Yangsheng Xu, Yongsheng Ou. Control of Single Wheel Robots
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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
List of Figures
XIX
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Brief History of Mobile Robots . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Problems of Stable Robots . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Dynamically Stable Mobile Robots . . . . . . . . . . . . . . . . . . 3
1.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Concept and Compromise . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Sensors and Onboard Computer . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Modeling in a Horizontal Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Kinematic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Modeling on an Incline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Motion on an Incline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Motion Planning on an Incline . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
XVI Contents
3 Model-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Linearized Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Path Following Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.3 Control Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Balance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.3 Line Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Control Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Vertical Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.3 Path Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Learning-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Learning by CNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Cascade Neural Network with Kalman Filtering . . . . . . . 74
4.1.2 Learning architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.3 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.4 Training procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Learning by SVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Learning Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Learning Control with Limited Training Data . . . . . . . . . . . . . . . 99
4.3.1 Effect of Small Training Sample Size . . . . . . . . . . . . . . . . . 102
4.3.2 Resampling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.3 Local Polyn
omial
Fi
ttin
g (LP
F) . . . . . . . . . . . . . . . . . . . . . 1
10
4.3.4 Simulations and Experiments . . . . . . . . . . . . . . . . . . . . . . . 112
5 Further Topics on Learning-based Control . . . . . . . . . . . . . . . . . 119
5.1 Input Selection for Learning Human Control Strategy . . . . . . . . 119
5.1.1 Sample Data Selection and Regrouping . . . . . . . . . . . . . . . 121
5.1.2 Significance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.3 Dependency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.4 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 29
5.2 Implementation of Learning Control . . . . . . . . . . . . . . . . . . . . . . . 135
5.2.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Contents XVII
6 Shared Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1 Control Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.1 Switch Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.2 Distributed Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.3 Combined Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Shared Control of Gyrover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.4 How to Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.5 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 61
6.5.1 Heading Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.5.2 Straight Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.5.3 Circular Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5.4 Point-to-point Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1.1 Concept and Implementations. . . . . . . . . . . . . . . . . . . . . . . 1 75
7.1.2 Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1.3 Model-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.1.4 Learning-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
List of Figures
1.1 Thebasicconfiguration of therobot ........................ 6
1.2 Communication equipment: radio transmitter (left) and
la
pt
op
sw
ith
wir
eless
Mo
de
m(
ri
gh
t).
....................... 9
1.3Hardwareconfiguration of therobot........................ 9
1.
4T
he
firs
tp
ro
to
ty
pe
of
th
er
ob
ot
...........................
. 10
1.5The second prototype of therobot ..........................11
1.6The thirdprototype of therobot ........................... 12
2.1D
efinition of coordinate frames and system va
riables.
....
....
13
2.2The simulationresults of arollingdisk without the flywheel. ... 22
2.3 Thesimulationresults of thesinglewheel robot. .............. 22
2.4 Thesimulation resultso
ft
ilting the
flywheel of the robotwith
˙
β
a
=73 deg/s ............................................ 23
2.5T
he expe
rimental
resultso
ft
ilting the
flywheel of the robot
with
˙
β
a
=73 deg/s ....................................... 23
2.6Critical torqueofarolling diskv.s. aclimblingangle .......... 26
2.7
Cr
it
ic
al
tor
qu
eo
ft
he
ro
bo
tv
.s.
ac
lim
bl
in
ga
ng
le
.............2
6
2.8Changeorientation....................................... 27
2.9Disk rolls on aplane ...................................... 28
2.10 Rolling up
of
adisk ....
....
.....
....
....
....
.....
....
....
.3
1
2.11 Rolling up
of
asingle wheel robot
....
....
....
.....
....
....
.3
1
2.12 Rolling down of adisk .................................... 32
2.13 Rolling down of asingle wheel robot ........................32
3.1The lateral descriptionofGyrover..........................34
3.2Schematic of thecontrol algorithms. ........................ 34
3.3Principle of linefollowing.(topview)
........................ 38
3.4Schematic of the controlalgorithm for the Y-axis. .....
....
... 38
3.5 Thesimulation results(
S 1) forfollowingthe Y-axis.
.......... 41
3.6The simulationresults (
S 2) forfollowingthe Y-axis.
.......... 41
3.7The simulationresults (
S 3) forfollowingthe Y-axis.
.......... 41
XX List of Figures
3.8 The simulation results for following the Y-axis with
˙
γ =10 rad/s............................................ 43
3.9The simulationresults forfollowing the Y-axis with
˙γ =3
0
ra
d/s
..
..........................................
43
3.10
The
sim
ulation
re
sults
for
fo
llo
wing
the
Y-axis
with
σ =2
0.
..
..
43
3.11 The simulation results for following the Y-axis with σ =40..... 44
3.12 System parameters of Gyrover’ssimplified model. ............ 45
3.13 Parameters in positioncontrol.............................. 51
3.14 Therobot’s path alongconnected corridors.................. 54
3.15 Theparametersinthe linetrackingproblem. ................55
3.16 Leaningangle β in balancecontrol.......................... 58
3.17 Precession angle velocity˙α in balancecontrol................ 58
3.18 Drivingspeed ˙γ in balancecontrol.......................... 59
3.19
Pa
rameters
of
˙
α,
β,
˙
β, ˙γ in
ba
la
nc
ec
on
tr
ol
.
..
.................
59
3.20 Leaning angle β in balancecontrol II........................ 60
3.21 Precession angle velocity˙α in balancecontrol II.............. 60
3.22 Drivingspeed ˙γ in balancecontrol II........................ 61
3.23 Pa
rameters of ˙
α, β,
˙
β, ˙γ in
ba
la
nc
ec
on
tr
ol
II
..
............... 61
3.24
Displacement
in X.
........................................ 62
3.25
Displacement
in Y.
........................................ 63
3.26
X-Yofori
gi
n.
........................................... 63
3.27 The joint-space variablesinposition control. ................. 64
3.28
Linetracking in Xdirection. ................................. 64
3.29
Linetracking in Yd
irection.
................................. 65
3.30
X-Yofinlinetracking. .................................... 65
3.31 Thejoint-space variables in linetracking. .................... 66
3.32
Function Tanh(.).
.......................................... 66
3.33
Function Uanh(.). ......................................... 66
3.34 Hardwareconfiguration of Gyrover. ........................ 67
3.35 Experimentinlinefollowingcontrol........................ 67
3.36 Camerapicturesinbalance control.......................... 68
3.37 Sensor data in balancecontrol..............................69
3.38 Tr
aj
ectories in
point-to-pointcontrol. ....
....
....
...
..
....
..
70
3.39 Sensor data in point-to-pointcontrol. ....................... 70
3.40 Trajectories in thestraightpathtest. ....................... 71
3.41 Sensor data in the straight path test. .......................71
4.1The cascade learning architecture. .......................... 76
4.2Similaritymeasure between
¯
O
1
and
¯
O
2
...................... 79
4.3Control data fordifferentmotions........................... 80
4.4Switchingsinhuman control of theflywheel. .................81
4.5 Similar inputs canbemappedtoextreme differentoutputsif
switchingoccurs..........................................82
4.6 Thepractical system andthe human controlfromadynamic
system......
....
....
....
.......
....
....
....
.......
......
88
List of Figures XXI
4.7 A learning controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.8 Gyrover: A single-wheel robot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.9 Definition of the Gyrover’s system parameters. . . . . . . . . . . . . . . . 97
4.10 The tilt angle β
a
Lean angle β of SVM learning control. . . . . . . 100
4.11 U
1
comparison of the same Human control and SVM learner. . . 1 00
4.12 Curse of dimensionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 01
4.13 Linear regression M=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.14 Polynomial degree M=3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.15 Polynomial degree M=10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.16 The RMS error for both training and test sets. . . . . . . . . . . . . . . . 108
4.17 Examples of the unlabelled sample generation, when k = 3. . . . . 110
4.18 Local polynomial fitting for lean angle β . . . . . . . . . . . . . . . . . . . . . 115
4.19 Comparison of U
1
in a set of testing data. . . . . . . . . . . . . . . . . . . . 116
4.20 Human control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.21 The CNN-new model learning control. . . . . . . . . . . . . . . . . . . . . . . 117
5.1 Clustering the data into a small ball with radius r . . . . . . . . . . . . 122
5.2 The local sensitivity coefficients of the first four significance
variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 SVM learning control results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Vertical balanced motion by human control, X
(1, 1)
. . . . . . . . . . . . 137
5.5 Control trajectories comparison for X
(1, 1)
. . . . . . . . . . . . . . . . . . . . 137
5.6 Vertical balanced motion by human control, X
(1, 2)
. . . . . . . . . . . . 138
5.7 Control trajectories comparison for X
(1, 2)
. . . . . . . . . . . . . . . . . . . . 138
5.8 Vertical balanced motion by human control, X
(1, 3)
. . . . . . . . . . . . 139
5.9 Control trajectories comparison for X
(1, 3)
. . . . . . . . . . . . . . . . . . . . 139
5.10 Tiltup motion by human control, X
(2, 1)
. . . . . . . . . . . . . . . . . . . . . 140
5.11 Control trajectories comparison for X
(2, 1)
. . . . . . . . . . . . . . . . . . . . 140
5.12 Tiltup motion by human control, X
(2, 2)
. . . . . . . . . . . . . . . . . . . . . 141
5.13 Control trajectories comparison for X
(2, 2)
. . . . . . . . . . . . . . . . . . . . 141
5.14 Tiltup motion by human control, X
(2, 3)
. . . . . . . . . . . . . . . . . . . . . 141
5.15 Control trajectories comparison for X
(2, 3)
. . . . . . . . . . . . . . . . . . . . 142
5.16 Vertical balancing by CNN model, trail #1. . . . . . . . . . . . . . . . . . 143
5.17 Vertical balancing by CNN model, trail #2. . . . . . . . . . . . . . . . . . 143
5.18 Vertical balancing by CNN model, trail #3. . . . . . . . . . . . . . . . . . 144
5.19 Vertical balancing by human operator. . . . . . . . . . . . . . . . . . . . . . . 145
5.20 Tiltup motion by CNN model, trail #1. . . . . . . . . . . . . . . . . . . . . . 146
5.21 Tiltup motion by CNN model, trail #2. . . . . . . . . . . . . . . . . . . . . . 147
5.22 Tiltup motion by human operator.. . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.23 Combined motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.24 Fluctuation in the lean angle made by the tiltup model. . . . . . . . 148
5.25 Tiltup and vertical balanced motion by CNN models. . . . . . . . . . 149
6.1 Switch mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Distributed control mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
XXII List of Figures
6.3 Combined mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 A detailed structure of behavior connectivity in Gyrover control.156
6.5 Subsumption architecture of shared control. . . . . . . . . . . . . . . . . . 157
6.6 Sensor data acquired in the heading control test, A = 0 . 2. . . . . . 163
6.7 Sensor data acquired in the heading control test, A = 0 . 8. . . . . . 164
6.8 Experiment on tracking a straight path under shared control. . . 165
6.9 Experiment on tracking a curved path under shared control. . . . 165
6.10 Experiment on point-to-point navigation under shared control. . 167
6.11 Trajectory travelled in the straight path test. . . . . . . . . . . . . . . . . 168
6.12 Sensor data acquired in the straight path test. . . . . . . . . . . . . . . . 169
6.13 Gyrover trajectories in the curved path test. . . . . . . . . . . . . . . . . . 170
6.14 Sensor data acquired in the circular path test. . . . . . . . . . . . . . . . 171
6.15 Gyrover trajectories in the combined path test.. . . . . . . . . . . . . . . 172
6.16 Sensor data acquired in the combined path test. . . . . . . . . . . . . . . 173
List of Tables
1.1 Table of different actuating mechanisms in Gyrover. .......... 8
2.1Variables definitions ...................................... 14
2.
2P
ar
ame
ter
su
sed
in
sim
ul
at
io
na
nd
e xp
er
ime
nt
s.
............. 21
2.
3S
ystem
pa
ra
mete
rs
.......................................
29
3.1The initial conditions forthe simulations with differentinitial
headingangles ........................................... 41
3.
2P
hy
sica
lp
ar
ame
ter
s.
...................................... 56
3.3Initial parameters in balancecontrol........................57
3.4 Target andgain parameters in
balance control
I.
...
....
....
..
57
3.5Initial parameters in positioncontrol........................ 62
3.6Target andgain parameters in positioncontrol. ..............62
3.
7I
ni
tial
pa
ra
mete
rs
in
lin
et
ra
ck
in
g.
..........................
62
3.8Target andgain parameters in line tracking. ................. 63
4.1Similaritymeasures between differentcontrol trajectories. .....79
4.2Sample human control data................................98
4.3The training data sample. .................................113
4.4The testingdatasample ...................................113
4.5The resultsoflearningerrorwith unlabelledtraining data.....113
4.6 Sample human control data................................114
5.1Gyrover’s sensor data string...............................130
5.2Sample of human control data .............................130
5.3The noisevariance σ informationfor variables ................130
5.4The significance ordertable ................................132
5.5 Theaverage “(linear)relative” coefficients ¯ρ in full system
variables ................................................132
5.6Partof¯ρ in the5variables ................................133
5.7¯ρ in the6variables .......................................133
XXIV List of Tables
5.8 Part of ¯
ρ in the4variables ................................133
5.9Similaritymeasures for verticalbalancedcontrol between
human andCNN model...................................136
5.10
Similarit
ym
easures
for
tiltup
con
trol
be
twe
en
hu
man
an
d
CNN
mo
de
l.
............................................
.1
40
5.11 Performance measures forvertical balancing..................142
5.12 Performance measures fortiltup motion. ....................144
5.13 Performance measuresfor combinedmotion. .................146
6.1Decision making of A =0. 25. ..............................160
6.2Decision making of A =0. 50. ..............................160
6.3Decision making of A =0. 75. ..............................161
1
Introduction
1.1 Background
1.1.1 Brief History
of Mobile Robots
Landlocomotion can be
broadly ch
aracterized as quasi-static or dynamic.
Quasi-staticequilibriumimplies that inertial (acceleration-related) effects are
small enough
to
ignore. That is, motions are slo
we
nought
hats
tatic equi-
librium can be assumed.Stabilityofquasi-staticsystems depends on keeping
the gravityvector through, the center of mass, within the vehicle’s polygonof
support determinedbythe ground-contactpoints of itswheels or feet. Energy
input is utilized predominantly in reactingagainst static forces. Such systems
ty
picallyhav
erelatively
rigid mem
be
rs, andcan be
con
trolled on
thebasiso
f
kinematic considerations.
In dynamiclocomotion,inertialforcesare significantwith respect to grav-
itational forces. Dynamic effects gain relativei
mportance when
sp
eed is high,
gravityisweak anddynamic disturbances (e.g. roughterrain) arehigh. Signif-
icante
nergyi
nput is requiredincontrollingsystem momentum,
andinsome
cases,incontrollingelastic energy storageinthe system.Asperformance lim-
its of mobile robotsare pushed, dynamiceffects will increasingly come into
play. Further, robotic systems that behave dynamically maybeable to ex-
ploit momentumtoenhance mobility, as is clearly demonstrated by numerous
hu
man-controlled
systems: gymnasts, dancers andother athletes; stuntb
icy-
cles andm
otorcycles;m
otorcycles on rought
errain; cars that
vault obstacles
fromramps; etc.
It is paradoxical that those factors whichproduce staticstabilitymay con-
tradict dynamic stability. Forexample, afour-wheelvehiclethatisvery low
andwide hasabroad polygonofsupport, is very stable statically,and can
tolerate large slopes without roll-over. However, when this vehicle passes over
bumps,dynamic disturbances at the wheels generate large torques, tending
to upset the vehicle aboutthe roll,pitchand yawaxes. In effect, the large
Y. Xu and Y. Ou: Control of Single Wheel Robots, STAR 20, pp. 1–12, 2005.
© Springer-Verlag Berlin Heidelberg 2005