Robot Dynamics and Control

303 Pages·2004·5.09 MB·English

Robot Dynamics and Control
Second Edition
Mark W. Spong, Seth Hutchinson, and M. Vidyasagar
January 28, 2004
2
Contents
1 INTRODUCTION 5
1.1 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 History of Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Components and Structure of Robots . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Symbolic Representation of Robots . . . . . . . . . . . . . . . . . . . 8
1.3.2 Degrees of Freedom and Workspace . . . . . . . . . . . . . . . . . . 9
1.3.3 Classification of Robots . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Common Kinematic Arrangements . . . . . . . . . . . . . . . . . . . 11
1.3.5 Robotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.6 Accuracy and Repeatability . . . . . . . . . . . . . . . . . . . . . . 16
1.3.7 Wrists and End-Effectors . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Outline of the Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS 29
2.1 Representing Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Representing Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Rotation in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Rotations in three dimensions . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Rotational Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Composition of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Rotation with respect to the current coordinate frame . . . . . . . . 40
2.4.2 Rotation with respect to a fixed frame . . . . . . . . . . . . . . . . . 42
2.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Parameterizations of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Roll, Pitch, Yaw Angles . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5.3 Axis/Angle Representation . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Homogeneous Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 51
3
4 CONTENTS
3 FORWARDKINEMATICS:THEDENAVIT-HARTENBERGCONVEN-
TION 57
3.1 Kinematic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Denavit Hartenberg Representation . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Existence and uniqueness issues . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Assigning the coordinate frames . . . . . . . . . . . . . . . . . . . . 63
3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 INVERSE KINEMATICS 79
4.1 The General Inverse Kinematics Problem . . . . . . . . . . . . . . . . . . . 79
4.2 Kinematic Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Inverse Position: A Geometric Approach . . . . . . . . . . . . . . . . . . . 83
4.4 Inverse Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 VELOCITY KINEMATICS – THE MANIPULATOR JACOBIAN 95
5.1 Angular Velocity: The Fixed Axis Case . . . . . . . . . . . . . . . . . . . . 96
5.2 Skew Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Angular Velocity: The General Case . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Addition of Angular Velocities . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Linear Velocity of a Point Attached to a Moving Frame . . . . . . . . . . . 102
5.6 Derivation of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6.1 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6.2 Linear Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.8 The Analytical Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.9 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.9.1 Decoupling of Singularities . . . . . . . . . . . . . . . . . . . . . . . 114
5.9.2 Wrist Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.9.3 Arm Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.10 Inverse Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . 119
5.11 Redundant Robots and Manipulability . . . . . . . . . . . . . . . . . . . . 120
5.11.1 Redundant Manipulators . . . . . . . . . . . . . . . . . . . . . . . . 120
5.11.2 The Inverse Velocity Problem for Redundant Manipulators . . . . . 121
5.11.3 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . 122
5.11.4 Manipulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 COMPUTER VISION 127
6.1 The Geometry of Image Formation . . . . . . . . . . . . . . . . . . . . . . 127
6.1.1 The Camera Coordinate Frame . . . . . . . . . . . . . . . . . . . . . 128
6.1.2 Perspective Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1.3 The Image Plane and the Sensor Array . . . . . . . . . . . . . . . . 129
6.2 Camera Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS 5
6.2.1 Extrinsic Camera Parameters . . . . . . . . . . . . . . . . . . . . . . 130
6.2.2 Intrinsic Camera Parameters . . . . . . . . . . . . . . . . . . . . . . 131
6.2.3 Determining the Camera Parameters . . . . . . . . . . . . . . . . . . 131
6.3 Segmentation by Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.1 A Brief Statistics Review . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.2 Automatic Threshold Selection . . . . . . . . . . . . . . . . . . . . . 136
6.4 Connected Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.5 Position and Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5.2 The Centroid of an Object . . . . . . . . . . . . . . . . . . . . . . . 144
6.5.3 The Orientation of an Object . . . . . . . . . . . . . . . . . . . . . . 144
7 PATH PLANNING AND COLLISION AVOIDANCE 147
7.1 The Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2 Path Planning Using Configuration Space Potential Fields . . . . . . . . . 151
7.2.1 The Attractive Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.2.2 The Repulsive field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2.3 Gradient Descent Planning . . . . . . . . . . . . . . . . . . . . . . . 154
7.3 Planning Using Workspace Potential Fields . . . . . . . . . . . . . . . . . . 155
7.3.1 Defining Workspace Potential Fields . . . . . . . . . . . . . . . . . . 156
7.3.2 Mapping workspace forces to joint forces and torques. . . . . . . . . 158
7.3.3 Motion Planning Algorithm . . . . . . . . . . . . . . . . . . . . . . . 162
7.4 Using Random Motions to Escape Local Minima . . . . . . . . . . . . . . . 163
7.5 Probabilistic Roadmap Methods . . . . . . . . . . . . . . . . . . . . . . . . 164
7.5.1 Sampling the configuration space . . . . . . . . . . . . . . . . . . . . 165
7.5.2 Connecting Pairs of Configurations . . . . . . . . . . . . . . . . . . . 165
7.5.3 Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.5.4 Path Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.6 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8 TRAJECTORY PLANNING 169
8.1 The Trajectory Planning Problem . . . . . . . . . . . . . . . . . . . . . . . 169
8.2 Trajectories for Point to Point Motion . . . . . . . . . . . . . . . . . . . . . 170
8.2.1 Cubic Polynomial Trajectories . . . . . . . . . . . . . . . . . . . . . 172
8.2.2 Multiple Cubics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2.3 Quintic Polynomial Trajectories. . . . . . . . . . . . . . . . . . . . . 175
8.2.4 Linear Segments with Parabolic Blends (LSPB) . . . . . . . . . . . 180
8.2.5 Minimum Time Trajectories . . . . . . . . . . . . . . . . . . . . . . 183
8.3 Trajectories for Paths Specified by Via Points . . . . . . . . . . . . . . . . 185
8.3.1 4-3-4 trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6 CONTENTS
9 DYNAMICS 187
9.1 The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.1.1 One Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . 188
9.1.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
9.2 General Expressions for Kinetic and Potential Energy . . . . . . . . . . . . 196
9.2.1 The Inertia Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.2.2 Kinetic Energy for an n-Link Robot . . . . . . . . . . . . . . . . . . 198
9.2.3 Potential Energy for an n-Link Robot . . . . . . . . . . . . . . . . . 199
9.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.4 Some Common Configurations . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.5 Properties of Robot Dynamic Equations . . . . . . . . . . . . . . . . . . . . 210
9.5.1 The Skew Symmetry and Passivity Properties . . . . . . . . . . . . . 211
9.5.2 Bounds on the Inertia Matrix . . . . . . . . . . . . . . . . . . . . . . 212
9.5.3 Linearity in the Parameters . . . . . . . . . . . . . . . . . . . . . . . 213
9.6 Newton-Euler Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
9.7 Planar Elbow Manipulator Revisited . . . . . . . . . . . . . . . . . . . . . 221
10 INDEPENDENT JOINT CONTROL 225
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10.2 Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.3 Set-Point Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.3.1 PD Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.3.2 Performance of PD Compensators . . . . . . . . . . . . . . . . . . . 235
10.3.3 PID Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.3.4 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10.4 Feedforward Control and Computed Torque . . . . . . . . . . . . . . . . . 238
10.5 Drive Train Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
11 MULTIVARIABLE CONTROL 247
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
11.2 PD Control Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11.3 Inverse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
11.3.1 Task Space Inverse Dynamics . . . . . . . . . . . . . . . . . . . . . . 253
11.4 Robust and Adaptive Motion Control . . . . . . . . . . . . . . . . . . . . . 254
11.4.1 Robust Feedback Linearization . . . . . . . . . . . . . . . . . . . . . 255
11.4.2 Passivity Based Robust Control . . . . . . . . . . . . . . . . . . . . . 259
11.4.3 Passivity Based Adaptive Control . . . . . . . . . . . . . . . . . . . 260
12 FORCE CONTROL 263
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
12.2 Constrained Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
12.2.1 Static Force/Torque Relationships . . . . . . . . . . . . . . . . . . . 266
12.2.2 Constraint Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
CONTENTS 7
12.2.3 Natural and Artificial Constraints . . . . . . . . . . . . . . . . . . . 270
12.3 Network Models and Impedance . . . . . . . . . . . . . . . . . . . . . . . . 272
12.3.1 Impedance Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 273
12.3.2 Classification of Impedance Operators . . . . . . . . . . . . . . . . . 274
12.3.3 Th´evenin and Norton Equivalents . . . . . . . . . . . . . . . . . . . 275
12.4 Force Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
12.4.1 Impedance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
12.4.2 Hybrid Impedance Control . . . . . . . . . . . . . . . . . . . . . . . 277
13 FEEDBACK LINEARIZATION 281
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
13.2 Background: The Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . 283
13.3 Single-Input Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
13.4 Feedback Linearization for N-Link Robots . . . . . . . . . . . . . . . . . . 295
8 CONTENTS
Chapter 1
INTRODUCTION
1.1 Robotics
Robotics is a relatively young field of modern technology that crosses traditional engineer-
ing boundaries. Understanding the complexity of robots and their applications requires
knowledge of electrical engineering, mechanical engineering, systems and industrial engi-
neering, computer science, economics, and mathematics. New disciplines of engineering,
such as manufacturing engineering, applications engineering, and knowledge engineering
have emerged to deal with the complexity of the field of robotics and factory automation.
This book is concerned with fundamentals of robotics, including kinematics, dynam-
ics, motion planning, computer vision, and control. Our goal is to provide a complete
introduction to the most important concepts in these subjects as applied to industrial robot
manipulators.
The science of robotics has grown tremendously over the past twenty years, fueled by
rapid advances in computer and sensor technology as well as theoretical advances in control
and computer vision. In addition to the topics listed above, robotics encompasses several
areasnotcoveredinthistextsuchaslocomotion,includingwheeledandleggedrobots,flying
andswimmingrobots,grasping,artificialintelligence,computerarchitectures,programming
languages, and computer-aided design. A complete treatment of the discipline of robotics
would require several volumes. Nevertheless, at the present time, the vast majority of robot
applications deal with industrial robot arms operating in structured factory environments
so that a first introduction to the subject of robotics must include a rigorous treatment of
the topics in this text.
1.2 History of Robotics
The term robot was first introduced into our vocabulary by the Czech playwright Karel
Capek in his 1920 play Rossum’s Universal Robots, the word robota being the Czech word
forwork. Sincethenthetermhasbeenappliedtoagreatvarietyofmechanicaldevices,such
as teleoperators, underwater vehicles, autonomous land rovers, etc. Virtually anything that
9
10 CHAPTER 1. INTRODUCTION
operates with some degree of autonomy, usually under computer control, has at some point
been called a robot. In this text the term robot will mean a computer controlled industrial
manipulator of the type shown in Figure 1.1. This type of robot is essentially a mechanical
arm operating under computer control. Such devices, though far from the robots of science
fiction, are nevertheless extremely complex electro-mechanical systems whose analytical
descriptionrequiresadvancedmethods, andwhichpresentmanychallengingandinteresting
research problems.
Figure 1.1: The ABB IRB6600 Robot. Photo courtesy of ABB
An official definition of such a robot comes from the Robot Institute of America (RIA):
Arobotisareprogrammablemultifunctionalmanipulatordesignedtomovematerial, parts,
tools, or specialized devices through variable programmed motions for the performance of
a variety of tasks.
The key element in the above definition is the reprogrammability of robots. It is the
computer brain that gives the robot its utility and adaptability. The so-called robotics
revolution is, in fact, part of the larger computer revolution.
Even this restricted version of a robot has several features that make it attractive in
an industrial environment. Among the advantages often cited in favor of the introduction
of robots are decreased labor costs, increased precision and productivity, increased flexi-
bility compared with specialized machines, and more humane working conditions as dull,
repetitive, or hazardous jobs are performed by robots.
Therobot,aswehavedefinedit,wasbornoutofthemarriageoftwoearliertechnologies:
that of teleoperators and numerically controlled milling machines. Teleoperators,
or master-slave devices, were developed during the second world war to handle radioactive
materials. Computer numerical control (CNC) was developed because of the high precision
requiredinthemachiningofcertainitems,suchascomponentsofhighperformanceaircraft.
The first robots essentially combined the mechanical linkages of the teleoperator with the
autonomyandprogrammabilityofCNCmachines. Severalmilestonesontheroadtopresent
day robot technology are listed below.