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Motion control of holonomic wheeled mobile robot with modular actuation [Elektronische Ressource] / vorgelegt von Ahmed El-Shenawy

196 pages
The University of MannheimInstitute of Computer EngineeringMOTION CONTROL OF HOLONOMIC WHEELED MOBILEROBOT WITH MODULAR ACTUATIONInauguraldissertationzur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaftender Universitaet Mannheimc 2010vorgelegt vonAhmed Khamies El-Shenawyaus Aegypten, AlexandriaMannheim15 April 2010∗The thesis of Ahmed Khamies El-Shenawy was reviewed and approved by thefollowing:Dekan:Professor Dr. Felix Freiling,Mannheim UniversityReferent:Professor Dr. Essam Badreddin,Heidelberg UniversityKorreferent:Professor Dr. Reinhard MaennerHeidelberg UniversityTag der muendlichen Pruefung: 15. April 2010.AbstractThis thesis proposes a control scheme for a new holonomic wheeled mobile robot.The platform, which is called C3P (Caster 3 wheels Platform), is designed andbuilt by the Automation Lab., University of Heidelberg. The platform has threedriven caster wheels, which are used because of their simple construction and easymaintenance.TheC3Phasmodularactuatorsandsensorsconfigurations. Therobot’sactua-tion scheme produces singularity difficulties for some wheel steering configuration,described as the following: When all wheels yield the same steering angle value,the C3P cannot be actuated in the direction perpendicular to the wheel velocityvector. The C3P has a modular sensing scheme defined by sensing the steeringangleandthewheelangularvelocityofeachcasterwheel.
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The University of Mannheim
Institute of Computer Engineering
MOTION CONTROL OF HOLONOMIC WHEELED MOBILE
ROBOT WITH MODULAR ACTUATION
Inauguraldissertation
zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften
der Universitaet Mannheim
c 2010
vorgelegt von
Ahmed Khamies El-Shenawy
aus Aegypten, Alexandria
Mannheim
15 April 2010∗The thesis of Ahmed Khamies El-Shenawy was reviewed and approved by the
following:
Dekan:
Professor Dr. Felix Freiling,
Mannheim University
Referent:
Professor Dr. Essam Badreddin,
Heidelberg University
Korreferent:
Professor Dr. Reinhard Maenner
Heidelberg University
Tag der muendlichen Pruefung: 15. April 2010.Abstract
This thesis proposes a control scheme for a new holonomic wheeled mobile robot.
The platform, which is called C3P (Caster 3 wheels Platform), is designed and
built by the Automation Lab., University of Heidelberg. The platform has three
driven caster wheels, which are used because of their simple construction and easy
maintenance.
TheC3Phasmodularactuatorsandsensorsconfigurations. Therobot’sactua-
tion scheme produces singularity difficulties for some wheel steering configuration,
described as the following: When all wheels yield the same steering angle value,
the C3P cannot be actuated in the direction perpendicular to the wheel velocity
vector. The C3P has a modular sensing scheme defined by sensing the steering
angleandthewheelangularvelocityofeachcasterwheel. Thisworkhasfourmain
contributions
1- developing a controller based on an inverse kinematics solution to handle
motion commands in the singular configurations;
2- modelingtheC3P’sforwarddynamicsoftheC3Pforthesimulationpurpose;
3- developing a motion controller based on an inverse dynamics solution; and
4- comparing the C3P with other standard holonomic WMRs.
In order to escape singularity condition, the actuated inverse kinematics solu-
tionisdevelopedbasedontheideaofcouplinganytwowheelvelocitiestovirtually
actuate the steering angular velocity of the third wheel. The solution is termed
as the Wheel Coupling Equation (WCE). The C3P velocity controller consists of
two parts: a) the WCE regulator to avoid singularities and adjust the steering
iiiangles to the desired value, and b) the regular PID controller to maintain the ref-
erence robot velocities with respect to the floor frame of coordinates. The solution
reaches acceptable performance in the simulation examples and in the practical
experiments. However, it generates relatively large displacement errors only dur-
ing the steering angles adjustment period.
The Euler-Lagrangian method is used for obtaining the forward dynamic and
theinversedynamicmodels. Theforwarddynamicmodelconsistsoftwoequations
of motion: the WTD (Wheel Torque Dynamics) to calculate the wheel angular
velocities with respect to the actuated wheels’ torques, and the DSE (Dynamic
SteeringEstimator) for calculating the steering angles and steering angular veloc-
ities corresponding to the angular wheels’ velocities and accelerations.
The inversedynamics solution definestheforces and torquesacting on each ac-
tuator and joint. The solution is used in the development of the C3P velocity and
position controllers. In comparison to the proposed inverse kinematics solution,
the inverse dynamics solution yields less displacement errors. Lyapunov stability
analysis is carried out to investigate the system stability for different steering an-
gles’ combinations. The steering angles’ values are considered as the disturbances
affecting the platform.
Finally, a comparison is made between the C3P and three other holonomic
wheeled mobile robots configurations. The comparison is based on the simulation
results in relation to the following aspects: a) mobility, b) total energy consumed
by each robot in a finite interval of time and c) hardware complexity. The C3P
platform shows its advantage in the aspects “b” and “c”.
ivTable of Contents
List of Figures ix
List of Tables xiii
List of Symbols xiv
Chapter 1
Introduction 1
1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Kinematics Modeling . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 Wheeled Mobile Robot Control Structure . . . . . . . . . . . 9
1.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 2
The C3P Kinematic and Dynamic Modeling 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Kinematic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 The C3P Kinematic Modeling . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Inverse and Forward Kinematic Solutions . . . . . . . . . . . 22
2.4 Robot Dynamics Modeling . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Nonholonomically Constrained System . . . . . . . . . . . . 25
2.4.2 Holonomically Constrained System . . . . . . . . . . . . . . 26
v2.4.3 The C3P Platform Constrained System . . . . . . . . . . . . 27
2.4.4 Euler-Lagrange Method . . . . . . . . . . . . . . . . . . . . 30
2.4.5 Kinetic Energy Equations . . . . . . . . . . . . . . . . . . . 31
2.5 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 The Wheels Torque Dynamics (WTD) . . . . . . . . . . . . 32
2.5.2 The Dynamic Steering Estimator (DSE) . . . . . . . . . . . 34
Chapter 3
Kinematics Based Motion Control 37
3.1 C3P Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Coupling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Singularity Indicator . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Wheel Coupling Equation Adaptation . . . . . . . . . . . . . . . . . 48
3.4 Velocity Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Position Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 4
Inverse Dynamics Based Motion Control and Analysis 59
4.1 Inverse Dynamics Solution . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Dynamics Based Motion Control Structure . . . . . . . . . . . . . . 61
4.2.1 Velocity Controller . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Dynamics Performace Examples . . . . . . . . . . . . . . . . 63
4.2.3 Position Controller . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 5
The Lyapunov Stability Analysis 78
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 The Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 6
Implementation and Practical Results 89
6.1 Platform Hardware Configuration . . . . . . . . . . . . . . . . . . . 89
6.2 Kinematics Based Controller Experiments . . . . . . . . . . . . . . 91
6.3 Dynamic Based Control Results . . . . . . . . . . . . . . . . . . . . 99
6.4 Experiments on C3P Stability . . . . . . . . . . . . . . . . . . . . . 111
vi6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Chapter 7
Comparing Different Holonomic WMRs 116
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Description of Holonomic Mobile Robots . . . . . . . . . . . . . . . 117
7.3 Comparing The C3P Vs Holonomic WMRs . . . . . . . . . . . . . . 119
7.3.1 Driving in 3 Degrees of Freedom . . . . . . . . . . . . . . . . 121
7.3.2 Driving in the Infinity Shape(∞) . . . . . . . . . . . . . . . 122
7.4 Performance Function Comparison . . . . . . . . . . . . . . . . . . 126
7.4.1 Mobility Aspect . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4.2 Energy Consumption Aspect . . . . . . . . . . . . . . . . . . 126
7.4.3 Hardware Complexity Aspect . . . . . . . . . . . . . . . . . 127
7.4.4 Cost Functional Calculation . . . . . . . . . . . . . . . . . . 128
7.4.4.1 Driving in 3DOFs . . . . . . . . . . . . . . . . . . 129
7.4.4.2 Driving in (∞) Shape . . . . . . . . . . . . . . . . 131
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Chapter 8
Conclusions and Future Work 134
AppendixA
Kinematics Modeling 137
A.1 The Velocity Generalized Wheel Jacobian . . . . . . . . . . . . . . 137
A.1.1 The Acceleration Wheel Jacobian . . . . . . . . . . . . . . . 138
A.1.2 Actuated Inverse and Sensed Forward Kinematics . . . . . . 139
Appendix B
The Dynamic Steering Estimator (DSE) 141
Appendix C
Inverse Dynamics Equations 143
C.1 The Inverse Dynamics Solution . . . . . . . . . . . . . . . . . . . . 143
C.2 The Inverse Kinematics for Castor Wheel Acceleration Variables . . 144
AppendixD
The Lyapunov Analysis 149
Appendix E
The Kinematics and Dynamics Modeling of Different Holo-
nomic Wheeled Mobile Robot 154
viiE.1 Kinematics Modeling of Holonomic Mobile Robots . . . . . . . . . . 154
E.1.1 Holonomic Caster Wheeled Robot (HCWR) . . . . . . . . . 154
E.1.2 Omni Directional Wheeled Robot (ODWR) . . . . . . . . . 155
E.1.3 Ramsis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
E.2 The Robots Dynamics Equations . . . . . . . . . . . . . . . . . . . 158
E.2.1 Holonomic Caster Wheeled Robot . . . . . . . . . . . . . . . 159
E.2.2 Omni Directional Wheeled robot . . . . . . . . . . . . . . . 161
E.2.3 Ramsis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
viiiList of Figures
1.1 Using WMR as explosives transporter [9] . . . . . . . . . . . . . . 2
1.2 Rigid body degrees of freedoms . . . . . . . . . . . . . . . . . . . . 2
1.3 Nonholonomic wheeled mobile robot . . . . . . . . . . . . . . . . . 3
1.4 a) Caster wheel, b) Conventional wheel, c)Omnidirectional wheel
[22] and d) Ball wheel [22] . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 A mecanum wheel mobile robot platform [28] . . . . . . . . . . . . 5
1.6 Using PCW in different WMR configurations: a) 4-wheeled plat-
form configuration [32], b) Stanford University PCW [38] c) 3-
wheeled platform configuration[36] . . . . . . . . . . . . . . . . . . 6
1.7 RNBC control structure . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 C3P platform construction . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 C3P Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 C3P Coordinates Conventions . . . . . . . . . . . . . . . . . . . . . 20
2.3 The C3P parts structure . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 C3P Dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Different steering configurations . . . . . . . . . . . . . . . . . . . . 38
˙ ˙3.2 Coupling between θ and θ . . . . . . . . . . . . . . . . . . . . . 39x x1 2
3.3 Open loop structure using C3P dynamic model . . . . . . . . . . . 43
3.4 C3P wheel configuration considering the results in Figure (3.5) . . . 44
3.5 The C3P simulation results from example (1); driving in p˙ =
T
[0.12(m/s) 0.12(m/s) 0(r/min)] . . . . . . . . . . . . . . . . . . . 45
3.6 C3P wheels configuration considering the results in Figure (3.7) . . 46
3.7 The C3P simulation results from example (2); driving in p˙ =
T
[0(m/s) −0.12(m/s) 0(r/min)] . . . . . . . . . . . . . . . . . . . 47
3.8 The geometric representation for achieving Ψ. . . . . . . . . . . . . 48
3.9 C3P simulation results with and without WCE regulator . . . . . . 49
3.10 C3P velocity controller structure . . . . . . . . . . . . . . . . . . . 51
3.11 C3P velocity controller structure . . . . . . . . . . . . . . . . . . . 51
3.12 C3P performance with and without the velocity controller . . . . . 52
ix3.13 Robot Position representation . . . . . . . . . . . . . . . . . . . . . 54
3.14 Robot trajectory for K = 20 and different K . . . . . . . . . . . 55φ er
3.15 Robot trajectory for δ = 5 and differentΦ . . . . . . . . . . . . . 55er e
3.16 Robot Position update . . . . . . . . . . . . . . . . . . . . . . . . . 56
o T o T3.17 Simulation result for driving fromp = [0 0 0 ] top = [3 3 −90 ] 57i g
o T3.18 Position update and trajectory for p = (3 3 −90 ) . . . . . . . . 58g
4.1 Dynamics Based Velocity Control Structure . . . . . . . . . . . . . 62
4.2 Position Control Structure . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 The Steering angles orientation for driving in X direction from con-
figuration (a) to configuration (b) . . . . . . . . . . . . . . . . . . . 64
4.4 The wheels velocities and acceleration for ramp input and driving
from singular condition . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 The robot velocities and acceleration for ramp input and driving
from singular condition . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Comparing dynamic and kinematic inverse solutions for deriving in
x direction from initial singular condition . . . . . . . . . . . . . . 67
4.7 The steering angles orientation for driving in y direction from con-
figuration (a) to configuration (b) . . . . . . . . . . . . . . . . . . . 68
4.8 Comparing dynamic and kinematic inverse solutions for mobility in
y direction from an initial singular condition . . . . . . . . . . . . . 69
4.9 Dynamics and Kinematics fusion block . . . . . . . . . . . . . . . . 70
4.10 Simulation results using Dynamics and Kinematics fusion . . . . . . 71
4.11 Robot Position representation. . . . . . . . . . . . . . . . . . . . . . 72
o T4.12 Positioncontrollerresultsfordrivingbetweenp = [0m,0m,0 ] toi
o Tp = [−4.5m,−4.5m,90 ] . . . . . . . . . . . . . . . . . . . . . . . 74g
4.13 The effect of the parameters K and K on the robot trajectory. . 75φ er
4.14 the effect of initial steering angles values on the C3P trajectories
for two different examples. . . . . . . . . . . . . . . . . . . . . . . . 76
5.1 The C3P open control structure . . . . . . . . . . . . . . . . . . . . 79
˙5.2 The values of V for equal uniform values of steering angles . . . . . 83
˙5.3 The values of V for non-equal uniform values of steering angles . . 85
˙5.4 3-DspaceV representationforrandomvaluesθ andθ , unstable:s s1 3
*, stable: * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 2-D Lyapunov space with random θ and θ values for differents s1 3
oK values at θ = 205 values, a)K =0.5, b) K =0.9, c)K =1.1,x s x x x2
d)K =1.5, unstable: *, stable: * . . . . . . . . . . . . . . . . . . . . 88x
6.1 The C3P practical prototype . . . . . . . . . . . . . . . . . . . . . 89
x