Geometric control of an under-actuated balancing robot

134 Pages·2015·6.76 MB·English

by  Cees Verdier

Geometric control of an under-
actuated balancing robot
Cees Verdier
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Delft Center for Systems and Control

Geometric control of an
under-actuated balancing robot
Master of Science Thesis
For the degree of Master of Science in Systems and Control at Delft
University of Technology
Cees Verdier
22nd July 2015
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of
Technology
The work in this thesis was conducted in collaboration with Alten Mechatronics. Their
cooperation is hereby gratefully acknowledged.
Copyright (cid:13)c Delft Center for Systems and Control (DCSC)
All rights reserved.
Delft University of Technology
Department of
Delft Center for Systems and Control (DCSC)
The undersigned hereby certify that they have read and recommend to the Faculty of
Mechanical, Maritime and Materials Engineering (3mE) for acceptance a thesis
entitled
Geometric control of an under-actuated balancing robot
by
Cees Verdier
in partial fulfillment of the requirements for the degree of
Master of Science Systems and Control
Dated: 22nd July 2015
Supervisor(s):
Dr.ir. G. Delgado Lopes
Ir. G-J. Heldens
Reader(s):
Dr.ir. G. Delgado Lopes
Prof.dr.ir. M. Wisse
Dr.ir. D. Jeltsema
Ir. G-J. Heldens

Abstract
Ball-balancing robots, or Ballbots, are under-actuated omni-directional mobile robots
that balance on top of a single ball. The under-actuated nature arises from the fact
that both position and attitude of the robot are actuated by the same actuators. This
thesis introduces a geometric approach to the control of ball-balancing robots.
In this approach, a new full 3D model is derived using screw theory. Based on this
model, a geometric observer and geometric controller are proposed. Two methodologies
are implemented, a computed torque controller and a sliding mode controller, that can
track the attitude of the robot on either the special orthogonal group SO(3) or the
2-sphere S2. Position control is achieved through the use of the relation between the
desired linear acceleration and the attitude of the robot. The resulting desired attitude
is tracked by one of the geometric attitude controllers.
Simulation results show the effectiveness of the proposed controllers. The proposed
sliding mode controller is shown to be more robust to model uncertainties. The posi-
tion controller is shown to be able to control the position and follow trajectories, but
overall global stability is not guaranteed. Recommendations are made to improve the
performance. For the geometric observer, stability is shown under a set of assumptions.
The work is concluded with a set of experiments on a real platform.
Master of Science Thesis Cees Verdier
ii
Cees Verdier Master of Science Thesis
Contents
Acknowledgements ix
1 Introduction 1
1-1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1-2 Alten Mechatronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1-3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1-4 Structure of the MSc thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Background 7
2-1 Brief history of ball-balancing robots . . . . . . . . . . . . . . . . . . . . 7
2-2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2-3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Mathematical preliminaries 15
3-1 Screw theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3-2 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3-3 Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Modelling of the robot 23
4-1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4-2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4-3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4-4 External forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4-4-1 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4-4-2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4-5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4-6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Geometric Nonlinear control 39
5-1 Controller requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Master of Science Thesis Cees Verdier
iv Contents
5-2 Attitude error function on S2 . . . . . . . . . . . . . . . . . . . . . . . . 42
5-3 Balancing controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5-3-1 Inversion of g (q˙ ,q¯) . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 r
5-3-2 Geometric computed torque control . . . . . . . . . . . . . . . . . 44
5-3-3 Geometric sliding mode control . . . . . . . . . . . . . . . . . . . 49
5-4 Position control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5-5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5-5-1 Attitude control . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5-5-2 Parametric uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 66
5-5-3 Unmodelled friction . . . . . . . . . . . . . . . . . . . . . . . . . 67
5-5-4 Position control . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5-5-5 Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . 71
5-5-6 Trajectory tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5-5-7 Extension to other systems . . . . . . . . . . . . . . . . . . . . . 81
6 Geometric observer 83
6-1 Available measurement data . . . . . . . . . . . . . . . . . . . . . . . . . 84
6-1-1 Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6-1-2 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6-1-3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6-1-4 Reconstructing a rotation matrix . . . . . . . . . . . . . . . . . . 86
6-2 Global geometric observer on SO(3) . . . . . . . . . . . . . . . . . . . . 87
6-2-1 Candidate Lyapunov function . . . . . . . . . . . . . . . . . . . . 88
6-2-2 Proposed observer . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6-2-3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6-3 Geometric ballbot observer . . . . . . . . . . . . . . . . . . . . . . . . . 90
6-3-1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6-4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Implementation 97
7-1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7-1-1 Inertia of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 98
7-1-2 Inertia of the body . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7-2 Linear approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7-2-1 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7-2-2 LQR controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7-2-3 Observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7-2-4 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7-2-5 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7-3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8 Conclusions and recommendations 109
8-1 Recommendations for future research . . . . . . . . . . . . . . . . . . . . 111
Bibliography 113
Cees Verdier Master of Science Thesis