Design, Development and Control of Mobile

Design, Development and Control of Mobile
Design, Development and Control of Mobile Biaxial
Inverted Pendulum
by
Jonathan Missel
Friday, August 7th 2009
A thesis submitted to the
Faculty of the Graduate School of the
State University of New York at Buffalo
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Mechanical and Aerospace Engineering
Copyright by
Jonathan Missel
2009
ii
Acknowledgements
First and foremost, I wish to thank my family for their unwavering support throughout my education. I have focused heavily on my education over the past several years,
and they have all made sacrifices on my behalf. My parents, Dan and Linda Missel, have been exemplary models of success, both as parents and in their respective
careers. Their dedication to selflessness has always been an inspiration to search for
broader meaning. My dad gets special thanks for his many hours spent helping design
and manufacture Vertigo, and my mom, for help editing this thesis. Thanks to my
sister and her husband, Becky and Charlie Fennie, for their perpetual encouragement,
sarcastic candor and comic relief. I would also like to extend my gratitude for their
engenderment of my favorite quartet of playmates, Danny, Eli, Ainsley and Mara.
Those kids are a blast, and their chaos is confoundingly soothing. I want to thank
my brother, Matt Missel, for his support, companionship, ideas and advice as we
grew up, and now pursue similar academic fields together. His mechanical wizardry
was of tremendous assistance in designing Vertigo. Thanks to all of my friends back
in Rochester for their understanding and encouragement as I have spent the majority
of my academic career a city away.
Thanks to several in academia, particularly my advisor Dr. Puneet Singla, for
being available and willing to help. It was his courage and insight that made gambling
on my endeavors end in success. It has been a pleasure working with him, and I now
iii
look forward to working under his former advisor as I continue on towards my Ph.D.
at Texas A and M University. I would also like to thank John Wadach, Dr. Tarunraj
Singh and Dr. David Forliti for their openness and inspiration on both personal
and academic matters. Thanks to all of my labmates over the years, those in Dr.
Singh’s lab, Kurt Cavalieri and Jack Lee for weathering my antics with a smile, and
maintaining an eagerness to assist. Special thanks to Shajan Thomas and Mark
Tjersland for their computer genius and immense contribution as they volunteered
their time and expertise to my research. I would like to thank cs for her prolonged
belief, uplifting humor and studio photography skills. Lastly, thanks and apologies
to anyone I may have inadvertently left out; I attribute this accomplishment to a
myriad of combined efforts, and I thank you all!
iv
Contents
Acknowledgements
iii
List of Figures
xiii
List of Tables
xiv
Abstract
xv
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Design
17
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.1
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2.2
Development . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.3
Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3
Actuation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
v
2.4.1
Onboard Computer . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4.2
Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.4.3
Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.4.4
Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.5
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.6
Design Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.7
Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3 Mathematical Modeling and Analysis
66
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.2
Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.2.2
Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.2.3
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.2.4
Ground-Based Derivation . . . . . . . . . . . . . . . . . . . .
75
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.3.2
State Observability and Controllability Analysis . . . . . . . .
79
3.3.3
Design Parameter Observability and Controllability Analysis .
86
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3.3
3.4
4 Communication Architecture
99
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Onboard Sensing Architecture . . . . . . . . . . . . . . . . . . . . . . 100
4.3
External Sensing Architecture . . . . . . . . . . . . . . . . . . . . . . 102
vi
99
5 Control
110
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2
Ground-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3
Sphere-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1
Power Optimal Control: Fixed Final Time, Unconstrained Input 117
5.3.2
Power Optimal Control: Free Final Time, Constrained Input . 120
5.3.3
LQR Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3.4
Stabilizing LQR . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.5
Tracking LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Estimation
137
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2
Continuous-Discrete Extended Kalman Filter
6.3
Application of Continuous-Discrete EKF . . . . . . . . . . . . . . . . 140
6.4
Sampling Time Analysis for C-D EKF . . . . . . . . . . . . . . . . . 148
7 Implementation
. . . . . . . . . . . . . 138
153
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2
Implementation Challenges . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3
Ground-Based Implementation . . . . . . . . . . . . . . . . . . . . . . 158
7.4
Balancing Implementation . . . . . . . . . . . . . . . . . . . . . . . . 169
8 Conclusions and Future Work
176
Bibliography
181
A Supplementary Concepts
184
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
vii
A.2 Directionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.3 Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.4 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A.5 Control Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.6 Pivot Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.7 Sphere Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.8 Weight Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
B Powering Vertigo and Qwerk
198
C Connection to Vertigo
201
D Changing Qwerk’s Embedded Code
214
E Maintenance
237
viii
List of Figures
1.1
(a) Stationary manipulator (b) Statically stable vehicles (c) Omnidirectional statically stable vehicle (d) Differential drive balancing vehicle (e) Bipedal walker. . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Statically and dynamically stable vehicles encountering terrain. . . . .
5
1.3
Anatomical planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1
Fixed biaxial inverted pendulum concept. . . . . . . . . . . . . . . . .
20
2.2
Front, 45 deg. and top view of omni-directional sphere actuation concepts, (a) Concept A: inverse mouse, (b) Concept B: omni-wheel Euclidian perpendicularity, (c) Concept C: omni-wheel Euclidian orthogonality, (d) Concept D: omni-wheel spherical perpendicularity. . . . .
23
2.3
Vertigo 1.0 assembly photograph. . . . . . . . . . . . . . . . . . . . .
40
2.4
Vertigo 1.0 motor-encoder-wheel exploded subassembly (CAD). . . .
44
2.5
Vertigo 1.0 motor-encoder-wheel compressed subassembly (photograph). 45
2.6
Vertigo design generation. . . . . . . . . . . . . . . . . . . . . . . . .
50
2.7
CNC mill machining Delrin legs for Vertigo 2.1. . . . . . . . . . . . .
55
2.8
Modifying omni-wheels for Vertigo 2.1 use (photographs). . . . . . . .
59
2.9
Vertigo 2.1 motor-encoder-wheel exploded subassembly (CAD). . . .
61
3.1
Planar sphere-based Vertigo model. . . . . . . . . . . . . . . . . . . .
70
ix
3.2
Ground-based free body diagram. . . . . . . . . . . . . . . . . . . . .
76
3.3
Rank of controllability matrix while varying θ and φ. . . . . . . . . .
80
3.4
Rank of observability matrix while varying θ and φ. . . . . . . . . . .
81
3.5
Inverse condition number of controllability matrix while varying θ and
φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.6
Inverse condition number of observability matrix while varying θ and φ. 83
3.7
Determinant of controllability matrix while varying θ and φ. . . . . .
84
3.8
Rank of controllability matrix while varying θ̇ and φ̇. . . . . . . . . .
85
3.9
Rank of observability matrix while varying θ̇ and φ̇.
. . . . . . . . .
85
3.10 Rank of controllability matrix while varying θ̇ and φ̇. . . . . . . . . .
88
3.11 Rank of observability matrix while varying θ̇ and φ̇.
88
. . . . . . . . .
3.12 Inverse condition number of controllability matrix while varying Vertigo mass and `.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.13 Inverse condition number of observability matrix while varying Vertigo
mass and `. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.14 Determinant of controllability matrix while varying Vertigo mass and `. 91
3.15 Inverse condition number of controllability matrix while varying Vertigo mass and ` with fixed rotational inertias. . . . . . . . . . . . . .
91
3.16 Rank of controllability matrix while varying sphere mass and sphere
radius.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.17 Rank of observability matrix while varying sphere mass and sphere
radius.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.18 Inverse condition number of controllability matrix while varying sphere
mass and sphere radius. . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.19 Inverse condition number of observability matrix while varying sphere
mass and sphere radius. . . . . . . . . . . . . . . . . . . . . . . . . .
x
94
3.20 Determinant of controllability matrix while varying sphere mass and
sphere radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.21 Inverse condition number of controllability matrix while varying sphere
mass and sphere radius with fixed rotational inertias. . . . . . . . . .
96
4.1
Onboard sensing communication architecture. . . . . . . . . . . . . . 101
4.2
Ground-based communication schematic. . . . . . . . . . . . . . . . . 109
5.1
Ground-based free body diagram. . . . . . . . . . . . . . . . . . . . . 113
5.2
Simulation results for a circle reference trajectory: norm-error 0.7821
5.3
Planar sphere-based Vertigo model. . . . . . . . . . . . . . . . . . . . 116
5.4
Gramian based control for 2π translation in 1.5 sec. . . . . . . . . . . 119
5.5
Gramian based control, u max = 55. . . . . . . . . . . . . . . . . . . 121
5.6
Gramian based control, u max = 45. . . . . . . . . . . . . . . . . . . 121
5.7
Gramian based control, u max = 20. . . . . . . . . . . . . . . . . . . 122
5.8
Gramian based control, u max = 15. . . . . . . . . . . . . . . . . . . 122
5.9
Plot of successful u max convergences and associated final times.
5.10 State response with stabilizing LQR controller.
115
. . 123
. . . . . . . . . . . . 126
5.11 Response with stabilizing LQR controller wrt vertical.
. . . . . . . . 128
5.12 State response with augmented stabilizing LQR controller. . . . . . . 129
5.13 Response with augmented stabilizing LQR controller wrt vertical. . . 129
5.14 State response with tracking LQR controller, heavily weighted rates.
132
5.15 Response with tracking LQR controller wrt vertical, heavily weighted
rates.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.16 State response with tracking LQR controller, heavily weighted angles.
133
5.17 Response with tracking LQR controller wrt vertical, heavily weighted
angles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xi
5.18 State response with augmented tracking LQR controller, heavily weighted
angles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.19 Response with augmented tracking LQR controller wrt vertical, heavily
weighted angles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.20 State response with augmented tracking LQR controller, realistic weighting.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.21 Response with augmented tracking LQR controller wrt vertical, realistic weighting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1
Planar measurement model for sphere-based Vertigo. . . . . . . . . . 141
6.2
C-D EKF, true state response.
6.3
C-D EKF, estimated state response.
6.4
C-D EKF, angle state errors.
6.5
C-D EKF, rate state errors. . . . . . . . . . . . . . . . . . . . . . . . 147
6.6
C-D EKF, true state response, fm = 25Hz, fs = 100Hz. . . . . . . . 149
6.7
C-D EKF, estimated state response, fm = 25Hz, fs = 100Hz. . . . . 149
6.8
C-D EKF, angle state errors, fm = 25Hz, fs = 100Hz. . . . . . . . . 150
6.9
C-D EKF, rate state errors, fm = 25Hz, fs = 100Hz.
6.10 Sampling time mesh.
. . . . . . . . . . . . . . . . . . . . . 145
. . . . . . . . . . . . . . . . . . 145
. . . . . . . . . . . . . . . . . . . . . . 147
. . . . . . . . 150
. . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1
Proportional two-dimensional step track.
7.2
Circle trajectory tracking using proportional controller with proportional yaw control: norm-error=2.3897.
7.3
. . . . . . . . . . . . . . . 160
. . . . . . . . . . . . . . . . 162
Circle trajectory tracking using PID controller with proportional yaw
control: norm-error=1.8976. . . . . . . . . . . . . . . . . . . . . . . . 162
7.4
Circle trajectory tracking without yaw control: norm-error=0.8614. . 163
xii
7.5
Circle trajectory tracking with proportional yaw control: norm-error=0.3866.
163
7.6
4 circle trajectory tracking laps with proportional yaw control.
. . . 165
7.7
Circle trajectory tracking with transformed control: norm-error=0.5402.
166
7.8
Circle reference trajectory tracking while using proportional yaw control for various retrogrades. . . . . . . . . . . . . . . . . . . . . . . . 167
7.9
Yaw control for 2π retrograde and 20π retrograde.
7.10 Balancing plank pitch with proportional control.
. . . . . . . . . . 169
. . . . . . . . . . . 171
7.11 Balancing plank pitch with PID control. . . . . . . . . . . . . . . . . 173
7.12 Balancing plank pitch with PID control by tracking Vertigo. . . . . . 173
A.1 Directionality (a) Bi-directionality (b) Bi-directionality with yaw (c)
Omni-directionality (d) Omni-directionality with yaw. . . . . . . . . . 185
A.2 Stages of translational motion. . . . . . . . . . . . . . . . . . . . . . . 187
A.3 Spherically orthogonal omni-directional actuation. . . . . . . . . . . . 189
A.4 Ground-based trace of translation with rotation. . . . . . . . . . . . . 190
A.5 Inverted pendulum on cart. . . . . . . . . . . . . . . . . . . . . . . . 193
A.6 Uncontrolled pivot drift. . . . . . . . . . . . . . . . . . . . . . . . . . 194
D.1 Wiring for access to Qwerk. . . . . . . . . . . . . . . . . . . . . . . . 217
xiii
List of Tables
2.1
Features of drive concepts. . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
Properties of Delrin. . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
xiv
Abstract
The limited versatility of existing experimental robotic platforms has left demand for
a single practical unit with broad application in dynamics and controls. To address
this, an autonomous biaxial robotic inverted pendulum is proposed that balances and
navigates on top of a sphere. The spherical base and innovative actuation techniques
permit omni-directional translation, and direct control of yaw. With this unprecedented motion, Vertigo, as it is called, is capable of modeling a myriad of other
systems, and offers great potential for original work. To maximize versatility it was
designed for reconfigurability, can accept spheres of different size and can operate
on the ground as a more traditional omni-directional vehicle. This thesis focuses on
the design, mathematical modeling, analysis, communication, control, estimation and
early stages of implementation for Vertigo and its various configurations. Methods
used in this development have general application, making the results and findings
broadly relevant.
xv
Chapter 1
Introduction
1.1
Motivation
The conforming nature of existing mobile robotic platforms has lead to an impressive
understanding of like systems, but this convention has done little for the progression
of unique or complex systems. By following this paradigm, designers have imposed
significant limitations on their creations. The most severe of these constraints confine
the maximum acceleration and deceleration, ability to withstand unexpected loading,
and directional control. In the past these obstacles have been individually overcome
with robots that either balance but only have two degrees of freedom, or have omnidirectionality but are prone to become dynamically unstable. Recent attempts to
completely liberate machines from such inabilities have led to the expensive and often unnecessary complications of bipedal walkers. These platforms have proved to be
unreliable and well beyond most budgets of those interested in implementing them.
For simplicity, many experimental platforms are designed to be linear, though nonlinear systems dominate the real world. Linear control methods are only guaranteed
to work on nonlinear systems over small realms, limiting the usefulness of these plat1
Jonathan Missel
Motivation
forms to test many practical applications. To form a genuine knowledge of dynamics
and controls, there is a need to develop versatile robotics that can model a diverse
range of other systems, and contribute original experiments without being overly
convoluted or costly.
There are many desired characteristics for robotic platforms, they are mostly
rooted in performance requirements and are challenging to satisfy wholly with a single unit. Specifications are dependent on objective; however, autonomy, exceptional
mobility, responsive dynamics and cost effectiveness are among the most commonly
sought after traits. Figure 1.1 shows a series of robots that give a rough idea of what
is available.
Figure 1.1: (a) Stationary manipulator (b) Statically stable vehicles (c) Omnidirectional statically stable vehicle (d) Differential drive balancing vehicle (e) Bipedal
walker.
2
Jonathan Missel
Motivation
Figure 1.1(a) shows a stationary robotic arm. These are useful in assembly applications, where their dexterity, accuracy and stamina often find them performing tasks
that would be too mundane, difficult or tedious for humans. As with the numerous
bench top robotics in existence, these are very precise and relatively easy to model,
but as stationary robots their applications are limited. To provide mobility, statically
stable vehicles, like the example in Figure 1.1(b), are often the default platform. To
be statically (mechanically) stable simply means that on a flat surface the robot will
not fall over under the acceleration of gravity when all control is terminated. These
units are easily understood, and well documented which makes their implementation
more straightforward; however, limits circumscribe their mobility. They commonly
have four wheels where two steer similar to a car, or have differential drives like a
wheelchair. In this way they only have two degrees of freedom because they can move
forward/backward and turn, but cannot move side to side. Accordingly, the mobility
of these systems leaves something to be desired, especially in close quarters. The
acceleration of these platforms is also limited when their center of gravity is above
the axles of their wheels. This is due to a likelihood of falling over. Similarly, extreme terrain can cause them to tip, or have wheels lift off the ground. Mitigating
these issues requires a low center of gravity and large base, making them even more
cumbersome.
Omni-directional vehicles, similar to the one in Figure 1.1(c), have made advances
by solving some of the directionality limitations. These robots can move in any planar
direction. Their rotation does not affect their direction of motion, so a complex path
can be followed without having to spend time and power rotating. Further explanation
of directionality can be found in Appendix A. Traditionally, these are also statically
stable, and therefore inherit the risk of becoming dynamically unstable.
Balancing (dynamically stable) differential drive robots, also called self-balancing
3
Jonathan Missel
Motivation
robots, have begun to take advantage of dynamic stability. These are variations of
inverted pendulums, as shown in Figure 1.1(d), and require unremitting control to
balance because of their instability. However, this ostensible curse of instability is actually advantageous at times. Studies [1] have shown that dynamically stable robots
are far superior at navigating terrain. This is because they are able to shift their
weight to keep their center of gravity above their wheels, as shown in Figure 1.2 [2].
The value in this ability is most clearly seen when unexpected loading is applied and
the controllers automatically compensate to maintain balance. Self-balancing robots
also have greater potential for agility. Controlling where their center of gravity is
allows dynamically stable robots to take advantage of their instability and let gravity
assist in the task of acceleration. This is similar to how jet fighters are intentionally aerodynamically unstable for greater performance. In light of these dynamic
prospects, the two-wheeled inverted pendulum configuration has been eagerly embraced as the focus of many research projects worldwide; however, this is a myopic
response to straightforward results. Such vehicles do not take full advantage of the
very feature that sets them apart, their dynamic stability. They are self-balancing
on one axis, but maintain dependence with static stability on the other. Since they
can still tip over in one direction, slopes must be dealt with head-on and directional
control is lost. They also have problems with unexpected external loading applied
in both directions, such as in the task of opening a door. Without prior knowledge
of the exact size and location of the door, two-wheeled self-balancing robots cannot
accomplish this. While they do have a niche, two-wheeled self-balancing robots can
be viewed as an incomplete idea because they maintain the disadvantages of their
constituents: they require active control to maintain stability, yet are still prone to
tipping over.
Bipedal walkers like Honda’s ASIMO (Figure 1.1(e)) have managed to exploit
4
Jonathan Missel
Motivation
Figure 1.2: Statically and dynamically stable vehicles encountering terrain.
dynamic-stability in both directions, and are omni-directional to boot. These machines have taken advantage of biomechanical design by letting nature come up with
the solution. Modeling robots after humans promises extremely agile performance;
ASIMO can run, carry objects, and even climb stairs. With continued development,
these platforms will even be able to outperform humans. However, all this grandeur
comes at a tremendous cost, both fiscal and in robustness. Each of the 48 ASIMO
units cost nearly $1,000,000 just to manufacture. This does not include the fund-
5
Jonathan Missel
Motivation
ing devoted to research and development, or any other aspect of its engenderment.
Bipedal walkers are also inundated with single-point-failures which foster vulnerability and unreliability. ASIMO is 120 lbs of wires, actuators, sensors, circuits and
structure, failure of any one of these components (with few exceptions) results in
complete malfunction. The unreliability and cost of bipedal walkers has pushed their
availability out of reach for most researchers, causing many facilities, such as MIT’s
Leg lab, to shut down.
The biaxial inverted pendulum was the iconoclastic idea that combined dynamic
stability and omni-directionality in a frugal and simplistic manner. Rather than
differential drive self-balancing robots, the conclusion that should have been drawn
from understanding dynamic stability is to balance on a sphere and liberate both
axes. By simply mirroring the effect of the two-wheeled inverted pendulum, this idea
is not entirely disparate, and accordingly seems logical; however, an ongoing literature
study revealed that this has only recently been done by one other group at Carnegie
Mellon University with Ballbot in 2006 [3], [4]. The purpose of the Ballbot project is
to investigate how to maintain reliable balance in robots to decrease their footprint
for improved navigability. This is meant to overcome the problem of statically stable
platforms having wide bases. Ballbot is a commendable project that has yielded
impressive results with its mobility. However, there is one remaining feature that
lingers, unaddressed: yaw control. Ballbot is omni-directional, so it can move and
change direction without having to turn, but it also has no choice, it is not able to
turn.
This thesis introduces a robot named Vertigo, a proposed solution to the final
obstacle in achieving a dynamically stable system with all three planar degrees of
freedom. As it was in the contributory stages, this evolution in mobility was an
exercise in innovative mechanical design, and was rooted in punctuations of dynamics
6
Jonathan Missel
Background
and control theory. In many ways, Vertigo is similar to Ballbot, with its primary
departure being the method of actuating the spherical base. It does so in a way
that fully liberates all three forms of planar motion by directly controlling yaw, in
addition to Ballbots two directional degrees of freedom. It was also designed to
maximize adaptability by accepting different sized spheres, reconfiguring to change
center of mass and inertial characteristics, and can even operate without the sphere
as a statically stable omni-directional ground vehicle. This allows Vertigo to carry
out maneuvers that no other vehicle can, giving it the flexibility to model and test
control theory for a wide range of other systems.
Rocket guidance, under-actuated spacecraft, and in-flight refueling are among the
challenging control problems that require extensive research and experimentation.
Vertigo’s mobility makes it capable of modeling these more expensive and complex
systems which would help expedite their development in a cost effective manner.
Vertigo also has the potential to make original contributions in the fields of rover
locomotion, and human and machine cognition among others. This new form of
mobility promises to affordably stretch the bounds of what ground vehicles are capable
of.
1.2
Background
Recognizing the heritage from which Vertigo evolved is an important part of fully understanding it and appreciating the legacy that it carries on. The inherent instability
of inverted pendulums has proved critical to their contribution toward technological
advancement. They have been used in such diverse applications as seismograph instrumentation [5], biological neuron modeling [6], and testing control methodology.
Although this configuration has been around for a long time, the development of
7
Jonathan Missel
Background
mobile inverted pendulums is in relative infancy. The geometry of these systems
has a strong likeness to that of the human frame, for this reason, terminology from
anatomy is frequently used. Figure 1.3 shows how the anatomical planes of the human
body are defined. This chapter briefly outlines the background history of Vertigo’s
most unique feature, dynamic stability. More specifically, it covers the progression of
mobile inverted pendulums and their directionality.
Figure 1.3: Anatomical planes.
As a robotic platform, this system got its start in 1986 when Professor Kazuo
Yamafuji from the University of Electro-Communications in Tokyo successfully implemented the first wheeled inverted pendulum [7]. It was driven by a two-wheeled
base, with both rotating in unison. In this way, it could not turn and is conceptually
analogues to balancing on a cylinder. Lack of emphasis on documentation and pub8
Jonathan Missel
Background
lishing his work has brewed debate; however, this is regarded by many as the first
autonomous inverted pendulum. In 1988, Professor Yamafuji came up with another,
very different idea for the two-wheeled inverted pendulum [8]. An arm at the top
of the pendulum was actuated to swing, providing a control torque that stabilized
the pendulum. The two-wheeled base was controlled as if it were a statically stable
device, and could turn like traditional differential drive vehicles. Since the wheels did
not dictate the pendulum’s motion, this device was really a fixed inverted pendulum
with reaction wheel control for balance that was mounted on a mobile base.
In 1994, a group in Japan successfully demonstrated the control of the first twowheeled, true inverted pendulum balancing robot that was able to turn. A differential
drive allowed each wheel to be driven independently, enabling the base to rotate while
its translational position governed the pendulum balance [9]. The ability to balance
eliminated the need for differential drives to require a third, passive, caster that
functioned solely as a static stabilizer. Later, they introduced the first one-wheeled
balancing design. It rolled on the major axis of an ellipsoid to balance in the sagittal
plane, and hinged at the “hips,” to generate a control torque to balance in the coronal
plane [10]. By balancing on an ellipsoid, the advantages of having a differential drive
were lost, and rather than being able to turn in place, the robot was limited to wide
arcs. This was also the first attempt at a mobile biaxial inverted pendulum; however,
only one axis was controlled by the base, while the other was analogous to the reaction
wheel control.
In 1999, inventor Dean Kamen started a company devoted to making clean transportation that focused on utilizing dynamic stability. The mobile inverted pendulum
experienced its greatest boost in popularity in late 2001 when this burgeoning company unveiled the Segway Personal Transporter to the public for the first time [11]. It
had a two-wheeled differential drive that utilized the same principles as its Japanese
9
Jonathan Missel
Background
predecessors, but its sensing and control systems differed. The most noteworthy aspect of the Segway is that it was designed to be a vehicle for humans, a very dynamic
and unpredictable payload. The attractiveness of zero turning radius, speed, and the
novelty of this design drew a lot of attention; however, it confined research to the
two-wheeled differential drive configurations despite its many drawbacks. Their most
significant shortcomings are coupling between yaw and directional control, and their
tendency to become dynamically unstable when perturbed in the coronal plane.
Not until 2006 were the full benefits of dynamic stability exploited in ground
robotics with Dr. Ralph Hollis’ “Ballbot,” created at Carnegie Mellon’s Robotic
Institute [3]. Ballbot is a true biaxial inverted pendulum that balances on a ball.
Its chief advantage is its omni-directionality: it does not have to turn in order to
change directions. It is also dynamically stable in both directions, so the same control
methodology could be applied in both the sagittal and coronal plane. Ballbot was
designed as a robot for researching motion in interactive human environments, so it
took on the rough dimensions of the human frame. Its smooth motion has potential
to assuage the resistance to assimilation of human and robotic cohabitation at home
and in the work place. Though it is very slender, it is human height and difficult to
transport, so testing requires a large space and is risky. This serves as evidence that it
was not intended to be an experimental platform. Despite all its grandeur, Ballbot’s
configuration sacrifices direct control in yaw, a critical aspect of many trajectories.
In addition, functionality is somewhat limited because it cannot be driven on the
ground without the sphere, and the sphere has to satisfy very exacting dimensional
tolerances. This is a result of the method used to actuate the sphere, which is similar
to that of the ball and encoder assembly in an old computer mouse, but inverted.
Two cylindrical rollers are used to drive the sphere in the x and y directions, but
in order to prevent conflicting control inputs on the surface of the sphere they must
10
Jonathan Missel
System Description
be perpendicular and placed on the equator (largest cross-sectional diameter). When
one direction is controlled, the sphere rotates about the point of contact with the
perpendicular roller. This method decouples directional control, but requires that
the sphere be a very specific size to ensure the friction is sufficiently large enough to
prevent slip, yet small enough avoid impeding motion. Since the actuation had to
be placed at the equator, it was not able to support the weight of the robot on the
sphere, and passive rollers had to be included in the design for this.
Though relatively new, the mobile inverted pendulum has quickly proven its usefulness. From its humble beginnings as a bidirectional bench top experiment, to its
most recent manifestation as the biaxial, omni-directional and fully autonomous Ballbot, this platform has made rapid and fruitful progress. With this thesis providing a
solution for its only remaining absence in motion, direct control of yaw, this platform
awaits a strong future giving the robotics, dynamics and controls communities new
tools for discovery.
1.3
System Description
This thesis proposes a solution to help solve the problem of limited mobility in robotics
and its trade off with cost. The idea is that relatively inexpensive dynamically stable
omni-directional motion can be achieved by developing a platform that balances,
navigates and rotates on a single spherical wheel. This original design has potential
for exceptional agility, and was developed to promote mass dimensional flexibility for
adaptability to wide-ranging experimentation. As a fresh take on motion, it offers new
avenues for research, and broadens the scope of ground vehicle navigation. Vertigo,
as it is called, takes the form of a biaxial inverted pendulum with yaw control. The
name was chosen for its blend of irony and accuracy. The word vertigo comes from
11
Jonathan Missel
System Description
the Latin word vertigin, meaning “dizziness,” and can be traced further back to the
word verto, meaning “I turn.” Today it is used as a medical term describing a form
of dizziness that includes the sensation of spinning or swaying, often resulting in loss
of balance. A symptom of balancing disorders seemed perfectly ironic for naming a
robot whose party piece is its ability to balance. Additionally, the meaning “I turn,”
form which the word was originally derived, is quite fitting because Vertigo’s ability
to rotate as a biaxial inverted pendulum is what makes it unique.
Vertigo was designed to be simplistic. With only four actively moving parts, all
of which are identical, the chance of mechanical failure is minimized; therefore, its
simplicity translates in to robustness. With a height of only 17 inches, and weight
of 5.6 lbs, it is easily transported, which is convenient for its role as a mobile experimental platform. The structural assembly was carefully designed with four legs that
support two interchangeable mounting plates to allow reconfigurability and fortify its
defenses against failure from impact. The combined influence of these traits granted
Vertigo with longevity and the ability to contribute experiments that pertain to vastly
different systems.
One of Vertigo’s applications is modeling under-actuated systems; these are systems that do not have direct control of all feasible degrees of freedom. Fully exploring
the dynamics of these systems is currently an active area of research; its mission is
to discover and carry out maneuvers that control more degrees of freedom then are
directly actuated. Requiring fewer actuators permits significant reduction in weight,
cost, and energy needed to perform tasks. In addition, when fully actuated systems
experience failure, these techniques may save the mission by using the remaining actuators to compensate for the loss. Laboratories in Japan are experimenting with
passive joints to help improve the dexterity of under-actuated probes [12]. Similarly,
Vertigo can be controlled to induce yaw rotation without direct control: a trajectory
12
Jonathan Missel
System Description
can be assigned that combines pitch and roll for a resultant rotation in yaw. This
maneuver is analogous to the way an airplane banks to turn, the rudder (primary yaw
actuator) plays little role in carrying out substantial net changes in direction. By design, Vertigo 1.0 is under-actuated because it does not have direct control of yaw, so
simulating this is straightforward. Vertigo 2.1 can also be configured or controlled to
be under-actuated by only employing direct control over any two of the three degrees
of freedom. With intelligent control, any combination of two of the four actuators
can be eliminated, and the system can still secondarily dictate all three degrees of
freedom.
Understanding how three degrees of freedom can be controlled by actuating only
one is somewhat abstract for this system. However, it can be explained with the aid
of a motorcycle stunt called a “circle wheelie,” as an illustrative example. In this
stunt, only the rear wheel of the motorcycle (used for propulsion) is on the ground,
while the front wheel (normally used for steering) is suspended. While maintaining
this stance, the bike and rider carve out circular trajectories by leaning to the side.
This is an extremely technical stunt because the precise control of three degrees of
freedom is completely dependent on the application of a single control input as the
rider manipulates the throttle to power the back wheel and directly maintain pitch.
When the bike naturally falls to one side, it turns in circles like a freely rolling coin.
To avoid falling over, the lean angle (roll) must be kept in check. This is done through
centripetal force by balancing speed with the roll angle while turning. To change the
direction of the circular turns, increasing speed will cause the roll angle to surpass
vertical and lean to the other side. Yaw is determined by how much of the turn is
completed before straightening out. Speed of rotation can be increased by taking on
a more aggressive roll angle with a shorter turning radius. To recap, pitch is directly
controlled through powering the rear wheel, and the coupled motion of roll and yaw
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Jonathan Missel
System Description
is governed by centripetal force and how long the stunt is carried out for. This is a
simplified explanation of the dynamics at work; phenomena like gyroscopic motion
and momentum are not considered. Additionally, the coupling between translation
and pitch is not stressed; when fully detailed, a total of four resultant degrees of
freedom are being managed. Though simplified, this is a close example of how Vertigo
2.1 could remain operational with actuator failure. As long as any two are intact, it
can be controlled analogously to a motorcycle on one powered wheel.
Even though it is possible to fully control an under-actuated system, it requires
high precision and is more time consuming to execute than having a fully-actuated
system; therefore, these methods are generally implemented when actuators fail unexpectedly, or when weight and cost make including more actuators unreasonable.
This is often the case with spacecraft. In space, they are free to move in six degrees
of freedom, but the cost of building and launching space vehicles makes reducing the
number of actuators desirable when possible. Also, replacing failed actuators is rarely
an option; therefore, these techniques may be exploited to save a mission using only
what remains operational.
Vertigo is a highly maneuverable platform, and is capable of modeling more than
just under-actuated systems. Some other examples with related motion to that of
Vertigo are: the control of in-flight refueling systems, quad-rotor helicopter hover
and planar translation, and actively controlled rockets. These systems are expensive
to operate, risky to test and require highly trained personnel in controlled settings to
ensure safety. Additionally, their development requires the progressive accomplishment of many stages that build up to a final result. A platform like Vertigo, that
can mimic the general motion of these systems, is ideal for addressing some of the
preliminary stages of development in an inexpensive, reusable and safe way. It can
also be reconfigured to best match the simulated dynamic response, and its small size
14
Jonathan Missel
System Description
and robust design make it practical for a single person to safely operate with minimal
risk of failure.
As an experimental platform, applications of Vertigo are not limited to modeling
other systems; it promises original contributions as well, particularly to the field of
rover locomotion. As previously discussed, a biaxial inverted pendulum has exceptional performance and agility because it takes advantage of dynamic motion, rather
than fighting it. With fully developed control, Vertigo can exploit omni-directional
motion with independent yaw, scale inclines and terrain in any direction, and can actively adapt to unexpected loadings. These qualities make it ideal for accomplishing
the objectives of many roving scenarios. Its fluid motion also has enormous potential
to bridge the gap for human and robotic coexistence. Building this cooperation is a
driving motivation behind Carnegie Mellon’s Ballbot project. Smooth robotic interaction with humans mandates that machines have high centers of gravity and small
footprints; however, this limits the maneuverability of statically stable robots due to
their likelihood of tipping. Developing dexterous robotics may help the interface between lifeless machines and skeptic humans. In doing so, the importance of humanlike
motion and dimensions is more than just psychological; cohabited environments expect both parties to operate in surroundings designed for the human frame. Though
the setting may be different, this is the primary objective of NASA’s Human and
Robotic Exploration program, for designing robots that assist astronauts [13]. These
machines must be built to work in environments meant for humans, without sacrificing functionality. The omni-directional motion and dynamic stability of Vertigo
means that it is better suited than most platforms to work side-by-side with humans
here on Earth.
Lastly Vertigo would make an exceptionally intriguing educational tool. There
are hundreds of platforms and experiments on the market to help teach dynamics,
15
Jonathan Missel
System Description
control and mechatronics, but most are limited in focus and fail to inspire students.
Vertigo tends to invoke curiosity to those first introduced to it, and is flexible enough
to accommodate work at all levels. Simple experiments can be set up to confirm and
illustrate concepts taught in the classroom. Demonstrations that alter the physical
properties and investigate the dynamics in comparison to theory would be an excellent
aid for proving concepts at an introductory level, and would not be difficult provided
a small bank of controllers. Additionally, its mobility and versatility offer much to be
explored in advanced controls and estimation research.
16
Chapter 2
Design
2.1
Introduction
Designing hardware is a notoriously challenging aspect of experimentation. It requires thorough knowledge of how mechanical components fit and work together once
assembled, and comprehension of underlying theory, so as not to corrupt the principles being tested. To simplify and insure the success of this process, previous work
of others often serves as a foundation to build upon. Designing this particular hardware was especially challenging and critical because much of the projects originality
stemmed from its mechanics. As a unique platform, there were few examples of pertinent work to guide its development; therefore, extra effort was made to ensure that
functionality, accuracy and robustness were built into the design for its long-lasting
success.
Many factors must be considered when conceiving and producing designs for a
brand new robotic system. Often, these factors have conflicting solutions, and decisions must be made based on priority of tradeoffs. This chapter details how these
decisions directed the conception, design and production of Vertigo. Starting with the
17
Jonathan Missel
Conception
main objective, the evolution of the final product is detailed in full. Along the way,
the challenges and logic behind every decision is explained, giving both the advantages and disadvantages of all considered options. The techniques and processes for
manufacturing are also included, as this knowledge is an important tool for efficient
design. Lastly, suggestions for improving or altering future designs are given.
2.2
Conception
This section discusses the engenderment of Vertigo as a concept. It starts by establishing the design objective, which is to be the motivation throughout, and the
reason behind every decision made in the project’s maturity. Various solutions to
the given objective are then detailed, and supporting arguments are presented for a
platform that balances on a sphere. Finally, the challenges implied by this solution
are explained, and set to be addressed in the remaining design.
2.2.1
Objective
The primordial objective that seeded this project was to create a new ground platform
with exceptional mobility, and mechanical simplicity. A machine capable of maneuvering with few constraints is attractive for experimentation and promises application
in many areas. Achieving this in a cleverly simplistic manner, with no unnecessary
parts, means that weight and cost can decrease while reliability and performance
increase. Simplicity also aids in repair, assembly, disassembly, manufacturing and
mathematically modeling. Therefore, the goal was to create an inexpensive testbed
that was mechanically efficient, and had as many degrees of freedom as possible.
18
Jonathan Missel
2.2.2
Conception
Development
To meet all aspects of the objective, a drive concept that provided mobility was first
generated, and then an efficient design was created to embody this concept. Looking
to the work of others, there were many existing testbeds that were highly maneuverable, but they were overly complex, or specialized in their application. Some were
detached all together from any practical application, and served only as experimental
amusement. Turning to the work of creation, it was clear that omni-directionality is
very common in nature and uncontrolled systems. This would be very advantageous
for improving mobility; however, it is notoriously difficult to accomplish properly in
robotics. Nature also features many variations of the inverted pendulum throughout
anatomical motion, and its purpose is often improved performance and agility. This
can be observed when an agile creature like the cheetah turns while at speed. Front
on, the animal carries the tall slender signature of an inverted pendulum. As it turns,
it leans into the corner to cancel the centripetal force from turning. By matching the
angle of lean with the aggressiveness of the turn, the animal experiences no lateral
forces and can turn much sharper. A statically stable system in these circumstances
would either tip over, or lose traction with the ground and slip. The inverted pendulum is also attractive because it is famous for being a benchmark problem in control.
Accordingly, an omni-directional inverted pendulum was decided on for the desired
design.
To make an omni-directional inverted pendulum, the bottom of the pendulum
would ideally be actuated in both directions independently. The first concepts brainstormed for this were fixed platforms that supported the base of a pendulum and
19
Jonathan Missel
Conception
actuated it in both directions. This was often done with perpendicular tracks or
rails, like those in Figure 2.1. These devices could control the pendulum in any
planar direction, but their motion was limited to the size of the chassis. A larger
range of motion would require a larger device, which quickly becomes cumbersome
and impractical. To overcome this restriction, a mobile vehicle would need to be used.
Figure 2.1: Fixed biaxial inverted pendulum concept.
Omni-directional vehicles had been done, but they were statically stable, with
wide footprints and low centers of gravity. Inverted pendulum vehicles also existed,
but they were not omni-directional; they could only move forward and backward and
had to turn when changing directions. The idea of placing a pendulum on top of an
omni-directional vehicle seemed promising at first, it would allow both features to
be tested as controls experiments. This is similar in principal to the fixed platforms,
20
Jonathan Missel
Conception
but would not be limited to the size of the base. The major drawback of this configuration is that it would not fully take advantage of the unstable dynamics of the
pendulum. Cornering is still mostly dependant on the abilities of the statically stable
base, with minor influence from the pendulum. Properly combining the two features
would mandate a new platform all together. To ensure that the inverted pendulum
dynamics were incorporated, it was necessary that the base only have one point of
contact with the ground. This base also needed to move in any direction and hold
as much symmetry as possible to maintain consistency in its dynamics. From these
criteria, a sphere was chosen to act as a single-wheeled base that a pendulum would
balance on.
2.2.3
Design Challenges
As with every new idea, there are many hurdles to overcome in the preliminary
stages of design. With balancing on a sphere as the updated objective, the dominant
struggle was developing a method for precisely controlling the sphere in two directions
while it simultaneously serves as a base for supporting the robot. For control in any
directions, multiple actuators must be interfacing the same surface while oriented in
different directions.
Another challenge in the design of this robot, was maintaining the highest possible degree of symmetry to preserve the foreseen simplifications it would permit in
the modeling and controls development. A symmetrical system should respond the
same in all directions, yielding consistently predictable motion. To retain symmetry,
all components (structure, electronics, actuators, etc.) must be designed, chosen and
incorporated with precise intention.
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Jonathan Missel
2.3
Actuation Method
Actuation Method
Pursuing such a unique drive made the method of actuation the most critical aspect to its success. Controlling one spherical surface in multiple directions is not a
straightforward task because multiple actuators are required to be in contact with
the sphere while avoiding conflicting control. By studying the geometry and motion
of a sphere, four solutions to this problem were generated; Figure 2 shows the front,
45 degree and top view for the four concepts. The commonality in these methods
is their omni-directional control, but there are many differences in their complexity,
practicality and performance.
The first idea, concept A, came from contemplating the motion of a basketball
spinning on a finger. If the finger is assumed to be a single point of contact, then the
sphere can rotate about it, and the point will be uninfluenced by the rotation. To
control the sphere’s rotation, a second point of contact, a roller in this case, must be
present. To avoid interference, the roller must be constrained to only apply control
in the direction the sphere would naturally rotate about the first point. Now, if both
points of contact are rollers, it can be deduced that omni-directionality can only be
achieved if the axis of both rollers are in a plane that intersects the largest diameter
of the sphere. Furthermore, control will be coupled for all orientations in that plane
unless the two rollers are perpendicular. Since the sphere must be placed on the
ground, a third point of uncontrolled contact is introduced. To account for this, the
plane must be parallel with the ground, so the equator of the sphere is intersected by
the plane. Figure 2.2(a) shows the geometry of this configuration. This is evidently
the inverse of how an old computer mouse operates, when the ball rolls, perpendicular
22
Jonathan Missel
Actuation Method
rollers on the equator of the ball decompose the rotation and encoders are used to
measure the component. Therefore, when used as an actuation method, only two
actuators are needed and they independently control a component of the rotation.
Figure 2.2: Front, 45 deg. and top view of omni-directional sphere actuation concepts,
(a) Concept A: inverse mouse, (b) Concept B: omni-wheel Euclidian perpendicularity, (c) Concept C: omni-wheel Euclidian orthogonality, (d) Concept D: omni-wheel
spherical perpendicularity.
Of the four designs, the inverse mouse drive is the easiest to understand, but there
are several drawbacks. Though it only needs two actuators, it cannot directly control
yaw rotation, so it only offers two degrees of freedom. The remaining drawbacks
are attributable to the mandate of control implementation at the equator. Since
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Jonathan Missel
Actuation Method
contact at the equator of a sphere does not support the weight of the vehicle above
it, additional passive points of contact are needed to hold the robot. Additionally,
just as it is with the computer mouse, at least one other point of contact must be
placed on the sphere to apply force against the rollers. Lastly, this dependence on the
sphere parameters means that each system has to be designed for a very specifically
sized sphere. This is undesirable, especially when such trivial things as temperature
and wear can change the diameter of the sphere enough to influence or incapacitate
the system.
The second solution, concept B, came from a different train of thought all together.
Traditional omni-directional robots overcome the problem of multiple actuators on
the same surface with omni-wheels. Based on this, Figure 2.2(b) shows how this
design’s Euclidian perpendicularity can be adapted to conform to the contours of a
sphere to give the same affect. As with the inverse mouse ball method, this has two
actuators oriented perpendicular to each other and in a single plane that is parallel
to the ground. However, by using omni-wheels rather than rollers, the actuator plane
is no longer constrained to intersect the equator of the sphere. This is because all
components of motion opposing the direction of control are passively permissible as
slip, and the perpendicular orientation of the wheels decouples the directional control.
Since control no longer needs to be applied at the equator, the sphere diameter is
irrelevant as long as it is large enough to make contact with both actuators. This
also means that the robot could be placed directly on the ground, analogues to a
sphere of infinite radius. Being able to drive the platform using two, completely
different methods is very attractive to the overall goal of experimental diversity and
adaptability. This method does however have disadvantages. As with concept A,
this needs passive contact with the sphere to help support the robot because the
two actuators cannot do this alone. These two actuators each control their respective
24
Jonathan Missel
Actuation Method
degree of freedom and do not allow for direct control in yaw, which is not pertinent to
omni-directionality, but would improve the mobility of the system. The remaining two
concepts solve the problem of including yaw along with omni-directional actuation.
Keeping to the theme of one actuator per degree of freedom, Figure 2.2(c) shows
how concept C proposes to completely decouple control in all three degrees of freedom
by using three orthogonal actuators. Here, roll is controlled by an omni-wheel on the
equator, pitch is controlled at the top of the sphere, and yaw is also controlled at the
equator, exactly opposite to the one that controls roll. Two variations exist for this
configuration, the yaw omni-wheel can be placed 90 degrees from where it is (either
direction) along the equator of the sphere. These configurations put an actuator at
each apex of a spherical quadrant, and orient them orthogonally in Euclidian space.
The omni-wheel orientation is critical to this approach because each location depends
on both the permitted slip from the wheels and the point of rotation idea from concept
A to decouple control. Together, these three exhaustively compose the entire set of
possible ways to independently control all three degrees of freedom using just three
actuated omni-wheels. If the position or orientation of these actuators is off even
slightly, control input will conflict and become coupled. Accordingly, this system is
highly dependent on geometry, and will only work with a specific size sphere which
is one of its flaws. Furthermore, this concept cannot operate on the ground. This
is because it relies on the geometry of the sphere to locate points of rotation when
decoupling control. Additionally, in all three configurations of concept C, passive
contact with the sphere is needed to support the robot. In the previous methods,
symmetry meant that control design was identical for both roll and pitch, so a planar
model could be used to design a single controller, which could be applied to both
axes. That is not the case here; all three degrees of freedom would require their own,
slightly different, controller to account for their varied locations and lack of symmetry.
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Jonathan Missel
Actuation Method
Concept D, shown in Figure 2.2(d), also includes yaw and omni-directional control.
This method uses four spherically orthogonal omni-wheels, and is not dependant on
the size of the sphere. Opposite pairs of actuators independently control roll and
pitch, while a combined effort of all control yaw. This configuration also solves one
of the shared problems of the other three that has not yet been mentioned. In every
other method, control was designed to act through the axis of yaw rotation of the
entire system; this passes through the sphere’s point of contact with the ground and
the center of mass of the robot. If the actuators are slightly misaligned, or the center
of gravity is not at the geometric center of the robot (from a top view), or if the
center of gravity shifts during operation, control will apply an additional undesired
torque that acts to spin the robot and couple its motion. By jointly applying control
from pairs of motors, input can be allocated to account for this if necessary.
All the previous concepts used one actuator per degree of freedom and are fittingly
classified as equal-actuated; since four actuators are used to control three degrees of
freedom, concept D is considered over-actuated. However, it is possible to use only
three actuators, and eliminate the excess. With three, all of the omni-wheels axes of
rotation would have to be oriented radially outward from the top view, and spherical
orthogonality would no longer be required. The only constraint on the projected angle between the motors is that the sum of any two must be greater than 180 degrees,
so this variation in itself can take on many forms. The sacrifice of only using three
is that no control components are decoupled. This is true for all forms. For example, if the actuators all have equal projected angular spacing with respect to each
other, then control is not applied in perpendicular directions. Even if two actuators
are positioned perpendicularly, the control of the third motor would still be coupled.
Additionally, yaw would be coupled with directional control because the perpendicular actuators would not apply control through the center of the sphere, causing a
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Actuation Method
resultant torque. With four actuators, only yaw is coupled, both directional control
components are completely decoupled, allowing for the previously mentioned planar
development. The major advantage of this over concept C, which also included yaw
control, is that any sized sphere can be used here, and it can be placed directly on
the ground while preserving all three planar degrees of freedom.
Concept # of
Actuators
A
2
B
2
C
3
D
4
# of Sym. Ctrl.
DOF
Sym.
SelfOmniSupport Wheel
Ground- Any
Based
Sphere
2
2
3
3
No
No
No
Yes
No
Yes
No
Yes
No
No
No
Yes
Yes
Yes
No
Yes
No
Yes
Yes
Yes
No
Yes
No
Yes
Table 2.1: Features of drive concepts.
Table 2.1 shows a comparison of significant features for the four drive concepts.
The gravity of these statistics is subjective, and should be established to make an informed choice. To maximize versatility, a system that can accept any sized sphere and
operate on the ground was desirable. This would also help with robustness against
uncertainty in the sphere size. Recognizing that yaw control is not required for omnidirectionality, and wanting to avoid over-actuation, concept B was initially chosen
for Vertigo over concept D. To solve the support issue, two passive omni-wheels were
arranged on the other side of the sphere; this also preserved the symmetry of the
system, eliminating that flaw. As the project was under way, and concept B was
under construction, a continued literature survey uncovered the recently published
paper on Carnegie Mellon’s Ballbot. This robot used the inverse computer mouse
drive detailed by concept A. Although Ballbot required a precisely sized sphere and
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could not be placed on the ground, it had the same degrees of freedom and number of
actuators as Vertigo. A drive for complete originality pushed the project into an early
design iteration that implemented concept D. This method had the same decoupled
directional control as Ballbot, but had the additional advantage of yaw control and
the ability to function on any size sphere or on the ground. This iteration came after
the first version, Vertigo 1.0, was built, so the development of both designs is included
in this chapter.
2.4
Components
With the objective and general method for accomplishment established, further development of the structure necessitated acquisition of all major components needed
to make the platform functional. To determine what components were required, the
various aspects of an implemented feedback control system were considered and found
to be: sensing, actuation, computation and power. Sensing gives feedback measurements to the control loop so that the appropriate input can be determined. To carry
out the control input, actuators are needed to influence the motion of the system. A
computer is required to send and receive signals in accordance with the control loop,
and possibly communicate with other computers. Finally, all of these processes need
a power supply to run. These components had to be acquired before manufacturing
the structure because its design was dependent on their size, weight and mounting.
This section explains how the components were deemed necessary, the criteria they
had to meet, and the specifications of the chosen items.
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2.4.1
Components
Onboard Computer
Intelligent robots need a brain to support their functions, and Vertigo is no exception. As the central hub of the control loop, the computer is charged with taking in
sensory signals, determining the appropriate control, and generating an output signal
for actuation. To determine the control, the computer must either apply its own
embedded controller, or transmit signals to and from an external computer that does
so. Therefore, the onboard computer should accommodate a broad range of inputs
and outputs, and be able to communicate with other computers. Microcontrollers
are capable of executing these tasks, but tend to be specialized, and are not easy
to implement without experience. There are a handful small computers designed for
general robotics use, and after careful consideration, Qwerk was chosen. Qwerk is
a compact board that was developed at Carnegie Mellon’s Robotic Institute and is
produced by Charmed Labs. It is a powerful little computer with impressive specifications.
Qwerk specifications [14]:
• Qwerk Overview
– Powerful robotics solution for university and hobbyist markets
– High-performance CPU with an excellent I/O feature-set for robotics and
mechinatronics applications
• Low-cost
• Hardware
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– 200 MHz ARM9 RISC processor with MMU and hardware floating point
unit
– 32 Mbytes SDRAM, 8 Mbytes flash memory
– Linux 2.6 installed
– WiFi wireless networking support
– WebCam video input support
– 4 Amp switching power supply, 90% efficient, 7 to 30 Volt input range
– Rugged aluminum enclosure
– 5.1” x 5.8” x 1.3”, 11.8 oz
• I/O
– 4 closed-loop 2.0 Amp motor controllers (supports quadrature encoder and
back-EMF “sensorless” position feedback as well as current sensing)
– 16 RC-servo controllers
– 16 programmable digital I/Os
– 12-bit analog inputs
– 2 Rs-232 ports
– I2 C ports
– Built-in audio amplifier with MP3, PCM and WAV audio support
– USB 2.0 host ports for connecting standard USB PC peripherals
– 10/100BT Ethernet port
This computer was chosen over its alternatives for its dense package of features,
and compact size. Qwerk met all desired specifications, and has additional features
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Components
that allow for future expansion and experimentation. It is able to accept a significant
voltage range, thus keeping many powering options open, and its wireless networking
interface is useful for stand-alone operation that will support autonomy.
2.4.2
Actuation
Much of the theoretical requirements for actuation were already covered by determining the drive method; however, selecting and building hardware to carry out these
requirements has its own set of challenges. Choosing the omni-wheels alone had a
great deal of tradeoffs. Due to the curvature of the wheel, gaps exist between the
rollers. With existing wheels, a continuous control surface is achieved by having two
rows of rollers, as seen in Figure 1.1(c) in the Motivation section. These rows are
staggered such that the gaps of one row align with the rollers of the other. Therefore, on a flat surface these wheels will rotate smoothly. However, this application is
controlling a sphere, and having two rows of rollers is not ideal because building a
system that makes contact with both rows restricts it to a single sphere size, voiding
one of the major advantages of this drive. Also, as the wheel rotates, and control
application oscillates between the two rows, the output is inconsistent. Accordingly,
a single row omni-wheel would have to be used for Vertigo. Unfortunately, the gaps
in single row wheels make the affective radius inconsistent, and vibration from rolling
on a discontinuous surface will add noise to the sensors.
In selecting single row omni-wheels, it was important that they have many rollers
around the circumference to minimize the size of the gaps between them. The rollers
should also be made of, or coated with, material that will provide traction on the various surfaces Vertigo will encounter. These criteria governed the verdict of choosing
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Kornylak Transwheel 2051-10mmX CAT-TRAK wheels. CAT-TRAK is their synthetic rubber coating that is applied to the rollers for added grip and significantly
improved performance.
Wheels specifications [15]:
• Standard 2” O.D. - 10mm Plain Bore
• Recommended max load:
– Steel bottom = 25 lbs = 11.3 kg
– Plywood surface = 7.5 lbs = 3.4 kg
– 200 lb test corrugated bottom = 5 lbs = 2.25 kg
• Weight = 1.00 oz
• 8 rollers
The 10mm I.D. was attractive because it is a common size for hardware, which
would simplify the process of finding fittings, and it was large enough to allow small
hardware such as setscrews to fit within its diameter. Having eight rollers is dense
for a 2 inch O.D. wheel, and the gaps are small because of this. Additionally, the
roller axels are made from stainless steel, so rust will not be a concern, and their light
weight will not appreciably hinder the response time of the motors.
The motors for Vertigo 1.0 were chosen primarily for their torque because this
is the control input in the equations of motion. The motors also had to match the
7.2-12 volt range of Qwerk’s output. Finally, Jameco Gear Head DC motors were
chosen for their compatibility with Qwerk and because they had one of the highest
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torque ratings in its class.
Motor specifications [16]:
• Rated voltage = 12 VDC
• Operating range = 6 - 24 VDC
• Current at maximum efficiency = 105 mA
• Speed at maximum efficiency = 8 rpm
• Torque at maximum efficiency = 3500 g-cm
• Gear ratio = 332:01:00
In Vertigo 1.0, passive contact is required for the drive mechanism; the same
omni-wheels were also used for this. To let them spin freely, two 10mm bearings
were press fit into their I.D.s. Spacers were made to position bearings flush with the
outer faces of each wheel. The bearings inner races had an I.D. of 4mm, so 4mm
partially threaded shoulder cap screws were used as axles, and threaded into brass
slugs. These slugs were made to match the mass and diameter of the motors, and the
cap screw axles were threaded in with the same offset as the motors driveshaft. All
this attention to detail served to maintain symmetry throughout the system.
2.4.3
Sensing
The nature of this system mandates access to very accurate measurements of both
the sphere and body dynamics for the feedback control loop. There are many types of
sensors that measure attitude and acceleration, and in general, they tend to increase
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in accuracy with price. Most inexpensive sensors are not accurate enough for this
application. Some attitude sensors depend on the acceleration of gravity for their
measurements, which is acceptable for static situations, but the dynamics of this
system would invalidate their readings. An Inertial Measurement Unit (IMU) is a
package of sensors that is specifically designed to measure the rates and acceleration
of dynamic systems. Attitude is easily backed out from this information, so an IMU is
well suited for this application. The IMU chosen was the Crossbow IMU400CD-100.
It is a six-axis measurement system designed to measure linear acceleration along
three orthogonal axes and rotation rates around three orthogonal axes. It uses three
accelerometers and three angle rate sensors based on a 3-2-1 Euler angle system to
make a complete measurement of the body dynamics. Methods such as dead reckoning can be used with this information to track the position.
IMU specifications [17]:
• Performance
– Update rate > 100 Hz
– Start-up time < 1 sec
• Angular rate
– Rate range = ± 100 ◦ /sec
– Rate bias < ± 1 ◦ /sec
– Resolution < .025 ◦ /sec
– Bandwidth > 25 Hz
– Random walk < 2.25 ◦ /hr1/2
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• Acceleration
– Range ± 4 g
– Bias < ± .012 g
– Resolution < .00006 g
– Bandwidth > 75 Hz
– Random walk < 1 m/s/ hr1/2
• Operating temperature = -40 to + 71 ◦ C
• Electrical
– Voltage = 9 to 30 VDC
– Current < 250 mA
– Power consumption < 3 W
– Output format: RS-232
• Weight < 1.4 lbs
From the specifications it is clear that this unit is extremely precise and does
not take much power to operate. It also fits within the voltage limits of Qwerk and
outputs the supported RS-232 data format, so it is more than adequate for measuring
the dynamics of the body. Additionally, the physical dimensions are close to a cube,
which helps maintain the symmetry assumptions in modeling.
For the sphere, the actuation method, detailed in the previous section, has effectively decomposed its rotation. Encoders that measure the rotation of the omniwheels would be able to provide information on the motion of the sphere with respect
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to the body. Encoders are inexpensive, compact, and very common, so implementation is well documented. Quadrature encoders are particularly attractive because of
their high precision, and they are supported by Qwerk. They have four slot patterns
in their encoder disks and combine the readings from all to generate very accurate
measurements. For this application, RE201 Kit Incremental encoders were chosen.
Quadrature encoder specifications [18]:
• 1 inch code disc
• Maximum line count of 4096
• 2 data channels in quadrature
• Standard 90◦ index pulse
• CMOS ASIC design
• 500 kHz frequency response
• 5 pin headers
In addition to these sensors, Qwerk supports electromotive force (EMF) feedback,
which is a measure of energy per unit charge that the motor provides to the circuit.
This is a function of rotational velocity for the motor, so EMF feedback is a sensorless way of measuring the motor rates. This method is not as precise as encoder
measurements, but it can be combined with low weighting to improve overall accuracy. Qwerk also supports video feed via webcam. To verify which specific devices
are compatible, the Terk website was consulted. Terk is an organization through
Carnegie Mellon that provides support and advisement for the Qwerk unit, and was
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Components
referred to throughout this project [19]. From their list of webcams, the Logitech
Communicate STX was selected for its performance. Not only is the inclusion of this
webcam a means of video surveillance, but with proper image processing, it is high
enough quality to back out attitude data. Further development of this could even
lead to real time obstacle avoidance.
Webcam specifications [20]:
• High quality VGA sensor with RightLightT M Technology
• Video capture: Up to 640 x 480
• Still image capture: Up to 1.3 megapixel with software enhancement
• Built-in microphone with RightSoundT M Technology
• Frame rate: 30 frames per second
• USB 2.0 certified
2.4.4
Power
Over the course of this project, Vertigo has been powered by a few different methods.
Manufacturer defects made finding a suitable power supply that lasted challenging.
The criteria for selecting a power supply were based on size, cost and the demands
of the equipment. For wireless autonomy Vertigo needed to be battery powered,
and lithium polymer (LiPo) batteries have the most attractive capacity, size and
cost combination for this application. Based on the specifications of Qwerk and the
other electronic components, it was determined that the power supply should provide
roughly 12 VDC to the system. The amperage required varies greatly depending on
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Structure
the usage, but an estimate of about 2.6 A/hr was determined from the component
specifications. With this knowledge, packs of 2 Zippy-R 4150 mAh, 11.1 V three cell
LiPo batteries were chosen. Wired in parallel with a Y connector, a pairs of these
power bricks should last for more than 3 hours of constant operation. Two of these
brick bundles were purchased so that a fully charged pair would always be on hand.
They take about 1.5 hours each to recharge, so even if Vertigo is constantly in use,
the other pack will have time to recharge.
A drawback of these batteries is that they can sustain damage if they drop below a minimum voltage. To address this, a voltage cutoff circuit was installed to
terminate power when the battery provides less than 9 VDC. Another drawback to
the battery is that they come in long brick shapes which, unless mounted vertically,
do not promote symmetry. This would impose a product of inertia that couples the
directional dynamics and causes departure from the planar model. To assuage this,
the two batteries could be mounted perpendicular to each other, forming an X. This
would improve the validity of the planar model by canceling the product of inertia.
Additionally, a computer power supply was hacked to provide a second option for
powering Vertigo. It plugs into a standard outlet and provides 12 VDC at 1.6 A to
Qwerk. This tether makes it more suitable for use when Vertigo is stationary, as it is
for programming and diagnostics. Further details on powering options can be found
in Appendix B.
2.5
Structure
With the majority of the components now specified, design of the structure could
proceed. This section details the various aspects of the assembly design for Vertigo
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Structure
1.0, with focus on the structural skeleton. For the purpose of discussion, Figure 2.3
shows a photograph of the assembled prototype. It will be referred to as the design is
explained, starting from the top at the webcam, and finishing with the sphere at the
bottom. The figure shows just one configuration of the assembly; variation will be
explained in the discussion. Throughout the development, SolidWorks was used to
design the robot in CAD; as components were ordered and received, the model was
updated with exact measurements. The time and effort to make precise working CAD
models is well worth the benefit; they are extremely convenient for design because the
new ideas can be tried and analyzed, with time being the only cost. The model was
used to fit all components and confirm that the design was feasibility before the final
prints went to the machine shop. Renderings from this model are used interchangeably
with actual photographs to help illustrate the various assemblies being discussed.
Perched on top of Vertigo is a very light weight plastic spherical webcam that
includes a cover to protect the lens, and an image capture button on the side. It came
with a mount for the sphere to snap in to, but it was designed to grasp computer
monitors, and was not suitable for adhering to Vertigo. Therefore, a new mount had
to be designed and built to withstand crashes, and possibly help protect the camera
as well. The final mount design scavenged the clip from the original mount so the
camera could be disconnected easily. This also protected the camera by allowing it
to break free in collisions rather than bear the impulse of the entire robot. A ball
and socket joint was incorporated into the design and handmade so that the camera
could be prescribed to any desired fixed pan and tilt angle. This joint has .005 in
of interference for the ball and socket which makes it stiff enough to maintain the
desired orientation through small bumps, but gives it the flexibility to deflect in major
collisions to discourage breakage of the mount or camera. Preventing deflection from
small impacts was important for reducing uncertainty if the camera is to be used
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Structure
Figure 2.3: Vertigo 1.0 assembly photograph.
for attitude sensing. The bottom of the mount was made to be flat, giving a secure
contact surface for fastening with double sided tape. It was also center drilled to
allow concentric mounting on a motor for actively controlling the pan. The metallic
cylinder under the camera mount in Figure 3 is a small motor used to control pan; it
has since been removed.
Continuing down the assembly, the yellow anodized aluminum cube is the IMU
that was specified earlier. At each of its corners, a bumper was attached as a safety
measure. These bumpers are marketed for child safety, and are intended to be fitted to
the corners of furniture. They work perfectly for this application and their holes lined
up with the screws in the top panel of the IMU, allowing them to be included without
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Structure
modification. The electrical connection on the front face is composed of a scavenged
DB-15 connector from a desktop computer, and was ideal for this assembly because
its pins exited downward, protecting them from being bent. A 4-pin connector was
then used to plug into the DB-15 outputs and cable into Qwerk. Attached to the side
of the IMU is the USB extension and Airlink wireless adapter that allows Qwerk to
communicate with other computers.
Further down is a black Delrin plate that the IMU attaches to. Delrin is a high
grade engineering plastic whose properties are given in Table 2.2 [21]. In addition to
these attractive characteristics it machines well, has a clean finish, is very durable,
resists warping, resists corrosion, and is in the same general price range as alternate
materials such as aluminum. For these reasons, Delrin was used for the entire skeletal
structure and motor mounts.
Properties
Density
Specific Gravity
Tensile Strength
at 73◦ F
Linear Thermal
Expansion Coeff.
ASTM
Test
Method
D792
D792
D638
Units
DELRIN
lbs/cu in
g/cu cm.
psi
0.0564
1.56
8,500
D696
in./in./-◦ F
4.5 X 10-5
Table 2.2: Properties of Delrin.
The top plate is supported by four identical Delrin legs that link it to the bottom
plate. The legs have cutouts to reduce excess weight and allow wires to be fed through
them. The top and bottom holes in the legs that mount to the two plates are coaxial,
and the hole patterns in both plates are identical so they can be interchanged for
reconfigurability. All structural components are fastened with corrosion resistant
stainless steel #8-32 socket head cap screws so it can be completely disassembled
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Structure
with only one tool. These cap screws are recessed in counterbored holes so they seat
flush; this maximizes the usable mounting surface area on the plates, and maintains
the clean lines of the skeleton. The bottom plate holds the battery pack which is
attached via hook-and-loop so it can easily be attached, removed, and relocated.
On the underside of the bottom plate is Qwerk, the onboard computer. To further
promote reconfigurability, mounting holes were made in Qwerk that mirrored those
of the IMU. Also, the holes in the top and bottom plates were completely threaded
through; this allowed the IMU to be mounted on either side of both plates, and
Qwerk on the top of the top plate, or the bottom of the bottom plate. This is true
for all structural configurations. The larger plate was sized to match the dimensions
of Qwerk to help protect it, and fit the curves of the legs in the shown configurations.
To further protect Qwerk, it was sent out to be anodized black; this reduces pitting
from corrosion on its aluminum casing.
In Vertigo 1.0, the legs not only connect the plates, but also serve as motor mounts.
For simplicity, this was all accomplished in one solid piece, so only six structural
components assemble to form the skeleton, four of which are the identical legs. The
drawback to combining the legs and motor mounts into one part is that the wheels
had to be offset from the center of the sphere. It was understood that this would
influence the motion slightly in practice, but theoretically it was not very significant.
The dynamics of this system ideally prefer control along the axis of symmetry and
through the center of gravity. To accomplish this, the structure would need to be
off set to accommodate the wheels. This would further weaken the assumption of
symmetrical inertias. Since the omni-directional wheels always act through the center
of the ball anyway, this is physically similar to having the legs in the plane of symmetry
and offsetting the wheels. Therefore, straight legs were just as affective and were made
to greatly simplifying their design and manufacturing. The biggest repercussion of
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Structure
straight legs is that the body will, in affect, be rotated with respect to the reference
frame that control is applied in, slightly coupling the dynamics. The lower portion of
the leg was designed to permit some flex to help all four wheels maintain contact with
the sphere, functioning similarly to the suspension of a car. Additionally, permitting
flex absorbs some of the shock from impact, which helps to protect the robot as a
whole. As an adaptable vehicle, determining exact specifications for this permitted
flexing was impossible because it required knowledge of the mass and inertia which
are variable. Therefore, from experience with the material, the legs were designed
to allow a nonspecific midrange amount of deflection, but were kept rigid enough to
avoid appreciable departure from the rigid body assumptions in the mathematical
modeling. In practice, this design proved to be successful.
The motor mounts were designed specifically to clamp tightly down on these particular motors. Accordingly, the diameters of the ballast motor blanks that support
the two freely spinning wheels were sized to that of the motors, and their length was
set to match the motors weight. The blanks accepted a 4 mm shoulder screw that
functioned as an axel for the bearings that press fit into the wheels. A Delrin spacer
was made to fit between the bearings and keep them flush with the outside surfaces
of the wheels to help them run true.
The motor-encoder-wheel assembly was challenging to design. The encoder wheels
were very fragile, and the assembly had to be as compact as possible for minimal
compromise of symmetry due to wheel offset. For the encoders to function properly,
the reader circuit had to be fixed with respect to the motor, and the encoder wheel
must rotate with the omni-wheel. Both wheels had to be fixed to the motor shaft,
and the encoder wheel had to line up with the optics slot in the circuit. Figure 2.4
shows the exploded CAD model of the final design solution for this assembly. The
key components to this design are the wheel hub and the encoder circuit mounting
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Structure
plate. The aluminum mounting plate screwed into preexisting holes on the face of
the motor. This would fix the circuit and gave the assembly a solid and consistent
stop for securing in the motor mounts. A hole in the plate allowed the driveshaft to
pass through the encoder assembly. The encoder circuit screwed into the thin plate
and was positioned such that the wiring harness plugged in along the side of the leg
to shield it from impacts.
Figure 2.4: Vertigo 1.0 motor-encoder-wheel exploded subassembly (CAD).
The encoder and omni-wheels were secured to the driveshaft with an aluminum
hub. The hub was fitted with a setscrew that tightened on the flat of the shaft. The
inner half of the hub accepted the encoder wheel, which was secured with a setscrew.
The hub was designed such that when it was seated at the bottom of the shaft, the
encoder wheel lined up perfectly with its slot. The hub was given a stop to consistently
locate both wheels from their respective ends, and allow clearance between the omni44
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Structure
wheel and encoder plate. The omni-wheel also needed to be fastened to the hub, but
its geometry was not ideal for fitting a setscrew. To address this, two mechanisms
were utilized for securing the wheel. For the first, the hub radius was made with .002
in of interference with the wheel I.D., forming a snug press fit. Secondly, this section
was made .005 in shorter than the thickness of the wheel, and threaded at the end
so that a screw and washer could clamp the wheel against the hub stop. Though
it increased the offset slightly, the encoder was mounted between the motor and the
omni-wheel to protect its delicate exposed wheel. For all of its intricacy, the final
design was very compact, as seen from the photograph of the compressed assembly
in Figure 2.5. As a final modification, flats had to be milled between the rollers of
the wheels to allow clearance with the curved surface of the sphere.
Figure 2.5: Vertigo 1.0 motor-encoder-wheel compressed subassembly (photograph).
Since the conception of this project, much thought and debate was directed toward
determining the ideal sphere for the robot to balance on. Though it was intended to
fit any size sphere, a choice had to be made for acquiring the first one. The design
controllability and observability analysis shed light on how this decision could be made
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Structure
to simplify the problem, but it was understood that this was an experimental platform,
so a degree of challenge was welcomed. It was concluded that a hollow sphere would
minimize rotational inertia and increase agility, and that a rubberized finish would
mitigate concerns for traction with both the omni-wheels and ground. Eventually,
a basketball was settled upon to launch the project because they satisfied all these
requirements, are inexpensive, come in standard sizes and are readily available. Once
acquired, the surface was lightly sanded to reduce the severity of the bumps and
groves in the ball.
In general, there were a few critical notions that guided this structural design: simplicity, durability, reconfigurability, symmetry and manufacturability. The simplistic
and open structure was intended to make assembly and repair straight forward. This
also gave fewer parts the opportunity to fail, and made visually analyzing problems
easier. Durability was secured by designing a compact and rigid skeleton made of
Delrin. Safety features like the camera mount, IMU bumpers, Qwerk mount and encoder assembly also help protect the robot and improve overall hardware robustness.
The primary service that reconfigurability provided was the ability to easily change
the center of mass and inertial properties of the robot without the use of dead weight.
As an experimental vehicle, it was important that the various constituents be flexible
to test which setup produced the best results and to make room for additional equipment. For this reason Qwerk was modified to accept the same mounting hole pattern
as the IMU. Also, the two plates were designed to be interchangeable, with universal
mounting hole patterns on both sides. This allows Vertigo 1.0 to be assembled in 12
unique configurations of the major components, each with different center of mass
and inertia. The design was also adaptable in that it had many mounting surfaces
to attach new equipment as needed. In any given assembly, one side of a plate is
completely empty, and the IMU and Qwerk were left exposed to serve as additional
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Structure
mounting surfaces.
Symmetry was stressed to permit simplifying assumptions in the mathematical
modeling. This was achieved by designing and purchasing symmetrical components,
and assembling them such that coronal and sagittal planes had optimally similar
inertias. The major departure from this was in the motor mount and assembly; this
offset the wheels, but was not severe and was consistent for all four. Therefore, the
angle could be determined and accounted for in the dynamics if deemed necessary.
Having the privilege of designing and building a system to best fit its equations of
motion was unusual, but welcomed.
Lastly, manufacturability and assembly was at the forefront of design for every
part. It is very common that engineers design parts to meet their mechanical objective
in an assembly, but their prints get sent back from the machine shop because they
are infeasible or impractical to manufacture. By understanding the processes used in
manufacturing the parts, cost, time and consistency can be improved with intentioned
design. All of Vertigo 1.0’s structural components were cut out on a CNC mill with a
1/8 in end mill. Therefore, 2-D designs were used to simplify production, and all inner
radii were limited to 1/16 in. Flat, or 2-D components are easier and less expensive
to design and manufacture because they only require that one two-dimensional profile
be sent to a CNC mill which can complete the entire cut in a single tool-path.
Knowledge of the manufacturing procedure was used in aesthetic design as well. To
create tool-paths for the mill to execute, the SolidWorks prints had to be exported into
MasterCAM, a computer aided machining program that can write programs for the
mill. In MasterCAM, splines are reduced from equations to a series of representative
tangential lines. There is a fixed number of lines per spline, so short or conservative
curves appear to be smooth, but long profiles do not and distract from the overall
appearance. This is because light catches the flats like a disco ball, and it is easy to
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Design Iteration
see that the shape is discontinuous. For this reason, the dramatic swooping curves of
Vertigo’s skeleton were composed of combinations of tangentially joined fixed-radius
arcs whose equations are readily passed between the programs and give a smooth
finish to the profile.
The method of assembly also had to be accounted for in the design of the encoder
mounts. The initial design was feasible to manufacture, and fit properly when assembled, but analysis of the CAD model revealed that it was not actually possible
to assemble. Fortunately, this was spotted in the early stages of design, before manufacturing, and served as a testament to the advantages of maintaining an accurate
and up-to-date CAD model.
2.6
Design Iteration
Before moving on to the Vertigo 2.1 design iteration, a quick recap of the project’s
evolution to this point is given. The discussion is complemented by the images in
Figure 2.6 which illustrate some of the major phases in Vertigo’s design generation.
Figure 2.6(a) shows the first CAD rendering that depicted the conceptual fundamentals and a primitive vision for the design. It established the idea of using an inverted
pendulum and balancing on a sphere, it even began to work out the orientation of
the actuators for omni-directionality. This sketch launched the project by helping to
explain its main ideas.
Once the project was accepted, the model in Figure 2.6(b) was created to help
determine what components were needed, and how they might be assembled. This
model established the actuation method and sensing equipment to be used by includ48
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ing components that were dimensioned based on the published specifications of the
manufacturer. At this stage, early ideas for efficient design and reconfigurability were
also being explored. As the components were ordered, and the design took shape,
the continuously updated working model evolved into that of Figure 2.6(c.1). This
included all components of Vertigo 1.0, modeled precisely to measurements. Most
importantly, this was the final design for Vertigo 1.0 that was analyzed and approved
for manufacturing. Figure 2.6(c.2) shows a photograph of the physical system after
manufacturing. Side-by-side with the CAD rendering, it is apparent how accurate the
model was, and why it could be used with such confidence for designing the structural
components.
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Figure 2.6: Vertigo design generation.
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After Vertigo 1.0 was designed and manufactured, the early stages of implementing
simple feed forward control highlighted some of its shortcomings. When operating
on the ground for extended periods of time, friction from the offset wheels tended
to result in a net rotation. Also, the motors were too slow in practice, even after
modifying the gear box to eliminate one of the gear reductions. It was decided that
these problems were not debilitating for preliminary testing and experiments, and
fixing them could be delayed until necessary. It was at this time Carnegie Mellon’s
Ballbot paper was encountered. Ballbot did not work on exactly the same principles
as Vertigo 1.0, but it was very similar. This pushed the project into an early design
iteration to make use of drive method D for actuating the sphere. This would give
Vertigo direct control in yaw, which Ballbot did not have. It would also eliminate the
possibility of motor offset interfering with the dynamics of the system. Since both the
motor offset issue and new drive method only pertained to the legs, the remainder of
Vertigo 1.0, for which the design proved to be very successful, was preserved. This
reduced the time and cost of redesign, and still allowed the Vertigo 1.0 legs to be
fitted if desired.
Vertigo 2.0 and 2.1 differ only in that new actuators and motor mounts were
fitted to Vertigo 2.1. With every other aspect redundant, only the design of Vertigo
2.1 is included here. As with the explanation of the original design, discussion of
the redesign is accompanied by visuals to better illustrate ideas. Figures 2.6(d.1)
and 2.6(d.2) are respectively the CAD model and photograph of the completed Vertigo
2.1 design. Features will again be explained from the top down, starting with the
first modification at the top of the legs.
The spines that protrude vertically from the tops of the legs were included to
give further protection to the components on top of the upper plate. They were
specifically sized to shield the wireless USB adapter, the IMU connector and the
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retro-reflective markers used for tracking with Vicon (see Chapter 4). Further down
the legs, larger cutouts were included to make wrapping and securing wires easier.
Attached to the underside of the top plate is a speaker which makes use of Qwerk’s
audio compatibility. This was initially included for amusement, but proved to be very
useful in debugging code, as different tones were scattered throughout implemented
programs to pinpoint errors.
To fortify protection in the highly chaotic early stages of implementation, a pneumatic rubber bumper (not shown) was stretched around the narrowest span of the
legs. It consisted of a nine inch lawnmower inner tube, inflated to half capacity. Rubber construction made it durable to resist puncture, and with its volume primarily
occupied by air, its mass and inertia did little to affect the system’s dynamics. The
bumper provided an inexpensive and affective buffer that cushioned most components
from impact.
Continuing down the robot, the next major design change was the pronounced
kinks in the legs. These were included for several reasons. In Vertigo 1.0, accessing
Qwerk’s ports was somewhat difficult. The USB ports were particularly tight, and
the plastic casings on the plugs had to be shaved down. Increasing the distance
between Qwerk and the legs not only freed up all of the electrical connections, but
it also protected the USB peripherals, whose plug ends were exposed in the original
design. The exact distance the legs extended was determined by the overall assembled
geometry of Vertigo. Together with the upper spines, the top plate, and the bottom
plate, these elbows were spaced so that at no point when Vertigo 2.1 is on its side will
the IMU or webcam contact the ground. This is not to say that these components will
never hit the ground if Vertigo falls off the sphere, or tipping over at speed, but it is
very affective at protecting them from minor falls. It is also beneficial for working on
Qwerk, allowing the body to safely rest on its side. Lastly, the angle of the elbow was
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determined in accordance with the full leg assembly. Excluding the motor mounts,
the new legs had to be made in two parts for manufacturing reasons. The holes that
mount the bottom plate had to be drilled and tapped from the underside of the legs.
This meant that the mill had to have clearance to maneuver, and the lower portion of
the legs would interfere if made of one solid piece. Therefore, the legs were designed
in two parts and the angle of the kink was assigned for ease of assembly and to form
a sturdy perpendicular joint. As before, all skeletal joints are fastened with the same
size screws to make assembly, reconfiguration and maintenance easier.
Making up the bottom half of this joint is the lower leg. Its function is to recover
from the wide conclusion of the upper leg, and locate the actuators. One of the
convenient aspects of having two-part legs is that if the desired angle of the actuators
changes, only the bottom section needs to be redesigned. This saves material cost and
time. In this case, the lower legs were designed to position the actuators sufficiently
far below Qwerk so as not to limit access to its electrical connections or prevent it
from being removed without disassembling the legs, but not so far down that the legs
were prone to vibration. Note that, similar to the original legs, these were designed
to permit deflection for improved contact with the sphere, and absorbing impact.
However, this was done in moderation to maintain the rigid body assumption.
Another concern that was solved by the lower legs was that colliding with walls
in ground-based mode could damage or dislodge the actuators if they bore the brunt
of the impact. For this reason, the lower legs tuck the actuators under such that
the sturdy leg knuckles impact first. In practice, this was also found to be true
when operating with the sphere; more often than not, the rotation from tipping over
had Vertigo impacting the ground horizontally, and the knuckles protected all other
components from damage by impacting first. The angle of the lower leg’s bottom
face determines the actuator’s angle of attack. These were designed to orient the
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omni-wheels near perpendicular with respect to the surface of this sphere; however,
this was not mandatory. Since the omni-wheels always act through the center of the
sphere, making contact at an angle is simply analogues to running on different sized
wheels. Accounting for this in the controller gains or equations of motion can easily be
done experimentally or geometrically. Designs for adjustable legs were generated for
consideration, but they were decided against due to their complexity, and durability
concerns. An adjustable joint is not as secure, and jarring from impact could easily
misalign them. This is not only inconvenient, but would make fixing all four joints
at exactly the same angle difficult.
Fusing motor mounts into the legs was no longer practical with the redesigned
actuation method. The new mounts had to secure the driveshafts of the motors in
the plane of the legs. Logically, this meant that the motors should be fixed to the
bottom of the legs, making Vertigo 2.1’s mounts vastly different than the originals.
In the final design, two clamps secured each motor, and were made specifically to fit
the motor diameters. Therefore, if new motors were ever used, only the clamps had
to be remade. Each clamp connected to the leg with only one bolt which permitted
them to pivot in the absence of a motor, but with a motor, their motion was fully
constrained. These particular motors changed diameter along their length; therefore,
the two clamps had to be made different sizes. They suspended the motors such
that, when assembled, the bottoms of the legs made perfect tangential contact with
the largest diameter of the motor. This formed a very stiff assembly, and transferred
most of the pressure away from the mounts; therefore, they did not have to support
the weight of the body which allowed them to be smaller and lighter.
To give the design some flexibility, the motors could be adjusted by sliding them
further in or out of the clamps. This let the posture of Vertigo be very quickly
altered to raise or lower the center of gravity, and take on a wider or narrower stance.
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If desired, the motors could even be completely reversed, keeping all of Vertigo’s
control contact very near the top of the sphere. Lastly, the clamps were designed
to concentrate most of their flexing at the bottom. This kept the upper half more
rigid, ensuring a secure and consistent fit to the legs. It also gave the assembly more
clearance to fit smaller spheres without rubbing on the mounts. All leg and motor
mount components were again made of Delrin and machined with a CNC mill as seen
in Figure 2.7.
Figure 2.7: CNC mill machining Delrin legs for Vertigo 2.1.
Drive method D utilized four motors, rather than two, which gave rise to another
design decision. Purchasing two more of the original motors would be less expensive,
but their shortcomings would still be present. These motors had plenty of torque,
but this was accomplished by gear reductions which sacrificed speed. Vertigo would
be able to balance and navigate with them, but with severely limited performance.
In a fruitless attempt to rectify this problem, the gearbox was modified to eliminate
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one of the reductions. This did increase their speed, but not enough. During this
modification, it was found that these high-torque motors had gears made of ABS
plastic, which later failed as a result of their inadequate construction. Opening up
the gearbox also revealed that they were unnecessarily bulky, with cavities of unused
space. Another drawback was that its method of gear reduction offset the output shaft
from the center which made mounting all four in precisely the same way challenging.
Lastly, the original motors required the intricate and fragile encoder assembly to
measure its rotation. This was unnecessary because it had since been realized that
motors and encoders could be purchased as a single unit. Faced with a long list of
cogent arguments against using the original motors, it was concluded that splurging
for four completely new motors was well worth the cost.
New motors for Vertigo 2.1 were carefully selected to eliminate the problems experienced with the previous set. They had to meet both the torque and speed requests of
the controller, as well as have embedded quadrature encoders, concentric driveshafts,
metal gearing, and be appropriately compact. To satisfy this exacting list, Faulhaber
2232V0050 gear-head DC motors with embedded quadrature encoders were eventually called upon after extensively researching the market.
Motor specifications [22]:
• Motors
– Nominal voltage = 6 V
– Terminal resistance < .8 Ω
– Max. output lower = 11 W
– Max. efficiency = 86 %
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– Max. speed = 8,000 rpm
– Max. current = .035 A
– Stall torque > 60 mNm
– Permissible torque = 10 mNm
– Speed constant = 1,200 rpm/V
– Back EMF constant ≈ .8 mV/rpm
– Mechanical time constant = 6 ms
– Rotor inertia = 3.8 gcm2
– Max. angular acceleration 120 103 rad/s2
• Planetary Gearhead
– Gear ratio = 20/1
– All metal geartrain and housing
– Backlash ≤ 1◦
– Preloaded bearings
• Encoders
– Liners per revolution = 512
– Supply voltage = 5.5 VDC
– Max. frequency = 160 kHz
– Code disc inertia = .09 gcm2
These outstanding motor packages offered plenty of torque and speed for this application, and were only half the volume of the 1.0 generation motor-encoder assembly.
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Additionally, their stainless steel planetary gears output the driveshaft concentrically
with the motor casing. At the opposite end, the embedded encoders were secured
up against the motor casing. They were completely enclosed and conveniently fitted
with a ribbon that not only bundled the four encoder wires, but tied in the DC motor
leads as well. The ribbon terminated with a port connection that allowed an organized harness to be made that extended the leads from Qwerk so the ribbon could
simply be plugged in. The location of the encoders worked perfectly with this assembly because it nestled them in the center with Qwerk where they were protected and
near to their electrical connections.
It should be noted here that between the 1.0 and 2.1 designs, it was discovered
that the current version of Qwerk’s firmware did not support the IMU or quadrature
encoders. However, the circuitry and terminals were all present, and the manufacturer
was in the process of developing new firmware that promised to incorporate the
seamless use of these devices. Therefore, design proceeded with these components,
knowing that in the future all forms of onboard sensing would be made functional. For
the time being, the responsibility of sensing would be placed on Vicon, and external
motion capture system that only required small retro-reflective markers be placed on
Vertigo. Detailed descriptions of the sensing methods can be found in Chapter 4.
The new actuation method also required single-row omni-wheels, allowing the
originals to bemodified for use. At their new orientation, the unmodified wheels
were still affective at controlling the sphere, but did not permit ground-based control.
This was because the rollers were set in a cumbersome thick-walled housing that
interfered with them contacting the ground. Tooling for these wheels, as with all
of the cylindrical parts and modifications, was done primarily on a lathe, as seen in
Figure 2.8. The objective of this modification was to maximize the exposure of the
rollers. Modifications from the 1.0 generation had already milled down the material
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in the roller gaps to eliminate interference with the curvature of the sphere. The
new modifications faced both sides of the wheels and put chamfers at the gaps. The
upper right corner of Figure 2.8 shows a side-by-side comparison of the original and
modified wheels. Since the spokes that support the rollers only act in compression,
and all stress is transmitted through the axels, the material above the spokes and on
the walls did not serve much structural purpose. In way of torsional strength, these
wheels were rated for much higher loads than those experienced in this application.
Therefore, slimming down the wheels did not compromise their integrity.
Figure 2.8: Modifying omni-wheels for Vertigo 2.1 use (photographs).
Joining these modified wheels to the new motors was the aluminum hub seen
in Figure 2.9. Unlike the original hub, this did not share the burden of securing
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and spacing an encoder wheel; therefore, it was made much smaller and with less
complexity which required fewer machine setups to manufacture. With the success of
the original fixture to draw upon, the wheels were held by a similar two-stage fixture:
press fit diameter, and clamped faces. The aluminum washers were made thicker to
discourage denting and bending on impacts. In the new design, the hub ends were
outward facing, so rounded cap screws were used to make it easier to handle and to
avoid damaging objects, floors or walls it crashed into. The hubs were secured with
setscrews onto the motor driveshaft.
This design mandated that it be assembled in a specific order, starting with the
hubs and setscrews, and then placing the wheels, and finally the clamping washers
and end cap screws. Disassembly had to be done in the reverse order. Though it was
not very complicated or time consuming, this requirement was made as a sacrifice
to reduce the overall size by burying the setscrew under the wheel when assembled.
To minimize play, the hub had to span as much of the driveshaft as is permitted;
however, the end screw had to be threaded into the other side of the same hole,
and needed its own space to thread in properly. Working out the spacing such that
both components could take advantage of the same concentric hole in the hub was
accomplished in both 1.0 and 2.1 design generations, but was more challenging here
because there was less room to work with.
The primary goal of this design iteration was to implement the new drive mechanism that controlled all aspects of Ballbot’s and Vertigo 1.0’s motion, with the addition of direct yaw control. This was successfully accomplished by utilizing actuation
method D. Just as importantly, this iteration was used as an opportunity to further
improve upon the overall design by taking simple and well reasoned measures that had
substantial impact on performance, usability and robustness. Though the new legs
were made in two parts, and the motor mounts were no longer merged with the legs,
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Figure 2.9: Vertigo 2.1 motor-encoder-wheel exploded subassembly (CAD).
both generations had the exact same number of unique modified and manufactured
parts. This was mostly because all four limbs of the new design were identical, where
the original had two with motors, and two with passive rollers. Many safety features
were added or improved upon, making Vertigo 2.1 an exceptionally robust piece of
hardware. The new legs offer enhanced protection for every component, including
electrical connections. Wiring was made easier with larger cutouts in the legs, more
clearance and better positioning of the encoder leads. The new legs were designed to
not only be durable enough to survive impacts, but were sized to protect the other
components.
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Small features like the speaker, rounded cap screws and filleted corners made
working with the robot much easier. By allowing the motors to be adjusted, even
faster flexibility in the inertial and center of mass properties was achieved. At least
one spare was made of every manufactured part to eliminate downtime if something
broke, and two sets of batteries were purchased so that experiments would never be
halted due to low power.
Aesthetics also played a role in the design, with knowledge that this may one day
be marketed as a commercial platform, it will need every edge it can get for sales. Vertigo’s open design balances simplicity and detail to show off its functionality without
appearing austere or purpose built, even though it is. The materials, geometry and
components were all chosen to form a durable system that matched its mathematical
model well. In summary, the new design managed to provide solutions to Vertigo
1.0’s flaws, maintain or improve upon all of its advantages, and avoid accruing new
drawbacks.
2.7
Future Improvements
As work progressed with building and implementing Vertigo 2.1, even more ideas were
formed that would alter or improve the hardware. In this section, these ideas are
covered, and the consequences of their application discussed. Some of these concepts
were devised prior to manufacturing the current design, but were omitted as their
tradeoffs did not make them sensible for inclusion at this early stage.
As discussed previously in the selection of omni-wheels, this system required that
only one row of rollers be used. For all existing wheels, this meant that gaps between the rollers would make the control surface discontinuous and add noise to the
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measurements. The wheels used were chosen to mitigate this problem, and were
successful to a degree, but the problem had not entirely vanished. This inspired a
handful of new design ideas for omni-wheels, all of which would perform better than
what was currently on the market, but one design stood out from the rest. The least
ambitious design would simply use a larger wheel with smaller rollers to minimize
the size of the gaps. This is really identical to the existing concept but with more
advantageous dimensions. The most promising idea directly confronted the need for
gaps. Currently, spaces exist because the spokes that support the roller axels have
to fit between the rollers, pushing them apart. In addition, these gaps widen radially
outward as the circumference increases, but the length of the rollers remains constant.
Therefore, even designs that did not have radially facing spokes, and eliminated the
inner spacing between the rollers would still have gaps at the control circumference.
To overcome this, two different types of rollers could be used, with one resembling
the existing rollers, and the other being larger with hollows at each end. Inside the
hollows would seat the spokes and gaps for the smaller rollers. The outer surface of
both types of rollers would be turned to the same radius as the assembled wheels.
This configuration allows the two types of rollers to be placed as close as tolerances
allow, forming an almost perfectly continuous surface.
Inclusion of these wheels would greatly decrease the demands on the estimator
because measurement noise from the drive mechanism would be nearly eliminated,
and prescribed control would more accurately be carried out, fitting the mathematical
model more accurately and improving predictions. The drawback to this design is
that they are extremely intricate and complex. To make just one wheel that had 6 of
each type of roller, 28 tiny parts would have to be designed and made from scratch;
that is about twice the number it takes to manufacture the whole of Vertigo 2.1. To
make four of these would be quite time consuming, but in the future, their benefit
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may be deemed worth the effort.
To this point, the 2.1 legs have only solved problems, not caused any; however,
this is not to say that problems cannot arise in the future as trajectories increase in
complexity, and the dynamics of the system are tested to their limit. The number
of legs is an example of this, and the decision to include four rather than three
was nontrivial. Three legs are all that is needed to support and control the robot
on top of the sphere with this actuation method, allowing for minimal weight and
number of parts. It would also make the system equal-actuated, with three motors to
control three degrees of freedom. However, there was a fundamental concern with only
having three vertical legs in our orthogonal universe: the configuration is limited to
being asymmetrical at best. Therefore, even if the actuators are applying orthogonal
control, the product of inertia, a consequence of omitting the symmetry of four legs,
would still couple their effect on the system. Accordingly, using only three legs would
not fit the planar and decoupled mathematical model, and controller design would
be more challenging. Another advantage to using three legs is that it would ensure
that all control contact be firmly secured to the surface of the sphere, increasing the
frictional force and reducing the likelihood of slippage. In the four legged design,
this issue was mitigated by holding tight tolerances when manufacturing the uniform
length legs, and designing the legs to flex slightly. This permitted flexing has not yet
been problematic, but may be in the future if more extreme control induces undesired
vibrations. If this does occur, the addition of simple struts that brace the lower legs
would restrict their motion and prevent oscillation.
Another possible issue is with the battery orientation. The battery pack is currently composed of two identical bricks that are bundled together to make removal
and installation convenient. Their diagonal orientation causes a dynamically coupling
product of inertia that has not yet been addressed, but may need to be in the future.
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The solution to this is fast and simple. Simply mounting them perpendicularly will
cancel out their products of inertia and improve the system’s dynamics.
Vertigo’s design gives it a high degree of flexibility in its center mass and inertia. However, this reconfiguration requires some disassembly which may be too time
consuming for a given experiment, depending on what is being moved. To make the
physical properties of Vertigo even more flexible, simple appendages with sliding and
interchangeable weights could be added. They could be designed to mount in several directions and support weights of various magnitudes in different locations. This
would help experimentally prove the theoretically suggested concept that it is easier
to balance with a higher center of gravity. It would also help explore the affects that
coupling inertias have on the system dynamics.
Lastly, Vertigo is currently initialized in Vicon by being placed on the ground as a
reference. The controller is then applied, and then the body placed on the sphere manually. This should be done quickly and accurately to prevent human interference with
the control, almost to the point where it is aligned a short distance above the sphere,
and dropped. If hands are trying to guide the body onto the sphere while the controller is also trying to adjust its position, the system oscillates and can even diverge
because both inputs are acting in the same direction, causing overshoot. Inclusion of
retractable, statically stabilizing legs may be a solution to avoid this. A simple design
where servomotors retract the leg extensions as soon as balancing control is applied,
and quickly extends them again when control is terminated would eliminate the need
for placing the body on the sphere, or making sure it is off before ending control. It
would also allow Vicon to initialize the system at its proper balancing height, and
no bias would have to be included to account for the added height from the sphere.
The Ballbot team has implemented a similar set of retractable legs with great success.
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Chapter 3
Mathematical Modeling and
Analysis
3.1
Introduction
The intent of this chapter is to derive the equations of motion for Vertigo and conduct an extensive controllability and observability analysis on these equations. A
trustworthy mathematical model is invaluable when designing reliable and affective
estimators and controllers. In development, accuracy and convolution must be balanced using reasonable assumptions to yield dependable equations that avoid excess
computational cost. Analysis of these equations is also important; it helps to verify
their accuracy and establish the objective of the control and estimation processes
that will be applied later on. This chapter derives the nonlinear equations of motion
for sphere-based Vertigo, and then shows how they are linearized for use in linear
estimation and control theory. Ground-based equations of motion are also derived,
and will be used for control in that configuration. Observability and controllability
analyses are then conducted for the states and design parameters of the sphere-based
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equations of motion. This development should give a quality model that is well understood and can reliably be used in the remainder of this thesis.
3.2
3.2.1
Mathematical Model
Introduction
Having a mathematical model of a system is required for most analysis, control and
estimation methods. In general, there are two approaches for obtaining a model:
system identification, and mathematical derivation. System identification is a broad
topic that is composed of a diverse collection of methods and algorithms; in general
they are all designed to extract dynamical models from measurement data. System
identification is most advantageous when applied to complex or flexible systems. This
is because it does not require knowledge of the system that is to be modeled. Deriving equations of motion mathematically does require knowledge of the system, so
it is most readily done for systems with well known dynamics, such as rigid bodies.
Derivation does not require previously collected data, so it can be done for theoretical
systems or in the preliminary stages of development, before data is available. Having
the equations before the system is even built means that they can be used in simulation as proof of concept. Additionally, their analysis can help detect unforeseen
problems that may be addressed early on or maybe avoided all together with design
modifications.
Vertigo’s model was derived mathematically because rigid body dynamics can be
assumed, and data was not available when first modeling the system. The equations
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of motion were developed as Vertigo was being designed, and they were analyzed to
help make design decisions. System identification methods can also be used to verify
and improve the model after implementation. Having an existing model will aid in
that process by focusing efforts appropriately.
In the derivation that follows, a planar model is assumed. This is a simplification of the actual system, but a reasonable one because Vertigo was intentionally
designed to be highly symmetrical so the motion will not only be the same in both
the sagittal and coronal body planes, but the product of inertia will not couple their
motion significantly. Using a simplified model helped accelerate the project by requiring management of simpler equations, and turned out to be a beneficial decision.
In the planar model yaw is not incorporated; however this is not debilitating because
control superposition can be applied in moderation for trajectories that do not demand extravagant yaw control (see Control Superposition section in Appendix A).
Once the project is further along and more complex trajectories are attempted, the
more inclusive dynamics can be exploited. Since Ballbot does not have control in
yaw, Carnegie Mellon’s team was also able to derive its equations of motion for a
planar model [3]. Their model will be followed here because it is very well reasoned
with logical assumptions, and it closely describes Vertigo’s planar motion.
A vast majority of introductory analysis, estimation and control techniques are
designed for use on linear systems. Nonlinear systems, such as Vertigo, can take
advantage of these methods with local accuracy if the system is linearized about the
point of interest. Vertigo’s nonlinear equations of motion will be linearized with a
first order Taylor series expansion approach.
Ground-based equations of motion are then developed for use in that configuration. The model used yields linear equations, so linearization is not necessary.
All three sets of equations proved to be sufficiently accurate for drawing conclusions
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about the characteristics of Vertigo’s motion, and were used for the analysis, estimation and control that follows. The reconfigurable nature of this platform means
that the parameter values may change slightly between the various simulations and
implementations, but the same model structures are used throughout.
Mathematical models will have errors based on measurement uncertainties and
simplifying assumptions. Examples of these include the inertias, friction coefficients
and the relation between signal input and actual torque that is applied by the motors. To minimize the impact of these errors, filters can be applied that account for
uncertainty, or even predict unknown parameters.
3.2.2
Model Derivation
Viewed by many as the most important step in control and simulation, modeling is
charged with balancing accuracy and practicality. Following the Ballbot team’s procedure, Lagrangian mechanics are used here to develop Vertigo’s nonlinear equations
of motion. This method starts with the identification of kinetic energy K = Kb + KB
and potential energy V = Vb + VB , where the subscript b represents the sphere, and
B represents the body of the robot. From the energies, the Lagrangian L can be
constructed and the Euler-Lagrange equations of motion calculated as follows.
L (q, q̇) = K − V
(3.1)
d
Lq̇ − Lq = T
dt
(3.2)
Where T is the input torque for the motors and q is the state vector.
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Figure 3.1: Planar sphere-based Vertigo model.
Figure 3.1 depicts the state and parameter convention that will be used throughout
this thesis for the sphere-based configuration. It is worth noting here that the angle
of the body φ is measured with respect to the angle of the sphere θ. This is done for
mathematical convenience; however, it makes some of the results and simulations less
intuitive to interoperate. For this reason many of the results include representations
that are easier to visualize in addition to the state results.
The following assumptions are made during this formulation:
1. Rigid sphere and body
2. No slip between ball and ground
3. No slip between ball and actuation
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4. Inputs are torques applied between roller and ball
5. Sagittal and coronal plane are symmetric and decoupled
6. Neglect friction in actuator rollers
7. Neglect dynamic friction
8. Origin is at the center of the sphere
9. Yaw coupling is negligible
The first step in the Lagrangian formulation is to define the kinetic and potential
energy for the body and sphere.
Sphere:
2
Kb =
Ib θ̇
+
2
2
mb rb θ̇
2
Vb = 0
(3.3)
(3.4)
Body:
mB
KB =
2
rb2 θ̇2
I 2 B
2
+ 2rb ` θ̇ + θ̇φ̇ cos (θ + φ) + ` θ̇ + φ̇ +
θ̇ + φ̇
(3.5)
2
2
VB = mB g`cos (θ + φ)
(3.6)
Where, mb and mB are the mass of the sphere and body respectively, Ib and IB are
the inertias of the sphere and body respectively, rb is the radius of the sphere, and
` the length from the body’s center of mass to the center of the sphere. The total
energies are considered for the derivation, and are calculated as follows. Total kinetic
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energy is K = Kb + KB , and total potential energy is V = Vb + VB . Defining the
system configuration vector as q = [θ φ]T , the Lagrangian formulation is continued
with,
L (q, q̇) = K − V
(3.7)
The input torque that is applied from the motors is defined by τ . The viscous
friction is modeled by defining


 µθ θ̇ 
D (q̇) = 

µφ φ̇
(3.8)
where µθ is the viscous damping coefficient for friction between the sphere and ground,
and µφ is the coefficient for friction between the sphere and body. The Euler-Lagrange
equations can then be expressed with this notation as


d
 0 
Lq̇ − Lq =   − D (q̇)
dt
τ
(3.9)
The derivative of this gives the equations of motion, which are rearranged to take the
form


 0 
M (q) q̈ + C (q, q̇) + G (q) + D (q̇) =  
τ
(3.10)
Where,


 Γ1 + 2mB rb `cos (θ + φ) Γ2 + mB rb `cos (θ + φ) 
M (q) = 

Γ2 + mB rb `cos (θ + φ)
Γ2
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(3.11)
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is the mass and inertia matrix with,
Γ1 = Ib + IB + (mb + mB ) rb2 + mB `2
(3.12)
Γ2 = IB + mB `2
(3.13)
The vector of Coriolis and centrifugal forces is defined by

2 
 −mB rb `sin (θ + φ) θ̇ + φ̇ 
C (q, q̇) = 

0
(3.14)
and the vector of gravitational force components is


 −mB g`sin (θ + φ) 
G (q) = 

−mB g`sin (θ + φ)
T
By defining u = τ , and x=
(3.15)
q q̇
, the equations of motion can be expressed
in nonlinear state space form as,

q̇

 


ẋ = 

 0 

 M (q)−1   − C (q, q̇) − G (q) − D (q̇)
u
3.2.3



 , f (x, u)


(3.16)
Linearization
Many control, estimation and analysis methods are for linear systems, so Vertigo’s
nonlinear equations must first be linearized. The linearization used here is based
on keeping only the linear terms of a Taylor series expansion about an operating
point. This is often referred to as Jacobian linearization because, for a state space
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formulation, ẋ = Ax+Bu, this can be reduced to taking the Jacobian of the nonlinear
equation with respect to the states, to obtain the A matrix, and with respect to the
inputs for the B matrix. This process takes on the form,
∂ ẋ ∂ ẋ ∂y ∂y A=
B=
C=
D=
∂x x = x
∂u x = x
∂x x = x
∂u x = x
u=u
u=u
u=u
(3.17)
u=u
where the barred values represent the point being linearized about. When the system
linearization is transformed to the origin, all states are zero, and the state space
becomes,

0
0
1
0



0
0
0
1

A= 
 −43.83 −43.83 −.0342 −.0878


79.31
79.31
.0439 −.1228

 1 0 0

 0 1 0

C= 
 0 0 1


0 0 0



 0 




 0 





B = 
 −43.9 







61.4
(3.18)

0 

0 

 D = [0]
0 


1
(3.19)
This model represents Vertigo with its completed communications architecture,
where all onboard sensing is assumed fully operational. Accordingly, C is the identity matrix because all states are measured. The extreme size of the symbolic state
space equations made them unreasonable for inclusion here, so the above model with
numerical values is given. Note that these values represent the parameters for a spe-
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cific configuration of Vertigo, and is linearized about the vertical equilibrium. As the
equation parameters and the point of linearization changes, so too will these values,
but the general results that this model yields are still valid.
3.2.4
Ground-Based Derivation
As mentioned before, Vertigo’s actuator orientation and axial symmetry means that
the full state space equation with x and y position and velocity is decoupled and
identically redundant for both directions. Therefore, control and implementation can
be carried out for a planar model, and applied to both planes independently. The
continuous equations of motion for the ground-based robot were put in standard state
space form q̇ = Aq + Bu, with the output taking the form y = Cq + Du. With the
T
state vector defined as q = x ẋ
and control as u = T , the model can be
expressed as,







 ẋ   0 1   x   0 
  + 
 =
T
2
ẍ
0 0
ẋ
mr

y=
1 0
(3.20)

 x 
 +
ẋ
0
T
(3.21)
Where r, m and T are the radius of the wheels, mass of the robot and imposed
torque of an individual motor, respectively. This model makes the assumptions that
all components are rigid bodies, dynamic affects can be neglected, and opposing
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Figure 3.2: Ground-based free body diagram.
frictions at the wheels are negligible. As a result of the dynamic simplifications, all
points of contact with the ground are assumed to carry equal portions of the total
normal force N. In reality, this is not true for two main reasons: all four wheels will
not always be in contact with the ground due to uneven terrain and the discontinuity
of the wheels, and as the body accelerates, the leading wheel experiences less normal
force than the trailing wheel (discussed further in the Implementation Challenges
section). Neglecting opposing frictional forces mainly assumes that the rollers on the
wheels do not have any appreciable resistance since they are designed to allow slip.
If it were deemed significant, modeling this friction would depend on the system’s
dynamics and estimated coefficients, making it highly inaccurate.
During some of the ground-based controller development, discrete time equations
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of motion (with Ts = 0.04) were used.







 ẋ   1 0.04   x   0.0240 
 =
  + 
T
ẍ
0 1
ẋ
1.1976

y=
1 0
(3.22)

 x 
 +
ẋ
0
T
(3.23)
The sampling time was chosen based on the average sampling time of the implemented
closed loop (see the Implementation Challenges section for more details). The discretized equations were obtained with MATLAB’s c2d command.
3.3
3.3.1
Analysis
Introduction
The objective of this section is to determine the conditions for which sphere-based
Vertigo can be controlled, and whether its states can be reliably determined if provided
knowledge of its inputs and outputs. Investigating these characteristics will help
build an understanding of the system and establish what should be expected of the
control and estimation processes that it will rely on for balance and navigation. To
accomplish this, an extensive controllability and observability analysis is conducted on
the linearized model. It is important to note that a linearized model may well capture
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the dynamics of a system near its equilibrium, but subject to large perturbations, the
inherent nonlinearities will decrease the validity of the simplification.
In this analysis, the observability O and controllability C matrices are calculated
respectively as,
T
O = C CA · · · CAn−1
C=
B AB · · · An−1 B
(3.24)
(3.25)
where n is the number of states. The system is observable if the observability matrix
has full column rank, and controllable if the controllability matrix has full row rank.
All four states will be varied over a range of values, and this calculation carried out
for the relinearized system at each point. The rank criterion for O and C only gives
insight as to whether the system is, or is not observable or controllable, it tells nothing
of a measure or degree of observability and controllability. To dig deeper into this,
the inverse of the condition number and determinants of these matrices are examined.
Many factors contributed to the structural development of this platform. Balancing machines are fundamentally dependant on the location of their center of gravity
(length of the inverted pendulum), so understanding the nature of this dependence
was important to the design process. Furthermore, Vertigo was intentionally created
to be reconfigurable, which allows its center of mass to be altered; so it was necessary
to determine how the physical changes should be expected to affect the performance,
making experiment design more predictable. Two main components assemble to comprise this system, the sphere, and the body (Vertigo). To explore the sensitivity to
these components, a controllability and observability analysis is done for both while
varying their dimensional and mass parameters. As before, the determinant and condition number are also considered for further evaluation.
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3.3.2
Analysis
State Observability and Controllability Analysis
For a system to be controllable, an input must exist that will transfer any state to
any other state in a finite amount of time. It assumes that the linear system has
infinite control magnitude available, and that any trajectory may be taken. This is
an extremely important preliminary calculation for system design to ensure that it is
possible for the objectives to be met. The controllability is determined by checking
the rank of the controllability matrix C.
rank(C) = rank( B AB · · · An−1 B )
(3.26)
If the controllability matrix has full row rank, then the system is controllable. In this
case there are four states, so the controllability matrix must have rank four.
If a system is observable, it means that for any initial state there exists a finite,
positive time where a record of the input and output is enough to uniquely determine
that initial state. To determine observability, the observability matrix O must be
found to have full column rank.
T
rank(O) = rank( C CA · · · CAn−1 )
(3.27)
Just as in controllability, for this system, full rank is four.
In MATLAB, the controllability and observability were calculated while varying all
four states, and the rank of the matrices at each point were stored in four-dimensional
matrices. Figures 3.3 and 3.4 clearly show that this system has full rank for all values
of theta and phi under consideration. This range of values was logically chosen based
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on the physical system’s characteristics. A range of ±1 radian is nearly ±60◦ , which
tests a comprehensive span of possibilities because by this point Vertigo would have
fallen off of the sphere it rests on.
Figure 3.3: Rank of controllability matrix while varying θ and φ.
Many of the results in this thesis are presented in terms of β = θ + φ, the angle
of Vertigo with respect to vertical, because it is more intuitive, and combines the
four states into just two values. However, this analysis does not lend itself to such a
representation because its results are not unique. For given rates, multiple combinations of angle states may give the same β, but combinations do not necessarily share
controllability and observability characteristics.
The rank values in this analysis only inform of whether or not the system is
controllable or observable. To further look into what is taking place as these variables
change, the inverse of the condition number was calculated and plotted for both
matrices. The condition number of a matrix is the largest singular value divided by
the smallest singular value. The singular values of a matrix A(mXn) , where m = n,
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Figure 3.4: Rank of observability matrix while varying θ and φ.
are the positive roots of the eigenvalues of AH A. Matrix A is non-singular if and only
if (iff) all of its singular values are greater than zero. It can be shown that if C can be
put in to Control Canonical Form (CCF), it is completely controllable; in order to do
so, it must be nonsingular because its inverse must be taken. This is also true of O
for the Observer Canonical Form (OCF). Therefore, the condition number provides a
sensitivity bound for the solution of linear equations of the matrix. By definition, the
condition number is always greater than or equal to one, so calculating the inverse of
the condition number gives an easy to visualize scaled measure between one and zero.
If the inverse of the condition number is close to one, the matrix is said to be “well
conditioned.” This means that the inverse can be computed with high accuracy. A
smaller inverse condition number means that the matrix is nearing singularity, and
computation of its inverse, or the solution to linear systems of equations is prone
to large numerical errors. If the inverse of the condition number is exactly zero,
that corresponds to a condition number of infinity, and the matrix is singular. Here,
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this would translate to not being controllable or observable for the controllability or
observability matrices respectively.
Figure 3.5: Inverse condition number of controllability matrix while varying θ and φ.
From Figures 3.5 and 3.6, it can be seen that the controllability and observability
characteristics are definitely affected by the state angles, an observation that could
not be made from the controllability and observability pots. For controllability, in
Figure 3.5, consider the profile curve for fixed φ at zero, and varying θ. θ is defined
with respect to φ, so fixing φ at zero means that the body and sphere will rotate
together. Therefore, it makes sense that there is a peak about the unstable equilibrium
point, because it would take less control input to move to another state. It is also
easy to understand that if the body is initially tipped at an angle, more control would
be required to work against gravity and resume balance. The rotation of the sphere
has nothing to do with the gravitational component of the system or controllability,
except for the fact that the angle of the body is defined with respect to it. This is why
the general shape of the profile remains the same along curves of fixed θ, and only
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Figure 3.6: Inverse condition number of observability matrix while varying θ and φ.
shifts to the right or left, giving the symmetrical nature of these plots. In Figure 3.6,
if the constant φ = 0 profile is considered again, the results show that the system is
more easily observed near the equilibrium. This suggests that, given the input and
output history, it is easier to determine the initial condition when it started closer to
vertical. Again, the angle definition is the reason for the symmetrical shape of the
plot in Figure 3.6. For both plots, the scaling shows that these peaks are small, so
the state variations do not have an overwhelming impact.
As explained before, the inverse of the controllability matrix must exist to be put
in to CCF; therefore, the determinant must be nonzero. To further illustrate this,
the determinant of the controllability matrix was calculated and plotted for analysis
as well. Figure 3.7 shows how these results agree with the findings of the inverse
condition number. This analysis was not done for observability because it does not
always take the form of a square matrix, and its determinant is not always calculable;
however, techniques exist that account for this.
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Figure 3.7: Determinant of controllability matrix while varying θ and φ.
To this point, the controllability and observability analysis has been devoted to the
θ and φ state combination as they are varied. The same analysis was done for all six
combinations of states pairs as the four states were varied. Figures 3.8 and 3.9 show
that as the two rate states θ̇ and φ̇ were varied, the controllability and observability
are still preserved.
When this rate analysis was extended to the inverse condition numbers, it was
found that for both observability and controllability, the value was a constant for all
rates. Accordingly, the mesh plot for the inverse condition number of the controllability matrix was a flat plane at 1.67 × 10−7 , and the inverse condition number of the
observability matrix was a flat plane at 1.17 × 10−6 . The plot of the determinant of
the controllability matrix was also a constant value at 1.6 × 1011 . Constant values
mean that these states do not affect the controllability or observability. This was
confirmed upon closer inspection of the A matrix, where it can be shown that these
values cancel each other out, and only the angle states can change the A matrix.
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Figure 3.8: Rank of controllability matrix while varying θ̇ and φ̇.
Figure 3.9: Rank of observability matrix while varying θ̇ and φ̇.
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Therefore, all of the state varying characteristics were captured in the angle state
analysis. As a result of the controllability and observability being invariant to rate
states, the four remaining state combination analyses yielded no new results, and
were not included for this reason.
3.3.3
Design Parameter Observability and Controllability Analysis
The physical parameters of a system can greatly affect the mathematical modeling.
Vertigo is a testbed for dynamics and controls experiments, and for that reason was
designed to be reconfigurable, permitting changes to the center of mass location and
magnitude. In addition, new design iterations are always underway, so it is important
to determine how these physical changes in the system affect its performance. The
two main components of Vertigo’s assembly are the sphere, and the body. To explore
the sensitivity to these components, a controllability and observability analysis was
done for both constituents, while varying their dimensional and mass parameters.
This analysis also looks into the determinant and inverse condition number for further evaluation. Performing this analysis sheds light on how the equations of motion
are affected by uncertainty in these parameters, which will help in the estimation and
filtering to come.
Vertigo Design Analysis
First, controllability and observability are checked by plotting the rank of these two
matrices while the design parameters, body mass of Vertigo and length from the
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center of the sphere to the center of mass of Vertigo `, are varied.
Figures 3.10 and 3.11 show that the system is both controllable and observable
for all parameter variations because the rank of both matrices is four at each point,
matching the number of states. Now the inverse condition number of these two
matrices is considered to evaluate the parameter’s affect on how well conditioned
they are. The contours in Figures 3.12 and 3.13 show the inverse condition number
as the parameters are varied. The vertical blue lines mark the parameters to which
Vertigo is currently configured.
Figure 3.12 reveals that as the length from the center of the sphere to the center
mass of the body increases, the controllability matrix monotonically becomes more
ill conditioned. To help understand this, consider the following scenario: Vertigo is
required to return to a vertical stature from an initial angle of 10 degrees. This plot
shows that more control effort would be required for the system to recover if the
length is greater. This makes sense because, for a fixed angle, the horizontal lever
arm creating a torque with gravity grows with this parameter. However, if this was
the only affect of increasing this length, the plots would decrease at a constant slope
because of the linear relationphip; this is clearly not the case. The reason is that the
rotational inertia also increases as ` increases, causing the body to fall more slowly.
Additionally, the lever arm from the control to the center of gravity also increases.
Both of these tend to decrease the amount of control effort required to recover. This
explains the concave characteristics with increasing length. It is important to note
that this fixed angle example assumes the same initial angle of departure, but in
practice larger rotational inertia cases take longer to fall, and will not experience as
much departure in a given period of time. So, when time is included, a longer ` is
actually perceived as “easier” to control.
For the same plot, if we inspect how the condition changes as Vertigo’s mass
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Figure 3.10: Rank of controllability matrix while varying θ̇ and φ̇.
Figure 3.11: Rank of observability matrix while varying θ̇ and φ̇.
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Figure 3.12: Inverse condition number of controllability matrix while varying Vertigo
mass and `.
Figure 3.13: Inverse condition number of observability matrix while varying Vertigo
mass and `.
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increases, a definite ridge can be seen where the matrix is best conditioned. Consider
again the initial angle recovery scenario. Increasing mass will increase the moment
with gravity, making recovery more difficult. However, linear inertia also increases, so
it has greater resistance to motion, making it easier to move the sphere underneath
it and recover from the initial angle. The increased mass is assumed to be at a point,
so the rotational inertia about the center of mass does not change, and the control
torque required to rotate it does not change. However, the rotational inertia with
respect to the center of the sphere does increase, and the body will fall more slowly.
The slower falling body, with no additional control requirements, acts to improve
controllability with increasing mass. These phenomena work against each other and
this ridge represents the point at which the increasing inertias are most dominant.
As marked by the vertical blue line, Vertigo’s current configuration is at an optimal
mass for its pendulum length. It is worth noting that just as the mass variations
generate an optimal ridge due to counteracting physical principles, so too does the
length variation. However, this ridge corresponds to a length that is within the radius
of the sphere and was therefore not a feasible design option and was omitted from
these results.
Figure 3.13 shows how the observability matrix’s condition is affected by varying
Vertigo’s design parameters. The range of inverse condition values spans only about
.0004, and no maxima or minima are present. Accordingly, the observability is not as
critical for design because it is not affected greatly. Just as before, the counteracting
principles result in a contoured surface.
The mesh of controllability matrix determinants in Figure 3.14 confirms the
results from the inverse condition number. As the length increases, the determinant
approaches zero monotonically (for this range). The contour of this surface ensues
from the same counteracting phenomena mention earlier.
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Figure 3.14: Determinant of controllability matrix while varying Vertigo mass and `.
Figure 3.15: Inverse condition number of controllability matrix while varying Vertigo
mass and ` with fixed rotational inertias.
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As proof of concept, this design analysis was also carried out with fixed rotational
inertia to isolate the principles at work; Figure 3.15 shows the inverse controllability
condition mesh for this. By comparing these results with the original findings, it is
clear that these design parameters most significantly impact controllability through
rotational inertia when mass is increased. Without the increasing inertia acting to
improve controllability, the matrix becomes more ill conditioned with increasing mass.
This shows that increasing rotational inertia reduces the required control effort for
recovery.
Sphere Design Analysis
In this section, controllability and observability are first checked by plotting the rank
of these two matrices while the sphere design parameters, sphere mass and sphere radius, are varied. Consistent with previous plots, the vertical blue line in the following
figures represents the current configuration of Vertigo’s parameters.
Figures 3.16 and 3.17 show that the matrices have full rank, so it is both controllable and observable for this range of sphere design parameters. Figure 3.18 and 3.19,
respectively, look further and show how the condition of the controllability and observability matrices are affected as sphere mass and sphere radius are varied. The
controllability matrix is shown to become better conditioned with both increasing
mass and radius. This is because as the radius gets larger, the distance from the
actuators to the ground increases, and it requires less control efforts to generate the
torque needed to move the sphere. This is similar to increasing `. As the mass gets
larger, the sphere has more inertia, making it harder to move; however, the control
efforts would then have more of a tendency to rotate Vertigo back to the top of the
sphere, rather than move the sphere under the body. In practice, there is a balance be92
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Figure 3.16: Rank of controllability matrix while varying sphere mass and sphere
radius.
Figure 3.17: Rank of observability matrix while varying sphere mass and sphere
radius.
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Figure 3.18: Inverse condition number of controllability matrix while varying sphere
mass and sphere radius.
Figure 3.19: Inverse condition number of observability matrix while varying sphere
mass and sphere radius.
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tween these two, and this is why the plot is contoured, thought it is mild. Figure 3.19
shows that the observability matrix becomes better conditioned with increasing mass
and radius. As with the body parameters, the observability condition does not experience any major changes over this considered range; however, the contour with
increasing mass is more dramatic then that of controllability.
Figure 3.20: Determinant of controllability matrix while varying sphere mass and
sphere radius.
The determinant of the controllability matrix mesh is shown in Figure 3.20. It
reveals that the determinant is not close to zero, and its highly nonlinear manner of
leveling off suggests that it will not approach zero near this range. Therefore, it is
safe to assume these results are valid for reasonable sphere properties.
The inertial influences were also isolated for the sphere design parameters; the
inverse condition for controllability with fixed inertia is shown in Figure 3.21. The
radius of the sphere appears in both body and sphere rotational inertia calculations,
and the influence from holding them constant tends to cancel out so no significant
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Figure 3.21: Inverse condition number of controllability matrix while varying sphere
mass and sphere radius with fixed rotational inertias.
change is observed. However, when the sphere mass is varied, it is clear that the rotational inertia of the sphere was the major contributor to the improved controllability
condition because this parameter does not appear in the body inertia calculations.
3.4
Conclusions
This chapter derived the equations of motion for Vertigo, and conducted an extensive
analysis to determine their controllability and observability characteristics. It began
with the derivation of nonlinear equations of motion for sphere-based Vertigo using
the Lagrangian formulation, which followed the derivation of the Ballbot team at
Carnegie Mellon University. Their model was chosen based on its accuracy and their
success with implementing it. Jacobian linearization was applied to these equations
to extract a linear state space model, allowing linear systems analysis techniques to
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obtain locally accurate results. The linearized model is also used throughout the
control and estimation portions of this thesis. The equations of motion for groundbased Vertigo were then derived. Their simplifying assumptions neglected nonlinear
affects, so their linearization was not necessary.
To understand the nature of the sphere-based equations, the controllability and
observability of all feasible state conditions was explored and it was found that for every combination of states, the system is both controllable and observable. To get more
insightful information about how the controllability and observability are affected by
these state combinations, the inverse condition number was analyzed at each point.
Smaller inverse condition numbers meant that the matrix is nearing singularity, and
computation of its inverse, or the solution to linear systems of equations, is prone to
numerical error. The determinant of the controllability matrix was plotted and gave
supporting evidence for the inverse condition number analyses. For both controllability and observability, it was found that the most accurate results can be obtained
near the vertical, unstable equilibrium. This is ideal because most of Vertigo’s motion
is confined to this region.
To further explore the characteristic of Vertigo, the controllability and observability analysis that was outlined for the state variations was done while varying physical
parameters. These results were used to draw conclusions about the design of Vertigo,
and learn what was to be expected for controls and estimation to come. It was found
that for the current length between the center of gravity of the body and the center
of the sphere, Vertigo is at an optimal mass for controllability condition. The same
analysis was conducted for the design parameters while holding the rotational inertias constant and the results compared with the previous analysis. This comparison
helped isolate the influence of the many factors that contribute to the controllability of this system. The range of values chosen for the body and sphere parameter
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variations were chosen based on physical constraints, yielding an inclusive analysis of
feasible options.
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Chapter 4
Communication Architecture
4.1
Introduction
In order to implement closed loop control methods on Vertigo, a network of communication had to be established. The main objective of this architecture is to send
sensory input to a processor for control law to determine the desired course of action. The control input must then be delivered to actuators to persuade the system
accordingly. The response is measured by the sensors and the loop is repeated. This
generic schematic can take on many forms, depending on the application, and does
so even for the various methods of controlling Vertigo.
This chapter explains the implemented communication methods for onboard sensing and external sensing. Communication for both 1.0 and 2.1 generations of Vertigo
was almost identical, so only the 2.1 communication is included here, and acceptations
for 1.0 are explained as they arise. Additionally, the sphere and ground-based modes
of Vertigo 2.1 had relatively similar communication networks, differing most dramatically in their applied control and estimation; therefore, an instance of ground-based
control is used to illustrate the method, and the variations in control and estimation
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are covered in their respective chapters.
4.2
Onboard Sensing Architecture
To measure the dynamics of this system, Vertigo was fitted with onboard sensors
that were intended to transmit measurements through Qwerk for feedback in the
control loop. These devices included the quadrature encoders, Inertial Measurement
Unit (IMU), back-EMF and webcam. Despite later discovering that Qwerk did not
actually support the majority of these sensors as advertized, they were not omitted
from the system because new firmware was being developed by the manufacturer that
promised to patch up these connections. Therefore, their physical properties were
included in all dynamic modeling to avoid new models having to be derived once
Qwerk is updated. This should accelerate the process of incorporating the sensors
when they are made functional. Of course, the sensing equipment was also included
in the hardware to match the dynamics of the model, this will permit all controllers
to be directly applicable after adaptation to fit the new communication methods.
Working with Qwerk was a long and finicky process; therefore, none of the control
or estimation was hardcoded onboard Vertigo, though it was possible. Instead, the
power and familiarity of MATLAB made it the desired program for implementation;
therefore, communication had to be established between Qwerk and MATLAB to
pass on sensory and control inputs. This in itself was a laborious task and is covered
later in this chapter.
For the onboard sensing architecture, Vertigo’s brain was rooted in Qwerk, and
served as a central hub for receiving and transmitting signals. As seen in Figure 4.1,
it received sensory input from the motor EMF feedback, quadrature encoders, camera
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Figure 4.1: Onboard sensing communication architecture.
and IMU, and wirelessly transmitted these signals to a PC where they were taken as
inputs in a Simulink or MATLAB control loop. The loop calculated the appropriate
control to correct the system’s motion and then wirelessly sent it back to Qwerk where
voltages were prescribed to the motors. When the system responded, the loop was
repeated. The intricacies of these lines of communication were rather complex and
will be detailed later; they were spoken of in generalities here because this method
had not yet been implemented, leaving no specific details to speak of.
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4.3
External Sensing Architecture
External Sensing Architecture
All implemented forms of Vertigo’s communication architectures enact the same general functions: sense, compute and actuate. They differ mostly in their means of
accomplishing these tasks. The schematic in Figure 4.2 depicts a specific instance
of ground-based control as an illustrative example of the methodology. To avoid
redundancy and excess detail, the major aspects of this description can be applied
to sphere-based control as well. Since onboard sensing was not available, Vicon, an
external motion capture system, was used for measuring the dynamics. Vicon works
by locating retro-reflective markers through triangulation of multiple infrared strobe
cameras. The infrared light emitted is a high frequency electromagnetic wave that
borders that of visible red light. Working with such short wavelengths allows for
highly accurate measurement of the position of each marker. The relative speeds of
ground vehicles does not produce a large enough shift for Vicon to register, preventing the Doppler effect from fully being exploited; therefore, all rate states had to be
calculated externally with numerical time derivatives.
With Vicon’s accompanying software, multiple reflectors can be made into an
object, and position and attitude are tracked as such. It is important to recognize
that the process by which Vicon defines the object is dependent on the markers;
however, the object’s height, size, attitude and origin do not necessarily reflect that
of the true body. Therefore, the actual center of mass may be slightly misrepresented
by the Vicon object’s origin. Attitude tracking has a similar concern. When the
object is defined, its Euler angles are automatically aligned with the inertial frame
of the system. However, the inertial frame is defined somewhat arbitrarily, making
orientation of the body for accurate initialization difficult. As a result, the attitude
of the body is often misrepresented by the Vicon object as well. These discrepancies
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were made small enough to be assumed negligible in most applications of groundbased control by intelligently positioning the markers and with carefully aligned initial
conditions. However, the precision exacted by sphere-based control necessitated that
estimation techniques be used. Unaddressed, these erroneous measurements plagued
the system with implacably whimsical responses that oscillated divergently.
Once an object was created and tracked, its Euler angles and position, with respect to the inertial frame, were transmitted to a manufacturer provided Simulink
plant for deciphering. From there, the sim() command was used in MATLAB to call
the Simulink plant and fetch the measurements for the feedback controller. Of all
processes executed in the control loop, this was the most time consuming, casting it
as the leading role in defining the measurement sampling time. In addition, it took
an inconsistent amount of time to acquire the data, ensuing a variable sampling time.
Furthermore, this communication periodically paused, causing the input to be held
constant, and the controller to fail in sphere-based mode. The cause of this lapse
was never confidently pinpointed, but its frequency during waking hours and scarcity
throughout the night suggested that it was an interference problem.
For ground-based, and the alternate architecture variations, MATLAB can be
thought of as the central hub for all communication. It received and sent signals, and
was the platform on which controllers and estimators were written and implemented.
Figure 4.2 shows how the trajectories were generated in MATLAB, and how errors
were calculated with the measurement information from Simulink. For this specific
example of control, a proportional controller was applied to the yaw error. The position error was fed into an LQR controller which determined the appropriate control
torque to be applied by the motors. Before this control signal could be sent to Vertigo
it had to be conditioned. First it was converted into a representative angular velocity for these motors; this was calculated from a calibration chart bundled with the
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motor specifications [23]. Next, the input was transformed into the body frame so its
orientation matched the Vicon object, which in turn approximated that of the actual
body. Finally, the control was converted into representative scalar values recognized
by Qwerk. Yaw input did not necessitate transformation into the body frame because
its measurements were made in Euler angles. In sphere-based control, the same was
true for all associated angle measurements; transformation was not required, only
scaling for input to Qwerk. In the above schematic, scaling the yaw input was jointly
accomplished by the proportional controller gain. The conditioned translational inputs were then superimposed with the yaw input, a convenience permitted by the
unique manner in which Vertigo is actuated. See Appendix A for more on control
superposition.
After the control input had been converted into a form that Qwerk was able to
implement in the body-fixed frame, it had to be delivered. Forming the link between
Qwerk and an external computer turned out to be one of the more challenging aspects of this project, taking the greatest amount of time. Eventually, two variations
of communication through Java were successfully established. These were first discovered by others working with the Qwerk unit, and further matured by the TeRK
group [24]. The simpler method required some formatting of Qwerk, but essentially
launched Java based graphical user interfaces (GUIs) that initiated wireless communication through a router and governed Qwerk’s various functions. These programs
were primitive in their facilities, but their specialized nature made them excellent for
troubleshooting and diagnostic work. The second method expanded on the principles
of the first, but rather than operating from a pre-coded GUI, it allowed original Java
programs to be written and run directly. This admitted more fundamental access to
Qwerk’s faculties, but coding was still done in a fairly high level language. JCreator
was used as the editing program for developing Vertigo’s Java code, and TeRK pro104
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vided a small library of commands that controlled the functions of Qwerk. Of these,
the most applicable to this project were the audio, battery voltage and DC motor
commands.
The audio commands come in three flavors. In the first, a tone was prescribed
in Hz and given a duration of time to be played for. This was used extensively to
assign audible signals throughout implemented code to help determine its progress.
MP3, PCM and WAV audio clips of 15 seconds or less could be sent with the second
method. These audio files had to be saved in the associated directory, and were
called by name and extension in the function. The third and final audio command
sent text and synthesized it into speech, enabling the robot to say any desired phrase.
This function was quite entertaining, but also proved valuable when coupled with the
battery voltage command. This partnership formed a safety net by sending an audible
alert to warn the user to replace the battery when its terminal voltage dropped low
enough to impede Vertigo’s response. This prevented crashing due to low power.
There were initial concerns with using the audio commands that the code would
pause whilst the sound was being played; however, there was no trouble in practice,
as the code continued to proceed while projecting sound. Additionally, overtaxing of
these commands simply built up a queue for the sound bites to be played in order of
assignment.
The DC motor commands were the most sought after functions in the library.
Their application required some manipulation because they were written specifically
to drive a particular two wheeled robot; however, a logic convention was established to
independently control all four motors. The functions accepted scalar values between
-50,000 and 50,000 to represent the angular velocity of the motors. Since this board
was built to work with a myriad of motors, these values had to be defined on a scale
and specified as unitless.
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To form the link to MATLAB, a third communication method had to be developed. It built on the progress of the first and second methods by dynamically adding
a classpath to establish access to the library of Java commands and a generic GUI
that provided a means of wirelessly connecting with Qwerk. This GUI also supported
the video feed applications that worked with Vertigo’s webcam. Making the last connection opened the door for the Java based Qwerk commands to be inserted directly
into MATLAB code. To initiate this union, a program was written that dynamically
added the Java classpath and opened up the GUI for connection to Qwerk. This
saved a variable in MATLAB’s workspace that gave access to the Java library. The
nature of this variable did not allow m-files that were not in the same directory, or
functions to call the Java commands. Additionally, it had “unsavable” material that
could not be saved to a MAT-file for reloading in other locations; therefore, all code
had to be implemented accordingly. Further explanation and instruction for all three
connection methods can be found in Appendix C.
With these connections made, Qwerk was finally able to receive control inputs.
Figure 4.2 only illustrated the use of the motor commands for Vertigo 2.1, and differed
from Vertigo 1.0’s schematic in that all four motors were controlled rather than just
two. Qwerk accepted the conditioned control signal as a representative final scalar
value for desired motor speed. A trapezoidal angular velocity trajectory was generated
and an embedded PID controller applied to track the profile. The output of the PID
was converted to voltage and sent to the motors. Back EMF from the motors was
measured and converted to a representative velocity to be feed back into the embedded
PID controller.
Qwerk used this embedded control method to regulate the inputs to the motors
and protect itself from saturation. The downside to this was that the user was unable
to directly control the motor inputs. Signals were always feed into the trajectory
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generator, and then tracked by the PID controller, this significantly added to the
response time. In addition, there was no documentation that addressed, or even
mentioned the problem. Therefore, when the motors were found to have an extremely
slow response, taking 20 seconds to accelerate between the positive and negative
rotational velocity extremes, it took several months to determine that conservative
profile trajectories and poorly chosen hardcoded PID gains were the culprits. This
egregiously lazy response time was about three orders of magnitude too slow for
inclusion in this control loop which cycled at about 25 Hz. This put it in urgent need
of mending.
As embedded features, correcting them meant acquiring source code for Qwerk,
deciphering the interactions of hundreds of uncommented C files, developing a method
for correcting the errors, rewriting the source code, compiling new firmware, deleting
Qwerk’s existing firmware, uploading the new image, and reconfiguring the connection settings. This was an extremely time consuming and sensitive process that had
potential to damage Qwerk and required that a virtual machine be installed to gain
root access to Linux. Even after developing a structured routine, this remained a two
hour long process when everything worked properly.
The time and high risk associated with this process made it important that the
problems be fixed with minimal iteration. Gains that were too high caused extreme
actuator saturation and the process became even more risky. The Ziegler–Nichols
tuning method was used to safely improve the PID gains, this gave much greater
weighting to the P and D gain and almost zero to the I gain [25]. Also, the embedded
trajectory generation files were rewritten to include an acceleration parameter that
defined the slope of the angular velocity profile. The initial configuration of this
code would have had an acceleration parameter of 0.6; this was changed to 200,
provoking the original gentle trapezoidal profile into an approximate step function.
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For this reason, the PID gains were tuned to best track a step function. The detailed
process and instructions for changing the embedded PID gains and velocity profile
are included in Appendix D.
Ironing out the kinks of this convoluted communication network meant working
with a total of 10 computer programs and five different coding languages. With such a
tortuous journey, maintaining dependable communication was problematic throughout implementation. However, the reward of connecting to an external computer
was being able to program all controllers and estimators directly from the familiar
MATLAB/Simulink environment.
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Figure 4.2: Ground-based communication schematic.
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Chapter 5
Control
5.1
Introduction
Vertigo is composed of two major constituents: the main body and the sphere. The
body includes the processor, power supply, DC motors, sensors and structure. The
sphere is the base which Vertigo was primarily designed to balance and navigate on;
however, it can also be omitted, and control implemented in a secondary groundbased configuration. Ground-based mode simply means that the body of the robot
is placed directly on the ground. In this configuration, the system acts as a more
traditional statically stable omni-directional ground vehicle, rather than an inverted
pendulum as with sphere-based mode. In both ground and sphere-based modes,
the four spherically orthogonal motors drive omni-directional wheels. This actuation
method, along with a symmetrical design, provides Vertigo with one of its most
convenient features, decoupled control in x, y and yaw. For planar translation, an
opposite pair of actuators controls the motion, while the other perpendicular pair
slips freely in the direction of motion due to the omni-wheels. Accordingly, the two
pairs of actuators are managed independently from one another; however, yaw control
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mandates that all four motors collectively work to rotate the robot.
The equations of motion reflect this decoupled behavior, and it is the objective
of this chapter to develop controllers that govern the response of these equations. It
covers the derivation and simulation of controllers for both ground and sphere-based
configurations. These controllers function as verification for the physical concept,
and build a foundation for the early stages of implementation by managing the basic
aspects of sphere-based motion: balance and translation.
5.2
Ground-Based Control
For the ground-based control of Vertigo, the primary focus was on developing Linear
Quadratic Regulator (LQR) control to track a prescribed trajectory. Simulations with
this controller were then set up to explore the omni-directionality of the system, and
create a standard benchmark trajectory for accuracy assessment on the implemented
control. Several other controllers, such as PID, were designed for the ground-based
mode, but most of their work was completed as part of their implementation and
will therefore be covered in that chapter. For the standard-candle LQR simulation,
a consistent measure of accuracy was established as the “norm-error,” and a goal for
required precision set.
norm(error) ≤ 1
norm(error) = |norm([norm(errx) norm(erry)]) |
This criterion for error calculation accounted for the accumulated error in both x
and y position, and in time. It did so by calculating the difference in position from
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the trajectory at each time step, normalizing the vectors of x error and y error, then
normalizing the vector of these values, and taking the absolute value. Because this
value was dependent on both space and time, a standardized trajectory had to be
used for evaluating performance to maintain uniform and comparable results. This
trajectory was chosen to be a circular path with a radius of 0.5 meters, centered
at the origin, that traversed the circumference once at an angular velocity of 0.3
radians/second. A circular trajectory was chosen to study the response of both sets
of actuators, and validate the concept of omni-directional motion. In addition, it
also gave many opportunities for experimental variation when implemented, such as
sinusoidal waves and ellipses.
During the LQR controller development, discrete time equations of motion (with
Ts = 0.04) were used. These were derived earlier in the Mathematical Modeling and
Analysis chapter, and follow the planar model of the ground-based configuration.







 ẋ   1 0.04   x   0.0240 
T
  + 
 =
1.1976
0 1
ẋ
ẍ

y=
1 0
(5.1)

 x 
 +
ẋ
0
T
(5.2)
The discrete time LQR tracking problem is designed for full state feedback; however, as seen from the equations, this measurement model does not have such luxuries.
In order to obtain full state measurement, numerical derivatives were taken to approximate the rate states for implementation.
By definition, LQR works by forcing the states to zero. In the tracking problem,
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Figure 5.1: Ground-based free body diagram.
it is desired that the states be forced to a reference trajectory, rather than zero. To
accomplish this, a new system is defined where the states are the errors between the
current and reference positions. Therefore, when the LQR controller acts to minimize
the new states, it is minimizing the error, and forcing the system to follow the desired
trajectory. The LQR controller was designed by taking advantage of MATLAB’s
built-in lqrd function. This solves for the optimal controller gains for a prescribed
system and cost weighting. The proper mathematical development of LQR tracking
is posed as a minimization problem for the performance index J.
kf
1X
J=
{[Cq(k) − qr (k)]T Q[Cq(k) − qr (k)]} + uT (k)Ru(k)
2 k=k
(5.3)
0
Where, q(k), u(k) and y(k) are the state, control, and output vectors with lengths of
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2, 1, and 2 respectively. Also, Q is a 2x2 dimensional positive semidefinite symmetric
matrices, and R is a 1x1 positive definite symmetric matrix. These two matrices are
defined to weight the cost function according to the desired performance. It is desired
that the error e(k) = y(k) − z(k) be minimized with minimum control effort, where
z(k) is reference trajectory. After solving the Riccati equation, the optimal control
input becomes,
u∗ (k) = −L(k)q∗ (k) + Lg (k)g(k + 1)
(5.4)
Lg (k) = [R + B 0 P (k + 1)B]−1 B 0
(5.5)
L(k) = [R + B 0 P (k + 1)B]−1 B 0 P (k + 1)A
(5.6)
q∗ (k + 1) = [A − BL(k)]q∗ (k) + BLg (k)g(k + 1)
(5.7)
g(k) = A0 {I − [P −1 (k + 1) + E]−1 E}g(k + 1) + C 0 Q
(5.8)
P (k) = A0 P (k + 1)[I + EP (k + 1)]−1 A + V
(5.9)
V = C 0 QC
(5.10)
E = BR−1 B 0
(5.11)
where,
The above equations show how the matrices Q and R can be viewed as weightings
for the state and control minimization. In general, only the ratio of these weightings
is of importance, so it is standard procedure to set R as the identity matrix, and
only manipulate Q. Once the optimal gains were determined, the command dlsim
was used in MATLAB to simulate the response. This simulation was done for both
the continuous (using lsim) and discrete equations with similar results; however, one
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of the focuses of the ground-based work was to investigate the response when treated
as a discrete system, so only those results are included here. Figure 5.2 shows the
results of the simulated discrete response for the standardized circular trajectory.
This trajectory was specified by assigning Xref = amp ∗ sin(w ∗ t) and Y ref =
−amp ∗ cos(w ∗ t), where amp is the amplitude and was set to 0.5 m, and w is the
angular velocity and was set to 0.3 rad/sec.
Figure 5.2: Simulation results for a circle reference trajectory: norm-error 0.7821
As seen in Figure 5.2, the simulation results correspond very well to reference
trajectory, having a norm-error of only 0.7821. This is within the design criteria of
norm-error < 1; however, it was expected that the physical response of the real sys115
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tem would have greater error due to the un-modeled, and unpredictable disturbances.
This is mostly because the simulation assumes that the yaw angle is perfectly held
at zero, without the need for additional control input to keep it there. In reality, this
would not be the case because external influences such as slip and uneven ground
would accumulate small yaw errors. Also, slip and response lag would further distance the system from the trajectory.
5.3
Sphere-Based Control
Figure 5.3: Planar sphere-based Vertigo model.
In sphere-based control, the two primary functions are balance and translation. In
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this section, several optimal controllers are designed to accomplish these tasks with
various criteria for optimization. The techniques that were successfully executed here
are for linear systems, so the linearized equations of motion that were derived earlier
were used for development. This means that their accuracy is only valid near the
unstable equilibrium, which was the point the equations were linearized about. The
controllers developed with the linear equations were simulated on the nonlinear equations of motion to ensure their applicability on the real system. The sphere-based
planar model, shown in Figure 5.3 is used here, and yaw control is dismissed for the
time being. A common condition that many of these optimal control methods require
is that the system must be controllable and observable. This was proven earlier in
the state controllability and observability analysis for all physically practical state
combinations.
5.3.1
Power Optimal Control: Fixed Final Time, Unconstrained Input
The first control method considered is Gramian based power optimal control with
fixed final time and unconstrained input. It is meant for linear systems and its
development begins with the controllability Gramian Wc .
Z
Wc (t) =
t
0
eAτ BB0 eA τ dτ
(5.12)
0
If Wc is nonsingular, the pair (A,B) is controllable. It can also be shown that
for,
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t1
Z
eA(t1 −t) Bu (τ )dτ
(5.13)
0
u (t) = −B0 eA (t1 −t) Wc−1 (t1 ) eAt1 x0 −x1
(5.14)
At1
x (t1 ) = e
x (0) +
0
a control defined by,
will transfer the system from x0 to x1 in time t1 . The development and proof of this
can be found in [26]. The input u (t) defines the minimal power control because, for
any input u (t), with the same state transfer in the same amount of time, it holds
that
Z
t1
0
Z
t1
u (t) u (t)dt ≥
t0
u0 (t) u (t)dt
(5.15)
t0
The proof of this is well known and can be found in [27].
This method requires fixed final time for which the controller is designed, and
does not impose any constraints on the trajectory, or input. Figure 5.4 shows the
results from a linear simulation for the Gramian based power optimal controller that
takes Vertigo from zero initial conditions, to xf = [2π − 2π 0 0]T in 1.5 seconds.
Physically, this means that the controller should move Vertigo from a stationary and
upright position, to a distance equal to one sphere circumference away while ending
balanced upright with no velocity.
As shown in Figure 5.4, the linear simulation of this implemented controller takes
all the states to their desired location in the prescribed time. To do this in only
1.5 seconds is an impressive claim; however, close inspection of the state trajectories
and control inputs show that this short time requires very extreme magnitudes and
would not be feasible with the implemented system. As stated before, this method
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Figure 5.4: Gramian based control for 2π translation in 1.5 sec.
imposes no limitations on input or trajectory, so for situations where final states and
time are fixed, and all paths and inputs are available, the power optimal Gramian
based controller is a viable option. However, for Vertigo, and in most other real world
applications, constraints bind the possibilities for performing a maneuver. To accommodate this, a technique had to be developed to adjust the final time to fit realistic
capabilities.
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5.3.2
Sphere-Based Control
Power Optimal Control: Free Final Time, Constrained
Input
In most real systems, actuators are limited in their input capabilities; attempting to
exceed these limits will lead to saturation and can cause damage to the hardware. For
the new motors on Vertigo 2.1, and the capabilities of Qwerk, according to the manufacturer specifications, a realistic limitation on input u is about 45 Nm. To account
for this, a numerical method was developed that permits imposing limitations on input. It was implemented in the simulation coding to take a user defined maximum
control input, and initial guess for final time. It then calculated the associated power
optimal Gramian based controller and determined the maximum control required to
carry this out. It defined an error between this and the specified allowable value, then
updated the guess for final time based on this error and a scaling factors. This is
effectively a free final time method that finds the minimum power controller to meet
the state and input constraints in the minimum time. This was carried out for several
specified maximum u values, and some interesting results were uncovered.
Figures 5.5 - 5.8 depict linear simulations of the Gramian based control with
decreasing allowable umax values. In Figure 5.5, umax = 55 and the minimum time
that this maneuver was able to be carried out in, while still meeting the u constraint,
was tf = 1.2635 seconds, and all of the final states were taken to the appropriate
values. Figure 5.6 represents the estimated maximum allowable input for Vertigo
with umax = 45, and the minimum time that this maneuver was able to be carried
out in was tf = 1.3749 seconds, and all of the final states were taken to the appropriate
values. In Figure 5.7, umax = 20, the minimum time that this maneuver was able to
be carried out in was tf = 2.0154 seconds, but the final state values do not meet the
specified constraints perfectly.
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Figure 5.5: Gramian based control, u max = 55.
Figure 5.6: Gramian based control, u max = 45.
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Figure 5.7: Gramian based control, u max = 20.
Figure 5.8: Gramian based control, u max = 15.
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Figure 5.9: Plot of successful u max convergences and associated final times.
In Figure 5.8, when umax is reduced to 15, the minimum time is tf = 2.5143, but
the states do not even approach their specified location. In fact, it was found that for
any values below umax = 15, the code would not converge, due to singularity errors.
Even for umax = 15, many attempts and manipulations of the error scaling factor
were required before it would converge. The reason is that, for values below 15, the
controllability Gramian approaches singularity, and the equation,
0
u (t) = −B0 eA (t1 −t) Wc−1 (t1 ) eAt1 x0 −x1
(5.16)
is not able to be calculated by MATLAB due to its ill condition. These results support
the basis of the inverse condition number presented in the state controllability and
observability analysis. Again looking at Figures 5.5 - 5.8, it is clear that, as the
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controllability Gramian approaches singularity with smaller allowable umax values,
the accuracy of the calculations reduced, and the linear simulation was not able to
reach the final states. This error grew as the condition of the controllability Gramian
degraded. This also made the code more sensitive to the initial guess for final time,
so the plot and included equation in Figure 5.9 were created and used to generate
close approximations for the initial guesses based on the empirical results.
5.3.3
LQR Overview
In feedback control, the actuating signal is dependent on both the reference signal,
and the output of the system. In state feedback control, the plant feedback output
contains at least one of the states, full-state feedback is when all of the states are
included as outputs. For state feedback, input u is defined as follows.
u = r − kx
(5.17)
With this definition, x represents the output states that are to be fed back into
the controller and plant. The state space representation ẋ = Ax + Bu y = Cx can
be rewritten with this input in the following fashion.
ẋ = (A − Bk) x + Br
(5.18)
y = Cx
(5.19)
This form reveals that k can be used to adjust the eigenvalues of A. It can also
be shown that both the rank and determinant of the controllability matrix do not
change with state feedback. This means that controllability is invariant under state
feedback, and the previous result showing that Vertigo is controllable in all cases also
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holds for feedback control.
Establishing that Vertigo is controllable for feedback control verifies that this
method can be used for controller design. To optimally generate the gains, the LinearQuadratic Regulator (LQR) method was used. In LQR, for a cost function J, where
Z
J=
∞
xT Qx + uT Ru dt,
(5.20)
0
the control u = −F x minimizes this cost, where F = R−1 B T P , and P is found
by solving the Riccati equation. From this cost function, it is clear that both the
states and input are reaching weighted minimums. These weightings can be altered
to best meet the specifications of a given problem, and are often chosen or tuned
experimentally when implemented on physical systems. The LQR method poses
additional conditions that must be met by the system in order be applied; these
are as follows: the linear state space pair (A, B) must be stabilizable, and there
must be no unobservable modes. The above state controllability and observability
analysis verifies that this system meets these requirements, securing the validity of
LQR control for this system.
Two different objectives were considered for LQR controller design to manage the
two types of planar motion that Vertigo is capable of. The first was for balance and
the second was for point to point translation. The resulting controllers were implemented in a linear simulation to verify their accuracy for theoretically controlling the
linear system. When it was established that this method was working properly, simulations were run with the linear control methods on the nonlinear system to mimic
the physical system. These nonlinear simulations are the focus of this section.
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Stabilizing LQR
For the balance of Vertigo, a linear full-state feedback controller was developed using
LQR methods. This process was executed and simulated in MATLAB. Figure 5.10
shows the results of this simulation when applied to the nonlinear system using MATLAB’s ode45 integration command. The initial conditions for this, and the remainder
of the LQR simulations, are bounded random values to ensure accuracy for all reasonable scenarios.
Figure 5.10: State response with stabilizing LQR controller.
It was the primary objective of this controller to balance Vertigo, by bring the
states from their initial conditions to ones where the sum of the two angle states is
zero. This signifies a vertical and balanced stature. If fully effective, all states should
be brought to exactly zero, as this is their minimum possible value. The plots of
the rates in Figure 5.10 show that this method was effective at bringing these states
back to near zero values; accordingly, the angle states leveled out as the rates were
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brought to zero. This controller also attempted to drive the angles back towards zero;
however, they over shot and settled about .2 radians away from zero. Offset angle
states with zero rate states means that the system settled into a balanced position
at a translated location that is away from the origin. Purely in terms of balance,
this is not a problem, because the system did not tip over. However, this error may
cause difficulty for implementation on the real system because one of the options for
the implemented control of Vertigo is to have two separate controllers acting. One
of these would be devoted to assuring balance, while the other controls translation.
If the balancing controller is doing its job, but causing large translational errors, it
will create more work for the second controller and will add to the overall error.
Another issue with this controller is that it took this simulation about 4 seconds for
the transient response to settle. This is respectable, but reducing this time is desirable
if possible because it will further improve overall performance and accuracy.
Though improvement is possible, expecting perfect results is impractical here because this is a linear controller design method being implemented on a nonlinear
system. As the states deviate further from the point of linearization, the nonlinear
affects of the system have an un-modeled influence on the response. In this simulation,
the linearization was taken about a vertical stationary position where all states and
control are zero. Therefore, pushing for greater performance increases the magnitudes
of the rates and control input which introduces more linearization error. A balance
must be reached that provides acceptable accuracy and sufficient performance.
The convention for defining states in this model make it difficult to visualize the
motion of the body with respect to an inertial frame. To make this clearer, a plot
was generated that sums the two angle and rate state pairs. As seen in Figure 5.11,
this definition represents the response of Vertigo in terms of angles with respect
to a vertical reference. This clearly shows that the body of the robot returns to
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Figure 5.11: Response with stabilizing LQR controller wrt vertical.
equilibrium, but requires about 4.5 seconds to do so.
To mitigate the undesired characteristics, the A matrix was augmented. This
effectively includes an additional component to the cost function that continues to
penalize the cost with time. Augmentation for this system assumed the following
form,
Aaug = A − λI
(5.21)
where λ = .005 and I is the identity matrix. This augmented A matrix was applied
in the LQR procedure to generate new gains. Conservative values were necessary for
the augmentation because it is used to manipulate the state space model, which is already a linearized representation of the full equations of motion. Small augmentation
prevented extensive departure from the real system and diminished validity of the
controller. The Q matrix weightings required retuning for the augmented controller
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Figure 5.12: State response with augmented stabilizing LQR controller.
Figure 5.13: Response with augmented stabilizing LQR controller wrt vertical.
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design. In the original LQR tuning, equal weighting was assigned to all states; however, the best results for the augmented system were obtained when the angle states
had greater weighting to ensure that they were forced to zero.
Figure 5.12 shows that the balancing LQR controller designed with an augmented
A matrix was much more effective at managing the states and increasing performance.
The steady state bias of the angles was reduced by an order of magnitude, and the
rates appear to reach zero much faster than they did with the original controller.
Therefore, the augmented controller yields less translational error. Figure 5.13 is the
response of the body with respect to the vertical inertial axis. This clearly shows that
the body stabilized in under two seconds, which is less than half the original time,
and required less control to do so.
5.3.5
Tracking LQR
After balance, the second type of motion that must be controlled for Vertigo is translation. The details of this motion are often conceptually counterintuitive, and are
discussed in Appendix A. With some cleaver manipulation, LQR techniques can also
be used to develop a tracking controller for translation. By definition, LQR is a
full-state feedback method that drives the states to zero; therefore, if the states are
redefined as the difference between the current and desired states, the controller will
act to minimize the error. In doing so, the current states are driven to the final
desired states, affectively tracking the reference states. To implement this, a new
control input is defined as,
u = r − k (x−xf )
which makes the state space
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(5.22)
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ẋ = (A − Bk) (x−xf ) + Br
(5.23)
y=C (x−xf ) .
(5.24)
As with the stabilizing LQR, this technique was implemented and simulated in
MATLAB on the nonlinear system equations. The simulation takes the system from
random initial conditions bounded near zero, to xf = [2π − 2π 0 0]. In other words,
it is taking Vertigo from near vertical rest, to vertical rest at a distance one sphere
circumference away.
Tuning the Q matrix for translation was more difficult than tuning for balance.
This is because the final angle states were far away from the initial conditions, but
the rates were not. Therefore, quickly minimizing the angles required large rates, but
this fought against the rate minimization. If weighting is placed more heavily on the
rate states, the response will be smooth and predictable, but will take longer, as less
aggressive rate trajectories will be permitted. Figures 5.14 and 5.15 show how the
LQR tracking response with heavily weighted rates followed a very smooth trajectory,
but took nearly 20 seconds to reach the final values. The conservative nature of this
response also means that less control effort was required; in only two seconds the
body was nearly vertical as it slowly crawled to the final states.
By increasing the weighting on the angle states, and reducing it on the rates, the
performance can be improved by allowing the angles to change more quickly to their
final values. Figures 5.16 and 5.17 show the response with these weightings, and it
is clear that the final states were reached more abruptly. However, this improvement
came at the expense of a more chattery response and greater demands from the control
input.
This is a tradeoff of performance to accuracy and power, and is common in con131
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Figure 5.14: State response with tracking LQR controller, heavily weighted rates.
Figure 5.15: Response with tracking LQR controller wrt vertical, heavily weighted
rates.
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Figure 5.16: State response with tracking LQR controller, heavily weighted angles.
Figure 5.17: Response with tracking LQR controller wrt vertical, heavily weighted
angles.
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trols. If thought of as a standard transient step response, less weighting on the
rates improves the rise time, but increases chatter and control effort. Determining
the proper tuning is case specific and finicky. This is where the augmented controller
shines, because it is time dependant and helps blend the advantages of both extremes.
Low weighting can initially be given to the velocities to improve the rise time, and as
time progresses, the augmentation penalizes the rate errors more heavily to smooth
out the response. To demonstrate this characteristic, the system was augmented with
λ = 1, and simulated with the same aggressive tuning as the previous simulation. Figures 5.18 and 5.19 show the response of the augmented LQR controller with heavily
weighted angle. By comparing this to the two un-augmented responses, it is clear
that the augmented response is far superior. It marries the smoothness, accuracy and
efficiency of the heavily weighed rate tuning, with the high performance of the heavily
weighted angle tuning. In addition, the magnitudes of the control and rates required
by the augmented controller are much more realistic than those in the un-augmented
response; however, they are still quite demanding.
To represent a more realistic scenario, the same augmented system was more
evenly tuned with rate weightings at 75% of the angle weightings. Figures 5.20 and
5.21 show these results. With realistic tuning, this maneuver is executed in about
10 seconds and there are low expectations of the rate and input magnitudes. This
simulation is thought to be fairly representative of what should be reasonably expected
when implemented on the physical system.
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Figure 5.18: State response with augmented tracking LQR controller, heavily
weighted angles.
Figure 5.19: Response with augmented tracking LQR controller wrt vertical, heavily
weighted angles.
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Figure 5.20: State response with augmented tracking LQR controller, realistic weighting.
Figure 5.21: Response with augmented tracking LQR controller wrt vertical, realistic
weighting.
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Chapter 6
Estimation
6.1
Introduction
As an unstable system, Vertigo requires very responsive and accurate control to operate successfully. In developing feedback control techniques, it was assumed that
perfectly accurate measurements were available for all states and times; however,
this is not realistic when considering implementation on the actual system. Once all
functions of Qwerk are made available, information on all states can be measured
directly. Until then, the Vicon system had to be used for external measurement, and
none of the states are measured directly. Even if full state sensing were available, all
instruments are rated with a level of accuracy, which means their measurements are
inherently erroneous. The focus of this chapter is to establish a method for extracting
reliable state information that can be fed back into the control loop.
Here the continuous-discrete extended Kalman filter was developed and simulated
for Vertigo. One of the dominant challenges in controlling this vehicle was the information circuit for closed loop feedback control. The current sensing method utilizes
Vicon (an external 8 camera motion capture system) for information on the robot’s
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position and attitude. Despite the celebrated precision of Vicon, its measurements do
not directly correspond to the states of the system. Mathematically transforming the
raw measurements to representative state measurements introduces the accumulated
uncertainties of the physical system parameters (lengths, mass, etc.), Vicon measurements, and initialization errors. Consequentially, half of the states are not measured
at all, and the others only have flawed indirect measurements available. To address
this problem, the continuous-discrete extended Kalman filter (EKF) was exploited.
This provided a method for full state estimation and noise filtering; however, the EKF
requires linearization of the nonlinear measurement model and equations of motion.
This added to the uncertainties and had to be included in the development of the
filter to mitigate its affects.
6.2
Continuous-Discrete Extended Kalman Filter
The Kalman filter is a recursive process that efficiently estimates the states of a
process from noisy measurements. The filter is based on linear dynamical systems
that are discretized in the time domain, and is very powerful in that it can estimate
past, present and future states while minimizing the mean of the squared error. In
this way, it is analogous to applying a standard discrete estimator, but optimally
placing the poles with a minimum variance approach.
To be implemented, the filtered process must be modeled in the frame work of the
Kalman filter, which assumes a zero-mean Gaussian model for both process and measurement noise. This framework further assumes that the model and measurement
noise are uncorrelated from each other and in time. Once modeled, the error covariances and states must be initialized. Based on this knowledge, an optimal expression
for computing the estimator gain is applied. State and covariance estimates are then
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updated using their current values, the calculated gain, and the measurement model.
Lastly, the states and covariances are propagated forward in time to be used as the
a priori estimates for the next time interval. The process is repeated for each time
step thereafter. This explanation is for the discrete linear system, and follows the
original development presented in 1960 when R.E. Kalman first published a paper on
the topic [28]. Further development expanded the filters application to other types
of systems.
The extended Kalman filter (EKF) is a variation that accommodates nonlinear
systems by using their linearized simplifications. When linearizing these equations,
the Jacobian is taken, and the point of linearization is reestablished at each time
interval with the current conditions. This method is used when applying the filter
to Vertigo’s nonlinear equations of motion and measurement model to improve and
complete data acquisition for its feedback control loop. Vertigo relies on external
sensing for dynamic measurements; however, this data contains errors and does not
provide direct measurements of the states. There are errors in the model relating
these measurements to the states. Furthermore, the mathematical model has errors
based on measurement uncertainties and simplifying assumptions; examples of these
include the inertias, friction coefficients and the relation between signal input and
actual toque applied by the motors. The extended Kalman filter is a well suited
tool for estimating Vertigo’s states and attenuating the measurement noise. More
specifically, this application employs the continuous-discrete EKF because Vertigo
is a continuous system, but the measurements are discrete. The full mathematical
development of this filter is detailed with its simulated implementation in the next
section.
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6.3
Application of Continuous-Discrete EKF
Application of Continuous-Discrete EKF
This section covers the development of the continuous-discrete extended Kalman filter
and its application to the Vertigo system. The general method for this filter is well
documented in many sources; for consistency, the convention and notation from [29]
will be used here.
To apply the filter, the model must take on the following form.
w(t) ∼ N (0, Q(t))
ẋ = f (x, u, t) + G(t)w(t),
vk ∼ N (0, Rk )
ỹk = h(xk ) + vk ,
(6.1)
(6.2)
Where ẋ and ỹk are respectively the state and measurement models. f is the system’s
nonlinear equations of motion, with


 θ 
 
 φ 
 
x= 
 θ̇ 
 
 
φ̇
(6.3)
as the state vector. h(xk ) converts the states into measurement space, and w(t) and
vk are zero-mean Gaussian white-noise processes. The G matrix allocates the process
noise and is defined as,
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

 0 0 


 0 0 


G=

 1 0 




0 1
(6.4)
where the first two rows are zeros because they represent kinematical relationships,
and it is always true that velocity is the time derivative of position. The equations of
motion implemented here are those previously derived for Vertigo. The measurement
model will be based on the Vicon data acquisition process until the onboard sensing is
supported. To relate the Vicon measurements to the states, geometric and kinematic
relations are required.
Figure 6.1: Planar measurement model for sphere-based Vertigo.
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The Vicon motion capture system works by locating the center of spherical retroreflective markers through triangulation with multiple infrared strobe cameras. With
the accompanying software, clusters of reflectors can be made into an object and
tracked as such. The system is then able to determine the position and attitude of
this object. All time derivative states must be calculated from these measurements
externally. Track the motion of the sphere was not possible with this method because
the markers interfered with the actuators and ground. If the entire surface of the
sphere were coated in retro-reflective material, Vicon would recognize it as a single
marker, and could track its location. However, it would have no inclination about
rotation, which is the desired state.
Figure 6.1 illustrates the planar Vicon measurement model. β is the angle of the
Vicon object with respect to vertical, and dv is its distance from the origin. These
readings can be related to the states of the system by defining,




θ+φ

 β  
h=

=
2rb θ − lv sin(θ + φ)
dv
(6.5)
where rb is the radius of the sphere, and lv is the distance from the center of the
sphere to the center of the defined object.
Vicon is a very accurate system, despite this there is a great deal of uncertainty in
these measurements, and it all stems from the way it defines objects. In Figure 6.1,
the object is shown to be centered in the upper half of the IMU, this is because
the IMU, and its mounting plate, are the most convenient and protected places to
attach markers. The difficulty is in knowing exactly how this object is defined. Vicon
does their best to simplify the procedure by creating objects with Euler angles that
initially align with the inertial reference frame (defined in the initial calibration and
setup of Vicon), and measuring distances with respect to the inertial origin. However,
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the accuracy of these measurements depends on how well the object is defined with
respect to the body it is trying to track, and more specifically, the orientation of its
actuators.
Vicon observes all of the markers selected to compose an object and chooses a
point to center the object. The exact location of this object with respect to the body
is unknown. Similarly, the attitude of the object is defined to correspond with the
inertial frame, and is based on the location of the markers at the time it is created.
However, if the body is not perfectly aligned with the inertial frame, this object will
not accurately represent the body’s attitude. Uneven ground and the discontinuous
wheels make it extremely difficult to address this misalignment. Together with errors
in the radius of the sphere, every term in the measurement model has uncertainty.
This means there will be significant errors in the measurement data that will have to
be accounted for as best it can by the noise parameter vk in the Kalman framework.
To commence the cyclic filtering procedure, initial values of the components to
be calculated must be defined. For Vertigo, the states and error covariances are
initialized as,
T
x̂(t0 ) = x̂0 =
0 0 0 0
P0 = E x̃(t0 )x̃T (t0 )
(6.6)
(6.7)
where the zero initial states represent vertical stature that is stationary at the origin,
and E [] is the expected value. The Kalman gain is calculated with the formula,
−1
−
− T
−
Kk = Pk− HkT (x̂−
k ) Hk (x̂k )Pk Hk (x̂k ) + Rk
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(6.8)
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−
In this equation, Hk (x̂−
k ) = hx (x̂k ) is the Jacobian linearized measurement model.
Based on the geometry of the model schematic, this works out to be,


H = hx = 

1
1
0 0 

2rb − lv cos(θ + φ) −lv cos(θ + φ) 0 0
(6.9)
When a new measurements is available, the Kalman gain can be used to update
the state estimate and covariance by
−
−
x̂+
k = x̂k + Kk ỹk − h(x̂k )
(6.10)
−
Pk+ = I − Kk Hk (x̂−
k ) Pk
(6.11)
If a measurement is not available, the estimate remains, and is used as the initial
condition for propagation forward in time. The propagated values are taken as the a
priory estimate for the next step, just as the initialized values were for the first step.
The equations to be propagated are,
˙
x̂(t)
= f (x̂(t), u(t), t)
(6.12)
Ṗ (t) = F (x̂(t), t)P (t) + P (t)F T (x̂(t), t) + G(t)Q(t)GT (t)
(6.13)
Where F (x̂(t), t) = fx (x̂(t)) is essentially the A matrix from the linearized equations
of motion. These covariance and state equations are nonlinear, so they must be
integrated forward in time. Since the covariance and state propagations are coupled,
the two equations must be integrated simultaneously. This was done using the ode45
function in MATLAB by combining both systems. Once propagation is carried out,
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Figure 6.2: C-D EKF, true state response.
Figure 6.3: C-D EKF, estimated state response.
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Application of Continuous-Discrete EKF
the cycle is repeated in the next time step.
This filter was constructed and applied in simulation to ensure it was properly
incorporated. Not all uncertainties are readily determined in the continuous-discrete
nonlinear problem, so tuning of the Q and R matrices was required to improve the
estimator results after it was applied. This was a sensitive procedure for the extended
Kalman filter because factors such as linearization errors are not easily quantified, so
discerning their influence took time. A baseline for this tuning was formed with
knowledge of the noise in the process and measurement; from there, refinement was
done by trial and error.
In the implemented simulation, a linear-quadratic regulator (LQR) was used for
control. Together, the LQR and Kalman filter solve the famous linear-quadraticGaussian control problem. The simulation generated the true data for the response
of a translation that equates to a 2π rotation of the sphere, the true data is calculated in the filter loop and only computes to the next time step. This is so it can use
the estimated states to calculate the LQR control, which mimics how the loop will
be implemented on the real system. Noise was added to the true data to generate
measurements, and the filter was applied to them. Each time through, the filter calculates its own control using the same LQR method that the true data was generated
with. For those simulations that run the filter between measurement updates, the
most frequent measurement was stored for plotting at that step.
In the first simulation, the sampling time was based on the actual average sampling
time of the Vicon MATLAB implemented communication loop, 0.04 seconds (25 Hz).
The states and covariance were directly propagated forward to the next measurement,
so every time step updated the estimate with measurement data.
Figure 6.2 and 6.3 respectively show the true and measured response to the
simulated continuous-discrete extended Kalman filter. From these results, it is clear
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Application of Continuous-Discrete EKF
Figure 6.4: C-D EKF, angle state errors.
Figure 6.5: C-D EKF, rate state errors.
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Sampling Time Analysis for C-D EKF
that the estimated response has effectively represented the true response and filtered
the noise. For further comparison, the errors between the true and estimated states
were plotted with their respective 3σ bounds. Figures 6.4 and 6.5 show that the errors
were small and well contained by the bounds; therefore, the filter was well tuned and
succeeded in improving upon the measurement data.
6.4
Sampling Time Analysis for C-D EKF
In some applications, if the measurement sampling time is well known, the filter
can be run multiple times between measurements. The objective of this is to try
and improve the accuracy of the estimate without requiring more measurements.
When executed, the update portion of the filter is not used unless a measurement is
available at that time step. In the absence of a measurement, the propagation from the
previous step is used as the current estimate, and the filter proceeds as normal. The
applied code was set up to separately define the measurement and output frequency.
To investigate how this may affect the estimate, a simulation was run using the
same realistic measurement frequency of fm = 25Hz, but with an increased output
frequency of fs = 100Hz, making the filter run four times per update.
Figures 6.6 through 6.9 show that propagating four times per measurement update, versus only one, in the continuous-discrete extended Kalman filter also provides
improved estimates over the measurement data. In fact, close comparison between
the two simulations shows that propagating multiple times not only improved the estimate, but slightly improved the transient time of the true response as well. Improved
performance is the goal of incorporating estimation techniques. It is important to
note that this simulation mandated retuning to converge with the increased output
sampling time. Modified tuning also affects the performance of the filter, so this
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Sampling Time Analysis for C-D EKF
Figure 6.6: C-D EKF, true state response, fm = 25Hz, fs = 100Hz.
Figure 6.7: C-D EKF, estimated state response, fm = 25Hz, fs = 100Hz.
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Sampling Time Analysis for C-D EKF
Figure 6.8: C-D EKF, angle state errors, fm = 25Hz, fs = 100Hz.
Figure 6.9: C-D EKF, rate state errors, fm = 25Hz, fs = 100Hz.
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comparison shows that, when both filters are properly tuned, it is advantageous to
have fs > fm .
It is clear that the number of propagations influences the filters accuracy, but
the measurement sampling time also has a significant impact. Understanding the
nature of this affect is important for maximizing the chances of success with the
fully implemented system. Therefore, a simulation was created that varied both the
measurement and output sampling time. As a measure of accuracy, the 3σ values
for the states were stored once it had settled for each combination of sampling times.
The values were taken 5 time steps from the end, rather than exactly at the end,
because this is a cyclic process, and breaking that to exit the simulation causes the
last 3σ values to be invalid.
Figure 6.10 is the mesh of the 3σ values for θ as the sampling times were varied.
Analysis of all four states yielded the same trends; therefore, only the results for
θ were included here as a representative set. The previous analysis showed that
increased output frequency improved the estimator; this is confirmed here, though it
is not obvious. There are several reasons for this, the most pronounced being that
its impact is outshined by that of the measurement frequency variation. Also, the
maximum and minimum values for both sampling times were chosen to encompass
practical values, and ones that could all be simulated with the same tuning. This
fixed tuning was acceptable for all points; however, if it were possible to specify
ideal tunings for all simulations, the influence of varied output frequency would have
been more apparent. Lastly is the affect of diminishing return. Running the filter
with fs = 2fm , will improve the results, increasing that to fs = 4fm will show further
improvement; however, a limit exists to how accurate a filter can be without receiving
more frequent measurements. Because the range of variation was restricted by tuning,
it was not possible to start this analysis from the baseline point of fm = 25Hz,
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Sampling Time Analysis for C-D EKF
Figure 6.10: Sampling time mesh.
fs = 25Hz, and expand from there. It was also not realistic for the two ranges to
overlap, because impractical points would exist where fs < fm . Therefore, it was
necessary that fs > fm for all points, so the more significant initial improvement of
running the filter more than once per update could not be captured.
It is clear from Figure 6.10 that there is a much more influential dependence on
the measurement sampling time. Lower 3σ values tend to signify a more accurate
estimate. Therefore, this mesh verifies the expected results that the estimate will
improve as the number of measurements is increased. Again, the same general results
were found with the other three states, so they were not included here to avoid
redundancy.
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Chapter 7
Implementation
7.1
Introduction
Vertigo is an original platform that was built from scratch; as such, it was necessary
to increment its development by introducing controllers from the ground up. Taking
small steps built a familiarity with the system that made troubleshooting easier as
the project progressed to more sophisticated coding. The goal here was to develop
methods for implementation, and apply fundamental programs for testing functionality and theoretical principles. These would serve as a foundation for future work
in balancing and navigating on the sphere. In many ways, early implementation mirrored the efforts of establishing the communication architecture. Accordingly, the
first controllers were implemented through the Java based Terk programs, and interfaced through the provided GUIs. Complexity grew as feed-forward programs were
coded in JCreator. These programs performed simple tasks, and were written to help
troubleshoot hardware. This practice developed an understanding of the library of
commands that accessed Qwerk’s functions, and would later be used to send input
signals to Vertigo. This Java interface was also used to calibrate measurements and
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Implementation Challenges
develop conversion factors for signals between components.
When it was discovered that the current version of firmware on Qwerk did not
actually support the onboard sensors, it was necessary to establish the external sensing
communication loop that tied in Vicon, Simulink, MATLAB, Java and Qwerk. The
variable stored for connecting to Qwerk did not allow controllers to be implemented
in Simulink; therefore MATLAB had to serve as the central hub for coding. With
most of the kinks ironed out from this network, basic control implementation began.
Many challenges threatened to jeopardize implementation; several were anticipated,
but the majority of them were unforeseen. These will be discussed in the remainder
of this chapter along with the progression of ground-based control, and balancing
control. Finally, attempts at sphere-based balancing control will be detailed with a
discussion of what prevented their success.
7.2
Implementation Challenges
Uneven Terrain
Vertigo has four support contacts in ground-based mode, so the slightest curvature in terrain, or inconsistency in actuator dimensions, will lead to uneven weight
distribution amongst the wheels. This has several negative effects on performance,
all of which pertain to friction. Friction is needed between the wheels and ground
to move the body. If a leg is not supporting any weight, it has no normal force to
generate its maximum frictional force. This is the primary cause of slip. When this
affect is large or prolonged, only one pair of motors will be properly applying control,
inducing undesired yaw. If both pairs of motors are slipping, time response will suffer
from the wasted control. When tracking a step, slip near the final destination may
prevent the robot from achieving the final destination. This is because error is small,
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so the applied control was insufficient to overcome the potential causing slip.
Inconsistent Loop Sampling Time
Vertigo’s loop of communication calls upon five different programs, two of which
require network connections, one being wireless. A delay in any of these systems will
cause inconsistent loop sampling time. Many implemented control methods make the
assumption of constant sampling time; this introduces error. The effects of erratic
sampling times were most detrimental to implementation of the Kalman filter, and
will be discussed later in this chapter. To best mitigate this problem, an average
sampling time was used and assumed to be constant. There are some cases where
the Vicon measurement system momentarily paused, and control was held constant.
This caused tracking controllers to deviate from their desired trajectories, and loss of
balance in sphere-based mode.
Dynamic Instability
Vertigo was intentionally designed with a high center of gravity to improve its
controllability and dynamic stability in sphere-based mode; however, as a statically
stable ground vehicle this made it prone to dynamic instability. This was problematic
for large accelerations, which caused Vertigo to lean or tip. When leaning, only one
or two wheels contacted the ground and yaw was induced. Recovery from this often
caused large overshoot, sometimes to the point of divergence or erratic behavior, depending on the controller. Frequently, these departures were caused by the paused
Vicon connection. In this case, dynamic instability was merely the mechanism of
failure, with poor connection as the cause.
Embedded Signal Processing
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Unfortunately, Qwerk was programmed with no means of directly controlling the
voltage to the motors. Control is prescribed with a scalar reference value that is sent
to Qwerk. This value is between -50000 and 50000, and represents a unitless angular
velocity factor. Qwerk uses this value as a final value in a trapezoidal trajectory
generator. It then implements a PID controller with EMF feedback to follow this velocity profile. This entire process is embedded, and the original trajectory generator
and PID gains had an extremely slow response; it took roughly 10 seconds to reach
full speed from rest. This is two or three orders of magnitude too slow when dealing
with a system that samples around 25 Hz. Mending this problem was discussed with
the communication architecture, but the issue was mentioned here because it was
discovered through implementation, and delayed progress by several months.
Yaw Control
If yaw angle was not controlled during tracking, the robot was free to rotate
under external influences. This meant forfeiting control of camera direction, and was
especially problematic when controlling from the inertial frame. Any misalignment
(from yaw rotation) with the two frames skewed all control application. This was only
a problem for ground-based control, as most sphere-based methods only used Euler
angles. Eventually, when translation is attempted, this will also become an issue for
sphere-based control. To address this problem, two strategies for proportional yaw
control were implemented.
The first method was accomplished by fixing the yaw angle. In this situation, a
proportional controller regulated the angle, most often driving it to zero. This way,
Vertigo was only prescribed translational motion. Simple proportional control was
found to be more than sufficient, because yaw requirements were not very demanding. For gain determination, it was increased until recovery from a
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overshoot. The gain just prior to overshoot was chosen.
The second control strategy was similar, but assigned a trajectory rather than a
fixed angle. This most often took a form resembling a retrograde orbit in ground-based
control. Before a yaw trajectory could be applied, control input had to be transformed
from the inertial to the body frame. This was done by multiplying the positions with
a transformation matrix that accounted for the yaw angle; therefore, position control
input was made statically independent from yaw at the time of measurement. To
perform the retrograde reference trajectory, a user defined angular yaw frequency
was multiplied by time, and translation was assigned separately. By applying a
proportional yaw controller, Vertigo was forced to follow the proposed trajectory.
Implementation of this strategy was difficult, because Vicon returned its angle
measurements between π and +π. Adding π to the measurement signal fixed the
problem for conservative trajectories, but beyond one rotation there was no distinction
between 2π and 4π. Many commonly used rotation transformation methods, including
MATLAB’s mod command, did not fully address this issue. This is because they
conditioned the measurement back to zero when it crossed 2π, and the error appeared
to be 2π. Methods were also attempted that conditioned the trajectory by resetting
it to 0 at 2π, rather than the other way around. These did not work either because
the error became −2π and the robot quickly completed a full rotation in the opposite
direction to pick up the trajectory again. Eventually, a measurement conditioning
method was developed that allowed any number of rotations. This bit of coding is
convoluted, as it works with three incrementing counters, the mod function and a
time delay; however, it effectively shifts the measurements to a continuous scale.
Including yaw trajectories was not without drawback. Its problem was similar to
the error from undesired yaw disturbances when controlled form the inertial frame.
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ing the translations. For slow trajectories, this effect was negligible because the
sampling rate updated yaw measurements reasonably fast. As yaw rate increased, so
did the error.
State Feedback
LQR is a full state feedback method; however, Vicon does not measure rates.
Therefore, half of the ground-based states were not measured for feedback, and none
were for sphere-based mode. Derivatives of the position had to be taken for rates;
however, implementation was a discrete process, so numerical derivatives were taken
as the difference in position from the current and previous time step divided by the
sampling time between the two. In addition, controllers had to be implemented in the
time domain. This was challenging because many continuous and discrete methods
for controller design are not carried out in the time domain, accordingly, either extra
measures had to be taken with their application, or they were deemed too disparate
to be applicable.
7.3
Ground-Based Implementation
Implementation began with the ground-based mode, because it was a safer and more
attainable starting point than sphere-based control. The first code to successfully
make the link between Vicon and MATLAB tested the position and attitude of the
Vicon object. Using the sim command, MATLAB extracted the Vicon information
through a manufacturer provided Simulink plant that was able to record measurements. This program was useful throughout the project for checking the coordinates
of the robot, and analyzing how well the object was defined by Vicon.
Next, a program was built to complete the full communication loop by sending
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commands to the robot, and using the Vicon information as feedback. This controller
tracked a one-dimensional user defined step function with if-statements and constant
magnitude control inputs. Broken down, the logic followed that Vertigo should be
moved forward if the difference between desired and measures position was negative,
and backward if positive. This program helped to start piece together the coordinate
and actuation sign convention for the loop. For programs that introduced new functions, if-statements were used rather than proportional control because they allowed
exact specification of input, preventing motor saturation.
Building off this, a two-dimensional proportional controller that tracked a step
function in both x and y was constructed. By converting the x if-control into a proportional gain that was multiplied by the position error, and then mirroring application to the y axis, this controller verified that Vertigo has decoupled omni-directional
motion on the ground. It was also used to confirm the inertial axis convention, establish the safest way to terminate control, and experiment with control input and gain
values to determine reasonable ranges. Running these experiments began to expose
some of the flaws in using the Vicon system. Nonspecific object definitions meant
that the center of the Vicon model did not exactly match that of the actual robot.
Also concerns about yaw control were seeded here, as minor but erratic rotational
drifting was observed for large step functions. Figure 7.1 shows how this controller
tracked a two-dimensional step from (0,0) m to (0.5,0.5) m. Plotted with time as the
z-axis, the proportional control is apparent as velocity is greater at the beginning,
and consistently decreases as the error reduces. Plots of the control input showed the
inverse of this trend as magnitude diminished with time.
To kick off yaw control, and establish the Vicon defined convention, if-logic was
employed again. Once directionality was confirmed, proportional control was introduced. It was discovered that the gain and resultant control magnitudes were much
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Figure 7.1: Proportional two-dimensional step track.
more conservative for yaw than translation. This was because the error to recover
from was limited to ±π, and control was cooperatively applied by all four motors.
One of the goals of implementation was to compare the simulated LQR controller
with the physical response. This simulation was discussed in the Control chapter, and
was specified to track a standardized circular trajectory with a 1 m diameter, traversed
at 0.3 rad/sec. To begin work on this, the proportional controller was applied to
both directions, and coded with a variable reference point. By specifying how the
reference should change with time, this was effectively modified to track a trajectory,
rather than steps. For generating the trajectory, the tic toc MATLAB function was
used to synchronize the code to real time. The trajectory was specified in the same
way as the simulation, with sine and cosine waves for the two axes. Variations of
these allowed easy specification of sinusoidal and elliptical trajectories as well. The
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proportional gain was chosen empirically; the results included in Figure 7.2 represent
the most successful gain found. With norm-error=2.3897, this controller fell short of
the norm-error=1 criteria, but was respectable for such a simplistic form of control.
To work out how time derivatives would be taken, a PID controller was implemented next. Numerical derivatives and integrations were used. This required that a
delay of one time step be induced to calculate the first time difference. The ZieglerNichols method was used to specify a starting point for the PID gains; they were
experimentally tuned from there. Figure 7.3 shows that increasing the complexity of
the controller to PID reduced the norm-error by 0.5 compared to the proportional
controller.
LQR can be thought of as a form of PD control, which is similar to both P and
PID control; however, LQR’s primary advantage is that it is a method for determining
the optimal gains, so tuning is ideally not necessary. Gains from the simulated LQR
controller were used here to maintain consistency. A progressive series of LQR controllers were written to manage various aspects of motion. The response for tracking
the standardized trajectory and the associated norm-errors are included for comparison with other controllers and the simulated results.
The first LQR controller was implemented similarly to the PID controller, with the
exception that it converted the input from torque (Nm) to angular rate (rad./sec.),
and then scaled it to a corresponding Qwerk input. This allowed mathematically calculated gains to be written directly into the code, and converted internally. Observing
the performance of this controller confirmed the need for controlling yaw. Therefore,
another controller was created that expanded LQR to include proportional yaw control. This was applied using control superposition (see Appendix A) in moderation
to gently hold the angle.
Figures 7.4 and 7.5 illustrate the importance of yaw control for this system. The
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Figure 7.2: Circle trajectory tracking using proportional controller with proportional
yaw control: norm-error=2.3897.
Figure 7.3: Circle trajectory tracking using PID controller with proportional yaw
control: norm-error=1.8976.
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Figure 7.4: Circle trajectory tracking without yaw control: norm-error=0.8614.
Figure 7.5: Circle trajectory tracking with proportional yaw control:
error=0.3866.
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track started at the bottom of the circle and progressed counterclockwise. Shortly
after starting, both experiments encountered an uneven section of floor which caused
slip in the y direction. In Figure 7.4, where no yaw control was applied, this slip
and recovery undesirably induced a slight yaw rotation. Because control was being
applied from the inertial frame, this rotation skewed the control input for the remainder of the trajectory track. In Figure 7.5, the same slip occurred, but the fixed
yaw control accounted for any undesired yaw that may have been induced, and the
track was quickly resumed. The yaw controlled response had a norm-error about 2.5
times smaller than the uncontrolled response. These results capture just one example; the uncontrolled response experienced slightly different yaw perturbations each
experiment, and its accuracy varied accordingly. If the accumulated disturbances
ever exceeded ± π2 , complete divergence occurred because control was applied along
the wrong axis.
To demonstrate some of the disturbances, Figure 7.6 shown the test trajectory
being traversed 4 times while holding yaw to zero with a proportional controller. By
completing the circle multiple times, it is clear where the track consistently departed
from the trajectory. These represent uneven ground that caused slip. The lower,
lower-left and right portions of the circle show examples of this. There are also two
pronounced departures near the top, these were caused by temporary communication
failure between Vicon and Simulink. While delayed, control was held constant, resulting in the signature tangential wander. The departure to the right happened on
an even part of the floor with surefooted actuation, so the recovery was accordingly
swift. The tangential departure to the left was on uneven ground that was prone to
slip, as seen by the other three tracks; this resulted in a slightly longer recovery time.
The only other common obstacle that is not seen here is a Vicon object tracking
failure. This happens when Vicon loses sight of one or more markers and does not
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Figure 7.6: 4 circle trajectory tracking laps with proportional yaw control.
have enough information to track Vertigo. Vicon then sets the object’s position to
the origin and a signature spike will be seen in the track plot that goes to the origin
and then back out to Vertigo’s location once it is observed again. This results in an
inaccurately high position error that cause the controller to send large inputs and
huge departures occur. The frequency of this problem was increased for balancing
controllers because changing roll and pitch angles were more prone to hiding markers.
One final modification was made to the ground-based LQR controller that helped
to prove the concept of omni-directionality with yaw. This was done by transforming the control signal from the inertial frame, into the body frame. This allowed
independent yaw and position trajectories to be specified. It also meant that yaw
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Figure 7.7: Circle trajectory tracking with transformed control: norm-error=0.5402.
control could be omitted, and inertial trajectories could be tracked accurately in the
body frame, despite yaw disturbances. Figure 7.7 shows the standardized trajectory
being traced with transformed input. With norm-error = 0.5402, it outperformed
the uncontrolled track; however, it could not quite match the response of controlled
yaw with untransformed input. This is because actively controlling yaw increased
the control when disturbances were encountered, which helped conquer them faster.
With uncontrolled yaw, no additional input was sent to aid the LQR, and the response suffered slightly. Experiments were run with transformed input and fixed yaw
control; however, they did not improve upon the yaw control significantly.
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(a) Circle trajectory tracking with propor- (b) Circle trajectory tracking with proportional yaw control for 2π retrograde: norm- tional yaw control for 6π retrograde: normerror=0.5486.
error=0.5805.
(c) Circle trajectory tracking with propor- (d) Circle trajectory tracking with proportional yaw control for 10π retrograde: norm- tional yaw control for 20π retrograde: normerror=0.7973.
error=1.3933.
Figure 7.8: Circle reference trajectory tracking while using proportional yaw control
for various retrogrades.
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To analyze how prescribing both trajectories affected performance, experiments
were run on the standardized test trajectory with an increasing number of retrograde
yaw rotations. Figure 7.8 demonstrates how the norm-error increased with the number
of retrograde rotations. As mentioned earlier, this is because control is held constant
between sampling times. As the body rotates, directional translation skews, and a
straight trajectory in the body frame becomes a curved path in the inertial frame.
The greater this curvature becomes, the further out the track will move radially
for retrograde rotation. If the yaw trajectory were in the other direction, the track
plot would drift inward as the number of rotations increased. The two conservative
trajectories did not yield much more error than the yaw fixed case, but in Figure 7.8(c)
and 7.8(d), where the rotations were much faster, there was appreciable error.
The plots of yaw measurement and trajectory for the smallest and largest rotations in Figure 7.9 show that the method devised for conditioning the measurements
works for any number of yaw rotations. These plots also include the errors that indicate how far off the trajectory the track is. Errors plateau at a small nonzero number
because there must always be some error for the proportional controller to produce
input signal. It is worth noting here that simply looking at how well the track fits
the trajectory is not sufficient enough information to make discerned conclusions. As
mentioned before, the time response is also important, so the norm-error values are an
accurate basis for comparison because they take time into account as well as position
errors.
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(a) Yaw control during 2π retrograde.
(b) Yaw control during 20π retrograde.
Figure 7.9: Yaw control for 2π retrograde and 20π retrograde.
7.4
Balancing Implementation
Working towards the goal of building a preliminary foundation to expedite future
work in balancing on the sphere, the next step was to conduct simplified balancing
experiments. As with ground-based control, the first balancing controller exploited
the safety and predictability of if-statements to establish the coordinate, error and
control signal convention. This if-control was designed to balance an 8 foot long 2x12
inch plank that pivoted at the center. By translating along the top of the plank,
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Vertigo was able to effectively control the pitch of the board. If the pitch error was
negative, Vertigo would move forward, and vice versa.
This program was a valuable step because it exposed the need to keep all forms
of motion in check, not just for performance, but safety. As the robot moved about,
small disturbances in y and yaw accumulated, eventually causing Vertigo to fall off
the board. This was successfully addressed by adding y and yaw components to
the if-control. The only thing preventing this controller from balancing the board
indefinitely was battery life, and Vicon connection lapses. These lapses caused Vertigo
to quickly run off the board before there was time to catch it. More protection was
clearly needed for balancing experimentation to safely continue; therefore, a 6 inch
lawnmower tire inner tube was stretched around the middle of Vertigo. Partially
inflated, this served well as a durable bumper for damping crash impacts.
Controlling board pitch became slightly more intelligent by converting to proportional control for x, y and yaw. The gains and magnitudes discovered in the
ground-based proportional control experiments accelerated the process of finding appropriate gains here. The proportional controller easily balanced the board; however,
the response was choppy, and certain gain values induced oscillatory overshooting that
resonated, causing the board to sway. This was not an issue for if-control because its
response was not consistent enough to resonate. Figure 7.10 shows the board starting
with one end on the ground and plots the response as Vertigo brings it to balance.
At 3 seconds, the board tipped to the point where it almost made contact with the
ground. It recovered, but oscillation persisted.
To attenuate oscillation, PID control was called upon. Inclusion of the derivative
component was especially beneficial, as it helped to damp the response. Figure 7.11
shows the significant improvement over proportional control. Around 5 seconds into
the experiment, Vertigo raised the board to the balancing point with very little over170
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Figure 7.10: Balancing plank pitch with proportional control.
shoot. This plot also demonstrates how much noise is added by the Vicon measurements. For this experiment, the plank was fitted with markers, and its angle used
for feedback to the controller. Therefore, any variation in the response in the first
4 seconds can be attributed to Vicon noise because the plank was resting on the
ground at a fixed angle. Though this noise is undesirable, tracking the board was a
significant improvement over the response when Vertigo was the Vicon object. Figure 7.12 shows how noisy the response was when Vertigo’s angles were tracked by
the same PID controller. This additional noise is partially due to uneven terrain, but
the dominant contributor is actuation. Large gaps in the omni-wheels gave Vertigo a
bumpy ride, so the measured angle at a given time did not represent net pitch well.
This noise was so dramatic that performance was affected by it.
Tracking the board was just a temporary solution to circumvent the true prob-
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lem: Vertigo needed better omni-wheels. This raised concerns about using Vicon as
feedback for balancing on the sphere because tracking a different object would not
be an option, and the improved wheel design would be extremely expensive and time
consuming to manufacture.
Balance in 2-dimensions began by applying the if-controller for the board to both
axes. The primary objective of this was to confirm the convention of the Vicon roll
angle. This simple controller was able to balance a flat plate resting on a sphere.
This was analogous to balancing the plank in 2-dimensions. The biggest challenge in
this was recovery from large disturbances, which is best described with an example.
If the pitch angle deviated to the point where the front edge of the board was on
the ground, roll angle was held constant despite the efforts of y control. Therefore,
Vertigo moved in the y direction to try to account for the small angle of the uneven
ground. By the time x control brought pitch back to balance, y had veered far enough
to deflect the roll angle to rest on the adjacent edge. This was not a problem when
starting from a balanced point and the objective of establishing convention was met;
therefore, further development was not necessary.
At this point, an informative collection of controllers had been created to prepare
for balancing on the sphere. Although it surpassed the objectives of this project, early
stages of sphere-based implementation were attempted. Building off the findings of
previous controllers and experiments made it easier to understand the origins and
severity of problems encountered in this. Proportional control was first attempted
with the sphere. With Vertigo’s angles as Vicon feedback, this controller applied
input in the correct direction for balance, but was insufficient. Gains that were too
low did not apply enough control to recover from any appreciable disturbance, but
gains that were large enough to bring Vertigo back to the unstable equilibrium would
overshoot to an unrecoverable angle. However, this controller was beneficial because
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Figure 7.11: Balancing plank pitch with PID control.
Figure 7.12: Balancing plank pitch with PID control by tracking Vertigo.
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it started to demonstrate the specific challenges of balancing on a sphere.
PID control was then implemented in a more sophisticated attempt at balance.
This gave better results; however, it could not sustain balance, only keeping Vertigo
on the sphere for about two seconds. During this time, oscillations would diverge to
an unrecoverable angle. Tweaking the gains to reduce overshoot limited the recovery
angle such that it was not able to cope with the small deviations from noisy measurements and external disturbances. Since the damping component of the PID controller
was most beneficial for balance, and tuning was difficult, LQR was attempted next
in hopes that its optimal gains would solve the problem.
For LQR control, many techniques were tried for implementation and defining the
Vicon object, but none were successful. LQR was appealing because it included the
damping aspect of the PID control, and it gave optimal gains for the mathematical
model; however, it is possible that model errors were sufficient enough to invalidate
the results for the physical system. Also, the framework of implementing LQR meant
that measurements had to be related to the mathematical states of the system. This
used a geometric measurement model that relied on how accurately the Vicon object
was defined to represent the physical system. Errors from this were apparent because
the performance of the controller quickly worsened as balance was attempted further
from the origin where the uncertainties were magnified.
Based on the success of the estimation simulations, the continuous-discrete extended Kalman filter seemed well suited to accompany LQR control and solve the
problems with noise, modeling error and measurement error. However, three main
issues prevented this from being successful. The first was with initialization. Manually placing Vertigo close enough to the filters initial conditions was difficult, and
often caused immediate failure. Also modifying the filter to account for larger initialization errors diminished its effectiveness. The second problem was that Vicon
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did not always define the object well enough. This had similar repercussions to the
initialization error. The third and most influential problem was the inconsistent loop
sampling time as signals navigated the tortuous communication architecture. EKF requires knowledge of the sampling time for tuning, and to propagate a priori estimates
for the next time step. When sampling time varies, propagations are erroneous, and
the filter tuning is compromised. Despite the shortcomings, these attempts were not
in vain. EKF is extremely powerful, and it is likely that continued work will enable
it to provide state estimates reliable enough for Vertigo to balance on the sphere.
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Chapter 8
Conclusions and Future Work
This thesis introduced an autonomous biaxial robotic inverted pendulum as a new
experimental platform for research in controls and dynamics. Designed to balance
and navigate on a sphere, it is theoretically capable of exceptionally agile motion.
This, along with its reconfigurability, enables it to model a wide range of other, more
complex systems. Its unique abilities also permit original work in rover locomotion,
and as an educational tool. Covered in this thesis were the design, mathematical
modeling and analysis, communication architecture, control, estimation and early
implementation.
As an original platform, proper hardware design was a major effort. The original
objective was to create a simplistic and robust testbed with minimal restrictions on
its motion. The first prototype, Vertigo 1.0, balanced on a sphere, and had omnidirectional motion. Actuation methods were the cornerstone for enabling its dynamics. A second design iteration, Vertigo 2.1, was then developed with new actuation
that offered the addition of yaw control to the existing omni-directionality and biaxial
inverted pendulum dynamics. Both of these generations required only a handful of
moving parts, and were successful in protecting onboard components from impact.
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However, working with them gave rise to ideas for improvement. Results from control
implementation showed that a need for new omni-wheels was the most urgent of these
improvements. The gaps between the rollers caused chattery motion, which affected
measurement quality. A new design for near seamless omni-wheels was generated, but
cost and time of manufacturing made them impractical to this point. In the future,
it would behoove the project to incorporate these new wheels.
Also proposed is a design iteration that fits Vertigo with only three legs. This
would make the Vertigo 2.1 drive equal-actuated while preserving all modes of motion,
and making more consistent contact with the control surface. The only drawback is
a sacrifice of independent actuation; however, motion would remain decoupled, so
control could be decomposed and applied in representative components relatively
easily.
The mathematical modeling of Vertigo proved valuable for its development. Its
accuracy was confirmed through the response of simulations and agreement of analysis
results with theory. Analysis of the equations focused on observability and controllability. These results were useful in confirming the validity of control and estimation
methods that would be applied. By looking into the inverse condition number of the
controllability and observability matrices, insight was gained as to how states influenced their condition. This also guided the hardware assembly configuration to help
achieve desired performance. Future work on mathematical modeling should focus
on further improving accuracy in matching the physical system, as this is a suspect
for implementation difficulty. Deriving the equations with various methods and assumptions should give an idea of how to better represent the dynamics. Also, system
identification techniques could be employed to extract parameters from the physical
response. These efforts would almost certainly enhance implementation success and
the validity of simulations.
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Establishing the communication architecture was a laborious process, requiring the
use of five different coding languages. Eventually, an external sensing loop was closed
where Vicon measurements were sent to Simulink, and then passed on to MATLAB to
be conditioned for feedback in to a control loop. The output of this loop was sent to
Qwerk through a Java based wireless connection. Qwerk received this control signal as
input to its own PID control loop that tracked a velocity trajectory. Output from this
embedded controller was sent to the motors, and their back-EMF measurements were
fed back to the PID loop as they carried out the control. In the future, this network
will be greatly simplified because a new version of Qwerk’s firmware that supports
the onboard sensing architecture will be released soon. Vertigo will no longer be
dependent on Vicon measurements; a laptop and wireless router will be the only
equipment needed. This means easy transportation and set up almost anywhere, and
Vertigo can be made fully autonomous. The new firmware also promises to support
passing of the embedded PID gains and trajectory parameters. This will enable
further tuning for improved response, and back-EMF measurements can be added
with low weighting to MATLAB feedback loops.
Development of control methods spanned a wide range of motion. Starting with
the ground-based configuration, a standardized test trajectory was developed to compare the results of simulated and implemented experiments. The majority of the
methods simulated in this thesis were optimal control techniques. These gave ideal
gains for respective specified cost functions, and removed some of the guesswork from
tuning. Several sphere-based controllers were developed for balance and translational
motion; simulation of these yielded exceptional results that confirmed Vertigo’s conceptual principles. The most sophisticated and successful was an augmented LQR
tracking controller that balanced and translated Vertigo according to a cost function
that effectively minimized weightings of states, power and time.
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Many other optimal control techniques were attempted for sphere-based control,
but acceptable results were not attained. These included an analytical power-time
optimal method, a power optimal shooting method for translation, and a gradient
based method for translation. These techniques are highly sensitive to tuning and
initial guesses, but are very close to converging properly; therefore, continued work
on them will likely be fruitful. Success is especially hopeful for the gradient based and
shooting methods, as they are complementary methods, and are both near completion.
If either one of these methods is worked out, its results can be used as initial guesses
for the other.
For estimation, the problem was put into the Kalman filter framework, and simulated. These results showed its success at providing estimates for unmeasured states
while attenuating noisy signals. It was then shown that increasing the number of loop
iterations between measurements updates improves the estimate further. To fully explore the impact of sampling times, the simulation was run at twenty five different
combinations of measurement and output sampling times. The magnitudes of the settled 3σ bounds were plotted as a mesh surface which showed more measurements and
loop iterations improved accuracy, but the affect was much more pronounced for measurement sampling. The EKF is not a true optimal filter because it applies a linear
method to a system that is by definition nonlinear; however, it can still be used with
exceptional results as demonstrated here. More work can be done to use this filter
to estimate unknown parameters of the system. This is an extremely advantageous
feature of the EKF because it adds functionality as a real time system identification
technique to update the dynamic model and improve performance. For Vertigo, the
unknown friction coefficients are likely to be the most beneficial parameter to estimate because they cannot be measured, and pertain to slip. Work on this has already
begun, but revealed that defining new states that included an unknown parameter
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Conclusions and Future Work
make the system unobservable, and the process is not applicable. Continued research
in this area is likely to uncover a method for circumventing this.
In accordance with the objectives of this research, the majority of future work
will be in implementation. A large bank of ground-based and balancing controllers
were successfully implemented to serve as a spring board for continued development
of sphere-based implementation. With the current communication network, further
work on the implemented EKF and LQR combination is most promising for balancing on the sphere. However, progress should flourish with the addition of improved
omni-wheels, and the updated Qwerk firmware when it is released. The onboard
sensing loop architecture should solve many of the timing and connection problems
currently faced. Also, as the mathematical model is improved, implementing theoretical results will be more straightforward. With the accumulated success to this point
and expected results of future work, the Vertigo platform promises a rich and prolific
future.
180
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Symp. on Robotics Research, 2005.
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SYSTEMS. CRC Press, 2004.
183
Appendix A
Supplementary Concepts
A.1
Introduction
The objective of this appendix is to provide cogent explanations for some of this
platform’s non-intuitive aspects. Many of the items addressed can be attributed to
Vertigo’s nonlinearity and unique design. Mathematical backing is provided accordingly throughout this thesis; however, this appendix focuses on clearly conveying the
ideas in plain English. The topics of directionality, translational motion, actuation,
control superposition, pivot drift, sphere properties and weight distribution are commonly misinterpreted by those first being introduced to this system. They are also
critical to the proper understanding of how it works, and therefore warrant discussion.
A.2
Directionality
The directionality of a system refers to its mobility and is often mislabeled in robotics
when describing motion. Categories for directionality are loosely defined, and many
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Directionality
vehicles do not fit nicely into a particular profile. This section provides simple definitions and explanations for common types of directionality, and establishes the
convention to be used throughout this thesis. Figure A.1 illustrates the progressively
expanding mobility that climaxes at omni-directionality with yaw, which describes
the motion of Vertigo.
Figure A.1: Directionality (a) Bi-directionality (b) Bi-directionality with yaw (c)
Omni-directionality (d) Omni-directionality with yaw.
Bi-directionality (Figure A.1(a)) simply means that an object is capable of moving
forward or backward, along a single axis. A cart on a track is an example of this,
where motion is constrained to one degree of freedom. Bi-directionality with yaw
(Figure A.1(b)) is a more common form of motion. In this case, the vehicle has
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Translational Motion
the additional ability to rotate (yaw). Differential drive vehicles, such as tanks and
wheelchairs, are examples of this. They often require only two driven wheels to jointly
control their two degrees of freedom, and have additional passive wheels or casters to
maintain their static stability. Nearly all mobile robots fall into these two categories.
Omni-directionality (Figure A.1(c)) is among the most commonly misused terms
because it is considered a “holy grail” for robotic motion. It refers to a vehicles ability
to move in any direction on a planar surface. This means that the vehicle does not
need to be facing in a particular direction to carry out a maneuver, but also does not
have the ability to intentionally control the direction it is facing. Controls in the x
and y directions are usually decoupled for these two degree of freedom vehicles. This
type of motion improves maneuverability and response time because the vehicle does
not have to spend time “steering” to avoid obstacles.
Omni-directionality with yaw (Figure A.1(d)) is the most ambitious form of planar motion, having all three possible degrees of freedom. It means that the vehicle is
able to move in any translational direction on a surface, and control its yaw angle independently. The primary advantage of this over omni-directionality is that yaw can
intentionally be controlled at all times, so sensing equipment can always be facing its
subject. Both ground-based and sphere-based Vertigo have omni-directionality with
yaw.
A.3
Translational Motion
The method by which Vertigo is able to navigate from point to point while balancing
is often misunderstood. The key to this is controlling the point of contact with the
ground with respect to the center of mass. In this way, it is very similar to the way
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Translational Motion
a human moves, and when explained step by step is rather intuitive. Figure A.2
illustrates the following description as Vertigo translates from point A to point B and
time passes.
Figure A.2: Stages of translational motion.
Translational motion of Vertigo, or any inverted pendulum, can be broken down
into seven distinct stages. The first stage is simply the initial condition, where Vertigo
is vertical and stationary at point A. Now, to move forward, the robot must match
its acceleration with the angle of lean. Accordingly, lean must first be initiated in the
second stage by shifting the point of contact with the ground backward. This stage
is frequently confused despite humans subconsciously doing the same thing before
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Actuation
walking by rolling weight back onto the heel. The third stage is acceleration. This
stage is essentially a recovery from the initial lean in the previous stage, and the
greater the angle of lean the greater the acceleration. This is why sprinters start a
race on all fours; as soon as their hands are lifted they are leaning at an extremely
large angle and can impart greater force to accelerate than someone starting upright.
Stage four is cruise. Here, the robot is moving at a constant velocity and is perfectly vertical (assuming no friction or drag). Everything is mirrored about stage four,
so the deceleration in stage five is carried out similarly to acceleration, except the lean
is in the opposite direction and the brakes are applied. Stage six acts to terminate the
lean. This can be carried out in two slightly different ways; the first is by applying
the brakes perfectly so that Vertigo pivots back to vertical with a velocity of zero
and exactly at the desired destination. The second is the opposite to lean initiation
in stage two, the final destination is overshot slightly as the motion is slowed, then
the recovery acts to resume vertical and find the final destination. The second way
is more time efficient because the first theoretically takes an infinite amount of time;
however, both are actually different manifestations of the same process. Stage seven
is the final state of the robot, where point B is reached, and all translation has ceased.
A.4
Actuation
The method used here for actuating the angles of an inverted pendulum is unique to
Vertigo. When balancing an inverted pendulum, the goal is to control the base, in
this case the sphere. What sets Vertigo apart is its actuator orientation and use of
omni-directional wheels that allow control to be applied to the same surface in various
directions without interference. Four spherically orthogonal motors allow almost any
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Actuation
size sphere (restricted only to a minimum size) to be used as a base for Vertigo. Taking
this to the extreme results in a sphere of infinite radius which is a flat surface; this
is why the ground-based configuration of Vertigo is possible. The omni-directionality
comes from the two inch diameter wheels that are inlaid with rollers around their
circumference. This configuration allows control in the direction of rotation of the
motors, and slip perpendicular to this which decouples the directional control.
Figure A.3: Spherically orthogonal omni-directional actuation.
Here, directional control is referring to the motion of the sphere’s surface with
respect to the robot’s body. “Controlling direction” is somewhat specious wording;
really, the motors are applying a torque which acts to rotate the ball. This rotation
causes Vertigo to lean (as mentioned in stage two in the previous section), and is how
the angles are controlled. By calling upon opposing pairs of motors, roll and pitch
are decoupled and independent. To control yaw, all four motors rotate in the same
direction. If the friction between the ground and the ball is sufficient, the robot will
rotate while the sphere remains stationary.
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A.5
Control Superposition
Control Superposition
Previously, it was shown that Vertigo’s directional control is completely decoupled
through actuation of opposing pairs of motors, but its yaw control requires the cooperation of all four motors. For the continuous model, the solution to this can be handled
by superposition. The same principles apply for both the sphere and ground-based
configurations of Vertigo, so only ground-based control is considered here because it
is easier to visualize.
Figure A.4: Ground-based trace of translation with rotation.
Figure A.4 shows the trace as ground-based Vertigo travels along the straight
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green trajectory and executes a
Control Superposition
π
2
yaw rotation with constant angular velocity. The
red X represents a simplified top view of Vertigo at its initial position. As its center
of mass progresses along the straight green line, and yaws at a constant rate, periodic
snapshots are taken and traces of the four corners of the X (the motor locations) are
drawn. These traces represent the paths that each individual actuator takes. Since
the actuators only have to control motion perpendicular to their axle, trajectory projections can be drawn onto the plane of the wheel for each respective actuator trace.
In this way, each actuator need only control the projected motion. For continuous
control, this is equivalent to superposing the control. More specifically, the control for
translation (transformed into the body frame) and yaw rotation can be determined
separately, then added together to compose a complete control profile for the maneuver. To transform the translation control into the body frame, information about
the yaw angle is needed. In practice, this is a problem because continuous knowledge
of yaw and continuously transforming control into the body frame are both impossible. The reason behind such difficulties is discretization. Yaw information comes
from sensors that have a discrete sampling time and transformation takes place in
coded loops which do not provide continuous output. This naturally brings up the
discussion of the discrete-time controlled system.
In the discrete-time control system, yaw is measured, control is calculated and
transformed, and voltage is applied to the motors, at every time step. Between steps,
the voltage to the motors is held constant. To demonstrate the problem, consider
ground-based Vertigo under discrete-time control with a rather large sampling time of
one second. Now suppose that translational control for forward motion is determined,
and yaw control was separately determined to follow a rotational trajectory. Now,
when those two inputs are superimposed and applied as voltages to the motors, that
signal will remain constant throughout the time step. Therefore, at the start of
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Pivot Drift
the step, the state for which control was calculated, the input will be appropriate;
however, at the end of the time step, one second later, the straight line control
will have curved as a function of the yaw rotation. For the second time step, the
translation control will try to bring the robot back to the trajectory, but again yaw
rotation will cause divergence and prevent the desired control from being carried out.
The end result for this scenario would be a track which resembles a wavy bias error.
The problem of translational control divergence is a function of the yaw rotational
velocity, and the sampling time of the system. In practice, fast sampling times and
conservative yaw trajectories help to mitigate these errors. For most practical applications, a yaw trajectory can be determined that allows superposition to be taken
advantage of with negligible error. If more evasive maneuvers are required, empirical knowledge of the predicted error can be applied to help compensate. This error
can also be determined mathematically; however, constant sampling time must be
assumed and this is not the case with Vertigo’s current control loop.
A.6
Pivot Drift
A famous benchmark problem in control theory is the inverted pendulum on a cart,
where the angle and position of the pendulum are controlled with an input force
applied to the cart, as seen in Figure A.5. In many ways, the dynamics of this system
are conceptually similar to those of Vertigo; both are inverted pendulums, and both
apply an input force (or torque) that controls the base. However, the feature that
differentiates these two problems is the sphere that Vertigo balances on. Balancing
on the sphere has two dominant affects on the dynamics that prevent the control
methodology from being used interchangeably between these two problems; the first
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Pivot Drift
Figure A.5: Inverted pendulum on cart.
is pivot drift, and the second is the rotational inertia of the sphere.
In the pendulum on a cart problem, the pivot point of the pendulum is always
known to be at the hinge on the cart. In Vertigo’s case, this point is not so easily
determined. Its center of rotation is a function of the body mass, sphere mass, inertia, rotational inertia, velocity and angular rates. When control is being applied, this
point is shifting wildly and is extremely difficult to determine. To help illustrate how
complex the pivot drift is, Figure A.6 shows Vertigo’s response to the most simplistic
scenario: an uncontrolled fall. The figure shows snapshots of Vertigo’s central axis at
ten degree intervals as it falls uncontrolled. Here it is assumed that the robot does
not ever fall off the sphere and that only the sphere is constrained by the ground.
The pivot path can be traced out by the inner arch that is formed by the tangential
lines. The reason for this path is that Vertigo’s center line must pass through the
center of the sphere, while at the same time, the center of the sphere is moving as
Vertigo falls over and it rolls along the ground. The translation and rotation generate a path that is not characteristic of any standard polynomial or trigonometric
function. This thought experiment is for the most straightforward motion possible,
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Pivot Drift
Figure A.6: Uncontrolled pivot drift.
as soon as control is applied the sphere and body of the robot do not rotate as one,
and the complexity is magnified.
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A.7
Sphere Properties
Sphere Properties
The properties of the sphere have been established as a critical component of this
system, this is the second dominant departure from the classic pendulum on a cart
problem. Exploring exactly how it influences the dynamics is consequently important.
Beyond features like material and texture, which were discussed previously in the
Design chapter, this section looks at how the physical properties of the sphere, such
as mass and diameter, affect the system.
If, for a moment, the mass and inertia of the sphere are somehow assumed constant, and only the diameter is considered, it is found that balancing on a larger
sphere is easier than balancing on a smaller one. This phenomenon is often falsely
attributed to a gear reduction in the control, where the actuator is the first gear, the
sphere is a middle gear, and the ground is the final gear, or pinion. This is not true
however. As with any planar multi gear linkage, the only gears that contribute to the
overall ratio are the first and last, or in this case the wheel and ground. The sphere
size has no affect on this because as the wheel rotates the sphere, that arc length is
conveyed exactly to the next gear (ground), resulting in identical translation across
the ground for any size sphere. The only way to change Vertigo’s gearing past the
motor’s gear head is to change the diameter of the wheels. That being said, a larger
sphere is still easier to balance on. The reason for this is apparent when considering
the uncontrolled dynamics. As Vertigo tips over, it is also rotating the sphere. A
sphere with a larger diameter will travel along the ground farther per degree of rotation. Since this translation is in the same direction as the control needed to balance,
a larger diameter sphere is naturally easier to stabilize. In addition, a larger sphere
will raise the center of gravity which is beneficial as will be discussed later in this
appendix.
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Weight Distribution
Inertia is a property of matter which states that more massive objects have greater
resistance to change in motion. Rotational inertia (moment of inertia) is a similar
principle that deals with resistance to change in rotation rate, and is a function
of mass and geometry. To investigate the affects of sphere mass, the size can be
assumed fixed, so both forms of inertia will increase with mass. If we consider an
extremely massive sphere, it will be extremely resistant to change in motion, which
means Vertigo will be able to balance much more easily. It would be analogous to
a fixed sphere, where Vertigo would simply have to climb to the top, and then no
more control would be necessary. The fatal drawback is that greater inertia requires
more control input to accelerate. That is, when navigation is desired, the system has
to work very hard to move the sphere. Now, a near massless sphere would require
very responsive control to balance, because there would be little inertial resistance
to change. However, navigation would use comparatively less control. Besides pivot
drift, a massless sphere would cause Vertigo to behave very similarly to the pendulum
on a cart because sphere inertias would have no effect.
It is worth noting here that Vertigo is designed to be a controls experiment, so it
is important to select a sphere with properties that balance the challenges of control
with practical feasibility.
A.8
Weight Distribution
Since inverted pendulums, by definition, have centers of gravity above their pivot
point, and are known to be unstable, it is a common notion that lowering the center
of gravity will make it easier to control. To investigate this idea, assume the cart
in Figure A.5 is fixed, making it a simple uncontrolled inverted pendulum. The
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Weight Distribution
equation of motion for the angle θ would then be θ̈ =
g
sin(θ),
l
where the length
of the pendulum is inversely proportional to the acceleration. Thus, an inverted
pendulum with a higher center of gravity will fall more slowly, giving the controller
more time to respond and making it easier to control.
These simple equations provide cogent evidence, but there is a more intuitive way
of understanding this concept. To control the inverted pendulum, it must be rotated
by a torque, T = F l. If the control input is a force F at the base, then the length l
is the lever arm for the torque. When the force is applied, both linear and rotational
motion will ensue at the center of gravity. A longer lever arm means less control
force is required to impart the same torque. Additionally, smaller F will result in
smaller linear motion, so the pendulum will sway less when balancing. This is easily
seen when balancing a baseball bat, or broom: it is much less challenging when the
heavier end is at the top. Similarly, the rotational inertia has an effect on the system.
If the mass is more concentrated, the rotational inertia is reduced, and less input is
required to rotate the pendulum.
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Appendix B
Powering Vertigo and Qwerk
Through the evolution of Vertigo, and as its demands changed, several powering
methods were used. Currently, only two remain in use.
• LiPo battery packs
• Tethered power supply
LiPo Battery Pack:
Zippy-R 4150 mAh, 11.1 V, 3s1p (X2)
This method is best for mobile powering. There are two identical battery packs,
allowing a charged pack to be on hand at all times. Both packs bundle two 4150 mAh,
11.1 V bricks that are charged separately (see charging instructions below). They are
connected in parallel so they deliver their original voltage for twice as long. The
batteries were fitted with Deans connectors that enable quick, secure and high performance connection to the Y harness that wires the two bricks in parallel. To protect
Qwerk, and insure that the minimum limits of the battery packs are not breached,
a 9.0V LiPo battery voltage cut-off circuit was fitted between the power supply,
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Powering Vertigo and Qwerk
and the power input for Qwerk. (http://www.rctoys.com/rc-toys-and-parts/DF-DIVBATTCUT/RC-PARTS-WIRELESS-VIDEO-BATTERY-ACCESS.html)
Battery Charging: For single 4150 mAh, 11.1 V, LiPo battery pack.
1. Plug single (must charge both of the packs individually) battery into TRITON
2 charger through the Equinox interface.
2. On the charger, click Battery Type to select LiPo charge, and then choose 4150
mAh, 11.1 V.
3. Click the Equinox interface box to switch to interface mode.
4. Press dial on charger and hold to initiate the charging sequence.
Charging takes roughly an hour and a half per pack. Always charge both bricks
before use. It is a good idea to charge the pack immediately after they run out to
ensure that it will be ready when needed.
NOTE: The charger has a safety timer that may expire before full charge is reached.
It will say why it stopped on the display. If it timed out, simply proceed as though
you were charging it again, and it will finish.
Tethered Power Supply:
Input: AC 100-230 V, 0.7-0.4 A, 50/60 HZ
Output: DC 12V, 1.6 A
This method is best for stationary powering (programming, diagnostics, etc.).
Programming of any type should really be done on this power supply to minimize
the risk of undesired loss of power which may potentially damage Qwerk’s hardware
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Powering Vertigo and Qwerk
or software. This supply was lifted off of an old computer. The wiring was cleaned
up, removing all unnecessary leads, and it was fitted with a 12 foot power tether that
connects directly to Qwerk’s power input. Also added was a master power switch and
green LED that indicates the unit is on.
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Appendix C
Connection to Vertigo
Method 1: Through TERK program launched from site
1. Go to www.terk.ri.cmu.edu/software/index.php
(a) Choose program and launch it
2. Close internet
3. Plug router in
4. Power Vertigo (Qwerk)
(a) Qwerk will automatically connect
i. LED 6 will blink when connection is established
ii. It is normal for LEDs 0, 1, 2 and/or 3 to be blinking or solid
A. I think LED 3 blinking means it is connected to another comp.
5. Connect computer to LAIRS router
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Connection to Vertigo
(a) Wirelessly or cable in
i. Password: “lairs”
(b) Can check the router status by typing 192.168.1.1 into web browser (username: leave this blank, password: ”admin”)
i. StatusLocal NetworksDHCP Clients Table
ii. Should see ip addresses of both the computer and Qwerk
6. In the launched software, click the [Connect] button
(a) Connect directly to a peer
(b) Type in ip address of Vertigo and connect
Vertigo specific instructions are presented first, then the Terk given instructions
for installation of programs and running code.
1. Open JCreator LE version 4.50.010
2. File/open workspace
(a) VERTIGO/QWERK/MyFirstVertigo/terk-client-MyFirstRobot/
(b) Choose workspace
3. Choose the program you would like to run
4. Compile the project by either hitting F7 or going to Build->’Compile Project’
in the top menu
5. Run the project by going to Build->’Execute File’ in the top menu. Note that
you must execute the file, NOT the project.
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Connection to Vertigo
6. Running the program will open a graphical interface. To connect to a robot,
click on the [Connect] button.
7. From here, follow steps 5 and 6 from the previous section (Through TERK
Program Launched From Site).
THESE ARE THE PROVIDED INSTRUCTIONS FOR INSTALLING
AND CONNECTING THROUGH JAVA
ROBOT CLIENT QUICKSTART
———————-
Purpose
——This document describes how to install, compile, and run the MyFirstRobot program.
It also briefly describes important other files in this folder.
Installation
————
* Create a directory for the MyFirstRobot program and its associated files.
* Unzip the terk-client-MyFirstRobot.zip file into the directory you just created.
* You will need to have Sun’s Java SE JDK 5.0 installed. If it’s not already
installed, you can download it at:
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Connection to Vertigo
http://java.sun.com/javase/downloads/index.jsp
Follow their instructions for installing it on your machine.
Compiling and Running the MyFirstRobot Program in JCreator
(RECOMMENDED METHOD)
——————————————————————————JCreator is a freeware Java IDE. A project file, MyFirstRobot.jcp, is available
in the MyFirstRobot folder with all of the correct configuration settings to
quickly begin compiling and running the MyFirstRobot program.
* Install JCreator LE from http://www.jcreator.com/download.htm
* Open the file MyFirstRobot.jcp. The JCreator IDE should start.
* In the File View of the IDE, open MyFirstRobot.java for editing by double
clicking on it.
* Compile the project by either hitting F7 or going to Build->’Compile Project’
in the top menu.
* Run the project by going to Build->’Execute File’ in the top menu. Note that
you must execute the file, NOT the project.
* Running the program will open a graphical interface. To connect to a robot,
click on the ”Connect” button. A connection dialog will pop up and you will have to
select whether you are connecting to the robot via the CMU-based relay (relay mode)
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Connection to Vertigo
or if you will connect over a local network by entering the robot’s IP address (direct
connect). For more information about these two modes of operating your qwerk, visit
http://www.terk.ri.cmu.edu/projects/qwerk-overview.php. Make your selection and
click ’Next’:
- If you entered direct connect mode, you will now be prompted to enter the IP
address of the robot. Do so and click ’Connect’, then click ’Finish’. You should now
be connected to the robot.
- If you entered relay mode, you will be prompted to enter your TeRK login and
password. Enter your login, click the ’Login’ button, and then click ’Next’ to go
to the next page in the connection wizard. You will see a list of available robots.
Highlight the appropriate robot by clicking on it once. Then click the ’Connect’ and
the ’Finish’ buttons. You should now be connected to the robot.
* Once you are connected to a robot, you can begin displaying an image stream
from the robot’s onboard camera by clicking on ”Start Video”. Once started, you
can pause video at any time by hitting ”Pause Video”, and capture images from the
video stream by clicking ”Save Picture”. To run your program, hit the ’Play’ button.
To stop your program, hit ’Stop’.
* Several other example files are available and visible in the file view. They
are Headturner.java, Photovore.java, hearingTest.java, and soundTest.java. You can
compile and run these as you did for MyFirstRobot.java.
Compiling and Running the MyFirstRobot Program from Command Line
—————————————————————-
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* Open a command prompt and set the current directory to the directory you
created during installation.
* Type ”compile” (no quotes) at the command prompt to compile the program. It
will take a few seconds to compile, and will print out any syntax errors that may occur.
* If there are no compilation errors, type ”run” (no quotes) to run the program.
* Running the program will open a graphical interface. To connect to a robot,
click on the ”Connect” button. A connection dialog will pop up and you will have to
select whether you are connecting to the robot via the CMU-based relay (relay mode)
or if you will connect over a local network by entering the robot’s IP address (direct
connect). For more information about these two modes of operating your qwerk, visit
http://www.terk.ri.cmu.edu/projects/qwerk-overview.php. Make your selection and
click ’Next’:
- If you entered direct connect mode, you will now be prompted to enter the IP
address of the robot.
Do so and click ’Connect’, then click ’Finish’. You should now be connected to
the robot.
- If you entered relay mode, you will be prompted to enter your TeRK login and
password.
Enter your login, click the ’Login’ button, and then click ’Next’ to go to the next
page in the
connection wizard. You will see a list of available robots. Highlight the appropriate
robot
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by clicking on it once. Then click the ’Connect’ and the ’Finish’ buttons. You
should now
be connected to the robot.
* Once you are connected to a robot, you can begin displaying an image stream
from the robot’s onboard camera by clicking on ”Start Video”. Once started, you
can pause video at any time by hitting ”Pause Video”, and capture images from the
video stream by clicking ”Save Picture”. To run your program, hit the ’Play’ button.
To stop your program, hit ’Stop’.
Important Locations in this Folder
——————————
javadocs Folder: Documentation for every method in the classes included in this
project.
API Reference.pdf: This file contains descriptions and sample usages of all methods used for controlling the Qwerk, accessing robot sensor data, and using the GUI.
Read this document or the javadocs before writing any programs!!
MyFirstRobot.java: Skeleton starter file.
MyFirstCreate.java: Skeleton started file for controlling the iRobot Create.
Examples: Example programs which demonstrate some of the available robot
control methods.
All examples are commented and contain a header with a description of the program’s operation.
MyFirstRobot.jcp: Jcreator project file. Double click to open JCreator and the
MyFirstRobot project.
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MyFirstRobot.jcw and .jcu: Other Jcreator files.
compile.bat: Batch file for compiling MyFirstRobot.java from the command line
run.bat: Batch file for running MyFirstRobot.java from the command line
clean.bat: Batch file for removing any files generated by using compile or run.bat
RobotClient: Folder containing classes for accessing and controlling the robot and
GUI
- RobotClient.java: Contains all methods for controlling and accessing the robot,
as well as the GUI
- RobotClientGUI.java: Specifies the properties of the GUI.
- SimpleRobotClient.java: Wrapper class for RobotClient.java - can be used interchangeably with RobotClient.java
- CreateClient.java: Class containing methods for controlling the iRobot Create or
iRobot Roomba. See the Create recipe at www.terk.ri.cmu.edu for more information.
RSSReaders: Folder containing classes for reading RSS feeds from the internet:
- RSSReader.java: Generic class for reading feeds
- WeatherReader.java: Class for reading weather data for any US city, courtesy
of www.wunderground.com feeds
TTS: Folder containing class for generatic speech from text
- TTS.java: Generic class for generating text to speech
==============================
Method 3: Through MATLAB
Instantiating Objects from Java Classes in Matlab
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NOTE: Steps 2 and on may not be needed if you are using a dynamic Java path
(see **).
Versions used: Windows Vista Business
Java - jdk1.6.0 06
Matlab 7.4.0 (R2007a)
1. Matlab comes with a version of Java that it uses instead of the one installed on
your computer. You can check the version with the version –java command
in command window. You should update Matlab’s java version to the one you
have been compiling your classes with.
(a) This can be done by right clicking on ‘My Computer ’ and going into ‘Advanced system settings’.
(b) Next, click on the ‘Advanced’ tab then click on the ‘Environmental Variables’ button
(c) Under the ‘System Variables’ section, add a new variable MATLAB JAVA
with the following value ‘C:\Program Files\Java\jdk1.6.0 06\jre’ (this was
it in my case) or a value that corresponds to the version of Java that you
are running.
(d) Enter version –java to ensure that the version is updated (restarting
Matlab may be required)
2. Enter edit classpath.txt into the command window and add the directory
where your class files are located in a new line.
C:\Users\Jon\Desktop\VERTIGO\QWERK\MyFirstVertigo
\terk-client-MyFirstRobot\
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1. Try constructing a java object:
Example of constructing a java object in Matlab using two methods:
Method 1:
> > j=javaObject(’AddNum’,3.2,4.2)
> > j.writeValue()
7.4 Method 2:
> > q=AddNum(3.4,1.3)
> > q.writeValue()
4.7 This was first tested this with a .class file generated from the javac compiler
from the command prompt. Files generated from an IDE such as JCreator or Eclipse
were then successfully tested as well. This method has not yet been tested with
stuffing the class into a package.
Sample class:
public class AddNum {
private double myValue;
public AddNum(double a,double b){
myValue=a+b;
}
public void writeValue(){
System.out.println(myValue);
}
}
Instructions After Initial Installation and Setup
1. Open MATLAB
2. Open vertigo qwerk folder (in MATLAB)
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3. Enter Ts = 0.1 in the command window
4. Run vertigo nonlinear.m
5. Open MyFirstRobot.m, make sure the directory locations are all correct (see
code below *)
(a) The classpath.txt file altered above in the initial setup, must be returned
to its original state (see reference **, a section copied from the rn.pdf in
the QWERK dir.).
6. Run MyFirstRobot.m, GUI should pop up just like other connection methods
(a) Click [Connect]
(b) Choose to connect directly to a peer
(c) Enter Vertigo’s IP address (something like 192.168.1.100)
(d) Click [Finish], then [Play], and minimize the GUI
7. Now all of the Terk commands can be used in MATLAB just as they were used
in JCreator for Java.
8. Run vertigo java vicon loop.mdl, found in the vertigo simulink directory
*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% setting the path to include required %dir javaaddpath(’C:\Users\Jon\Desktop
\VERTIGO\QWERK\MyFirstVertigo\terk-client-MyFirstRobot’);
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% setting the path to terk.jar also. terk.jar is where the motor,
% connection and other commands are located javaaddpath(’C:\Users\Jon\Desktop
\VERTIGO\QWERK\MyFirstVertigo\terk-client-MyFirstRobot\terk.jar’);
%add text to speech functionality
%javaaddpath(’C:\Users\Jon\Desktop\VERTIGO\QWERK\MyFirstVertigo
\terk-client-MyFirstRobot\TTS’)
%javaadpath(’C:\Users\Jon\Desktop\VERTIGO\QWERK\MyFirstVertigo
\terk-client-MyFirstRobot\RobotClient’);
%applicationTitle=java.lang.String(’Woof’);
%ipaddress=java.lang.String(’192.168.1.100’);
%myRobot = RobotClient.RobotClient(’applicationTitle’);%,ipaddress);
%myRobot = RobotClient.SimpleRobotClient(’trial’);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
**
Java Interface Adds Dynamic Java Class Path (excerpt from rn.pdf in the
QWERK dir.)
MATLAB loads Java class definitions from files that are on the Java class path.
The Java class path now consists of two segments: the static path, and a new segment
called the dynamic path. The static path is loaded from the file classpath.txt at the
start of each MATLAB session and cannot be changed without restarting MATLAB.
This was the only path available in previous versions of MATLAB. Thus, there was no
way to change the Java path without restarting MATLAB. The dynamic Java class
path can be loaded at any time during a MATLAB session using the javaclasspath
function. You can define the dynamic path (using javaclasspath), modify the path
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(using javaaddpath and javarmpath), and refresh the Java class definitions for all
classes on the dynamic path (using clear java) without restarting MATLAB. See the
function reference pages for more information on how to use these functions.
The javaclasspath function, when used with no arguments, displays both the static
and dynamic segments of the Java class path:
javaclasspath
STATIC JAVA PATH
D:\Sys0\Java\util.jar
D:\Sys0\Java\widgets.jar
D:\Sys0\Java\beans.jar
DYNAMIC JAVA PATH
User4:\Work\Java\ClassFiles
User4:\Work\Java\mywidgets.jar
You can read more about this feature in the sections, “The Java Class Path”
and “Making Java Classes Available to MATLAB” in the External Interfaces
documentation.
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Appendix D
Changing Qwerk’s Embedded Code
Special thanks to Mark Tjersland for all his help in finally cracking this.
This document goes through how to change Qwerk’s embedded code. More specifically, the hardcoded gains for the PIDV controller, and the trajectory generation
function that the PIDV controller tracks. This was necessary to attain the responsiveness demanded by Vertigo. It details how to acquire the code, manipulate it,
compile it, delete the existing code, replace it, and reinitialize Qwerk. It also lists all
the required programs for these steps and how they can be acquired.
This procedure is loosely based on the outline given in the Qwerk Development
Guide (QDG) which can be found at:
http://terk.svn.sourceforge.net/viewvc/terk/embed/trunk/share/docs/QwerkDevelo
pmentGuide.html#BuildingtheQwerkSource
The relevant excerpts can also be found at the end of these instructions.
Programs:
For this procedure you will need root access to in Linux so the SENS accounts
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through the school will not work. You may need to install a virtual machine on your
computer. This can be done by installing Sun xVM VirtualBox (free online). Also,
ubuntu-6.06.1-desktop-i386 is needed. More current versions will not compile the
code correctly. You will also need to install the drivers for the USB-Serial adapter.
Remember the port that the adapter is plugged into when you do this because it will
always need to be plugged into the same port when connecting to and altering the
embedded Qwerk code.
Procedure:
Compiling Firmware:
Open Sun xVM VirtualBox (NOTE: Remove all flash drives, SD cards and
external hard drives. Also, the blue USD to serial adapter must be plugged in to the
port it was in during installation of drivers. See figure below for current location on
the editing computer.)
Click [state]. It will boot up then ask for username and password.
Username: jwmissel (these specifics will change with the user)
Password: password
In Linux, go to applications accessories terminal.
In terminal type in: cd terk/embed/trunk/share/src/terkapi
Then type: gedit Client.cpp & This is case sensitive. Executing this command
will open up a new window with Qwerk’s source code in it. You will now need to
find the lines with the gains for the motor controller and change them as desired.
To make this faster, click [find] and search for the keyword “gain.” After changing
them click [save].
NOTE: The original gains were (100, 0, 500, 0). These, with the original tra215
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jectory, required about 10 seconds for the motors to accelerate. Much too slow of
a response; to mitigate this, the P and D gains should be increased and the I gain
should remain low. The combination (500, 0, 1000, 0) was too high and caused spastic
motor behavior. This was lowered until they worked properly again at (150, 0, 600
,0). Here the response took about 6 seconds. Still too slow.
For changing the acceleration type:
cd terk/embed/trunk/share/src/libqwerk then gedit qemotortraj.cxx. Then
multiply the acceleration by a constant to dilate it’s affect. It is currently multiplied
by 200 to achieve a slope that is near to a step function. [save] The following section
picks up in the QDG under the section: Building the Qwerk Source.
In terminal, run the commands (one at a time) under Building the Qwerk Source:
cd ˜/terk/embed/trunk/share
make
cd ˜/terk/embed/trunk/cirrus-arm-linux-1.0.4
make edb9302 (NOTE: may take about 8 mins to run)
cd ˜/terk/embed/trunk/cirrus-arm-linux-1.0.4/edb9302
for i in ramdisk.gz zImage optfs.out; do
(sudo cp –backup=t $i /var/www); done
Then just type in the password: password DOESN’T ALWAYS ASK FOR THIS
At this point the new firmware has been compiled, but is not yet installed on
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Qwerk.
Installing Firmware:
All programs and windows from above should still be open.
Cable-in to router: Both Vertigo (Qwerk) and your computer MUST be cabled in!
Connect to serial port: You must connect your computer to Qwerk through the
serial port (UART1 on Qwerk). Plug the USB end of the blue USB Serial Adapter
into your computer. Connect the serial end to the gray serial to RJ11 (phone jack)
adapter, then connect that to the UART1 jack on Qwerk. See figure below for visual
reference, the red line shows the lines of connection under consideration.
Figure D.1: Wiring for access to Qwerk.
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Type sudo minicom in the Terminal window. This should bring up a Welcome
to minicom 2.1 prompt.
Turn off Qwerk.
Click on Terminal (to make it the active window).
Turn on Qwerk AND hit [ctrl]+[C] when you see text appear in Terminal. There
is a one second window you have to do this in. Hitting it a few times to make sure
you got it won’t hurt anything.
Press [enter] if instructed.
If this part has been executed properly, it would end up giving you the RedBoot>
command prompt in terminal. This is where you will enter the following commands.
In terminal, run the commands in the QDG under Loading the Linux Images:
load -r -v -b 0x1000000 -m http /optfs.out
load -r -v -b 0x800000 -m http /ramdisk.gz
load -r -v -b 0x80000 -m http /zImage
fis create -b 0x800000 -l 0x300000 ramdisk
Answer y to the posed question for continuation.
fis create -b 0x80000 -l 0x180000 -e 0x80000 zImage
Answer y to the posed question for continuation.
Then run the command for version 1.1 Qwerk (the first option in the QDG document):
fis create -b 0x1000000 -l 0x300000 optfs
Answer y to the posed question for continuation.
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Type reset in Terminal. NOTE: Don’t hit [ctrl]+[C] this time because you want it
to reboot.
Wait and press [Enter] when asked.
At this point, the firmware has been replaced and Qwerk is just like it was out of
the package, except of course for the embedded code that have been changed.
Now you have to change its connection settings back to where they were for communication.
Change Qwerk’s connection settings:
With everything still plugged in and connected as it was before, enter Qwerk’s IP
address into the web browser. Under General Configuration select DirectConnect. Then click [Save]. Then under Wireless Configuration, check Set the
SSID Manually and make the settings as follows:
Network name (SSID): LAIRS
Data encryption: WEP
Network key: B4A8BFD5F4
Max tune to retry this SSID: 60
Max unsuccessful attempts: 2
Click [Save and Restart].
Reboot Qwerk. After reboot, led 6 should be blinking, indicating that you are
connecting directly. If led 7 is blinking, it means that it is trying to connect through
a relay server. This is what was just changed in the previous step.
Everything should be back to normal and running properly with the new gains
and trajectory.
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Qwerk Development Guide (QDG): Only relevant sections are included
here.
This document describes all the steps required to develop software for the Qwerk.
Blocks with a grey background represent what you’ll see in the command prompt.
Bold text within those blocks is text you enter.
Building the Qwerk Source
Compile the share tree first. The share tree contains all the TeRK-specific code.
You can safely ignore all the ”might be used uninitialized in this function” warnings
reported during the build.
$ cd ˜/terk/embed/trunk/share
$ make
make -C ./src
make[1]: Entering directory ‘/home/YOUR USERNAME/terk/embed/trunk
/share/src’
make -C IceE-1.0.0 all
make[2]: Entering directory ‘/home/YOUR USERNAME/terk/embed/trunk
/share/src/IceE-1.0.0’
making all in src
make[3]: Entering directory ‘/home/YOUR USERNAME/terk/embed/trunk
/share/src/IceE-1.0.0/src’
making all in IceE
make[4]: Entering directory ‘/home/YOUR USERNAME/terk/embed/trunk
/share/src/IceE-1.0.0/src/IceE’
rm -f ../../include/IceE/BuiltinSequences.h BuiltinSequences.cpp
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slice2cppe –ice –include-dir IceE –dll-export ICE API -I../../slice ../../slice/IceE
/BuiltinSequences.ice
mv BuiltinSequences.h ../../include/IceE
rm -f ../../include/IceE/FacetMap.h FacetMap.cpp
slice2cppe –ice –include-dir IceE –dll-export ICE API -I../../slice ../../slice/IceE
/FacetMap.ice
...
˜/terk/embed/trunk/share
Writable filesystem successfully created!
$
Building the share tree may take a while, but should eventually finish with the report ”Writable filesystem successfully created!”. The writeable filesystem it created
is a file named optfs.out and is located in the ˜/terk/embed/trunk/cirrus-arm-linux1.0.4/edb9302 directory.
The build also copies over the opt directory from which the optsfs.out image was
created (this is necessary since, as we’ll see next, the build for the cirrus-arm-linux
tree also creates the optfs.out image by repacking the opt directory that it finds within
its ˜/terk/embed/trunk/cirrus-arm-linux-1.0.4/edb9302 directory).
Now build the cirrus-arm-linux tree. This builds three filesystem images for the
Qwerk:
$ cd ˜/terk/embed/trunk/cirrus-arm-linux-1.0.4
$ make edb9302
make[1]: Entering directory ‘/home/YOUR USERNAME/terk/embed/trunk/cirrusarm-linux-1.0.4/edb9302’
Creating root filesystem...
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Creating linux source tree...
Configuring linux...
Building linux modules...
Configuring module-init-tools...
Building module-init-tools...
...
Building linux zImage...
Copying linux...
Creating redboot source tree...
Building redboot...
Installing redboot...
Building IceE...
Building firmware...
Building r/w filesystem...
make[1]: Leaving directory ‘/home/YOUR USERNAME/terk/embed/trunk/cirrusarm-linux-1.0.4/edb9302’
$
This will generate three important files: zImage, ramdisk.gz, and optfs.out. All of
these are in the edb9302 directory under the cirrus-arm-linux tree.
Building the cirrus-arm-linux tree will likely take a while, but luckily it’s not something you’ll have to build very often. The code in the cirrus-arm-linux tree rarely
changes, so you’ll most likely build it once to create the zImage and ramdisk.gz images and then you’ll just do builds of the share tree (which does change often) which
creates the optfs.out image.
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Note that although the cirrus-arm-linux build generates a (new) optfs.out image, all
it’s really doing is repacking the opt directory which was copied into ˜/terk/embed
/trunk/cirrus-arm-linux-1.0.4/edb9302 by the share tree build above. It repacks the
image by calling the buildoptfs.sh script. In any case, it should be the same as the
one created by the share tree build.
We’re now almost ready to download the image files to the Qwerk! We’ll first need to
put the images in the Apache’s web root directory so your machine can serve them
to the Qwerk. To do so, copy the three image files to the /var/www directory. You
can either do so in the usual way, or by doing the following which creates backups of
any images which currently exist (this doesn’t matter for this initial copy, of course,
but you may find it nice to use in the future):
$ cd ˜/terk/embed /trunk/cirrus-arm-linux-1.0.4/edb9302
$ for i in ramdisk.gz zImage optfs.out; do
(sudo cp –backup=t $i /var/www); done
$
Network Setup
Configure your network as described in the Network Setup section of the Upgrading
Firmware document on the TeRK web site. Since your computer will be serving the
firmware images to the Qwerk, you must have your network configured as shown in
figures 4 or 5 of the Network Setup document.
Installing Firmware Images
To install Linux on the Qwerk, we’ll connect to it via the serial cable using Minicom,
load in three images, and then create a boot script.
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Installing and Configuring Minicom
We first need to install minicom:
$ sudo apt-get install minicom
Reading package lists... Done
Building dependency tree... Done
Recommended packages:
lrzsz
The following NEW packages will be installed:
minicom
0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded.
Need to get 155kB of archives.
After unpacking 913kB of additional disk space will be used.
Get:1 http://us.archive.ubuntu.com dapper/main minicom 2.1-10 [155kB]
Fetched 155kB in 4s (34.0kB/s)
Selecting previously deselected package minicom.
(Reading database ... 86463 files and directories currently installed.)
Unpacking minicom (from .../minicom 2.1-10 i386.deb) ...
Setting up minicom (2.1-10) ...
Now run minicom
$ sudo minicom
Welcome to minicom 2.1
OPTIONS: History Buffer, F-key Macros, Search History Buffer, I18n
Compiled on Nov 5 2005, 15:45:44.
Press CTRL-A Z for help on special keys
We’ll now configure minicom to be able to talk to the Qwerk. Minicom should be
set to use 57600 baud, 8-N-1 (8 data bits, no parity, 1 stop bit), and no software or
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hardware flow control. Type CTRL-A Z to bring up the main menu:
———————————————————————–
|Minicom Command Summary |
||
|Commands can be called by CTRL-A <key> |
||
|Main Functions Other Functions |
||
|Dialing directory..D run script (Go)....G |Clear Screen.......C |
|Send files.........S Receive files......R |cOnfigure Minicom..O |
|comm Parameters....P Add linefeed.......A |Suspend minicom....J |
|Capture on/off.....L Hangup.............H |eXit and reset.....X |
|send break.........F initialize Modem...M |Quit with no reset.Q |
|Terminal settings..T run Kermit.........K |Cursor key mode....I |
|lineWrap on/off....W local Echo on/off..E |Help screen........Z |
||scroll Back........B |
||
|Select function or press Enter for none. |
||
|Written by Miquel van Smoorenburg 1991-1995 |
|Some additions by Jukka Lahtinen 1997-2000 |
|i18n by Arnaldo Carvalho de Melo 1998 |
———————————————————————–
Enter O to configure minicom:
——-[configuration]——|Filenames and paths |
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|File transfer protocols |
|Serial port setup |
|Modem and dialing |
|Screen and keyboard |
|Save setup as dfl |
|Save setup as.. |
|Exit |
—————————-|
Now use the arrow keys to highlight ”Serial port setup” and type ENTER to choose
it. You should now see the following menu:
——————————————|A - Serial Device : /dev/tty8 |
|B - Lockfile Location : /var/lock |
|C - Callin Program : |
|D - Callout Program : |
|E - Bps/Par/Bits : 38400 8N1 |
|F - Hardware Flow Control : Yes |
|G - Software Flow Control : No |
||
|Change which setting? |
——————————————First type A to set the serial device. The device name to use depends on your
computer. For example, one of my machines only has one serial port, so I set the
device to /dev/ttyS0. Another (a MacBook running Ubuntu under VMWare Fusion)
doesn’t have any serial ports, so I use a GWC UC320 USB-to-serial adapter which
Ubuntu recognizes as /dev/ttyUSB0. The name of your serial port device may be
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different. When you’re done editing the device, type ENTER to save it.
——————————————|A - Serial Device : /dev/ttyS0 |
|B - Lockfile Location : /var/lock |
|C - Callin Program : |
|D - Callout Program : |
|E - Bps/Par/Bits : 38400 8N1 |
|F - Hardware Flow Control : Yes |
|G - Software Flow Control : No |
||
|Change which setting? |
——————————————Now type E to set the Bps/Par/Bits. Doing so will open a new menu that should
look like this:
————[Comm Parameters]————||
|Current: 38400 8N1 |
||
|Speed Parity Data |
||
|A: 300 L: None S: 5 |
|B: 1200 M: Even T: 6 |
|C: 2400 N: Odd U: 7 |
|D: 4800 O: Mark V: 8 |
|E: 9600 P: Space |
|F: 19200 Stopbits |
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|G: 38400 W: 1 |
|H: 57600 X: 2 |
|I: 115200 Q: 8-N-1 |
|J: 230400 R: 7-E-1 |
||
||
|Choice, or <Enter> to exit? |
——————————————
Enter H to select 57600 baud, then Q to select 8-N-1. As you do so, you should see
the line at the top which displays the current settings change. When you’re done,
type ENTER to close the Comm Parameters menu. You should now see that the
Bps/Par/Bits value has changed:
——————————————|A - Serial Device : /dev/ttyS0 |
|B - Lockfile Location : /var/lock |
|C - Callin Program : |
|D - Callout Program : |
|E - Bps/Par/Bits : 57600 8N1 |
|F - Hardware Flow Control : Yes |
|G - Software Flow Control : No |
||
|Change which setting? |
——————————————Finally, type F to turn off Hardware Flow Control. The settings should now look like
this:
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|A - Serial Device : /dev/ttyS0 |
|B - Lockfile Location : /var/lock |
|C - Callin Program : |
|D - Callout Program : |
|E - Bps/Par/Bits : 57600 8N1 |
|F - Hardware Flow Control : No |
|G - Software Flow Control : No |
||
|Change which setting? |
——————————————Type ENTER to exit this menu.
Now that we’re back at the configuration menu, use the arrow keys to highlight the
”Save setup as dfl” option and type ENTER to select it.
Now use the arrow keys to highlight the ”Exit” option and type ENTER to select it.
Finally, quit minicom by typing CRTL-A x.
Minicom is now configured to talk to Qwerks!
Connecting to the Qwerk Using Minicom
Run minicom:
$ sudo minicom
Welcome to minicom 2.1
OPTIONS: History Buffer, F-key Macros, Search History Buffer, I18n
Compiled on Nov 5 2005, 15:45:44.
Press CTRL-A Z for help on special keys
We’re now ready to turn on the Qwerk. Once we do so, it’ll spit out some system
info and then pause for 1 second before executing its boot script. We don’t want it
to execute the boot script yet, so we’re going to type CTRL-C to abort it.
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Go ahead and turn on the Qwerk, but type CTRL-C immediately afterwards.
It should look something like this:
+Ethernet eth0: MAC address 00:dc:6c:7d:6b:25
IP: 192.168.1.120/255.255.255.0, Gateway: 192.168.1.1
Default server: 192.168.1.103, DNS server IP: 192.168.1.1
RedBoot(tm) bootstrap and debug environment [ROMRAM]
Non-certified release, version v2 0 - built 13:09:18, Mar 21 2006
Platform: Cirrus Logic EDB9302 Board (ARM920T) Rev A
Copyright (C) 2000, 2001, 2002, Red Hat, Inc.
RAM: 0x00000000-0x02000000, 0x00041fa8-0x01fdd000 available
FLASH: 0x60000000 - 0x60800000, 64 blocks of 0x00020000 bytes each.
== Executing boot script in 1.000 seconds - enter ˆC to abort
ˆC
RedBoot>
You should now see a ”RedBoot>” prompt.
Pay attention to the IP, Gateway, and Default Server IP addresses. The IP address
is the address of the Qwerk, the Gateway is your router, and the Default Server is
the HTTP server (i.e. your machine). If the IP addresses are correct, then you can
skip down to the Loading the Linux Images section. Otherwise, follow the directions
in the Configure the Qwerk For Your Network section to set them for your network.
Configure the Qwerk For Your Network
If the IP, Gateway, and Default Server IP addresses (displayed in minicom when you
first power on the Qwerk) are incorrect, then you won’t be able to download the
images from your HTTP server. The IP address is the address of the Qwerk, the
Gateway is your router, and the Default Server is the HTTP server (i.e. your ma230
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chine). I usually go ahead and set the DNS Server IP Address, too, which should in
most cases just be the same as the Gateway IP Address.
Execute the fconfig command to set the IP, Gateway, and Default Server IP addresses.
Start by entering true for whether the script should run at boot, and then enter the
script exactly as shown below. Once you’ve entered the boot script, enter an empty
line to terminate the script editing mode. Continue on through the other values
and enter the correct IP addresses for the Qwerk, Gateway, and Default Server for
your network. Accept the default values for all the other options. Make sure to enter yes (y) when prompted whether to update the RedBoot non-volatile configuration.
In the example shown below, values you need to enter exactly as shown are shown in
black, bold type. Values that you need to enter which might be different for your
network are shown in red, bold type. You should use the defaults for all other values.
RedBoot> fconfig
Run script at boot: true
Boot script:
.. load -r -v -b 0x800000 ramdisk.gz
.. load -r -v -b 0x80000 zImage
.. exec -r 0x800000 -s 0x600000
Enter script, terminate with empty line
>> fis load ramdisk
>> fis load zImage
>> exec -r 0x800000 -s 0x600000
>>
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Boot script timeout (1000ms resolution): 1
Use BOOTP for network configuration: false
Gateway IP address: 192.168.0.1
Local IP address: 192.168.0.3
Local IP address mask: 255.255.255.0
Default server IP address: 192.168.0.2
DNS server IP address: 192.168.0.1
Set eth0 network hardware address [MAC]: false
GDB connection port: 9000
Force console for special debug messages: false
Network debug at boot time: false
Update RedBoot non-volatile configuration - continue (y/n)? y
... Erase from 0x607c0000-0x607c1000: .
... Program from 0x01fde000-0x01fdf000 at 0x607c0000: .
RedBoot>
Now either enter reset or power-cycle the Qwerk so that your changes take effect.
Either way, be ready to type CTRL-C to get back in to the RedBoot> prompt.
Loading the Linux Images
First, we’ll load three images from the HTTP server (running on your machine) into
the Qwerk’s RAM.
Enter the following three load commands (the range of hex addresses displayed may
be different):
RedBoot> load -r -v -b 0x1000000 -m http /optfs.out
|
Raw file loaded 0x01000000-0x011e099b, assumed entry at 0x01000000
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RedBoot> load -r -v -b 0x800000 -m http /ramdisk.gz
\
Raw file loaded 0x00800000-0x0095c6d4, assumed entry at 0x00800000
RedBoot> load -r -v -b 0x80000 -m http /zImage
/
Raw file loaded 0x00080000-0x0019d12b, assumed entry at 0x00080000
RedBoot>
Now we’ll initialize the flash filesystem. Answer yes (y) when it asks if it’s ok initialize.
The output should look like this (don’t worry if the numbers in the output differ):
RedBoot> fis init -f
About to initialize [format] FLASH image system - continue (y/n)? y
*** Initialize FLASH Image System
... Erase from 0x60040000-0x60fc0000: ............................................................
... Erase from 0x60fe0000-0x60fe0000:
... Erase from 0x61000000-0x61000000:
... Erase from 0x60fe0000-0x61000000: .
... Program from 0x01fdf000-0x01fff000 at 0x60fe0000: .
RedBoot>
Now we’ll load the ramdisk and zImage images into flash (don’t worry if the numbers
in the output differ):
RedBoot> fis create -b 0x800000 -l 0x300000 ramdisk
... Erase from 0x60040000-0x60340000: ........................
... Program from 0x00800000-0x00b00000 at 0x60040000: ........................
... Erase from 0x60fe0000-0x61000000: .
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... Program from 0x01fdf000-0x01fff000 at 0x60fe0000: .
RedBoot> fis create -b 0x80000 -l 0x180000 -e 0x80000 zImage
... Erase from 0x60340000-0x604c0000: ............
... Program from 0x00080000-0x00200000 at 0x60340000: ............
... Erase from 0x60fe0000-0x61000000: .
... Program from 0x01fdf000-0x01fff000 at 0x60fe0000: .
Now we’ll load the optfs image into flash. The command to load the optfs image
differs slightly depending on which Qwerk revision you have. The Qwerk revision is
printed on the board between the analog inputs terminal and the top of the enclosure.
It should read either ”Qwerk 1.1A (c) 2006” or ”Qwerk 1.2A (c) 2006”. Determine
which version you have and then use the appropriate command below to load the
optfs image into flash.
If you have a revision 1.1 qwerk, use this command (don’t worry if the numbers in
the output differ):
RedBoot> fis create -b 0x1000000 -l 0x300000 optfs
... Erase from 0x603c0000-0x606c0000: ........................
... Program from 0x01000000-0x01300000 at 0x603c0000: ........................
... Erase from 0x607e0000-0x60800000: .
... Program from 0x01fdf000-0x01fff000 at 0x607e0000: .
RedBoot>
If you have a revision 1.2 qwerk, use this command (don’t worry if the numbers in
the output differ):
RedBoot> fis create -b 0x1000000 -l 0xb00000 optfs
... Erase from 0x604c0000-0x60fc0000: ........................
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... Program from 0x01000000-0x01b00000 at 0x604c0000: ........................
... Erase from 0x60fe0000-0x61000000: .
... Program from 0x01fdf000-0x01fff000 at 0x60fe0000: .
RedBoot>
The difference is simply the value for the -l argument, which specifies the length.
Revision 1.1 qwerks have 8 megs of flash RAM, but revision 1.2 qwerks have 16 megs
of flash RAM. Using a value of 0xb00000 for the revision 1.2 qwerks enables the use
of the extra 8 megs of RAM.
Testing the Linux Installation
Now either enter reset or power-cycle the Qwerk so that your changes take effect (but
don’t type CTRL-C, because we want the Qwerk to boot up into Linux). You should
something similar to the following:
RedBoot> +Ethernet eth0: MAC address 00:dc:6c:7d:6b:25
IP: 192.168.0.3/255.255.255.0, Gateway: 192.168.0.1
Default server: 192.168.0.2, DNS server IP: 192.168.1.1
RedBoot(tm) bootstrap and debug environment [ROMRAM]
Non-certified release, version v2 0 - built 13:09:18, Mar 21 2006
Platform: Cirrus Logic EDB9302 Board (ARM920T) Rev A
Copyright (C) 2000, 2001, 2002, Red Hat, Inc.
RAM: 0x00000000-0x02000000, 0x00041fa8-0x01fdd000 available
FLASH: 0x60000000 - 0x60800000, 64 blocks of 0x00020000 bytes each.
== Executing boot script in 1.000 seconds - enter ˆC to abort
RedBoot> fis load ramdisk
RedBoot> fis load zImage
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RedBoot> exec -r 0x800000 -s 0x600000
Using base address 0x00080000 and length 0x00180000
Uncompressing Linux.......................................................................... done, booting
the kernel.
Wait a few seconds while the kernel boots. You should eventually see a message
(shown below) that prompts you to type Enter to activate the console. Do so, and
you should see:
Please press Enter to activate this console.
BusyBox v1.00 (2006.06.28-21:45+0000) Built-in shell (ash)
Enter ’help’ for a list of built-in commands.
˜#
Congratulations! You successfully installed the firmware on the Qwerk, and it’s ready
to use!
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Appendix E
Maintenance
This appendix lists the routine maintenance suggestions for Vertigo’s up keeping.
Most of these suggestions and timeframes are based on experience working with the
system and assume daily use.
*Set screws on wheel hubs should be tightened weekly when in use. 1/8 in hexkey loosens the wheels retaining washer, then a .05 in hex-key fits the set screw. Be
careful not to strip the hex or threads on these small screws.
*Keep an eye on battery life and how well they are charging.
*Watch for wheel erosion.
*Periodically check that all bolts and screws are securely fastened.
*Periodically check that none of the motors have shifted in their mounts so that
all four wheels contact a flat surface evenly. To mount the motors evenly, make the
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front face of the gear-head flush with the face of the front motor mount. Align them
on a flat surface, applying pressure to keep both in contact with the surface as you
tighten the mounts.
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