating the jet arrangement for the CCSV design.
Table 4.1: Summary of Maneuvering Primitives.
DOF
+U
-u
+v
+r
First Jet
+Jet 1
-Jet 1
+Jet 2
+Jet 1
+Jet 1
-r
-Jet 1
+W
+Jet 3
-w
-Jet 3
+Jet 3
-q
4.2
-Jet 3
Second Jet
+Jet 2
-Jet 2
-Jet 2
-Jet 1
-Jet 2
+Jet 2
+Jet 4
-Jet 4
-Jet 4
+Jet 4
Pump Valve Design
We have shown how 8 water jets can be arranged to achieve the 5 directions of
motion we desire. The challenge is achieving all of these jets within a shell that
58
is compact and smooth. It is simply not practical to use 8 independent pumps for
generating each of the 8 jets.
Pumps are heavy, bulky, and consume energy.
In
fact, with the exception of batteries and cameras, they tend to be the largest and
most difficult to package component. Therefore, there is a clear need to minimize the
number of pumps while still achieving the jet configuration outlined in Fig. 8-1.
The first thing to emphasize is that two jets can be shared by a single pump when
the two jets do not need to be activated simultaneously.
need to activate
For example, there is no
+Jet 3 and - Jet 3 at the same time because the forces and moments
would just cancel out. This means that we could easily reduce the number of pumps
by using a fluidic valve to switch a jet between the positive and negative direction
(Jets 3 and 4 are prime candidates). There will be cases however, when Jets 1 and
2 need to be activated at the same time. For example, activating
+Jet 2 and -Jet
2 simultaneously creates positive sway motion, +v, while activating
+Jet 1 and -Jet
1 simultaneously creates negative sway motion, -v.
This is an example where the
fluidic valves we have proposed serve as an ideal solution.
The fluidic valves can
switch the jet direction without creating large reaction moments. In addition, the
fluidic valves can switch a powerful jet at a high frequency (~
4.2.1
3Hz).
Pump Design
In the previous chapters we focused on the most common centrifugal pumps with
only one output port for the high velocity jet. Reversing the impeller direction simply
leads to poor performance.
Encyclopedia Britannica 1996
Figure 4-2: A diagram from The Encyclopedia Britannica illustrating a common type
of centrifugal pump design.
59
However, some studies by our group have explored altering existing pump designs
in order to achieve forces in two directions. Pump impeller design is an entire field in
itself, and this work is not intended to be a thorough exploration of pump design. The
pumps produced by TCS use a symmetric impeller which sucks the water through the
center and then eject it tangentially. This means that the impeller itself can function
when rotating both clockwise or counterclockwise. A photograph of this impeller is
shown in Fig. 4-3. This design was developed by TCS.
Micropump M400
Figure 4-3: A close up photograph showing the symmetric impeller.
We can exploit this unique feature for robot design. Specifically, we explore designing output nozzles so that the reversal of the impeller can be used to create a
second jet. The most logical manifestation of this would be to use two exit nozzles
in line with each other, allowing us to generate a negative and positive jet. However
simple studies show that this arrangement does not work well for our designs. As Fig.
4-4 shows, when the impeller spins counterclockwise, most of the flow is directed out
of Exit 1. However, as the figure shows, a substantial amount exits through Exit 2.
We refer to this issue as "back-flow", and this results in substantial force degradation.
This result was also observed in experiments.
However if the control ports are placed adjacent to each other and 90 degrees
apart, the issue of back-flow vanishes and the pump actually sucks in fluid from the
secondary port, causing performance to actually improve. As Fig. 4-5 shows, this
arrangement allows the generation of two powerful exit jets using a single centrifugal
pump. Reversing the impeller direction causes the jet to change direction completely.
We refer to this design as an "orthogonal dual output" pump design.
60
Figure 4-4: A CFD simulation illustrating an in-line dual-output pump design.
Exit 1
Exit 1
it
eE
(b) Clockwise Impeller Rotation
(a) Counterclockwise Impeller Rotation
the
Figure 4-5: A CFD simulation illustrating the 90 degree pump. Note how when
impeller direction is reversed, the jet switches exits completely without any back-flow.
Experimental data confirms the insights gleaned from this CFD analysis. Force
data for three pump designs are provided in Fig. 4-6. The three pumps designs are
the standard single output design described in Chapter 3 (red), the in-line dual output
port design (black), and the in-line orthogonal design (blue). The figure illustrates
clearly how the in-line dual output port pump suffers from poor performance. The
an
most interesting result is that the orthogonal dual output port design provides
extra direction and generates larger forces.
This makes it an ideal candidate for
robot designs.
4.2.2
Reduced Actuation Design
Reducing the number of pumps so that size, weight, and complexity is minimized
is one of the top design priorities. We refer to the design of a multi-DOF vehicle with
minimal pumps as "Reduced Actuation Design" (RAD). The use of fluidic valves
allows us to reduce the number of pumps from 8 to 4. Using dual-output port pumps
can reduce this number to 2. However, implementing this approach requires care
61
0
+Standard One-Way Pump
+ Orthogon Al Dual-output Pump
n 9 + In-line Du 3-output Pump
0. 15k
0.1
loolo
oVolo
0.05F
C
i
V
8
10
12
p
Figure 4-6: Experimental data comparing 3 pump design methodologies.
because only two maneuvering jets can be created at a time.
We combine 2 fluidic valves with one orthogonal dual-output pump to create what
we call a Bi-axis Actuation Unit. This propulsion unit can generate two sets of in-line
force (4 total) using only a single pump. The pump impeller direction can be used to
switch the exit nozzle between two outputs, which in turn are switched using a fluidic
valve. A photograph of a fully assembled BAU is provided in Fig. 4-7.
Figure 4-7: Photograph of a BAU prototype. Two fluidic valves are combined with
an orthogonal dual output port pump to generate forces in 4 directions.
Since only two jets can be activated at a time, the pumps and valves must be
oriented in a way that eases implementation and control. A key issue is that switching
62
the pump back and forth rapidly is undesirable due to reaction moments on the vehicle
as well as power spikes when the motor is momentarily at zero speed. We deal with
pump switching by dividing the motions into two families.
Movements within a
"family" will not require the pumps to reverse direction. Switching between families
will require reversing the direction of both pumps. We found that an intuitive way to
create the families is to divide the motions in the horizontal plane versus motions in
the vertical plane. We call these two families "swimming" and "diving" respectively.
Swimming is comprised of motions within the vehicle xy plane. These motions are
associated with Jets 1 and 2. Since the vehicle is designed to be passively stable in
pitch and roll, the vehicle xy plane is usually parallel to the Earth horizontal plane.
In diving mode the robot moves within the vertical plane. The robot uses Jets 3 and
4 to translate up and down as well as pitch up and down. While this structure tends
to restrict three dimensional motions, it makes driving the robot via a camera feed
very intuitive because the robot xy plane is almost always aligned with the Earth
horizontal plane. As a result, human operators are less likely to become disoriented.
Three dimensional motions can be performed by combining the swimming and diving
modes and by switching between the two periodically.
We use the swimming and diving motion families to design and orient our two
BAU units. The embodiment of our reduced actuation design is shown in Fig. 4-8
which shows the bottom half of the robot design. This chamber is allowed to fill with
water which helps keep the vehicle design symmetric and stable. BAU 1 uses Pump
1 to generate Jets 1 and 3 in both the positive and negative directions. Clockwise
rotation of Pump 1 is used to generate Jet 1, and counterclockwise rotation of pump
1 is used to generate Jet 3. Similarly, BAU 2 uses Pump 2 to generate Jets 2 and 4.
A summary of the pump valve combinations is provided in Table 4.2. One important
consideration is that the sway motions are generated using a single pump. The fluidic
valves are used to switch the jet back and forth so that the vehicle can achieve sway
motions without significant oscillations. For the scope of this thesis we refer to this
vehicle as the CCSV-RAD prototype. We use this slightly awkward title to differential
this prototype from a few others that are discussed in Chapter 10.
63
BAU 1
Jet 1
-Jet 1
+Jet 1
Jet 2
BAU 2
Figure 4-8: A rendering showing the pump-valve system for the CCSV-RAD prototype.
Table 4.2: Summary of Pump and Valve Combinations for Vehicle Motions.
DOF
+u
-u
+v
-v
+r
-r
+w
-w
+q
-q
Pump P1
Pump P2
First Jet
Second Jet
CW
CW
OFF
CW
CW
CW
CCW
CCW
CCW
CCW
CW
CW
CW
OFF
CW
CW
CCW
CCW
CCW
CCW
+Jet I
-Jet 1
Jet 2
Jet 1
+Jet I
-Jet I
+Jet 3
-Jet 3
+Jet 3
-Jet 3
+Jet 2
-Jet 2
N/A
N/A
-Jet 2
+Jet 2
+Jet 4
-Jet 4
-Jet 4
+Jet 4
4.3
CCSV Prototype
4.3.1
Mechanical Design
A full CCSV prototype was constructed using a combination of off the shelf pumps
and motors and custom designed components. Two TCS M400 Submersible micropumps were used for the two main pumps. For the valve switching motors, 4 TCS M200
64
submersible pumps were disassembled and used. These small micropumps were very
useful due to their small size and the fact that they were already designed for submersion in water. These components are shown in the photograph of the BAU in Fig.
4-7.
Due to the unique shapes of the valves and external shell, this type of design is
greatly eased by 3D printing manufacturing techniques.
Almost all of the custom
components were fabricated using the d'Arbeloff Laboratory's Dimension SST 1200es
3D printer. The printer prints parts using ABSPlus material which is strong enough
to be used for load bearing components and has been used in the past for a variety
of robotic applications. The downside to this printing technique is that the material
is porous and therefore unsuitable for watertight components. A watertight chamber
for the electronics and battery was designed as part of the top half of the robot. Since
this chamber is filled with air it makes sense to place it in the top of the robot in order
to achieve roll and pitch stability. This watertight chamber was printed using a 3D
Systems Viper Stereolithography machine. The original parts were printed using the
SI-40 material. Recently we have started to use the SI-60 material which is less brittle
and more transparent. We initially used the machine at MIT's Bioinstrumentation
Laboratory and are grateful to them for their assistance. For later prototypes, we
used Empire Prototypes, a rapid prototyping company in Massachusetts.
Two photographs of the CCSV-RAD prototype are shown in Fig. 4-9. The watertight chamber is clearly visible in the center of Fig. 4-9-a, the dull yellow material
contrasts with the bright yellow of the ABS plastic. A detailed view of the maneuvering system is provided in Fig. 4-9-b. The BAUs are clearly visible, and it is important
to emphasize how the construction of these geometries would have been very difficult
without 3D printing technologies. Additional photographs are provided in Fig. 4-10
and illustrate the jet exits as well as the vehicle dimensions.
4.3.2
Electrical Design
An electrical systems is needed to properly control the robot vehicle and allow it
to operate without any external tethers. As Fig. 4-11 shows, the robot must therefore
carry its own power supply (battery) and communicate wirelessly with a base station.
65
Pump2
BAU2
(b) Maneuvering Syste m
(a) Top View
Figure 4-9: Photographs of the CCSV-RAD prototype. A fully assembled prototype
is shown (a) along with a closer view of the maneuvering system is shown (b).
(b)Front View
(a) Top View
Figure 4-10: Photographs of the outside of the CCSV-RAD prototype.
66
This eliminates interference from a power or communication tether, and is the ideal
use scenario for the robot system.
Base Station
Robot
Figure 4-11: An overview of the electrical components for the CCSV system.
Off the shelf components were used for the battery, radio system, microcontroller,
inertial measurement unit (IMU), and motor controllers. The microcontroller is a
16MHz Arduino Mini (5V) available from Sparkfun Electronics.
The IMU was a
MinIMU digital IMU sold by Pololu Robotics. Multiple Pololu Qik 2s9v1 dual serial
motor controllers were used for pump and valve control. An Easy Radio system was
used for wireless communication (available from Active Robots). The 433MHz radios
were used for the experiments in this study. It should be noted that this frequency
lies within the amateur band in the United States and therefore requires a HAM
technician license to use. This frequency is not allowed for commercial use. Finally,
Lithium Polymer (LiPo) batteries were used to power the robot. These batteries
have very high energy density and are frequently used for small aerial vehicles such
as drones and quadrotors. Great care must be taken when using these batteries, and
it is strongly recommended to use fuses and
/
or Positive Thermal Coefficient (PTC)
devices to protect against accidental short circuits. Much of the electrical system
design was done in collaboration with two undergraduate students, Meagan Roth,
and Martin Lozano. Their senior theses provide more detailed descriptions of the
electrical system design and component selection [46], [47].
A printed circuit board (PCB) was developed to combine all the electrical com67
ponents in a compact and robust manner. This board contains all of the electrical
components and connections for the CCSV-RAD prototype. This "brain board" was
originally developed on a prototyping board by the author, and was then incorporated into a printed circuit board by Meagan Roth and Martin Lozano. The electrical
schematic (Eagle Software) is shown in Fig. 4-12. Extensive effort was expended by
Martin Lozano to miniaturize this board, test, and populate it. A clever "stacked"
design involved placing larger electronic components hanging over smaller ones. This
saved considerable space. The circuit layout and a photograph are provided in Fig.
4-13. This figure shows the small size of the brain board and illustrate the stacked
configuration. This board has been extremely useful for prototyping and been used
for several other prototypes.
6000Thf.
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LTT?
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Mir
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LAND
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AN1
Figure 4-12: Eagle software schematic layout for the robot brain board PCB.
4.3.3
Waterproofing Components
Creating resealable waterproof chambers is one of the most challenging implementation issues for the design of small underwater robots. Waterproof enclosures are
commercially available at a variety of sizes from companies such as Hammond Manufacturing. However, these boxes are not necessarily optimized for the shape of robots
or the shapes of the multitude of electronic components. The use of large boxes can
waste space and make the robot larger than necessary. Our approach to waterproof68
-
:1.9-
1.65"
Figure 4-13: Images showing the brain board layout and its fully assembled form.
ing prototypes is three-fold. First, as mentioned above, we use stereolithography 3D
printed chambers which can be made into custom shapes. Second, we insert small
holes in the plastic chamber through which wires are passed. These holes are designed
to be only slightly larger than the wire diameter in order to ease sealing. Penetrating
epoxy is then used to seal the wire holes. In this thesis we have used penetrating epoxy
from American Synthetics. These prototypes have maintained integrity to depths of
a few feet. Thorough pressure testing is an important area of future work.
Third
and finally, the chamber must be resealable so that the battery can be recharged,
the software updated, and the components checked. This is accomplished by using
a metal cap with a soft rubber gasket that creates a watertight seal. The caps sold
by Hammond Manufacturing have performed well for this task. A photograph that
illustrates the resealable waterproof chamber is provided in Fig. 4-14. In this case
the waterproof chamber was made quite small in order to facilitate testing and experimentation. More recent prototypes have the entire top chamber composed of one
SLA printed piece.
4.4
Full Robot Summary
For the sake of simplicity in future chapters we summarize some key parameters
on CCSV-RAD prototype.
The vehicle is spheroidal in shape, the radius on the
major axis, a, is equal to 73mm, the radii along the minor axes, b and c, are equal
and are 54mm. The resulting mass of the vehicle is 0.9kg. The use of a spheroidal
69
Figure 4-14: A photograph showing the inner components of the robot. Note the
metal cap, this is used for sealing the watertight electronics chamber.
70
shape greatly eases the calculation of moments of inertia and added masses. Since the
pumps and other heavy components are concentrated near the center of the vehicle,
we assume the center of mass lies at the volumetric center of the spheroidal shape.
The added mass coefficients can be estimated simply by using tables in [48].
A
photograph of CCSV-RAD prototype performing a diving maneuver by using Jets 3
and 4 is shown in Fig. 4-15.
To the best of our knowledge, no other underwater robot or vehicle that is completely appendage free and uses jets to achieve motions in 5 directions. Many multiDOF underwater robots exist, but very few appendage free ones have been proposed,
designed, and verified. We believe this novel prototype and others like it can serve
as a valuable test-bed for many new ideas in the field of underwater robotics. In the
following chapters we will examine a few of these such as precision control, closed
loop stabilization, and vehicle performance.
Figure 4-15: A photograph showing the CCSV-RAD prototype performing a diving
maneuver.
71
72
Chapter 5
Dual Pump-Valve Control
In this chapter we analyze the pump-valve system from the perspective of precision feedback control. In the following we describe our proposed approach to dual
pump valve control and propose three different control techniques: pump speed Control, valve PWM Control, and hybrid control. Each combined pump-valve control
technique is discussed in detail with the pros and cons examined. Since both valve
PWM and hybrid control involve using the fluidic valve to switch at high speeds, we
place emphasis on exploring the effects of valve switching on vehicle oscillations. We
conclude the chapter with simulations and experiments which illustrate the control
techniques and the value of our analysis tools.
5.1
Nonlinear Pump and Valve Dynamics
In Chapter 2 and Chapter 3 we showed how the statics and dynamics of the pumpvalve system are nonlinear. These issues are not uncommon and also exist in many
propeller thruster systems. Such nonlinearities can lead to poor control performance
[19], [49]. In the past, substantial research has focused on using advanced feedback
control techniques to compensate for the nonlinear thruster behavior. For example
the authors in [50] used adaptive sliding mode control, while those in [51] used a fuzzy
sliding mode based approach. The authors in [49] use model based feedforward and
feedback controllers, and the authors in [52] employ model reference adaptive control.
While these advanced control system designs have been shown to work quite well,
our pump-valve system has the unique property that the pump and valve can be con-
73
trolled independently. Specifically, the valve can be switched back and forth quickly
and without creating large reaction moments on the vehicle body. This is a substantial difference from propeller thrusters which can create large reaction moments when
their direction is reversed. This means that relay control and pulse width modulation
(PWM) control techniques can be employed. These techniques are based on using
ON/OFF switchings to modulate the force level rather than analog control [53]. Such
techniques have been especially well suited for systems that use discrete valves rather
than analog ones which can be much larger and more complex. Some examples include spacecraft [54], hydraulics [55] and airjet thrusters [56]. In addition, ON/OFF
control techniques can be used when the dynamics are poorly understood [57]. This
case is most relevant to our application, because while we can perform analog control
by using pump speed or pump voltage control, we have unpredictable and nonlinear
pump behavior at low speeds.
5.2
Combined Pump-Valve Control
The centrifugal pump plus fluidic valve system is a multi-input (pump voltage,
valve voltage), singe output (force) system. Our objective is to design a system that
coordinates the pump and valve so that the force, F, is produced in response to
the single controller output, u, in a manner that is predictable and appropriate for
feedback control. This system is illustrated in schematic form in Fig. 5-1.
S
Purnp Control
-*Pump
Controller
F
Vehicle
Valve Control -*Fluidic Valve
Figure 5-1: A diagram illustrating the principal of combined pump-valve control with
a single input from the controller and and a single force output.
5.2.1
Summary of Pulse Width Modulation
Since the following sections focus on the idea of pulse width modulation, it is
valuable to quickly summarize the concept. Pulse width modulation is an extremely
common technique for modulating a square wave and has been used for a range of
applications from DC motor speed control to remote control servomotors. The key
74
idea is that a square wave of frequency fPWM (period of TPWM) alternates between a
high level and a low level. A variable called the duty cycle, d, can be used to quantify
the duration of the high and low levels. The duty cycle is a percentage that ranges
from 0 to 100. As Fig. 5-2 shows, the high duration can be quantified as TPwMd
while the low duration can be quantified as TpwM(1 - d). A duty cycle, d of 100
percent corresponds to always high while a duty cycle of 0 percent corresponds to
always low.
Td
High
U
Low
t
TPWM
Figure 5-2: A diagram illustrating the components of a PWM Signal.
PWM
PWM
Duty
Duty
100%
100%
PWM
Duty
100%
i)
50%
500%
0%.
Pump
Output
____
u
Pump
Output
0
U
0
A Z
U
Output
U
A
0
II)
- - -
I
Dead
- --
-
.iii)
U
U
Fine
Zone
/000,
U
F,
FA
F
-
0%1
PWM
Zone
(a)
Pump Speed Control
(b)
(c)
Valve PWM Control
Hybrid Control
Figure 5-3: Dual pump-valve control algorithms: (a) pump speed control, (b) valve
PWM control, and (c) hybrid control. Each sub-figure includes i) PWM Duty Cycle,
ii) pump output, and iii) resultant force, all in relation to the controller output, n.
75
5.2.2
Pump Speed Control
The first control algorithm is called "Pump Speed Control" and uses the absolute
value of u to determine the pump while using the sign of u to determine the valve
direction. This means that the duty cycle is either 0 percent, or 100 percent. The
primary advantage of this system is its simplicity.
Analyzing and implementing a
linear control system using pump speed control is very intuitive and straightforward.
In addition, using the valve to reverse the jet means that symmetric performance can
be achieved which is frequently challenging with propellers.
Two key drawbacks exist with this technique. First, dead zones can reduce the
ability to generate small forces.
Figure 5-3-a shows the presence of a small dead
zone in order to reflect the actual system performance outlined in Chap. 3. These
dead zones can exist due to static friction, electrical issues, and even fluid dynamics.
These dead zones can lead to steady state errors or oscillation in the closed loop
system response. Secondly, even when the closed loop system functions properly and
causes the error to converge to zero, small deviations can cause the error to change
sign rapidly. For our pump-valve system this causes the switching motor to chatter.
This chattering not only wastes power but it can also cause substantial wear and tear
on the valve motor. A common solution to chattering is to add an error dead zone,
but this can degrade accuracy and make performance difficult to model and predict.
5.2.3
Valve PWM Control
The second control algorithm is called "Valve PWM Control." In this case the
pump is run at a constant speed known to be outside of the dead zone. To modulate
the output force, the valve is switched at a high frequency with a duty cycle proportional to the controller output, u. The output force, F, is modulated by switching
the valve at a frequency that is high compared to the vehicle dynamics.
A visual
explanation of valve PWM control can be found in Fig. 5-3 which shows the i) PWM
duty cycle of the valve, ii) the pump output, and iii) the resultant force, F. Valve
PWM control can be used to eliminate the the steady state errors caused by the dead
zone. In addition, if the PWM frequency is low, the wear on the valve motor due to
76
rapid switching can be substantially reduced. Finally, the duty cycle can be used to
linearly modulate the output force meaning that this method fits well into the context
of linear control.
An important disadvantage to this approach is the tension that exists betweens
speed of response, power consumption, and oscillations. Using a high pump speed
means larger forces and faster responses, but also means higher power consumption.
In addition, a high pump speed means that the valve must be switched at higher
frequencies in order to reduce oscillations. This increases power consumption further.
The oscillations and power consumption can be reduced by running the pump at a
lower speed, but this reduces the force levels.
5.2.4
Hybrid Control
The third control algorithm is called "Hybrid Control," and combines both pump
speed control and valve PWM control. The key idea is that for control outputs, u,
greater than a certain threshold value it is desirable to make use of the full capacity of
the pump but at small signal levels it is more appropriate to avoid nonlinear behavior
by using valve PWM. Hybrid control is an attempt to directly address the tradeoffs
between speed of response and and steady state oscillations
/
power consumption
associated with valve PWM.
The algorithm consists of two modes: pump speed control for large u, and valve
PWM control for small u. As Fig. 5-3-c illustrates, if the input lies within the "fine
PWM zone," (A < u < A), the valve PWM control is used, but with a relatively
small pump output, F. The fine PWM zone is sized so that it safely encompasses all
pump speeds where unpredictable behavior such as dead zones are not present.
5.3
5.3.1
Heading Control Case Study
Heading Control Dynamics
For the control analysis we will focus on a simplified 1 DOF example. Specifically
we will analyze the heading angle. For the systems analyzed in this entire thesis,
the heading can be thought of as the same as the yaw angle, V). This is because the
vehicle is neutrally buoyant and is rarely pitched up or down. Therefore the gravity
77
vector is almost always aligned with the vehicle z axis. Heading control is critical for
many applications as it is necessary for navigation, maneuvering around obstacles,
The vehicle designs we discuss in this thesis
and aiming sensors such as cameras.
are capable of turning in place without translating. Figure 5-4-b shows how the dual
jets can be activated to create a pure yaw moment. This means that the idea of
decoupling the yaw dynamics from the other motions is realistic and realizable.
Y
(b)
(a)
Figure 5-4: Illustration of vehicle heading angle (a) and a diagram showing the capability of the vehicle to turn in place using dual jets (b).
We examine the case of stationary yaw turning. This means that u, v, w, p, and
q are all assumed to be zero. Performing a moment balance about the z axis leads to
the following differential equation.
(Izz - Ni) i-(t) - N,.,r(t) Jr(t) I - Nr(t) = Ni (W
51
These simplified dynamics emerge from the assumptions on the fluid flow as well
as the vehicle shape. The symmetry of the vehicle shape means that there are no
couplings between rotations and translations due to the moment of inertia or due to
drag. For the fluid forces we assume quadratic drag and added inertia due to rotations
in the fluid.
In the above equation, Nr,. represents a quadratic drag moment, N,
represents a linear damping moment, and N, represents an added inertia due to the
78
surrounding fluid. Note that due to our conventions, N,, and N, are negative. Even
at small maneuvering speeds, the Reynolds number for the flow is approximately
1000. This means that we can neglect viscous effects. The yaw moment created by
the jets is represented as Ni, and we simplify the expression by using C, to represent
the drag moment associated with rotations about the z axis.
Izi + Czr |rI = Nin (t)
5.3.2
(5.2)
Response to PWM Signals
The goal is to examine the system response to valve PWM. We neglect the fast
valve dynamics and treat the output of valve PWM as a series of positive and negative
step inputs of height No.
For the sake of simplicity we create a couple intermediate variables, a, and b.
a=
No
-
Iz'
b=
C
(5.3)
-
Iz
Therefore eq. 5.2 can be rewritten.
dr(54
d- = a +br(t) 2
(5.4)
dt
This expression is valid when r(t) < 0 and mod(t, TPWM) < d -TPWM. The solution is
obtained in two steps. First, separation of variables and integration are used to find
an expression for r(t) based on an initial velocity of ro. Second, the expression for
r(t) is integrated with respect to time giving an expression for Vk(t). The solutions
are given in eq. 5.5 and eq. 5.6.
r (t) = ebtan
cos (tan- 1/(ro ))
log
iD .+tant on
-1d(yo) )(vv u
ro
(5.5)
(5.6)
(
(t) =
Va-bt + tan-'(4
Due to the absolute value function in eq. 5.2, the above dynamic equation (5.4)
79
No
-No
Table 5.1: Summary of Solution Cases
r<0
r>0
Case 0: co = 1, cb=l
Case 1: cal =1, cb=-1
Case 3: Ca3 = -1, Cb3 = Case 2:
Ca2 =-1,b2= -1
and its solutions (5.5) and (5.6) may take different forms depending on the sign of
r(t) and the moment input No, -No.
There are four cases which are all listed in Table
1. Using the sign variables cai, cs, we can rewrite the dynamic equations.
dt
=r~t
-Cai a + cbi b ri (t)2,
i = 0, ---
(5.7)
The solutions to these cases are similar in structure but depending on sign variables
may involve hyperbolic tangents and hyperbolic cosines.
5.3.3
Modeling PWM Induced Vehicle Oscillations
Two of the three pump-valve control methods that we have proposed involve the
use of PWM valve switching.
This means that the steady state response of the
control system will have oscillations due to the valve switching back and forth at a
set frequency while the pump operates at a constant speed. A key issue is the size
of the output (yaw) oscillations that result from this method. If the oscillations are
small, then vehicle performance is not degraded. However, sizable oscillations can
prevent the vehicle from properly carrying out its missions.
In this section we use
mathematical analysis to determine the relationship between the yaw oscillations of
the vehicle in relation to the valve PWM frequency, fPWM, and the input moment
magnitude, No (which is approximately proportional to the pump operating speed).
We assume that the yaw oscillations are dominated by the fundamental frequency
of the input (fpwM). Simulation results shown in Fig. 5-5 illustrate that this is a
valid approximation for the types of parameter values we are considering. The steady
state oscillations include all four dynamic cases described previously, and they are
each shown in Fig.
5-5.
Each cycle is composed of four regions corresponding to
each dynamic case: #i(t), ri(t), ti; i = 0, ... , 3. At the boundaries between the two
regions the trajectories smoothly connect, having the same value for the angle and
angular velocity. Since this system is nonlinear, the duty cycle, d, is undetermined
80
Table 5.2: Summary of Region Boundary Conditions
Case (i)
to
tf
ro
rf
cai
cbi
0
0
To
rmin
0
1
TO
T1
T1
0
rmax
T2
TPwM
rmax
2
3
T2
0
0
rmi
1
1
1
-1
-1
-1
-- 1
1
and treated as an unknown. Similarly, the peak angular velocities, rmin, and r.a.
are also treated as unknowns. This means that a total of 9 unknown variables exist:
A
0
, Ai
1,
A02 , A'03 , ri, r 2 , T, T1 , T2 . The two integrations over each region provide
8 equations. The 9th equation comes from requiring the net change in angle over an
entire cycle to be zero.
3
Z Ai
=0
(5.8)
i=O
The system of 9 equations was solved numerically for values of 1 z, Cz, No that
correspond to a realizable prototype.
The solution values were then fed back into
a Matlab differential equation solver in order to simulate the dynamics over a full
cycle. The results of this simulation are provided in Fig. 5-5. To compare oscillation
level we use Oa which is half the peak to peak amplitude.
This figure also shows
the nonlinear nature of the oscillations; note how in this case the minimum value
for V) over a cycle is not the same as the maximum value over the cycle, creating an
asymmetric oscillation.
5.3.4
Approximate Closed Form Expression
The approach outlined previously allows us to develop exact solutions to the simplified nonlinear yaw dynamics.
However, they also require a numerical solver to
solve a set of 9 coupled nonlinear equations. This means that this approach is computational expensive and reduces physical intuition.
In order to evaluate various
approaches, numerous computations would need to be run. It is useful to also have a
simplified closed form approximation that can be used to estimate the yaw oscillations
without requiring complex computations.
We will use the same metric outlined previously of 0,I', or half the peak to peak yaw
oscillations. If we look closely at the response of the nonlinear system in Fig. 5-5, we
81
20
, 10
.. Ni(t)/No
(t)
-
30
Case 0
Case 1
N, (t)>0
N. (t)>0
r(tI)>
r(t)<o
2
\
0'
T~
Case 3
Case 2
-10-
:N(t)<0 N(t)<0
%b%
20 -
r(t)<0
r(t)>o
-30
0
0.2
-
0.4
0.6
0.8
1
tITPWM
Figure 5-5: A plot of simulated yaw oscillations. The 4 solution cases are each labeled.
see that V),, can be estimated by focusing on the change in angle associated with case
1. The integration period for this case is T*. Case 1 simplifies the analysis because the
initial angular velocity is zero rather than some unknown quantity. The integration
period, T*, is unknown but we will approximate it. If our system were linear, this
period would be quadrant: T*
-
TPwm
We will use this linear approximation for our
analysis in this section.
OTO
+
0e
T2
ri (t) dt
(5.9)
The expression for r, (t) is:
r1 (t) =
tanh (Va (t - To))
(5.10)
The closed form solution is therefore:
7P (t) ~
log cosh (VrT*)
(5.11)
We can then substitute for a, b, T* in order to obtain an expression that provides
physical intuition based on the key system parameters.
Oka
~
log
VNOCz TPwm)
cosh
82
(5.12)
We can use this expression to develop substantial information about the oscillation
response. Two dimensionless parameters are clearly visible from eq. 5.12. The first
quantity, 6-, serves to modulate the amplitude. Similarly, the argument of the cosh
function provides a nondimensional quantity that includes the dependence on time.
This quantity can be used to determine a characteristic time, tc, based on the forcing
amplitude, No, the drag coefficient, Cz, and the combined inertia, Iz.
tc
=
(5.13)
Iz
This characteristic time, tc, can be used to create a nondimensional frequency, f'
f
(5.14)
= fPwule
Iz ,(5.15)
1PW
fl =.
f'=~w m VN-~z
The nondimensional frequency, f' can be used as a very simple tool for estimating
the oscillation magnitude. If f' is less than unity, the oscillation amplitude will be
large, while if f' is large, then oscillations will be small. Figure 5-6 provides an illustration of the oscillation amplitude, 02, as a function of the PWM frequency, fPWM.
First, this figure shows that the simplified nonlinear closed form approximation outlined in eq. 5.12 matches the full numerical solution very closely. This means that
the approximation we have outlined can serve as an effective and accurate tool for
predicting the oscillation amplitude. Additional, Fig. 5-6 shows how the oscillation
amplitude grows very rapidly for frequencies, f', that are below 1. This means that
the nondimsional frequency described in eq.
5.15 can be used as a quick way to
evaluate and compare various designs or parameter values.
Finally, we can also explore the linear case.
Linearizing the dynamics for sta-
tionary turning causes the quadratic drag term to vanish.
Computing this linear
oscillation amplitude, VkLin, is very straightforward, and the result is provided in eq.
5.16.
83
25 %
-Closed Form Approximation
U Full Nonlinear Solution
20 --
15-
-
>-10
5-
1.5
2
1.5
2.5
Figure 5-6: A comparison of the full nonlinear solution with the closed form nonlinear
approximation.
1-aTin W
32
N
Iz
(5.16)
This linear approximation is quite good for cases when the combined inertia, Iz,
is larger than the drag coefficient Cz. For smooth robots such as those described in
this thesis, Iz ~ 4Cz. This means that for smooth omni-robots, treating the vehicle
as an inertia is an effective tool for simplifying oscillation analysis.
However, since the linear analysis does not consider drag there will be cases where
it does not provide accurate results. For more general cases such as robot with large
drag inducing components such as fins or propellers, the linear analysis can break
down. This means that the analysis techniques described in the preceding sections
have relevance depending on the needs of the designer and the characteristics of the
vehicle design.
5.4
Controller Designs
The CCSV-RAD prototype described in Chapter 4 is used to design our orientation
control systems.
The inertia and drag properties as well as other relevant vehicle
properties are summarized in 5.3. A block diagram of the heading control system is
provided in Fig. 5-7.
84
Table 5.3: Summary of CCSV-RAD prototype yaw dynamics properties.
Parameter
m
Iz
Cz
GF
LF
TA
ydes +
Yaw
-
Controller
Value
0.9[kg]
0.0017[kg - m 2
3.7 x 10-4[kg/m]
0.0125[N/V]
0.05[m]
0.033[s]
N
Pump-Valve
Controller
Vehicle
--
Figure 5-7: A block diagram of the heading controller.
5.4.1
Selecting PWM Amplitude
The amplitude of the PWM moment output is controlled by modulating the pump
speed.
Since the pumps have speed control circuits, this speed is modulated by
modulating the operating pump voltage, V. This voltage level can range from 0-12V.
Experimental results have shown that the dead zones usually exist in the range of
0-5V. In order to remain safely outside of these unpredictable dynamics, we choose to
operate at 7V. This voltage level provides a steady state yaw moment of 0.012Nm.
The PWM amplitude can be tuned according to the application requirements.
5.4.2
Selecting PWM Frquency
Once the PWM amplitude, No, is known, the analysis tools described in this
chapter can be used to select the best PWM frequency. For our vehicle prototype,
the nondimensional frequency, f', is 1.05Hz. This means that in order to avoid sizable
yaw oscillations, we must operate at a frequency above 1.05. To choose an exact value
we will first set a target oscillation level that we want to achieve. For this study we
select a target of V),, = 1.5*. This choice is entirely arbitrary and can be adjusted
based on various user or application requirements.
85
The analysis in eq. 5.12 can be used to quickly and simply generate design curves.
A design curve for the CCSV-RAD prototype is provided in Fig. 5-8, it shows the
predicted oscillation amplitude,
4',
as a function of the PWM frequency, fpwM. Hor-
izontal lines can be drawn across various oscillation levels to determine the minimum
frequency. Usually it is best to operate at this minimum frequency in order to minimimize power consumption and wear and tear on the valve. For our case study of
a = 1.50, this plot reveals that a minimum frequency of 2.5Hz is required. Similarly,
if we relax our requirements to 4*, we can operate at a frequency of 1.5Hz instead.
10
8-
-
6cc
fpWM1.5Hz
F~1.5*
f,,, =2.5Hz
8.5
1
1.5
22.5
3
f PwM [Hz]
Figure 5-8: A plot illustrating how a design curve can be used to tune the PWM
frequency.
5.5
Simulation
A simulation environment was developed using Matlab and Simulink. The simulation environment models the full nonlinear yaw dynamics and includes a simplified
model for pump-valve dynamics. The pump valve dynamics are approximated using
linear first order dynamics. We assume speed control over the pump, with a gain,
GF used to relate pump voltage to the force created by the
jets, and the moment
coefficient can be contributed by multiplying by the length of the moment arm, LF
(eq. 5.17). The factor of 2 in this equation is due to the presence of dual maneuvering
jets.
86
(5.17)
GM = 2LFGF
As we have described, there exist several nonlinear effects that can complicate
control. These include the variation of rise times with the input, un-developed fluid
dynamics at low impeller speeds, and dead zones due to electrical and mechanical
issues. For these simulations we do not attempt to replicate all of these effects but
instead focus on the presence of dead zones because they have clear consequences for
precision feedback control. As we illustrate in Fig. 5-9, we assume a simplified static
model where there exists a dead zone, 6. We approximate the relationship between
the pump voltage and the pump force as linear. This is not exactly correct (it is
quadratic), but this does not affect control performance greatly.
F
GF
VP
Figure 5-9: Illustration of the simplified pump-valve static model.
The pump-valve dynamics are approximated in a similar way. As previous chapters have shown, the dynamics are nonlinear, and the rise time is dependent on the
force level. However, for the sake of this study we will approximate the pump-valve
system with simple first order dynamics. Since experimental results have shown that
both the pump and valve have similar dynamics, we use a single time constant, rA,
to approximate the pump valve dynamics.
TA
dF (t)
dt + F (t) = GFVP (t)
dt
87
(5.18)
5.5.1
Pump Speed Control
Pump speed control involves using the pump speed to control the magnitude of
the yaw moment while the direction of the moment is controlled by the fluidic valve.
A simple PD controller was designed using gains that balance implementation issues
with control performance.
Two simulations are provided in Fig.
5-10.
The first
simulation is of the controller performance under ideal conditions (no dead zone). As
the figure illustrates the controller has no overshoot and converges within 2 seconds.
A second simulation included a dead zone, 6 = 3V. As Fig. 5-10 shows, this small
change results in a steady state error.
This is due to the fact that the controller
commands small forces within the dead zone, but the pump-valve outputs zero force.
Larger dead zones lead to larger errors. Increasing the controller gains can reduce the
steady state error, but at the expense of increased overshoot.
200
150-
0 100-
50-PD Pump Speed Control
--- PD Pump Speed Control with deadzone
-. d(t)
1
2
Time [s]
3
4
5
Figure 5-10: Simulated response with and without a dead zone.
There are two common ways to reduce the effect of the dead zone. The first is
to simply introduce an integral term to the controller.
the error is driven to zero.
As the error accumulates,
However, the introduction of an integrator increases
overshoot and reduces stability. This is illustrated clearly in Fig. 5-11 which shows
how the integral control reduces the steady state error but at the expense of increased
overshoot. Figure 5-11 also illustrates the chattering effect discussed previously. As
88
the error goes to zero, small amounts of noise cause the error to change sign at high
frequency. This occurs at t ~ 7s and is highlighted in Fig. 5-11-b.
2
200
High Frequency Chattering
Large overshoot
1
150
O0.-
0 100
-
-1
50
-d(t)
-V(0
00
5
Time [s]
510
10
(b)
(a)
Figure 5-11: Simulated response with PID pump speed control with deadzone. The
integral action slowly eliminates steady state error (a). Note the presence of chattering
in the valve signal (b).
The second way to remove a dead zone is to characterize it and use feedforward
compensation to cancel it. This approach has been put into practice on numerous
occasions ([51] is just one example), and is generally effective. The downside to this
approach is that it is reliant on a very accurate model.
If the dead zone is not
characterized properly, a dead zone can still exist, or the robot will have trouble
remaining stationary. Practical considerations make this more challenging, as various
conditions such as deposits, wear, or sags in the battery voltages can all cause the
dead zone to behave differently.
5.5.2
Valve PWM
We can use PWM control to improve the control performance. We use the specific
design amplitude and frequency outlined previously (V = 7V, fPWM = 2M5Hz. Figure
5-12 illustrates how the use of valve PWM control can improve performance. Figure
5-12-a shows how the 0 time response now has an improved rise time and settling
time when compared to the PID pump speed controller discussed above. The output
of the pump-valve system is shown in Fig. 5-12, and reveals the on/off nature of the
force output. When an error of zero is reached, the duty cycle becomes approximately
89
50 percent. This means that simply maintaining a heading consumes power from both
the pump and the valve switching. Therefore, the improvement in accuracy comes
with a significant energy cost. As a result, it is advisable to only use valve PWM
control for very specialized applications where such accuracy is required. For the sake
0.2
-
0......o.
200
0PWM
am"
at 2.5 Hz
0.1
Improved settling time
150
Switching
Reduced overshoot
0
0 100 --
50 -
065
-0.1
-VEd(t)
-- Pump Speed Control
-Valve PWM Control
10
Time [s]
1
-0.26
50
Time [s]
(b)
(a)
Figure 5-12: Simulated response with PD valve PWM control. The improvement in
the overshoot and settling time is shown in the angle trajectory (a) as well as the
force output of the pump-valve system(b).
of comparison, a second simulation was performed with different PWM parameters.
In this case the PWM frequency,
fPWM, was set to 1.5Hz in order to reduce power
consumption. The results, shown in Fig. 5-13, reveal that the oscillation amplitude
increases from 1 to 2.6*. This is a good illustration of why the analysis techniques
in this chapter are important. A comparison with our model predictions reveal that
the closed form nonlinear approximation (Fig. 5-13) provides reasonable accuracy.
5.5.3
Hybrid Control
We can use hybrid control to provide additional improvement to the PWM controller. Specifically, we modulate the pump speed outside of the "fine PWM zone,"
but for small commands the pump runs at V and uses valve PWM to modulate the
force. This allows the controller to use a larger range of pump speeds, specifically
those larger than those associated with V. This means that larger forces can be applied to respond to large commands or disturbances. As Fig. 5-14-a shows, the use of
90
y~1.2'
182-
\j
1 5178--.80\
100
,
150
-
S176
174
Md(t)
... fPWM =1.5Hz
50
172
4.5
fPWM 2.5Hz
0
2
4
Time [s]
6
8
(a)
~.2.45
5
-Vd(t)
... fPwM -1.5Hz
fPWM = 2.5Hz
5.5
6
6.5
Time [s]
(b)
Figure 5-13: Valve PWM using 2 different PWM frequencies. Note how the oscillations for fpwM = 1.5 are significantly higher than those for fPWM = 2.5Hz.
I
hybrid control improves the rise time when compared to the valve PWM control. The
hybrid control method is shown in Fig. 5-14-b, where the pump speed is modulated
at large errors but for smaller errors, PWM is used.
5.6
Experiments
Finally, we conclude this chapter with an experimental evaluation of the 3 pumpvalve control algorithms. We use the CCSV-RAD prototype to examine yaw control.
Custom software was written for the onboard microcontroller in order to implement
the control algorithms.
Wireless communication was used to log the data using a
custom Matlab script. The experiments were performed in the d'Arbeloff Laboratory
using a small tank. For safety the vehicle was made slightly buoyant. All experiments
described in this chapter were performed with the vehicle at the water surface. An
illustration of this experimental system is provided in Fig. 5-15 which shows a series of
frames taken from videos of the experimental trials. This video was taken for recording
purposes only and does not provide the resolution for good angle measurements. The
black lines are used to provide a reference for the robot angle since the robot itself is
symmetric.
91
0.4
200
Improved rise time
0.2
150
Higher forces for
improved transient
response.
-1000
50
-0.2
""""V(t)
00
.Valve
2
PWM Control
Time [s]
4
Valve PWM Mode
Pump Speed
Control
-Hybrid
Mode
6
~0'40
2
Time [s]
4
6
(b)
(a)
Figure 5-14: Simulated response comparing valve PWM with hybrid control. Note
the improvement in the rise time (a) as well as the amplitude adjustment in the force
output (b).
5.6.1
Pump Speed Control
The first controller that we assessed was the pump speed controller. A PD controller was used due to concerns about the stability reduction associated with integral
control. As with the our simulations, the pump speed controller clearly has problems
with the dead zone. The dead zone causes over shoot and large steady state errors.
The steady state error slowly reduces but actually oscillates at a very low frequency.
The experimental data in Fig. 5-16 shows clearly the negative consequences associated with using pump speed control.
5.6.2
Valve PWM Control
We use valve PWM control to eliminate the negative effects of the dead zone.
Here again the valve PWM control provides a clear improvement by quickly reducing
the steady state error to below 10. A closer examination of the figure reveals that yaw
angle oscillations are barely visible. Figure 5-17 provides a close up view of the valve
PWM response.
Despite the oscillations in the response, the errors remain below
approximately 10.
92
Figure 5-15: A set of frames from the hybrid control experiment. The dashed black
lines are a reference to illustrate the angle tracking. The dot is used to indicate the
front of the robot, and the arrows indicate the direction of the vehicle angular velocity.
200
150
Improved steady state response
with valve PWM.
(D
0 100
50
-d(t)
0
0
2
4
Time [s]
6
--- Pump Speed Control
-Valve PWM Control
10
8
Figure 5-16: Experimental data from the CCSV-RAD prototype showing the performance improvements from using valve PWM control.
93
181
180.5--
--
180-
--
-
>179.5
179178.5
PWM Control
I-Valve
5
6
7
Time [s]
8
9
10
Figure 5-17: Zoomed in view of the valve PWM control experimental results.
5.6.3
Hybrid Control
Lastly, the hybrid controller was evaluated experimentally and compared with the
results from valve PWM control. As Fig.5-18 shows, the hybrid controller provides a
substantial improvement in the rise time. The transition between hybrid control and
valve PWM control is very smooth and is not noticeable in the experimental results.
The steady state response of the hybrid controller is identical to that of the valve
PWM controller.
5.7
Summary
This lengthy chapter provided a complete description of achieving precision feedback control with our novel pump plus fluidic valve system. We outlined two control
schemes (valve PWM control and hybrid control) that exploit the high speed switching of the valve in order to avoid the nonlinear pump dynamics. A heading control
case study was analyzed, and a set of tools were developed for estimating the yaw oscillations that arise from the valve PWM switching. These analysis tools are general
and can be applied to other servo problems such as position control. Finally, these
concepts were evaluated in depth using simulations in order illustrate the performance
tradeoffs. Experimental results using a fully functional prototype robot confirm the
efficacy of these control approaches and show that orientation control with degree
94
200-
150Hybrid control provides faster rise time
compared with valve PWM.
C-)100 --
50'uNd(t)
0
0
-Hybrid Control
-- Valve PWM Control
2
4
Time [s]
6
8
10
Figure 5-18: Experimental data from the CCSV-RAD prototype showing how hybrid
control can be used to improve the transient response.
level accuracy is possible.
95
96
Chapter 6
CCSV Directional Stability
While in Chapter 5 we examined stationary turning, in this chapter we will explore
the CCSV dynamics when the vehicle is moving at a cruising speed, Uc. These dynamics are actually quite interesting due to the presence of the Munk moment and
they hydrodynamic instability that results from it. We will outline a well known 6
DOF nonlinear model, linearize about a planar trim state, and then use the linearized
vehicle dynamics to inform both vehicle design and controller design. A feedback controller for the yaw angle, ,,
6.1
is developed, simulated, and verified using experiments.
Vehicle Dynamics
The hydrodynamic equations of motion for the maneuvering of a general 6DOF underwater vehicle are complex, coupled, and nonlinear [17]. However, for the spheroidal
vehicles that we describe in this work, the equations of motions simplify considerably
due symmetry.
1. The inertia matrix is diagonal (cross terms are zero). This is due to the symmetry of the shape.
2. The dominant hydrodynamic forces are from drag, added mass, and Munk
moments.
Due to the symmetry of the vehicle, added mass and drag cross
terms are zero. This is a common assumption [17], [58].
3. We assume that the dominant drag is quadratic.
Reynolds number (~
40, 000).
97
This is based on the large
4. The center of mass is positioned at the volumetric center of the vehicle.
5. The actuator dynamics are sufficiently fast and can be neglected from preliminary modeling and controller design.
In practice, the center of mass will never be exactly at the volumetric center. We
place it slightly below the geometric center in order to create passive stability for pitch
and roll. We assume that this distance is very small compared to the other dynamic
parameters, and this allows us to simplify the governing dynamics considerably.
The equations of motion are developed by taking the sum of the forces and moments at the vehicle center of mass. The same body fixed coordinate system (xyz)
outlined in previous sections, is used. To assist the reader, the coordinate frame is
shown in Fig. 6-1. The XYZ coordinates are used to denote a stationary coordinate
system, while the xyz coordinate system is fixed to the vehicle body. The quantities
u, v, w are used to denote translational velocities about the x, y, z axes respectively,
while p, q, r denote angular velocities about the x, y, z axes.
X
Xr
q
yv
+w
Figure 6-1: An illustration of the body fixed coordinate frame.
Ocean and aerospace engineering texts such as [48] and [59] can be used to obtain
the added mass and drag coefficients. Ocean engineering convention is used to denote
the added mass and drag. For example, -Xu is used to represent the quadratic drag
associated with translations in the surge, or x, direction. Similarly, -Xi,
is used to
denote the added mass associated with translations along the x axis. To denote the
98
jet forces, we use the notation Fji to represent forces from Jet 1. The same notation
is used for Jets 1-4.
The equations of motion are provided in eqs. 6.1-6.6.
m [du + qw - rv] = (FJ1+ F 2 )cos (7J) +X
(6.1)
+Xuu ul(
m
[
+ (-F
+ ru - pw = Y,,
1
+ FJ2) sin(y)
(62)
+Yv lvi
[dw
m
+ pv - qu]
Zdw
Z
+ FJ3 + F4
IxxP + (Izz - Ivy) rq = K
-
q
=
+ KPp p|
dt
(6.4)
-FJ
+ (FJ3 - F4) LFy
IYA + (Ixx - Izz) pr = M4
(6.3)
(65)
+Mqqq I|I
Izzr + (Ivy - Ix) pq = Nj1 - N2(66)
+N, - + Nr
Irl + NM
The yaw moments from the jets, NJ1 , NJ 2 , are denoted by
Ni1 = FJ1LFz
(6.7)
NJ2 = F2LFz
6.1.1
Munk Moment
A key factor in the vehicle dynamics is the Munk moment. The Munk moment
was first observed for dirigibles in the 1920s, and is the tendency of streamlined bodies
to rotate so that they are oriented perpendicular to the flow. This effect stems from
the asymmetric distribution of stagnation points on the vehicle body.
This leads
to local pressure gradients that create a pure moment. A simplified illustration is
provided in Fig. 6-2, which shows how the high and low pressures, P at each end of
99
the vehicle body create a pure moment. This moment tends to destabilize the vehicle,
and prevent it from moving straight.
LowP
High P
High P
Figure 6-2: An illustration of the Munk moment and how the stagnation points create
a turning motion on streamlined shapes.
The Munk moment can act to create pitching moments, KM, as well as yaw
moments, NM. The moment is a function of the free stream velocity, U,,, and the
angle between the flow and the vehicle. For pitching moments we use the angle of
attack, a, and for yaw moments we use
#.
The moment is also related to the shape
of the vehicle. Many simplified models use the difference between the added masses
associated with translations to approximate the Munk moment.
Mm = U' cos (a) sin (a) (-Z,, + Xi)
NM
6.1.2
=
U2 cos (#) sin (0) (-Ye + X&)
(6.8)
(6.9)
Countering the Munk Moment
Since the Munk moment leads to destabilization, methods have been developed
for countering it. Perhaps the most common technique, is to use hydrofoils placed at
the tail of the vehicle to create a lift force , L, and a drag force, D, at the tail. The
lift force creates a restoring moment that counters the Munk moment. For example,
the yaw moment, NL, from a lifting force created by a hydrofoil at the tail can be
approximated using a common expression from aerodynamics. Where CL represents
100
the coefficient of lift, AF represents the area of the foil, p represents the fluid density,
and LF represents the distance between the fin and the vehicle center.
NL ~~-pAFLFU002
2
L/
(6.10)
dfi
As this expression illustrates, it is possible to design fins that can counter the
Munk moment for all speeds (as long as the sideslip angle remains small).
This
technique is particularly useful because hydrofoils can be designed to have very small
drag coefficients for small sideslip angles, and this is a form of passive stabilization
which requires no feedback or power consumption.
This is why many AUVs and
torpedoes have fins at the tail.
However, while tail fins work quite effectively for forward motions, there are several
drawbacks. The first is that while fins stabilize forward motions, they destabilize the
vehicle if the vehicle direction is reversed.
As Fig.
6-3 shows, when the vehicle
direction is reversed, the lifting moment, NL, now acts to turn the vehicle further.
While this is not necessarily a problem for many AUVs or torpedoes, it is a problem
for the types of inspection robots we hope to create.
U
LF
U.
F
LF
If direction is reversed, fins act to
destabilize the vehicle.
Fin acts to stabilize forward motions.
Figure 6-3: A diagram illustrating how fins can provide both passive stabilization or
destabilization based on the direction of the vehicle.
Additionally, fins can reduce maneuvering performance by creating induced drag
during aggressive turns and by creating an asymmetric shape. This asymmetry can
lead to coupled motions such as coupling between translations and rotations that can
inhibit precision motions. Finally, fins are external appendages that increase the risk
of tangling on obstacles and fins can be damaged from collisions with the environment.
101
For all these reasons, the use of tail fins is highly undesirable. This is a good example
of how existing techniques are not suitable for these types of new robot systems.
Clever design can be used to stabilize the pitch axis against the Munk moment.
Placing the center of gravity below the center of buoyancy means that pitching motions can be countered by gravity. Therefore, no external appendages such as fins
are necessary to stabilize pitch. This effect can be quantified by estimating the z
offset, z, required to stabilize the vehicle against the Munk moment. A moment
balance between the pitching Munk moment, Mm, and the gravitational restoring
torque leads to the following expression for z,.
ZS
U (-Zj, + XI)
mg
The result for ze is dependent on the cruising speed, Uc, the added masses, and
the mass, m, of the vehicle. The z offset, z., can be nondimensionalized by dividing
by the radius of the vehicle, R. For spheroidal shapes which are not highly elongated,
this ratio, a is very small (0.022 for CCSV prototype 1). This signifies that the offset
needed to stabilize the pitch moment is very small compared to the other dimensions
of the vehicle. A value of - that starts to approach 0.5 means that the vehicle will
be very challenging to stabilize this way and will probably require tail fins or some
other method. If the vehicle shape is elongated, or if it is propelled at higher speeds,
- would rise substantially.
R
6.1.3
Closed Loop Stabilization of Yaw
If we assume that the vehicle pitch and roll angles remain relatively small, the
gravity vector will remain nearly parallel to the vehicle z axis.
As a result, yaw
motions cannot be stabilized using gravity. This means that some other method must
be used. There exist similar challenges in aeronautical engineering. For example, high
performance fighter aircraft are designed with Relaxed Static Stability (RSS) or even
instability in order to reduce size and weight [60, 61]. These vehicles achieve superior
performance and weight reduction by stabilizing through the use of feedback control
rather than large external stabilizers.
102
Similarly, work has also been done on directionally unstable ocean vehicles. The
most prominent examples are large oil tankers which are slightly unstable [58]. Considerable work has explored using rudders to stabilize such vehicles, and thrust vectored jets can also be used for feedback control. Our case is slightly different, where
we use two fixed jet thrusters to control the vehicle. The vehicle heading is stabilized using differential activation of Jet 1 and Jet 2 to create motions that counter
the Munk moment without the use of fins. This is an area that is still somewhat
unexplored and requires further examination.
6.2
Linearized Planar Dynamics
6.2.1
Linearized Equations of Motion
We can use this rigid body hydrodynamic model to design a stabilizing control
system for the vehicle. Since these equations are nonlinear and coupled, it is difficult
to develop physical intuition or design a controller. By linearizing the dynamics about
a planar equilibrium trajectory, we can simplify the dynamic equations and then use
well known control systems theory.
We choose a straight line longitudinal motion as our trim trajectory. We assume
the vehicle is moving forward at a cruising speed, Uc, and all other velocities and
angular velocities (v, w, p, q, r) are assumed to be small. The pitch and roll angles
(q$, 0) are also assumed to be small. We are then left with 6 linearized equations of
motion in eqs. 6.14-6.19. In these equations we use the symbol A to denote small
perturbations about the straight line equilibrium state, and the Munk moments are
linearized using the following approximations for
4
-
#
and ci.
-v
(6.12)
Uc
Aa =
-
(6.13)
Uc
= (AFJ1 + AFJ2 ) cos ('YJ) + 2X,,U lAul
(m - Xj') d (Au)
dt
103
(6.14)
I) dt
(M - (mY)d(Av)
= (AFJ2 - AFjj) sin (yj) - mUcAr
dAw
(Z,; + m) dt = AFJ3 + AFJ4 + AqUc
d Ap
(I., - Kp,) dt
dAq
(IVY - M4) qdt
-
- MgzA0
(-AF 4 + AFj 3 ) LF,y + UcAw (-Y, + X&) - mgzAG
(I~z - N.) d (Ar)
dt
=
-UcAv (-Yi, + Xi) + ANJ1 - ANJ2
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
A quick examination of these linearized equations revealed that they can be broken
up into 4 groups. Yaw and sway motions are coupled due to the Munk moment and
due to centrifugal forces. Similarly, pitch and heave are coupled with each other.
Surge motions are completely decoupled from the other 5 degrees of freedom. Roll
motions are also completely decoupled. Since our concern is the heading of the robot,
we focus on the sway-yaw dynamics. These sway-yaw dynamics are a common result
that is well known in the Ocean Engineering field, and substantial work has been
done on using this type of model to design heading controllers [58].
FU .
~
J2
Figure 6-4: An illustration of the key parameters for the planar vehicle model.
104
6.2.2
State Space Representation
For the sake of achieving yaw stability we will therefore only focus on the coupled
yaw-sway motions (eqs. 6.15, 6.19). A diagram illustrating the relevant variables for
the planar dynamics is provided in Fig. 6-4. The yaw rate, r, is the time derivative of
the yaw angle, 0, and the angle -y represents the angle of Jets 1 and 2. For dynamic
analysis and control of the sway-yaw system, we use use state space form, where
our three states are Av, Ar, AV). The state space form of the linearized dynamics is
provided in eq. 6.22. We put these equations in the context of closed loop control by
using the pump voltages for Jet 1 and Jet 2, AV, AV 2 as our inputs. We assume that
the forces and moments are proportional to the pump voltages within the linearized
regime. We use the variables GF, GM to represent the force and moment coefficients
respectively.
AFjj= GFAV1
AV
d
Ar
dtIz-N
A
=
IY1
Uc(-
0
0
0
+xi)
(6.20)
AN
1 =
GMAVl
AF
2
=
GFAV 2
ANJ 2
=
GMAV 2
-mur
0
Av
0
0
Ar
1
0
Alp
(6.21)
+
-GF sinyj
GFsinyj
GM
-GM
Izz-N,
I..z-N
0
1
M-Yo
0
L
A
2
(6.22)
To simplify our analysis we combine several parameters together. We use m. to
represent the combined fluid and rigid body mass along the y axis, and use I, to
denote the combined inertia for rotations about the z axis. Similarly, we use Am, to
represent the difference between the added masses. For streamlined vehicles, Am, is
positive.
MY = m -
105
(6.23)
IF
A
Iz
Izz - N.
Ama = -Y, + X.
6.2.3
(6.24)
(6.25)
Linearized System Characteristics
The state space model can be used to provide insights into vehicle design. The
eigenvalues of the state transition matrix can be used to examine the open loop
system dynamics.
The eigenvalues provide the system poles, Pi, P2, P3, which are
shown below.
Pi
0
=
P2
P3=
-
2
U AmM
(6.26)
mI
These expressions for the poles provide a number of valuable insights into the
system. First, p3, is positive and real, meaning that it is an unstable pole. This result
serves as a good reality check, it shows that the Munk moment which destabilizes the
vehicle motions creates unstable behaviors when linearized. The magnitude of p3 is
related directly related to Ama, and U. This means that if the vehicle is streamlined
and moves quickly, the poles move farther apart along the real axis and become more
unstable.
The state transition matrix is not the only part of the state space model that
provides valuable insights. The input matrix also provides very interesting results.
Specifically, the input matrix shows the importance of the jet angle, -yj. If the jets
were not angled (-yj = 0), the vehicle would have no control authority in the sway
(v) direction. The controllability matrix, C would be of only rank 2 (full rank = 3),
meaning the system is uncontrollable. This critical coupling between the sway and
yaw motions is not necessarily obvious. A designer who simply assumed that control
authority was only necessary for yaw motions would end up with a system that is
uncontrollable with poor performance regardless of controller design.
Instead, by angling the jets we achieve control authority over yaw and sway simul-
106
taneously. This is the key example of a control configured approach to underwater
vehicle design.
Design decisions have clear ramifications for control performance,
and jet angle is one of the most critical parameters for our system. In the following
sections we will show just how valuable the jet angle is and how poor selection can
degrade control performance
6.3
Feedback Controller Design
Now that we have a linearized model, we can explore using feedback control to
stabilize the system. Full state feedback, which is a powerful technique, is not the most
practical for this application. Measuring the sway velocity, v, is a challenging task, the
velocities are small making the use of differential pressure difficult. Accelerometers
are another option but they are sensitive to coupled motions and drift. An observer
can also be constructed, but this introduces additional computational and analytical
complexity to the problem.
A simpler approach is to use only the yaw rate, r, and the yaw angle, V). These
measurements can be obtained using affordable and simple off the shelf components
such as compasses or Inertial Measurement Units (IMU). This is a well known technique which is used for ships that do not have the the sensors for full state feedback
[58].
Therefore, we convert our system into a SISO system where the input to the
system is a voltage, AV, and the output is the yaw angle, V;. Since our physical
system has two inputs, AV, and AV2 we have to develop a way to convert it to a
single input system. We observe that for both the the sway and yaw dynamics, it
is the difference between the two jet voltages (AV - AV2 ) that acts to create a net
force or moment. Therefore, the system can be treated as having only a single input.
.
We therefore propose the following relationship between AV1 and AV 2
AV2 = -AV
(6.27)
Now that we have a single input, we now have to select our output of interest.
Our trim state involves a vehicle moving in a straight line, so it makes sense to treat
107
the heading angle, AO, as our output. We can then convert our dynamics in eq. 6.22
to a single transfer function.
_
2 (sGum, + Am.UcGF sin 7-j)
s (mIs2
- U2maM
(6.28)
)
&0(s)
AV,1(s)
Im(s),
Im(s) AVg(s)
AV,(s)
Z,
P2
Re(s)
ZC
Zj
P
P3
Pi
Re(s)
(b) Root Locus Plot
(a) Pole-Zero Diagram
Figure 6-5: A pole-zero plot (a) and root locus plot (b) for the SISO vehicle maneuvering system.
6.3.1
Open Loop Performance
We use the transfer function in eq. 6.28 to analyze the system dynamics using
pole-zero analysis. The pole zero plot for the SISO system is provided in Fig. 6-5. The
zero, z, is a key result of our analysis and plays a critical role in the dynamics of the
system. If the zero is located on the left half plane, (- > 0), the zero helps to stabilize
the vehicle. If the zero lies on the right half of the complex plane, (7 < 0), then the
system becomes non-minimum phase, which makes it hard to control. Finally, if the
zero is at the origin, a pole-zero cancellation occurs and the full state space system
becomes uncontrollable.
The zero location can be tuned using design tradeoffs on
both the moment coefficient, Gm, as well as the jet angle, -y.
=
-GFUc Am. sin -
(6.29)
(GMm
We can interpret this critical behavior using physical arguments. If we consider the
situation outlined in Fig. 6-6, where the vehicle receives some negative disturbance
to the sway velocity, v, resulting in a change in the sideslip angle, 6. If the jets are
108
not angled, the vehicle can create a restoring moment but does not counter the sway
velocity. The only mechanism that can drive the sway velocity to zero is drag, which
is neglected in the linear model. However, even in the case when drag is included,
the control performance is poor (Fig. 6-9).
If the jets are instead angled inwards, as in Fig. 6-6-b, the jets can provide both
a restoring moment and a y force. In this case, the net force on the vehicle, Fe, has
a -y component. This y force acts to reduce the sway velocity while also restoring
the vehicle angle. By reducing the sway velocity, the jets act to directly reduce the
Munk moment as opposed to simply trying to cancel its effects.
In addition, if the jets are designed poorly and angled outwards as in Fig. 6-6-c,
the y component of the resultant force is now positive. Now the jets act to increase
the sway velocity (and Munk moment). This behavior is manifested with a zero on
the right half plane. This type of zero greatly complicates control performance.
U
l6
NM
U
U,
NM
N
(b)
Jets angled inward
Restoring force in y direction
Jets angled outward
Restoring force in wrong direction
c
(a)
Straight Jets
No restoring force in y direction
(c)
Figure 6-6: A schematic diagram showing the role of jet angle in vehicle control
performance.
6.3.2
Controller Design
We can use the SISO model to design the appropriate stabilizing control system.
Specifically, we want a control system that can track a desired heading, 1/d(t). We can
begin by using qualitative analysis techniques because the relative pole-zero locations
do not change dramatically for physically reasonable parameter values. The pole-zero
diagram in Fig. 6-5 shows that simple proportional controllers will never stabilize the
109
system.
However, a PD controller can be used, as illustrated by the root locus plot in Fig.
6-5-b. We use the variables K 1, K2 to represent the derivative and proportional gains
respectively.
AV 1 (t) = KT
d
(?Pd (t) - V)(t))
K 2 (?d(t) - V(t))
(6.30)
We choose to locate the controller zero, zc, so that it is faster than the plant
zero, z 1 . As long as ze is faster than z 1 , z, will dominate the system response. An
interesting result is that the closed loop sway dynamics,
"
, have the same poles as
the heading control dynamics. As a result, v will converge to zero even in the absence
of direct feedback. Steady state disturbances can lead to errors, but impulsive ones
will be rejected.
6.4
Comparison with Conventional Approaches
As mentioned previously, the most common approaches to heading control are to
use either a rudder or a vectored jet. Interestingly, rudder and thrust vectoring based
control systems have a very similar pole-zero structure to the one shown in Fig. 6-5-a.
Both a rudder based and thrust vectoring based systems have a left half plane zero
that helps stabilize the system. In fact, PD heading control based on limited feedback
has been shown to be effective for rudder-based stabilizing systems.
Similarly, if the rudder is placed improperly (at the nose rather than the tail), the
zero ends up in the right half of the complex plane and makes the vehicle very hard to
stabilize (similar to having jets angled outwards). It is important to note that both
rudders and vectored jets use a position controlled servomechanism to manipulate the
rudder/jet. These systems have their own dynamics such as slew rate limits. Such
slew rate limits are known to complicate feedback control and can reduce feedback
control performance [601.
6.5
Implementation
The control design techniques outlined in this chapter we applied to the CCSVRAD prototype described in Chapter 4. A summary of the relevant vehicle properties
110
Table 6.1: Summary of Physical Properties for CCSV-RAD Prototype.
Parameter
m
Xi,
XuU
Y&,
YV
IZZ
Ni.
Nrr
Uc
Value
0.90[kg]
-0.27[kg]
-3.5[kg/m]
-0.54[kg]
-3.17[kg/m]
0.0015[kgm 2]
GM
-0.00015[kgm 2]
-0.0006[kgm 2]
0.18[m/s]
0.011[N/V]
5.42 x 10 4 [Nm/V]
'Yj
300
K1
10[Vs/rad
20[V/rad]
GF
K2
is provided in Table 6.1. The tables in [48] and [59] were used to estimate the vehicle
drag and added mass. The simplicity and symmetry of the spheroid shape greatly
eased modeling the vehicle.
6.5.1
Controller Design
A PD controller was designed to stabilize the system. The controller was designed
so that PD zero, ze, was located at s = -2rad/s which was substantially faster than
the plant zero (zi = -0.34rad/s). The closed loop poles, PCL,1, PCL,2, PCL,3 are shown
in Fig. 6-7. Note how a closed loop pole (PCL,1) moves towards z, as the controller
gain is increased. This pole is the slowest pole, and will therefore dominate the closed
loop system response.
The numerical values for the controller gains, K1, K2, that
achieve these closed loop dynamics are provided in Table 6.1.
6.5.2
Vehicle Simulations
A full nonlinear simulation environment was created in
MATLAB,
and this software
was used to simulate the vehicle dynamics for a variety of cases. The first test was
to simulate the open loop vehicle dynamics. The vehicle was assumed to be traveling
along its equilibrium trajectory at a constant speed, U.
the sway velocity was imposed at time t = 0.
111
As Fig.
A small perturbation to
6-8 illustrates, without
A y(s)
p
AV (s)
P2 = -1.84
Im(s)
Ze P2
PCL,3
PCL,2
Z,
P1
PCL,1
P3
Re(s)
=0
p 3 =1.84
z, =-0.34
c ==
PCL1
-049
.4
PCL,2 =
-2.47
PCL.3 = -3.74
Figure 6-7: A pole zero plot for the closed loop SISO system. The closed loop
poles (PCL,1, PCL,2, PcL,3) are shown as are the zeros (zc, zi) and the open loop poles
(P1, P2, P3). Note how the system is stabilized using PD control.
feedback control, the vehicle is immediately destabilized and can no longer maintain
a straight trajectory. The vehicle trajectory with respect to a fixed coordinate frame,
(XI, YI, Z1 ) is shown in Fig. 6-8-a. The yaw angle trajectory diverges and grows
rapidly without bound (Fig. 6-8-b).
A second simulation was performed to explore the ramifications of jet angle and
uncontrollability. For this simulation the jet angle was changed from 300 to 0*. A
controller was designed and simulated using a small initial perturbation to v. As Fig.
6-9 shows, the system is stabilized but performs poorly. The yaw angle, i0 does not
converge to zero, and the sway velocity, v, decays very slowly.
Finally, the simulation tools were used to evaluate the stabilizing PD controller
for the CCSV-RAD prototype(7 = 30*). The response of the closed loop system to
the initial sway velocity is shown in Fig. 6-10. The figure illustrates clearly how the
feedback controller we have designed stabilizes the system and brings the heading
error to zero. Also, Fig. 6-10-b shows how the sway velocity also goes to zero despite
having no feedback. In order to evaluate the validity of the linearized dynamics, the
linearized dynamics are plotted in red. The linearized model and the nonlinear model
match quite well for the conditions evaluated in these simulations.
112
- Robot Trajectory (Open Loop)
-- Desired Trajectory
0.4
500
-- d(t)
400
0.2
'300
----------------------
0-200
1001
-0.2
u0.2
0
0
0.8
0.6
0.4
X, [im]
--------------4
3
2
1
Time [s]
5
(b) Yaw Angle
(a) XY Trajectory
Figure 6-8: Simulated response for the open loop performance of the robot vehicle.
The trajectory in fixed global coordinates,(XI, Y1), is provided in (a) and the yaw
angle is provided in (b).
0.2
5-
0.19-u(t)
;0.18
-
4
-3-
U.1
'6
''
or
'0
2
6
4
Time [s]
6
2
4
2
4
6
Tme [s]
S2-
Time [s]
-dt
8
;-0.04
8
-0.0%
10
--
8
10
10
vd(t)
10
Figure 6-9: Simulated results illustrating the poor control performance when the jet
angle, f, is set to zero.
113
0.185
.. 1
1$
-
-
-yd(t)
-Linearized Model
-Nonlinear Model
4
'l0.18
S0.175!
0.17
-3
All .......
-Linearized Model
-Nonlinear Model
10
8
6
4
Time [s]
0
0
1
-0.02
-
2
0
2.
4
6
Time [s]
8
10
-0.04
2
-Linearized Model
- Nonlinear Model
4
6
8
10
Time [s]
(b) Vehicle velocity components
(a) Yaw Trajectory
Figure 6-10: Simulations of the closed loop system response. The angle (a) and
velocity (b) results of both the linearized model and the full nonlinear model are
shown.
6.5.3
Vehicle Experiments
Simple experiments were performed using the CCSV-RAD prototype. These tests
were performed in a small tank in the d'Arbeloff Laboratory at MIT. Custom feedback
control code was written for the robot's microcontroller. Experimental data was
obtained by transmitting and recording the vehicle angle measurements using the
wireless radio link. In addition, video data was recorded using a hand-held digital
camera. These videos were frequently taken very quickly and can have some jitter.
Therefore, their main purpose is for recording the trajectory rather than very precise
measurement of vehicle kinematics. Videos from these experiments can be found
online at [62]. The experiments in this chapter were performed with the vehicle at
the surface. the vehicle was designed to be slightly positively buoyant for safety
reasons.
Straight Test
The straight test simply involved placing the robot in the tank and commanding
it to move at a constant speed while maintaining a fixed heading. An experiment
with the controller turned off (open loop) was first performed in order to clearly
114
illustrate the nature of the vehicle instability. The open loop experiment is shown in
Fig. 6-11-a. As the video data illustrates, the robot is unable to maintain a straight
heading and instead simply spirals. This matches the predictions of the simulation.
The angle trajectory of the open loop system is shown in Fig. 6-12-a. This figure
shows how the yaw angle quickly diverges and grows rapidly.
A second set of experiments were preformed with the control system running
(closed loop). Video trajectory of one of these experiments is shown in 6-11-b. As
the figure illustrates, the PD controller improves the vehicle performance drastically,
allowing it to maintain a very straight heading. Figure 6-12-b shows how the yaw
angle is maintained with very small errors (~
10).
(b) Closed Loop Control
(a) Open Loop Performance
Figure 6-11: Experimental video data showing the vehicle trajectories for both open
loop control (a) and PD closed loop stabilization (b).
Disturbance Test
Since the controller was designed using linearized approximations for a highly nonlinear system, it is important to examine how the system behaves when the linearized
dynamics are no longer valid. This is especially important for open loop unstable
systems were the failure of the control system can lead to catastrophic failure. For
these experiments, the vehicle was commanded to maintain a straight line trajectory
and allowed to reach a steady state velocity. The vehicle was then subjected to a
substantial disturbance. For the sake of simplicity, the robot was simply disturbed
115
with the hand of one of the experiments.
As Fig. 6-12 shows, this disturbance is quite large and causes the robot to deviate
by about 75'. This type of disturbance causes the vehicle dynamics to deviate substantially from the linearized case: the vehicle diverges by a large angle, the forward
velocity slow considerable, and the actuators saturate.
Nonetheless, the vehicle is
able to recover and return to its original heading within
-
50
-Closed
20
Loop control
-----------
-- L(t)n
40
0
0---pn Loop
30
-20
20
40
-
/
-60-
10
2
0
3s.
-
4
2
Time [s]
4
--
-80-
-- ...
0
6
avd(t)
2
4
Time [s]
6
(b) Disturbance Rejection
(a) Straight Test
Figure 6-12: Experimental data showing the vehicle angle trajectory for both a
straight test (a) and a disturbance rejection test (b).
6.6
Summary
This chapter outlined techniques to stabilize an appendage-free, spheroidal underwater vehicle. These vehicles are hydrodynamically unstable and generally have been
stabilized using external appendages such as tail fins. We instead stabilized using
feedback control. A linearized model was used to design a feedback control system
and the model also provided valuable insights into how CCSV design can be used to
influence vehicle performance and control.
116
Chapter 7
CCSV-RAD Prototype
Performance
The successful development and implementation of a stabilizing directional control
system in Chapter 6 enables full experimental evaluation of CCSV-RAD prototype.
Characterizing this novel design allows analysis of the vehicle performance and the
identification of unique abilities that could be exploited for a variety of applications.
This chapter will use experimental results to illustrate the unique performance of our
robot design. Four of the of five directions of motion will be illustrated clearly with
video and numerical data. Finally, the chapter will conclude with an experimental
comparison between the open loop unstable CCSV vehicle and an open loop stable
design (stabilized using a tail fin). These results illustrate the value in using relaxed
static stability techniques.
7.1
Vehicle Performance
The vehicle performance was evaluated using a variety of testing tanks at MIT. A
combination of angle data and video data was used to evaluate the robot performance.
Several standard tests of the vehicle speed and maneuverability were performed, and
the results are summarized in Table 7.1. A common technique in Ocean Engineering
is to nondimensionalize the length scale by dividing by the vehicle body length (BL).
Therefore, we provide the measurements in both SI units as well Body Length units.
All of the experiments except for the heave translations were performed at the surface
117
Table 7.1: Summary of Vehicle Performance.
Parameter
Forward Surge Velocity
Reverse Surge Velocity
Stationary Turning Rate
Stationary Turning Radius
Turning Radius at Speed (1.2 BL/s)
Sway Velocity
Heave Velocity
Value
0.175[m/s}(1.2BL/s)
0.175[m/s](1.2BL/s)
360[Degrees/s]
0[m]
0.15[m](1.03BL)
0.077[m/s] (0.52BL/s)
0.157[m/s(1.08BL/s)
of the tank. The CCSV-RAD prototype is designed to be slightly positively buoyant
for safety reasons and will sit at the surface in the absence of forces in the z direction.
7.1.1
Surge Performance
The most obvious experiment is to simply evaluate the ability of the robot to
move forwards. The closed loop controller designed in Chapter 6 was used to track
the heading. An experiment was performed to examine the ability of the robot to
move both forwards and backwards. The robot was commanded to move forward in
a straight line, but at time t = 3s was commanded to reverse direction. Video data
of this experiment is provided in Fig. 7-1-a. As the video data shows, the closed loop
control allows the robot to move straight forwards, and then reverse itself without
deviating substantially from its path. This is a maneuver that a robot stabilized using
tail fins would have trouble performing. The forward and backward velocity can be
estimated from the video data, and is shown in Fig. 7-1-b. The forward and backward
velocity is approximately 0.175m/s(1.2BL/s). Moving at 1 body length per second
is a common benchmark for underwater vehicles, and our prototype achieves that
metric. In addition, it is worth noting how the vehicle returns to a point very near
to its starting point (- 20mm) using only heading control. This illustrates that the
robot is capable of moving in the surge direction without coupled motions, which will
be very valuable from the perspective of precision maneuvering and navigation.
7.1.2
Sway Performance
One of the most interesting innovations in the CCSV-RAD design is the use of the
angled jets and fluidic valves to achieve sway motions without adding extra jets. This
118
0.21
--Vx
....
VY
0.1-
-0.1-0
0.~
2
4
6
8
Time [s]
(b) Velocity time profiles
(a) Vehicle trajectory
Figure 7-1: Video (a) and numerical (b) data illustrating the forward and back performance of the CCSV-RAD prototype.
motion is extremely valuable for inspection operations, because it enables horizontal
scans and allows the vehicle to be able to make fine adjustments in both x and y
independently. The sway motion is achieved using the Valve PWM control techniques
described in Chapter 5.
One of the pumps (depending on the vehicle direction) is
run at a constant voltage while a control system modulates the duty cycle in order
to maintain a desired yaw angle, 0. When the angle error is zero, the duty cycle is
approximately 50 percent and the vehicle translates sideways. This motion is enabled
by the performance of the fluidic valves which are able to switch quickly and without
creating large moments. Video data of a sway translation experiment is shown in
Fig. 7-2-a. As the video trajectory shows, the vehicle is able to maintain a relatively
straight sideways motion. The vehicle does have some x translations, but they are
relatively small. As Fig. 7-2-b shows, the vehicle translates less than O.lm in the
fixed X axis compared with 0.6rn in the Y axis. This means that the net vehicle
heading is about 70 rather than the desired 00. This performance is good, but it can
be improved further by including sensors such as sonar that can measure distances
and provide feedback on the vehicle position.
As we discussed in detail in Chapter 5, the use of valve PWM can result in oscil-
119
0.1
0
----- -....
-
-0.1)(
F-0.2--0.3-0.4-
-0.5
-0.6
-0
2
4
6
Time [s]
8
10
(b) Position profile
(a) Vehicle trajectory
Figure 7-2: Video (a) and numerical (b) data illustrating the sway translation capability of the CCSV-RAD prototype.
lations. In this case the oscillations are mainly manifested as yaw oscillations, with
the vehicle appearing to "wiggle" as it moves sideways [62]. If these oscillations are
too large, the robot will move in a zig-zag manner and may not perform a horizontal
scan properly. Identifying which level of oscillations is "too large" requires a study
of the application that is beyond the scope of this work. Therefore, the valve PWM
parameters were chosen arbitrarily for these experiments, with fpWM = 1.5Hz, and
Va varying between 0.6 and 0.8 of the maximum. Some examples of the angle tracking data is provided in Fig. 7-3. An example of the angle oscillations from a purely
straight sway translation is shown in Fig. 7-3-a. The oscillations at f = 1.5Hz are
clearly visible, but the oscillations remain relatively small (-3' to 3). When in sway
mode the vehicle can still control its angle and can servo to maintain a desired yaw
angle, Vjd. Figure 7-3 shows the angle data from the robot first turning 1800 and then
settling into pure y translation mode.
7.1.3
Stationary Turning
The ability to turn in place is extremely important for many applications. The
ability to orient the vehicle without changing the XY position is very valuable for
inspection tasks where it is necessary to precisely place cameras or other sensors. The
120
4A
IZ-
-- (t)
(t)
100 -)
------- ----
2-
- --- --- -- --- ---
80
.5;
a)
0
0
6040-
-2
20-
0
2
4
Time [s]
6
~0
8
(a) Straight Sway Motions
1
2
3
4
Time [s]
5
6
(b) Turning while moving sideways
Figure 7-3: Angle trajectory data for pure sway translation (a) as well as angle
adjustments while moving sideways (b).
ability to turn in place also makes remote operation using a video feed much easier.
The turning ability of CCSV-RAD prototype is showcased in Fig. 7-4. As Fig. 7-4-a
shows, the vehicle center only deviates a small amount despite turning at a relatively
rapid rate. The 0 data from the experiment is shown in Fig. 7-4-b, and illustrates
that the CCSV can not only turn in place but can achieve very rapid turning rates
of ~ 360/s.
0
-200
-400
0 -600
-800
-1000
-1200
0
1
Time [s]
2
3
Figure 7-4: Video (a) and numerical (b) data illustrating the turn-in-place capability
of the CCSV-RAD prototype.
121
7.1.4
Turning at Speed
While the ability to turn in place is important, sometimes more dynamic maneuvers are required. Being able to turn quickly and sharply while moving quickly is
valuable for efficiently traversing a complex environment, or for quickly reacting in an
unstructured environment. We evaluate the turn-at-speed capability by commanding
the robot to move forward and allowing it to reach a steady state speed. At time
t = 3s the robot is commanded to turn 900. We explore 90' turns because they are the
largest turns that will likely be required. This is because the CCSV-RAD prototype
is capable of reversing thrust and moving backwards which provides a more effective
way to reverse momentum than an 180* turn. The data from the turning experiment
is shown in Fig. 7-5. The video data in Fig. 7-5-a illustrates that the CCSV-RAD
prototype is capable of impressive turning performance. The video trajectory, labeled
in red, shows a very sharp turn. The V) data in Fig. 7-5-b shows that the vehicle can
quickly alter its angle and reaches its final value in about 2s.
100
80
60-
-
20
>'
20-
-20 --
0
d(t)
1
2
4
3
Time [s]
5
6
7
(b) Angle Tracking
(a) Vehicle trajectory
Figure 7-5: Video (a) and angle tracking (b) data illustrating the turn-at-speed ca-
pability of the CCSV-RAD prototype.
A common metric for evaluating the turn-at-speed performance of a vehicle is to
use the turning radius and then compare it to the overall length of the vehicle [15].
We estimate the turning radius be examining the video trajectory of the robot. A
zoomed in view of the vehicle trajectory is shown in Fig. 7-6-a, and provides an
122
estimate of about 1.03 body lengths. The turning radius is essentially a function of
how quickly the momentum of the vehicle can be rotated. The velocity time trajectory
was estimated using the video data. The velocities in the fixed XY frame, Vx, Vy,
are shown in Fig. 7-6-b. In this case it is likely that the hydrodynamic instability
of the vehicle provides some assistance because the vehicle already has a tendency to
turn so that it is 900 to the flow.
Our turning radius of 1.03 body lengths while moving at 1.2 BL/s compares
favorably with some conventional AUVs, although it should be noted that our vehicle
is a small test prototype rather than a large, fully instrumented, autonomous robot
[15]. The "gold standard" for turning radius while moving at speed is perhaps the
performance of the Finnegan robotic turtle, which achieves a turning rate of 0.77
Body Lengths when moving at 0.5 BL/s [15].
0.2
0.15m [1.03BL]
3
-0
0.2
4
5
Time [s]
-0
Time [s]
(a) Turning Radius
(b) Velocity Profiles
Figure 7-6: Video (a) and velocity (b) data illustrating the turn-at-speed capability
of the CCSV-RAD prototype.
7.1.5
Heave Translations
So far our results have focused on motions in the XY plane, or "swimming"
motions as we titled them in Chapter 4.
The vehicle is also capable of "diving"
motions. The heave translation capability has of the CCSV-RAD prototype has been
verified and the experimental data is shown in Fig. 7-7. For these experiments we
123
used a transparent tank at MIT's Ocean Engineering Teaching Laboratory (OETL).
The vehicle mass was adjusted carefully to achieve nearly neutral buoyancy.
The
vehicle was then commanded to translate to the bottom of the tank and then return
to the surface. These experiments were open loop without any feedback on depth.
The video trajectory of this experiment is shown in Fig. 7-7-a. This video data was
also used to measure the vehicle trajectory. The vehicle Z trajectory is shown in Fig.
7-7-b. This data can be used to estimate the heave velocity, which is about 0.157m/s
(1.08BL/s). Note that the vehicle surfaces slightly faster than than it dives. Despite
our best efforts, the vehicle still has slightly positive buoyancy. This means that the
vehicle will slowly rise to the surface if Jets 3 and 4 are not in use.
Achieving complete neutral buoyancy presents an important challenge for this
robot design. This vehicle achieves its small size by only using 2 pumps, and therefore
cannot maintain a depth while also moving forward or turning. The best option is to
include a small buoyancy control system that uses a diaphragm or piston to provide
fine control over buoyancy. This could provide precise z control and the diving modes
could then only be used for rapid heave translations.
0.4
0.350.30.25
N
0.2
0.15
0.1
0.05
0
2
4
Time [s]
6
8
(b) Position profile
(a) Vehicle trajectory
Figure 7-7: Video (a) and velocity (b) data illustrating the diving capability of the
CCSV-RAD prototype.
124
7.1.6
Pitching Motions
The lack of neutral buoyancy also hinders the exploration of the pitching capability. Preliminary open loop studies have shown that by activating Jets 3 and 4 in
opposite directions can generate pitch angles of
450. However, these experiments
were performed with the vehicle at the surface, which leads to coupled motions and
relatively limited results.
The exploration of pitching motions therefore remains a
promising area of future work.
7.2
Switching Between Motion Families
In order to achieve three dimensional motions, switching between the motion
families is required. This means that pumps 1 and 2 must be reversed. The dynamics
of switching the pump direction will determine which three dimensional trajectories
are feasible and which are not. A simple experiment was performed to examine the
switching performance of the pump. A pump with in-line 1800 outlets was attached to
our force testing apparatus, and the pump was commanded to change direction while
operating at its maximum speed. Although this is a slightly different design from the
orthogonal output design, the switching dynamics should not deviate substantially.
The force data from this experiment is shown in Fig. 7-8, and illustrates that the
pump can reverse the direction of the force in approximately 100ms. This means that
if the vehicle is moving at maximum speed, (0.2m/s), it will drift about 2cm before
the robot enters diving mode. The vehicle will also continue to drift a bit until drag
slows its forward motion completely. It should be also noted that the switching of
the pump direction exerts substantial reaction moments on the vehicle, causing its
orientation and heading to change.
7.3
Exploring Passive Yaw Stability
In the past section we have explored the performance of our CCSV-RAD prototype
and shown some interesting features.
Perhaps the most important feature of this
design is the use of feedback control to achieve stability without altering the smooth
and symmetric spheroidal shape. Nonetheless, a relevant question that emerges is
"is it really better than using a fin?" where passive stability would prevent concerns
125
0.15
0.1
0.054
01
-0.05-.
-0.1
-0.15-
-+A~0.1s
-F [N]
--- Switch Signal
__
10.
0.2
0.4
0.6
0.8
1
Time [s]
1.2
1.4
1.6
1.8
Figure 7-8: Pump reversal dynamics.
about stability, controller design, and bounds on performance.
In this section we
perform a short experimental exploration of the use of fins and their consequences.
7.3.1
Hydrofoil Modeling
Many slender shapes can create a lifting force when inclined into a flow, and even
a flat plate can be used to generate lift forces. Airfoils (air) and hydrofoils (water)
are shapes that are optimized to minimize drag and maximize lift. An illustration of
a symmetric foil shape is shown in Fig. 7-9, it shows how inclining the foil at some
angle, 3 into the flow creates a lift force, L, and a drag force D. The lift force, L,
acts perpendicular to the free stream, while the drag force, D, is parallel to the free
stream. There are many types of foil shapes and designs, but symmetric ones such as
the one shown in Fig. 7-9 are most appropriate for stabilizing an underwater vehicle
because their lift and drag curves are the same for both positive and negative angles,
P. The approximations for lift and drag forces are well known and provided below
in eqs.
7.1 and 7.2. In these expressions, CL and CD represent the lift and drag
coefficients respectively, while AF represents the area of the foil.
L = -pAFCL (3) U2
2
126
(7.1)
(J
L
Figure 7-9: A simple diagram illustrating the hydrodynamic forces on a hydrofoil.
1
D = -pAFCD (p)U2
2
(7.2)
An example of the lift and drag coefficient shapes can be found using known tables
such as those on [63]. Data for a NACA 0015 airfoil (which is symmetric) is shown in
Fig. 7-10. This data was obtained from [63]. This data illustrates several key points.
First, at small angles, the lift force can be linearized (labeled as "linear region"), and
the drag force can be neglected. This leads to the very common approximation for
the lift force.
1
dC
L ~~-pAFU2dL /C
2
do
(7.3)
Another important lesson is that at higher angles, the lift force degrades and the
drag force increases dramatically due to separation of the flow over the foil.
This
means that aggressive maneuvers can lead to a substantial increase in drag which will
degrade performance.
7.3.2
Sample Tail Fin
In order to achieve passive stability we design a tail fin to cancel the yaw moment.
We first begin by creating linearized expressions for both the Munk moment, NM,Lin,
and the yaw moment crated by the fin, NFin,Lin-
1
NM,Lin = 1 (-Yi, + Xt) U.2#
127
(7.4)
1.5
Lin
%
--
1
0.5
~ar rgion0
-0.5
-1 .. C L
-
-1.5
.cD
-150
-100
-50
0
P [Deg]
50
100
150
NACA 0015 Airfoil, Re = 360000
Figure 7-10: Lift and drag coefficients for a NACA 0015 airfoil.
NFin,L2n
U2 dCL
1
-
(7.5)
pAF LFU0 d
We define a new variable, K, to measure the passive stability of the design. If K is
greater than 1, the system is stable, as the lift moment is larger than Munk moment
for all speeds at small angles. If , is less than 1, the system will be unstable because
the Munk moment will exceed the lift moment for all speeds at small angles.
d
NFin,Lin
(L
F
(-Yi, + XA)
NM,Lin
We use this analysis to create our own sample tail fin. Our fin has a chord, Cfin,
span, bfin, and is placed a length, LF, behind the vehicle center of mass. We create
our own symmetric airfoil. For our foil we simply assume that
d
dfl
is approximately
equal to 5. This is only a rough estimate, but is close to the coefficient for most airfoil
examples (for a NACA foil, dd/3
~ 4). This is merely intended as a simple experi-
mental evaluation for a fin, and a more thorough examination would involve carefully
designing the foil shape and measuring its lift characteristics. The parameters for the
fin are provided in Table 7.2, and all values are exact except for d,d/8
128
which was based
Table 7.2: Summary of Physical for Sample Tail Fin.
Value
Parameter
LF
0.073[m]
Max. Camber
0.0094[m]
.027[m]
CFin
0.038[m]
bFin
AF
0.001[M 2]
dCL
d3
5
on a rough guess.
Based on this fin design, we have a value for r, of 1.4. This means that the fin
is a bit over-designed but not tremendously so. It should be noted that we used a
relatively large estimate for the lift coefficient, so the value for r, is likely a bit lower
in reality. A photograph of the fin is shown in Fig. 7-11, note how the fin is relatively
small compared to the robot.
(a)
OL Stable: Stabilized using
a passive tail fin.
(b) OL Unstable: Stabilized
using feedback control.
Figure 7-11: Photographs comparing the OL stable prototype (a) and the OL unstable
prototype (b).
7.3.3
Experimental Evaluation
The performance of the fin was tested and compared with the performance of our
CCSV-RAD prototype. For the sake of simplicity, we use the nomenclature outlined
in Fig.
7-11.
We use "Open Loop (OL) Stable" to refer to the design passively
stabilized using a tail fin, while we use "OL Unstable" to refer to our design approach
129
which stabilizes the vehicle by using feedback control.
Straight Test
We first evaluated the role of the fin by testing the ability of the OL stable vehicle
to move forward in a straight line without the use of feedback control.
The jets
were set to provide a straight motion at a constant speed. As Fig. 7-12 shows, the
fin successfully stabilizes the vehicle and enables it to move in a relatively straight
manner without the use of any feedback control.
Figure 7-12: Video data illustrating the stabilizing effect of the tail fin. The vehicle
moves relatively straight without any feedback control.
Stationary Turning Comparison
The second test was again open loop, and involved testing the ability of the vehicle
to turn when Jets 1 and 2 were activated so that a pure yaw moment was created. The
video trajectories shown in Fig. 7-13-a reveal that the presence of the fin significantly
degrades the ability of the vehicle to turn in place. The drag on the fin causes the
vehicle to turn in an arc rather than in place.
In addition, the yaw rate, r, data
shown in Fig. 7-13-b illustrates that the turning rates are substantially lower for the
OL stable robot. The OL unstable robot achieves turning rates about 4 times faster
than those of the OL stable robot. This shows that the use of a passive stabilizer has
major performance consequences for low speed maneuvering.
130
-OL Unstable
-- OL Stable
200
Reverse
Command
-100
-200
-300
0
2
4
8
6
Time [s]
10
12
(b) Yaw rate time response
(a) Vehicle trajectories
Figure 7-13: Video (a) and yaw rate data (b) data comparing the stationary turning
performance of the two vehicle designs.
Turning at Speed Comparison
A third test was performed to examine the turning at speed performance. In this
case we need the feedback controller to be active in order to evaluate the OL unstable
design.
We therefore used the same exact stabilizing controller for the OL stable
design. The exact controller gains should not affect performance greatly because a
90* turn causes the controller to saturate for most of the motion. Both designs were
commanded to move forward at the same speed and then commanded to turn 900
at time t = 3s. The video trajectories in Fig. 7-14-a illustrate that the OL stable
design again performs poorly. The sharp turn creates substantial drag on the fin,
and the OL stable vehicle drifts substantially instead of achieving a sharp turn. The
i performance, shown in Fig. 7-14-b is similarly poor. The OL stable design turns
much more slowly and has a substantially larger rise time. Also, note that for passive
fins, the fin tends to align itself with the flow even if that is not always desirable.
This is illustrated by the fact that the controller for the OL stable system has trouble
tracking the desired angle accurately.
131
100
8060
I
-
40
20-OL Stable
I
-OL Unstable
0
0
2
4
Time [s]
6
(b) Yaw angle time response
(a) Vehicle trajectories
Figure 7-14: Video (a) and angle tracking (b) data comparing the turning at speed
performance of the two vehicle designs.
7.4
Summary
In this chapter we have shown experimental data which is designed to measure the
performance of our CCSV prototype vehicle. This vehicle design and performance
is quite unique, so obtaining numerical values for the performance is very valuable.
This section showcased the unique abilities of the CCSV, including turning in place
at high rates, forward and backward motions, and sway and heave translations. We
also provided a thorough comparison between our CCSV approach and a more conventional, passively stabilized approach. We used experiments to show how while a
passive fin eliminates the need for a stabilizing controller, it also severely reduces the
maneuvering performance of the design.
132
Chapter 8
General Design Approaches
Now that we have designed, implemented, and evaluated a new type of underwater
robot, we can examine the broader ramifications of these results. Key questions that
arise are what is the nature of the uncontrollable behaviors, and what is the best
shape to use.
In this chapter we return to more general modeling techniques to
explore these important questions. The results of this chapter can serve as general
design tools for robot designers interested in using a CCSV type approach.
8.1
General Properties of the CCSV Design
The first step before performing general analysis is to clearly define the CCSV
design. We define our CCSV as having the following properties.
1. A spheroidal shape with a major dimension, a, and minor dimensions, b = c.
2. Passive pitch and roll stability from gravity.
3. Water jet propulsion for a completely smooth outer shape.
4. Diamond jet configuration for xy maneuvering. The diamond configuration is
also used in lieu of additional sway actuators.
Essentially, for the sake of this chapter we will focus on how to best design vehicles
that are spheroidal, completely smooth, and with jets configured as in Fig. 8-1. These
types of vehicle are unique and there is value in exploring the general ramifications
of their shape, dynamics, and control performance.
133
-Jet 2
-jet 1
+Jet 4
+Jet 3
A
X
LF'Z.
LF,
r
%
*L
Z
-
W
1
+Jet
4
-Jet 3
+Jet
-Jet 4
Figure 8-1: A diagram illustrating the jet arrangement for the CCSV design.
8.2
Controllability of the CCSV Design
One of the key features of the CCSV concept is the use of feedback control to
achieve stability and perform maneuvers such as aggressive turns. The use of feedback
control is only truly valuable if the system is actually controllable.
We discussed
controllability briefly in Chapter 6 when describing the role of the angled jets. In this
section we will explore controllability more formally.
We can generalize any linearized system using state space form, where X represents the state vector, A represents the state transition matrix, B represents the
input matrix, and u represents the input vector.
(8.1)
X = AX + Bu
For the CCSV planar dynamics. We have the following state space model.
d I
dt
I
r
r
fl
_____l+
=
LAPJ [
-
~
0
Avl
0
0
Ar
1
0I Ay50
m-Y
*
__
I.-N.
0
+
0
AV
[
-2GFsin-fj
my
2GM
1
0.
]AV
(8.2)
For the sake of simplicity, we substitute in the variables a1 , a 2 , bl, and b2 . All of
134
these intermediate variables are assumed to be greater than zero.
AV
dAr
Westart
LAO
0
at kL+
m
0
-a 2
-a,
0
2b2 0V1(8.3)
1
0
0
AV
r
0 J
"J
-2b,
+
22
AV(83
0J
We start by examining the controllability matrix, C.
-2b
C =
2a1b2
-2aia 2 b1
2b 2
-2a 2b1
2a 2 a1 b2
0
2b2
-2a 2 b1
(8.4)
If C is of full rank, then the system is controllable.
examining the determinant of C.
We can evaluate this by
If the determinant is zero, then the system is
uncontrollable.
det C = 8a 2b1 (aib2 - a2 b2)
(8.5)
The determinant of the controllability matrix provides some very valuable insights
into the vehicle performance. First, if b1 = 0, then detC = 0. We refer to this as
"input uncontrollability" and illustrates the importance of having control authority
over the y direction. As we discussed in Chapter 6, if the jets are not angled, the
system is uncontrollable and leads to poor performance.
Similarly, if a2 = 0 the system is uncontrollable.
This presents an interesting
result. The most plausible way for a 2 = 0 is for the difference in the added mass
to be equal to zero. Therefore, spherical shapes would be uncontrollable, a form
of "shape uncontrollability."
The fact that a spherical shape is uncontrollable is
very important. There is a tendency to gravitate towards spherical shapes for these
types of applications due to their symmetry, lack of a Munk moment, and perceived
improvement in turning performance. However, this analysis reveals that the Munk
moment actually helps provide controllability by creating coupling between the yaw
and sway. An illustration of this is provided in Fig. 8-2 which shows how the V) and
135
v response is poor and does not converge. The v response is particularly noteworthy,
because the sway response decays extremely slowly despite the presence of angled
jets. A dedicated y thruster or pump would be needed to provide adequate control.
0.02'
0.2
0.0150
0.01
I
/
0.005t
IiE
.- 0.0)4
Ud
:3
5
Time [s]
m ---------
----------
-0.02-
-V
-w
> -0.06-
Time [s]
-- Vd
5
Time [s]
0.0%
10
10
10
Figure 8-2: Simulations illustrating the poor control performance of a spherical shape.
8.3
Designing for Control Performance
As the previous section on controllability shows, the choice of vehicle shape and the
design of vehicle jets plays a critical role in control performance. Designs that seem
obvious can actually result in poor performance.
Controllability as defined above,
does not serve as an ideal metric for vehicle design because of its binary nature. A
jet angle of 10, or a vehicle shape that is only slightly elongated will still have poor
performance but will still pass the controllability test.
Therefore, there is a well
recognized need for other ways to analyze control. Two potential options are using
the determinant of the controllability matrix [64] or using the ratio of the largest and
smallest singular values of the controllability matrix [65].
8.3.1
Determinant of Controllability Matrix
The use of the determinant of the controllability matrix to predict the control
performance is attractive due to its relative simplicity. However, one issue is that this
value can vary based on the choice of the state variables, something that is discussed
in [66]. We show the determinant of the controllability matrix, C in eq. 8.6. Note
that to calculate this, we attempted to reduce deviations by nondimensionalizing the
136
states and elements of the state space vectors.
C
= -Idet
C1 = 8L -L 6 V3GF3 sin (Uc
J
U0
U0
2 TAM,
3
z
(mma2
MSn
Amasin2Jz
y
(8.6)
In this expression we use Vo to represent the maximum voltage, and U0 to represent the maximum cruising speed, and L to represent a characteristic vehicle length
scale. Note that the expression in eq. 8.6 does not really provide any real physical
understanding. It is unclear what the quantity represents, and it appears to penalize certain factors such as the moment of inertia extremely heavily. Therefore it is
unclear what value would be provided by attempting to optimize C.
8.3.2
Vehicle Zero
Attempting to optimize controllability is challenging and lacks physical intuition.
Therefore, some other metric for assessing control performance should be explored.
One option is to return to our SISO analysis from Chapter 6. The SISO analysis
provided expressions for the open loop poles and zeros which have clear physical
significance.
Using the zero, zj, to assess performance appears appropriate for a
number of reasons.
First, the location of z, does not change with the feedback control gains and is
instead only dependent on the vehicle properties. Second, the zero location provides
valuable insights into control performance. If the zero is at the origin (s = 0), then
the system is uncontrollable, and if the zero lies on the right half of the complex
plane, the system is nonminimum phase and hard to control. Finally, the location of
the zero places fundamental limits on the vehicle performance because it will always
attract a closed loop pole. The expression for the vehicle zero, zj, is provided in eq.
8.7.
=
-GFUAMa sin y
(8.7)
8GM.m
Since the zero location is so critical to control performance we will use it as our metric
for controller performance. Specifically, we want to explore how to design the vehicle
137
jets and shape so that z, is negative and as large as possible.
Using Zero Location to Inform Vehicle Design
8.4
This section focuses on how a vehicle can be designed to maximize z1 . For sim.
plicity, we assume that GM = LFzGF. This simplifies the expression for z 1
-UcAma sin-yj
(8.8)
LF,Zmy
In this expression, the only parameter that is design independent is Uc.
We will
assume that the operating speed, Uc, is chosen based on the application and will not
vary with design. This leaves parameters that are dependent on the vehicle shape:
LF,z, Ama, my, and a parameter that is dependent on the jet angle: sin -yj.
8.4.1
Jet Angle
We will first examine the selection of the jet angle, yj. The jet angle is illustrated
in Fig. 8-3, which illustrates the key design parameters. If we wanted to optimize the
size of the zero, we would choose a value for -yj that approaches 900. However, the
choice of this parameter also affects the forward propulsion of the robot. The x thrust
force is dependent on cos-yj. A choice of -yj = 90' would result in a robot that could
no longer move forward. Therefore the choice of the jet angle is not clear cut, and
instead requires a tradeoff between forward speed efficiency and control performance
(as well as sideways efficiency). A choice of 45' is a good balance, but still degrades
the forward performance by about 30 percent. Without clear guidelines, we suggest
the following approximation that uses the desired speeds in the x and y directions,
-
U0 ,V 0
lb~
Figure 8-3: A diagram illustrating the key design parameters, a, b, and -yj.
2v~
U0
138
(8.9)
Our jet angle of yj = 300 is associated with a ratio of uo = 4vo which seems like a
valid ratio. Jet angles between 15' - 450 are all reasonable and the exact value will
depend on the application and the use scenario.
8.4.2
Vehicle Shape
While increasing the angle at which the jets are angled inwards can increase the
size of z1 , there are performance tradeoffs that prevent us from using large jet angles. In fact, in order to achieve suitable forward speeds, the jet angle must remain
relatively small. However, z, can also be tuned by adjusting the vehicle shape. Specifically, my, Ama, and LF,z are all dependent on the vehicle shape. In this subsection
we will analyze the effect of the vehicle aspect ratio (a) on control performance.
Specifically we will assume a fixed mass m and explore the ramifications of either
stretching the vehicle shape out or shrinking it to a sphere.
The following analysis fixes all parameters that are not dependent on the aspect
ratio: U, m, -yj. We then parameterize the remaining variables in terms of the vehicle
dimensions.
The values for the added masses can be obtained from a design table
listed on page 147 in [48]. We estimate Am, and my by fitting polynomials to the
data in the reference.
LF',
Ama = m
MY = m
- a
0.015 (,)2-
0.185
(-)
(8.10)
1.07
- 0.713
+ 1.033)
( )
2.02)
(8.11)
(8.12)
Using these results, we can create curves for z1 based on the aspect ratio. We
start by examining the case that matches our existing design (m = 0.9kg).
The
results of this analysis are plotted in Fig. 8-4, which shows how the value of zi for
various aspect ratios. We use aspect rations greater than 1, so the actual value of z,
is less than zero. We normalize this way so that we can compare with other positive
139
I
parameters such as p3. For the sake of comparison, the unstable pole location, p3 was
analyzed in a similar manner. As Fig. 8-4 illustrates, the behavior of p 3 and z1 is
strikingly similar. Both are zero for aspect ratios near 1, and then peak near aspect
ratios of 2 before tapering off.
However, the most relevant results are for z, which provide a clear optimal point
around Rb ~ 2.3.
This means that for the best zero location, an aspect ratio of
approximately 2.3 is desired. Also important is the slope of the curve. The value
for z, increases quite quickly between aspect ratios of 1 and 2, implying that in this
region, performance can be substantially improved by elongating the shape. At larger
aspect ratios, the zero location becomes slower but decays at fairly low rate.
---plmax(p 3
1.2
-z
-----
1 -
1
/min(z)
0.6-|#
-~2.3
0.6
-
b---
.
-
0.8 -
0.4b
0.2
0
-0.2
I
12
3
6
5
4
a
Figure 8-4: A plot illustrating the dependence of z1 on the aspect ratio for m
Note that for these aspect ratios, z, is actually negative.
=
0.9kg.
By substituting in the above simplifications into eq. 8.8 we can obtain an expression for z, that is only dependent on a and b.
0.015(2 - 1.07
z1
-Uc sin -yja
n (0.185(})
()
+ 1.033)
20)(8.13)
a
2
-
0.713
(
) + 2.02)
What the expression in eq. 8.13 shows, is that the optimal aspect ratio is mass,
140
speed, and angle independent. This means that regardless of our choice for other
parameters, there will be one aspect ratio that provides the best zero placement, and
this ratio is approximately 2.3.
8.5
Examining Vehicle Performance
The simulation environment used to simulate the vehicle dynamics in Chapter 6
again becomes useful for assessing the result of our zero analysis. While maximizing
the speed of z1 makes intuitive sense, what are the ramifications for closed loop vehicle
performance? We use simulations to quickly compare a variety of designs to illustrate
the value of optimizing the zero location.
We evaluate various aspect ratios with a simple angle servo experiment where the
vehicle is commanded to turn 250 while moving at a constant speed U, = 0.2m/s. The
sway performance and the control effort are shown in Fig. 8-5. The sway response
in Fig. 8-5-a clearly illustrates how a nearly spherical design (a = 1.01b) has poor
turning performance despite technically being controllable. The purpose of Fig. 8-5-b
is to show the consequences of an increased aspect ratio. The primary consequence
of a large aspect ratio is a larger moment of inertia that requires larger control effort
to maneuver. As Fig. 8-5-b shows, the control effort does increase for larger aspect
ratios, but not substantially. It is also interesting to note that above aspect ratios of
2.3 (the optimal zero location), the sway performance does not appear to noticeably
improve.
These results provide clear illustrations of why choosing an appropriate vehicle
shape is essential.
A shape that is nearly spherical is technically controllable, but
in practice will still perform poorly. The question that arises is how elongated to
make the shape? Our zero analysis predicted that an aspect ratio of approximately
2.3 would be best, and these simple simulations confirm that performance improves
substantially by using such an aspect ratio. However, the performance degradation
associated with longer aspect ratios is relatively small. Historically, elongated shapes
have been preferred due to their reduced drag. In the following section we will explore
the consequences of using larger aspect ratios.
141
20
-a/b =1.01
-a/b= 1.35
-a/b =2.3
a/b=4
-0.0215
a/r10
>-0.04-
-a/b
-a/b
1
2
3
Time [s]
<
5
.3
-a
P
-0.06
0
= 1.01
= 1.35
4
5
1
2
Time [s]
4
5
(b) Control Effort
(a) Sway Response
Figure 8-5: Simulation data illustrating the control performance for various aspect
ratios vary from 1.01 to 4.
8.6
Additional Performance Considerations
A key question that arises from the analysis in the preceding section is: why
not simply make the shape very elongated? The performance degradation at higher
aspect ratios is relatively minor, and an elongated shape has greatly reduced drag.
For the scope of this work we have used the common quadratic drag model. We can
estimate the x drag, FD,,, using the drag coefficient CD,,, as well as the frontal area,
b2 , and the speed, Uc.
1
U
FD,x = -PCDxb2 Uc
(8.14)
The drag coefficient, CD,-, can be obtained from tables in [59]. Elongating the
vehicle shape is very effective because an elongated shape has a reduced drag coefficient due to reduced separation of the flow. It is well known that the fluid flow on
a smooth sphere separates near the rear, causing an increase in the drag. The effect
of elongating an ellipsoidal shape is shown in Fig. 8-6. Also, elongating the shape
reduces the frontal area, which means that there are two mechanisms reducing the
drag. This is why in Fig. 8-6 the drag force, FD,,, reduces dramatically as the aspect
ratio is increased.
142
1.
4'
CDx
D,,1
1.2--
II~D,X)
b
0.80.60.4-
-
0.2-
a
Figure 8-6: Drag coefficients and forces for various aspect ratios.
Thus far if we simply consider the results in Fig. 8-5 and Fig.
8-6, we tend
to conclude that a highly elongated shape (large aspect ratio) would be the most
appropriate. This is not altogether surprising, because marine engineers having been
making underwater vehicles that are highly elongated such as torpedoes, gliders and
AUVs, for 50
years. However, for our applications of maneuverability at all speeds,
additional considerations must come into play.
8.6.1
Low Speed Trning
In the past chapters a great deal of emphasis has been placed on stationary turning
motions. This is one requirement that clearly does not benefit from a highly elongated
shape. An elongated shape means a larger moment of inertia which makes turning
performance more sluggish. In order to examine this effect, we create another metric,
a, that we refer to as the instantaneous angular acceleration.
.61 wpdTr
Iz
-
VoGFa
(8.15)
Iz
Using this expression we can again preform shape analysis and examine how the
turning performance changes with the vehicle aspect ratio. In Fig. 8-7, we plot three
143
metrics together, FD,,, z1 , and a. Note that FD,. is different from the other two
metrics in that a low value is good. The drag force is normalized using the drag force
on a sphere,
1.6
FD,x,sphere.
The plot in Fig. 8-7 shows clearly that an elongated shape
I.Iz
/ max(zd
1.4
a/ max(a)
-F
1.2-
FDx
/F
D,x,sphere
0.8
0.60.4
0.22
3
a
4
5
6
Figure 8-7: A figure showing all 3 performance metrics and the role of aspect ratio.
has detrimental consequences from the perspective of stationary turning performance.
Even though a longer vehicle can create larger moments, the moment of inertia increases with the square of the length. This means that if stationary turning dynamics
is important (and for our application it is), then simply elongating the shape infinitely
is not the best approach. This is a somewhat unsatisfying result, because it shows
that under ideal condition, there is no one optimal shape. Instead, under ideal conditions, the best shape will depend on how heavily the designer prioritizes efficiency
against stationary turning.
8.6.2
Practical Considerations
In the concluding paragraph of the previous section, we emphasized the "ideal"
nature of the analysis. Specifically, we had idealized the situation by assuming that
the aspect ratio can be made infinitely long. This is clearly not practical, as the
hardware we have shown in Chapter 4 would clearly not fit into a very long, slender
tube. This is a key issue in underwater robotics, because despite the desire to elongate
144
the shape, many components such as motors, batteries, and cameras cannot be made
very long and thin. This is especially true for inspection robots where the need for
forward facing sensors creates a lower limit on the vehicle radius, bmin.
This inconvenient fact forces us to reconsider our above analysis and frame it
from a more practical perspective. What if we assume there is some minimum radius,
bmin, which is required to fit the hardware.
This means that in order to achieve
higher aspect ratios, it is necessary to add mass to the vehicle. Therefore we have
now relaxed our constraint of constant mass, and we say that below a certain value for
b, we have to add mass to the vehicle. The question that arises is: Is it worthwhile to
add mass in order to achieve an elongated shape?. From the perspective of drag, the
answer is clearly yes, as the drag coefficient is independent of mass and still reduces
for longer shapes.
However, for control (zi), and turning (a), the answer is less obvious. In Fig. 8-8
we again examine our performance metrics but now prevent the radius, b, from being
below a certain threshold. This study was performed using parameters that match
the CCSV-RAD prototype.
1.6
I
-z
1.4-
/max(z)
a / max(a)
-FD/FD,
1.2-
x, sphere
1
0.8
0.60.4
2
3
a
4
Figure 8-8: A figure showing all 3 performance metrics.
radius, b, is imposed.
5
6
In this case a minimum
This plot looks quite different from the one in Fig. 8-7. The clearly visible kink
145
in the curve near ! = 2.5, shows how the addition of mass causes z1 and a to degrade
much more quickly. In addition, the drag force levels off and higher aspect ratios do
not provide much improvement. We can examine a more pronounced case, which we
call "spherical preference" where the aspect ratios above about 1.3 require additional
mass. This case is plotted in Fig. 8-9, and shows that now our optimal zero location,
z 1 , has also shifted from 2.3 to 1.9.
1.6
Ia
/ max(a)
1.4-
F/ FD, x, sphere
/max(z
-z
.-
)
1.2
10.8-0.6-
0.4
1.5
2
'25
a
3
'35
4
'45
5
Figure 8-9: A figure showing all 3 performance metrics. In this case a minimum
radius, b, is imposed, and aspect rations above 1.3 are penalized.
We return to our simulation environment to examine the ramifications of these new
results. What we care about in particular is the control performance while the vehicle
is moving at some constant speed. The simulated response to a turn command of 25*
is shown in Fig. 8-10. The results are quite interesting. The spherical performance
remains poor, but adding mass to achieve an aspect ratio of 1.9 shows a marked
improvement in the maneuvering performance. However, aspect ratios beyond this
show a clear degradation in performance.
The implication of this analysis is that
the "goal" aspect ratio should be around 2, even if this means adding mass to the
vehicle. If the minimum radius is small, then increasing the aspect ratio to 2.3 will
improve both control performance and longitudinal efficiency. However in the more
146
realistic case where the minimum radius is rather large, then aiming for an aspect
of ration of 2 should dramatically improve control performance without degrading
turning performance too much. If stationary turning is critical, ratios near 1.5 would
be better because they only degrade the turning performance by about 20 percent.
0-a/b =1.01----:0
--- - ---- -------..
--.
-a b 1
-a/b = 1.89
0.4 -a/b =2.3 .------ --------- -------- --------a/b = 4
- ---------0.3 --------- --------- -------
0.2
-
0.5
-
-
- -
-
0.1-
- -
--
-
S-0.06
--------
0.2
a/b= 1.01
- -- ---
0
S
004
--
---------
------
-0.02
-0.08
-0.1
-
- ----------04
0.:
-a/b=1.89
-a/b=2.3
a/b =4
02(b
0.8
x0m
3en
Time [s]
4
5
Figure 8-10: Simulated data showing the XY trajectory (a) and the sway response
(b) for various aspect ratios when a minimum radius is imposed.
8.7
Summary
This section has provided some very valuable general insights into vehicle design. We have shown that the vehicle zero, z1 can be used as a simple metric to
predict control performance. Optimizing the value of this zero can improve control
performance, but should not be done blindly because there are ramifications for other
metrics such as efficiency and stationary turning. While our analysis in this section
does not provide an absolute answer for all cases, it does show that there is a range of
values for the jet angle, y, and the vehicle aspect ratio that should provide superior
performance.
To conclude, in the absence of rigid requirements we can advise that the jet angle
be set between 150 and 45*. Similarly, we can advise that the vehicle aspect ratio be
set somewhere between 1.3 and 2.3. These values are all rather approximate and a
designer can use their discretion and knowledge to choose exact ones. However, we can
147
be fairly confident that any parameters in this range should yield good performance.
Finally, we should point out that while this analysis is simple, the conclusions
are far from obvious.
The angling of the jet angle at all is a unique feature, and
the choice of the vehicle shape is similarly counter intuitive. Spherical shapes seem
ideal in many ways but clearly do not perform well. In fact, our own initial prototype,
which was designed before this analysis, has a relatively small aspect ratio of 1.35 (due
to the desire to conserve mass). Some exciting future work will focus on exploring
experimentally, any performance benefits of elongating this design.
148
Chapter 9
Fundamental Control Performance
Limitations
The fact that we are stabilizing an unstable system using feedback control means
that there will be fundamental limitations on performance when implementing the
controller.
Specifically, Bode's Integrals can be used to predict the peak control
performance based on the idea of "available bandwidth" [60]. In this section we will
use the techniques outlined in [60] to examine the performance limitations that exist
for the CCSV concept and use these results to examine the general ramifications of
our closed loop stabilization technique.
9.1
Bode's Integrals
In words, Bode's integral states that the integral of the log of the sensitivity
transfer function is equal to 7r times the sum of the real part of any unstable poles.
This integral is evaluated over the available bandwidth, Qa. This can be summarized
mathematically in eq. 9.1, where S(jw) represents the sensitivity transfer function,
and p represent the open loop unstable poles.
In (S(jw)dw = r ERe (p)
0
(9.1)
PEP
Since the sensitivity transfer function is not always a common technique, we will
provide a short review. For our system, we can draw the SISO control system block
149
diagram with 3 components, the controller, Ge(s), the actuator, GA(s), and the plant
Gp(s). Figure 9-1 provides an illustration of this block diagram. Using this notation,
Ge (s)
F
G,(s)
G,(s)
V(S)
v(s)
Figure 9-1: A block diagram of the SISO yaw control system.
we can write the sensitivity transfer function, S(jw).
S(jW) =
1
1 + G, (s) GA (s) Gp (s)
(9.2)
The maximum sensitivity, max(S), can be used to predict performance such as
the phase margin, 0b, and gain margin, Gm, using the following approximations [67].
m
> 2sin- 1 (2mclx(S)
2max(S)
Gm ;>
(9.3)
ma()(9.4)
max(S) - 1
As these expressions show, the maximum sensitivity is a key factor in determining
performance, and making it as low as possible. With this in mind, the ramifications
of the Bode Integral in eq. 9.1 become clear. An open loop unstable plant places
lower limits on the integral, meaning that with some finite limited bandwidth, the
achievable control performance must decrease. The available bandwidth also emerges
as a critical parameter.
Smaller bandwidth means that the sensitivity cannot be
spread as "thinly" and therefore results in larger values.
9.2
Sensitivity Analysis
We can design a relatively simple sensitivity plot for the system by emulating the
approach in [60]. This prototype sensitivity function involves very low sensitivities
(Q1) at low frequencies in order to deal with steady state disturbances. The sensitivity
is then allowed to increase before going to zero beyond the available bandwidth, Q2a.
150
While this shape is not optimized or always easily realizable, it provides an example
Prototype design for sensitivity.
Low sensitivity at DC (steady state).
Figure 9-2: A prototype sensitivity function that should provide good control performance.
of an ideal controller. An ideal controller design would achieve a sensitivity function
similar to that provided in Fig. 9-2. An advantage of this prototype function is that
it is easy to compute the maximums sensitivity. We use the expression provided in
[60]. This expression is based on the size of the unstable pole, which is p3 for our
system.
Sma = exp (rP3
9.3
+ Q1
(9.5)
CCSV Performance
We now apply these concepts to the specific case of our CCSV design. Recall that
we already have an expression for p3-
P3=
Umm
(9.6)
The next step is to compute the available bandwidth. This bandwidth represents
the frequency at which our linearized models break down due to neglected nonlinear
effects such as fluid dynamics or time delays. For the case of these underwater robots
discussed in this thesis, the most likely limitation on the bandwidth stems from the
actuator model. For example we can use the propeller model outlined in [19] and
apply it to our specific system.
However, simple experiments have shown that a
151
linearization of the model begins to break down at frequencies around 12Hz. This
is not a criticism of the model at all, in fact this propeller model is one of the most
popular and widely cited ones because of its simplicity and accuracy. However, no
linearization will be accurate over all frequencies. This result makes intuitive sense
as well, switching a propeller speed at high frequencies can excite vibrations in the
shaft or blades, and the fluid dynamics are also unpredictable.
If we use this available bandwidth of 12Hz we can now start to examine control
performance. A common metric in the military is a minimum phase margin of 45*.
This means that S,,a, must be below 1.31 if Qa = 75rad/s and Q 1 = 6.28rad/s. Now
the only free parameter is the pole location, p 3 . Since the pole is highly dependent
on speed, we can calculate the speed limit for our design. A plot of Sm"2 for various
cruising speeds, Uc, is shown in Fig. 9-3.
1.8
1.6E 1.4CD
1.2
1
0.2
0.6
0.4
0.8
1
Uc [m/s]
Figure 9-3: A plot showing how the Bode Integral analysis can be used to find the
vehicle speed limit.
The results in Fig. 9-3 shows that if we tried to operate our CCSV-RAD prototype
at speeds faster than
-
0.4m/s, the maximum phase margin would be less than 45*
and the control performance would be poor. Our operating point is at about 0.2m/s,
which provides a much better response. This analysis can be repeated for a variety
of designs and can be used during the basic design phase to evaluate whether certain
requirements are truly realizable.
152
9.4
A General Tool
The general value of the Bode Integral analysis is clear, it can be used to examine
systems from simple inverted pendulums to high performance fighter aircraft [60]. The
specific results for underwater systems is somewhat restricted by the uncertainties
relating to propeller dynamics. Further study is required on propeller dynamics in
order to develop good guidelines for what the "available bandwidth" of such systems
should be. It is pretty clear that the propeller
/
pump dynamics are the limiting factor
for CCSV type systems. This is because they are the only mechanical components,
and the electrical components such as the microcontroller or digital sensors are capable
of comparatively high bandwidths (- 300rad/s).
However if we assume that the propeller dynamic models break down around
12Hz, we can use maximum sensitivity of 1.31 to compute the maximum value,
p3,,,
for the pole location. For our system this yields P3,m,. < 4.48. We can again
apply the techniques from Chapter 8 to examine the role of shape.
P3 =
ULmAma = Uc
13
IzMV
b
(')
-
(9.7)
With this expression we can show that the unstable pole, P3 can also be broken into
two lumped terms. The first term,
1,
is independent of shape and illustrates the
role of the vehicle speed and mass. The second term, P(b) is a function of the vehicle
aspect ratio. This means that we can examine the role of shape independently, and we
can quickly examine design concepts by using their speed, mass, and by approximating
their shape as a spheroid. This could be a valuable design tool.
In addition, this technique can be used to analyze the role of shape. A plot of
P as a function of the aspect ratio is provided in Fig. 9-4. This figure illustrates
how the pole location peaks for an aspect ratio of about 1.74 and then decays to
approximately 5 for larger aspect ratios. The results in this plot can be used for a
variety of tasks ranging from assessing design decisions to analyzing existing vehicles.
The fact that this plot provides the reasonable range of values for P means that
conservative analysis can be performed with knowledge of only the speed, U, and the
153
mass, m.
An interesting question that arises from this work is: what types of underwater
vehicles could also be stabilized with control rather than fins? From both Fig. 9-4
and eq. 9.7 we can see what types of designs would be appropriate for closed loop
stabilization, and which ones would be difficult to stabilize. The most difficult vehicles
to stabilize will still be ones designed to go very fast. While the aspect ratio effect
can cause the pole location to double, the pole is dependent linearly on the speed,
Uc. Therefore, vehicles such as torpedoes which are designed to go fast will likely be
very challenging to stabilize using feedback control. For example, a quick study of
existing torpedoes implies that without fins their poles would be around 15rad/s, or
3 times faster than the limit, p3,,..
Vehicles where closed loop stabilization holds
promise are ones that move relatively slowly or are very heavy. Ocean exploration
and surveying vehicles are a couple examples.
10
86
Aspect ratio for largest p 3
4
a~1.74
b
2
1
2
3
4
5
a
6
7
8
10
Figure 9-4: A plot showing the values for P for various aspect ratios.
154
Chapter 10
Additional Vehicle Prototypes
The bulk of this thesis focused on a single prototype that used an elegant reduced
actuation design to achieve motions in 5 directions using 2 pumps and 4 fluidic valves.
We showed how this prototype can achieve impressive maneuvering, multi-DOF motions, and precision control.
However, there exist two primary drawbacks to this
approach: 3D motions, and efficient propulsion. As we discussed in previous chapters, the use of reduced actuation design resulted in a vehicle that cannot perform
"swimming" and "diving" motions simultaneously. This has practical consequences,
especially with regard to tracking vertical features such as vertical welds. In addition,
the efficiency of the pump-valve system is rather low (- 0.2). This means that if the
vehicle must traverse long distances, large batteries are required. Recently we have
devised two new approaches that attempt to resolve some of these issues.
10.1
4-Pump CCSV
10.1.1
Jet Design
For this design we changed our requirements to include 3D motions. This means
that the vehicle should be able to dive or pitch while performing xy motions simultaneously. This requirement emerged from the practical challenges of inspecting vertical
features. For inspecting vertical welds, the heading must be maintained even as the
vehicle dives. This was impossible with the original prototype.
To achieve this new design, we returned to the 1800 pump that we discussed
155
and discarded in Chapter 4. If you recall, we discarded this approach due to issues
with backflow causing reduced force output. While this is problematic, we reasoned
that for applications where diving motions are less prevalent, the z jets can have
reduced force output. We therefore use 1800 pumps to provide pitching and diving
motions. For the xy motions we also make a modification. We remove the second
fluidic valve from the BAU and in doing so create a jet that is approximately twice as
powerful as the one that goes through the valve. This jet can then be used to achieve
more efficient propulsion for long distance motions.
For shorter distance motions
and precise control, we still use two fluidic valves. An illustration of this design is
provided in Fig. 10-1. As Fig. 10-1 shows, Pumps 3 and 4 provide "diving" motions,
while Pumps 1 and 2 provide "swimming motions." In addition, the high efficiency
(HE) exits can be used to achieve greater longitudinal speed or efficiency. Note how
these exits are also angled in order to achieve suitable feedback control performance.
Precision motions such as sway translations or forward and back motions can be
achieved by using the fluidic valves (green) that are coupled to Pumps 1 and 2.
Figure 10-1: A rendering illustrating the maneuvering system for the 4-Pump CCSV.
156
10.1.2
Vehicle Prototype
Using this design a full prototype was implemented using components similar to
those discussed in Chapter 4. The new 4-pump prototype is larger than the CCSVRAD prototype (1410 grams vs. 900 grams). This allows space for four pumps, and
also enables extended battery life and cameras for visual inspection. A photograph of
the internal maneuvering system for the 4-pump prototype is provided in Fig. 10-2.
The 4-pump prototype is equipped with two cameras, a higher quality one camera
for recording and a lower quality for a live feed. The low quality video is transmitted
wirelessly using a wireless transmitter. The live feed camera is equipped with an LED
light array for illuminating dark regions. A photograph of the robot performing an
inspection task using these cameras is provided in Fig. 10-3.
Figure 10-2: A photograph of the maneuvering system for the 4-Pump CCSV prototype.
The 4-pump prototype was recently used for demonstrations at EPRI headquarters
in Charlotte, North Caroline. As part of the demonstration, the robot was used to
perform a mock inspection of a vertical weld in the tank.
157
The robot was able to
Figure 10-3: A photograph of the outside of the 4-Pump CCSV prototype. The lights
on the camera are clearly visible.
Figure 10-4: An image taken from the onboard recording camera during a vertical
weld inspection.
158
successfully track the weld even as it was diving in the tank. A low quality screen
capture from the recording camera is provided in Fig. 10-4.
10.2
Propeller Based Design
10.2.1
Vehicle Design
Some slightly different research focused on exploring the broader applications of
pump-valve systems.
Specifically, we wanted to explore how pump-valve systems
can be incorporated to smooth, streamlined, robots that are designed for efficient
motions. We formulated a propeller based design that uses a high efficiency propeller
for longitudinal motions and two BAU units for sway, heave, pitch, and yaw. The
BAU's enable multi-DOF motions while still enabling a smooth, low drag shape.
While the use of propellers and the reduction in symmetry makes this design less
effective for infrastructure inspection or cluttered environments, such a design could
have potential for ocean or marine applications where there is a dual need for long
distance travel and multi-DOF maneuvering capability.
A rendering of our vehicle design is provided in Fig. 10-6. This figure shows how
two BAU units are now used to provide motions to supplement the propeller at the
tail of the vehicle. The BAU's are placed in the nose and tail, and allow an external
shape that is completely smooth. A photograph of the fully assembled prototype is
provided in Fig. ??. This figure illustrates both the maneuvering system as well as
the smooth outer shape.
10.2.2
Vehicle Performance
For the control of the propeller based design, we again used feedback control rather
than placing fins at the tail of the robot. Since this vehicle is more streamlined and
moves slightly faster (~
0.2m/s) than the other designs, the design is highly unstable.
An illustration of the open loop performance of the vehicle is provided in Fig. 10-7-a.
Note how the yaw angle, 0, diverges rapidly.
The use of feedback control allows
heading stabilization, but there still exist oscillations in the response. In addition,
passively stabilizing pitch using gravity is also challenging due to the shape and speed
of the vehicle. Pitching motions are noticeable and tend to degrade heading control.
159
It
1-
I
Figure 10-5: A rendering illustrating the maneuvering and propulsion system for the
propeller based design.
Pump-valve System
95mm
200mm
Figure 10-6: Photographs illustrating the maneuvering system as well as the outer
shape of the propeller based robot prototype.
160
A disturbance test was also performed using this system, and the results are provided
in Fig.
10-7-b. The data in Fig. 10-7 shows that the controller can reject sizable
disturbances but is vulnerable to oscillations in its steady state response. Future work
will focus on improving this control performance.
180 -Open Loop
160 _.yWdy
-tt
60
Loop
-
140 -Closed
80-V(t)
120
-40-
'a;100
180
60
> 20-
40
0
20
0 0
2
-- ---4
---
-
6
8
Time [s]
10
12
2
(a) Straight Test
5
Time [s]
10
15
(b) Disturbance Rejection
Figure 10-7: Heading angle data for both the straight test (a) and the disturbance
rejection test (b).
The turning performance of the propeller based design was also evaluated. Since
this vehicle is asymmetric, 1800 turns will sometimes be required. Data from a 1800
turn while moving forward is shown in Fig. 10-8-a. This figure illustrates that the
stabilizing controller can be used to achieve dynamic turning motions as well.
In
addition, we examined the ability of the robot to perform stationary turns. As Fig.
10-8-b shows, the robot is capable of performing rapid and precise stationary turns.
This type of motion is very valuable for operating in more confined environments.
10.3
Summary
In this chapter we have shown how the pump-valve system can be broadly beyond
reduced actuation designs.
mented, and evaluated.
Two additional designs have been formulated, imple-
These two designs show that the pump-valve systems and
control concepts that we have outlined in this thesis can serve as a valuable tool for
underwater vehicle design for a variety of applications and missions. Our group continues to explore developing new types of smooth, multi-DOF robots, with the most
161
200
200
---
150
(t)
150-
100
100
I
0
50
>- 50
0
0
--- d(t)
2
4
6
Time [s]
8
"'1)
10
2
2-4
4
-V
6
Time [s]
(a) High Speed Turning
8
(b) Turning in place
Figure 10-8: Heading angle data for both turning at speed (a) and stationary turning
(b).
recent work being a small ball shaped vehicle propelled using 2 valves and a single
pump [68].
162
Chapter 11
Conclusions
This thesis has presented a new approach to the design of highly maneuverable underwater inspection robots, which we have titled Control Configured Spheroidal Vehicles
(CCSV). These vehicles use powerful water jets to achieve motions in 5 directions and
achieve high performance and robustness by having a completely smooth spheroidal
shape. The vehicle prototype outlined in this thesis is the first of its kind due to both
a novel pump-valve propulsion system, elegant jet design, and careful use of active
stabilization. We believe the tools, technologies, and lessons provided in this thesis
are generally applicable and will contribute to the design, control, and analysis of
new types of underwater systems.
11.1
Summary of Thesis Contributions
We summarize the key contributions of this work.
" We proposed a new type of underwater robot, the CCSV, that is completely
smooth and uses water jets to motions in 5 directions.
To the best of our
knowledge, no such robots currently exist.
" We identified a new type of water jet propulsion system that combines powerful
centrifugal pumps with fluidic valves. We characterized this system using basic
principles and developed tools for the design, analysis, and optimization of such
devices.
" We used this new pump-valve system and combined it with unique orthogonal,
163
dual-output port pumps in order to create Bi-axis Actuation Units (BAU). We
outlined a smooth robot design that achieves multi-DOF motions using only
two pumps through the use of reduced actuation design and a clever diamond
jet configuration. We constructed a fully functional, tether-free prototype that
can serve as a test bed for analysis, dynamics, and controls. This prototype is
the first of its kind.
" We examined precise orientation control for the CCSV-RAD prototype by developing 3 pump-valve control algorithms. We showed that high speed valve
switching and Pulse Width Modulation (PWM) can be used to avoid pumpvalve nonlinearities such as dead zones. We developed several analysis tools for
PWM controller design, and verified these principles using both simulations as
well as the CCSV-RAD prototype system.
" We explored the CCSV-RAD concept by examining the significance of hydrodynamic instability. We used linearized models to outline insights that can be
used to educate vehicle design. A stabilizing feedback controller based on angle
and rate measurements was outlined and shown to achieve impressive control
performance for both yaw and sway. This controller was analyzed with simulations and then implemented on the CCSV-RAD prototype. Experiments were
used to show how the control system can achieve stability even in the face of
very large disturbances.
" We performed a variety of experiments with the CCSV-RAD prototype in order
to illustrate its unique performance.
The forward and backward speed was
measured, and the unique sideways mode was shown. The turning capability
was showcased with both stationary and high speed turning tests. Finally, a
passively stabilized prototype was designed and used to provide a conventional
comparison to the CCSV-RAD prototype. These experiments clearly illustrated
the value of the CCSV concept and illustrated how the performance drawbacks
of fins can be avoided through the use of clever design and feedback control.
" We analyzed the CCSV design from a more general perspective in order to
164
understand how to create the best overall design. Control systems principles
were used to explore the role of jet angle and shape. Specifically, the jet angle
and vehicle shape were examined from the perspective of optimizing open loop
zero placement. Simulations illustrated how zero placement plays a critical role
in closed loop performance. Ranges for both the jet angle (15 -450)
and vehicle
aspect ratio (1.3 - 2.3) were provided. These results can be used to create a
host of new CCSV designs for a variety of scales and applications.
* Lastly, we assessed the performance limitations of underwater vehicles which are
stabilized through feedback control. We used Bode's Integrals to examine both
the speed limits for our CCSV designs but also to explore the ramifications of
feedback stabilization for a variety underwater systems. We showed how feedback control holds promise for certain types of vehicles but also identified conditions where feedback stabilization will lead to excessively poor performance.
11.2
Future Work
This area remains an area of exciting research, and there exist many directions
for future work. We divide the future work into two sections; the first is focused on
practical problems that need to be addressed before the CCSV and robots like it can
be deployed into actual infrastructure environments. The second is section describes
exciting research topics that remain relatively unexplored and could have a significant
impact.
11.2.1
Practical Considerations
The most critical areas for further development on the CCSV designs are 1) depth
control, and 2) design for high pressures ( 40m depth).
As we have mentioned in
previous chapters, closed loop depth control is a necessity due to the challenges with
designing a vehicle that is exactly neutrally buoyant. Depth control has been achieved
for a variety of underwater vehicles, and will likely involve using a small pressure sensor to measure hydrodstatic pressure. Designing the prototypes for higher pressures is
more challenging but is also a well known field. Static analysis must be used to design
the waterproof housing so that it either can withstand large pressure differentials, or
165
the electronics must be designed in such a way that an air filled housing is no longer
a necessity. This will likely increase the size of the robot because the chamber walls
will have to be thicker, and the wire routings will require careful assessment.
11.2.2
Potential Research Directions
There also exist several exciting research directions that emerge from this work.
The first is the continued design of CCSV robots to fully incorporate the lessons
outlined in the latter chapters of this thesis. For example, an experimental assessment
of various aspect ratios would provide further knowledge on the role of the vehicle
shape. In addition, pump design remains a very important and interesting area. The
dual output pump was designed through experiments, and a more careful analysis of
pump dynamics and impeller geometry should lead to even better performance.
Localization and sensing are two other important challenges. While we have shown
that the CCSV can achieve impressive control performance, our experiments were
limited to orientation control due to challenges with measuring xy positions. Current
work underway in our research group could provide improved techniques for measuring
the position of the robot relative to certain features. This could prove very valuable
for positioning and focusing sensors. In addition, the practical experience of piloting
the prototype vehicles remotely illustrated very clearly how difficult it is to pilot the
vehicle without any information about the robot position or orientation with respect
to known features. Localization and mapping tools could be used to greatly improve
piloting and control.
Finally, communication remains a largely unresolved technical challenge. The goal
is to have wireless robots that can operate without a tether. Many visual inspections
are done using remote control, so a live video feed is essential. Transmitting video
data through water remains a big challenge. Optical communication is seen recent
developments, but such systems remain relatively large.
Acoustic communication
provides long range, but does not provide the data rates necessary for video. Miniaturizing optical communication technologies would go a long way towards enabling
robotic underwater visual inspections.
166
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