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Modern Robotics Mechanics--Planning and Control 平板电脑版本.pdf
Foreword by Roger Brockett
Foreword by Matthew Mason
Preface
Preview
Configuration Space
Degrees of Freedom of a Rigid Body
Degrees of Freedom of a Robot
Robot Joints
Grübler's Formula
Configuration Space: Topology and Representation
Configuration Space Topology
Configuration Space Representation
Configuration and Velocity Constraints
Task Space and Workspace
Summary
Notes and References
Exercises
Rigid-Body Motions
Rigid-Body Motions in the Plane
Rotations and Angular Velocities
Rotation Matrices
Angular Velocities
Exponential Coordinate Representation of Rotation
Rigid-Body Motions and Twists
Homogeneous Transformation Matrices
Twists
Exponential Coordinate Representation of Rigid-Body Motions
Wrenches
Summary
Software
Notes and References
Exercises
Forward Kinematics
Product of Exponentials Formula
First Formulation: Screw Axes in the Base Frame
Examples
Second Formulation: Screw Axes in the End-Effector Frame
The Universal Robot Description Format
Summary
Software
Notes and References
Exercises
Velocity Kinematics and Statics
Manipulator Jacobian
Space Jacobian
Body Jacobian
Visualizing the Space and Body Jacobian
Relationship between the Space and Body Jacobian
Alternative Notions of the Jacobian
Looking Ahead to Inverse Velocity Kinematics
Statics of Open Chains
Singularity Analysis
Manipulability
Summary
Software
Notes and References
Exercises
Inverse Kinematics
Analytic Inverse Kinematics
6R PUMA-Type Arm
Stanford-Type Arms
Numerical Inverse Kinematics
Newton–Raphson Method
Numerical Inverse Kinematics Algorithm
Inverse Velocity Kinematics
A Note on Closed Loops
Summary
Software
Notes and References
Exercises
Kinematics of Closed Chains
Inverse and Forward Kinematics
3RPR Planar Parallel Mechanism
Stewart–Gough Platform
General Parallel Mechanisms
Differential Kinematics
Stewart–Gough Platform
General Parallel Mechanisms
Singularities
Summary
Notes and References
Exercises
Dynamics of Open Chains
Lagrangian Formulation
Basic Concepts and Motivating Examples
General Formulation
Understanding the Mass Matrix
Lagrangian Dynamics vs. Newton–Euler Dynamics
Dynamics of a Single Rigid Body
Classical Formulation
Twist–Wrench Formulation
Dynamics in Other Frames
Newton–Euler Inverse Dynamics
Derivation
Newton-Euler Inverse Dynamics Algorithm
Dynamic Equations in Closed Form
Forward Dynamics of Open Chains
Dynamics in the Task Space
Constrained Dynamics
Robot Dynamics in the URDF
Actuation, Gearing, and Friction
DC Motors and Gearing
Apparent Inertia
Newton–Euler Inverse Dynamics Algorithm Accounting for Motor Inertias and Gearing
Friction
Joint and Link Flexibility
Summary
Software
Notes and References
Exercises
Trajectory Generation
Definitions
Point-to-Point Trajectories
Straight-Line Paths
Time Scaling a Straight-Line Path
Polynomial Via Point Trajectories
Time-Optimal Time Scaling
The (s,) Phase Plane
The Time-Scaling Algorithm
A Variation on the Time-Scaling Algorithm
Assumptions and Caveats
Summary
Software
Notes and References
Exercises
Motion Planning
Overview of Motion Planning
Types of Motion Planning Problems
Properties of Motion Planners
Motion Planning Methods
Foundations
Configuration Space Obstacles
Distance to Obstacles and Collision Detection
Graphs and Trees
Graph Search
Complete Path Planners
Grid Methods
Multi-Resolution Grid Representation
Grid Methods with Motion Constraints
Sampling Methods
The RRT Algorithm
The PRM Algorithm
Virtual Potential Fields
A Point in C-space
Navigation Functions
Workspace Potential
Wheeled Mobile Robots
Use of Potential Fields in Planners
Nonlinear Optimization
Smoothing
Summary
Notes and References
Exercises
Robot Control
Control System Overview
Error Dynamics
Error Response
Linear Error Dynamics
Motion Control with Velocity Inputs
Motion Control of a Single Joint
Motion Control of a Multi-joint Robot
Task-Space Motion Control
Motion Control with Torque or Force Inputs
Motion Control of a Single Joint
Motion Control of a Multi-joint Robot
Task-Space Motion Control
Force Control
Hybrid Motion–Force Control
Natural and Artificial Constraints
A Hybrid Motion–Force Controller
Impedance Control
Impedance-Control Algorithm
Admittance-Control Algorithm
Low-Level Joint Force/Torque Control
Other Topics
Summary
Software
Notes and References
Exercises
Grasping and Manipulation
Contact Kinematics
First-Order Analysis of a Single Contact
Contact Types: Rolling, Sliding, and Breaking Free
Multiple Contacts
Collections of Bodies
Other Types of Contacts
Planar Graphical Methods
Form Closure
Contact Forces and Friction
Friction
Planar Graphical Methods
Force Closure
Duality of Force and Motion Freedoms
Manipulation
Summary
Notes and References
Exercises
Wheeled Mobile Robots
Types of Wheeled Mobile Robots
Omnidirectional Wheeled Mobile Robots
Modeling
Motion Planning
Feedback Control
Nonholonomic Wheeled Mobile Robots
Modeling
Controllability
Motion Planning
Feedback Control
Odometry
Mobile Manipulation
Summary
Notes and References
Exercises
Summary of Useful Formulas
Other Representations of Rotations
Euler Angles
Algorithm for Computing the ZYX Euler Angles
Other Euler Angle Representations
Roll–Pitch–Yaw Angles
Unit Quaternions
Cayley–Rodrigues Parameters
Denavit–Hartenberg Parameters
Assigning Link Frames
Why Four Parameters are Sufficient
Manipulator Forward Kinematics
Examples
Relation Between the PoE and D–H Representations
A Final Comparison
Optimization and Lagrange Multipliers
Bibliography
Index
MODERN ROBOTICS
MECHANICS, PLANNING, AND CONTROL
Kevin M. Lynch and Frank C. Park
May 3, 2017
This document is the preprint version of
Modern Robotics
Mechanics, Planning, and Control
c Kevin M. Lynch and Frank C. Park
This preprint is being made available for personal use only and not for further
distribution. The book will be published by Cambridge University Press in
May 2017, ISBN 9781107156302. Citations of the book should cite Cambridge
University Press as the publisher, with a publication date of 2017. Original
figures from this book may be reused provided proper citation is given. More
information on the book, including software, videos, and a feedback form can
be found at http://modernrobotics.org. Comments are welcome!
Contents
Foreword by Roger Brockett
Foreword by Matthew Mason
Preface
1 Preview
2 Configuration Space
2.1 Degrees of Freedom of a Rigid Body . . . . . . . . . . . . . . . .
2.2 Degrees of Freedom of a Robot . . . . . . . . . . . . . . . . . . .
2.2.1 Robot Joints
. . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Gr¨ubler’s Formula . . . . . . . . . . . . . . . . . . . . . .
2.3 Configuration Space: Topology and Representation . . . . . . . .
2.3.1 Configuration Space Topology . . . . . . . . . . . . . . . .
2.3.2 Configuration Space Representation . . . . . . . . . . . .
2.4 Configuration and Velocity Constraints . . . . . . . . . . . . . . .
2.5 Task Space and Workspace
. . . . . . . . . . . . . . . . . . . . .
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Exercises
3 Rigid-Body Motions
3.1 Rigid-Body Motions in the Plane . . . . . . . . . . . . . . . . . .
3.2 Rotations and Angular Velocities . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
3.2.1 Rotation Matrices
3.2.2 Angular Velocities
. . . . . . . . . . . . . . . . . . . . . .
3.2.3 Exponential Coordinate Representation of Rotation . . .
3.3 Rigid-Body Motions and Twists . . . . . . . . . . . . . . . . . . .
i
ix
xi
xiii
1
11
12
15
16
17
23
23
25
29
32
36
38
38
59
62
68
68
76
79
89
ii
Contents
3.3.1 Homogeneous Transformation Matrices
. . . . . . . . . .
3.3.2 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Exponential Coordinate Representation of Rigid-Body Mo-
89
97
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4 Wrenches
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.8 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Forward Kinematics
137
4.1 Product of Exponentials Formula . . . . . . . . . . . . . . . . . . 140
4.1.1 First Formulation: Screw Axes in the Base Frame
. . . . 141
4.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Second Formulation: Screw Axes in the End-Effector Frame148
4.1.3
4.2 The Universal Robot Description Format
. . . . . . . . . . . . . 152
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.6 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5 Velocity Kinematics and Statics
171
5.1 Manipulator Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 178
5.1.1
Space Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 178
5.1.2 Body Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 183
5.1.3 Visualizing the Space and Body Jacobian . . . . . . . . . 185
5.1.4 Relationship between the Space and Body Jacobian . . . 187
5.1.5 Alternative Notions of the Jacobian . . . . . . . . . . . . 187
5.1.6 Looking Ahead to Inverse Velocity Kinematics
. . . . . . 189
5.2 Statics of Open Chains . . . . . . . . . . . . . . . . . . . . . . . . 190
5.3 Singularity Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4 Manipulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 201
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.8 Exercises
6 Inverse Kinematics
219
6.1 Analytic Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . 221
6R PUMA-Type Arm . . . . . . . . . . . . . . . . . . . . 221
Stanford-Type Arms . . . . . . . . . . . . . . . . . . . . . 225
6.1.1
6.1.2
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iii
6.2 Numerical Inverse Kinematics . . . . . . . . . . . . . . . . . . . . 226
6.2.1 Newton–Raphson Method . . . . . . . . . . . . . . . . . . 227
6.2.2 Numerical Inverse Kinematics Algorithm . . . . . . . . . 227
6.3
Inverse Velocity Kinematics . . . . . . . . . . . . . . . . . . . . . 232
6.4 A Note on Closed Loops . . . . . . . . . . . . . . . . . . . . . . . 234
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 236
6.8 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7 Kinematics of Closed Chains
7.1
Inverse and Forward Kinematics
7.1.1
7.1.2
7.1.3 General Parallel Mechanisms
245
. . . . . . . . . . . . . . . . . . 247
3×RPR Planar Parallel Mechanism . . . . . . . . . . . . . 247
Stewart–Gough Platform . . . . . . . . . . . . . . . . . . 249
. . . . . . . . . . . . . . . . 251
7.2 Differential Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 252
Stewart–Gough Platform . . . . . . . . . . . . . . . . . . 252
. . . . . . . . . . . . . . . . 254
7.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.6 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.2.1
7.2.2 General Parallel Mechanisms
8 Dynamics of Open Chains
271
8.1 Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . 272
8.1.1 Basic Concepts and Motivating Examples . . . . . . . . . 272
8.1.2 General Formulation . . . . . . . . . . . . . . . . . . . . . 277
8.1.3 Understanding the Mass Matrix . . . . . . . . . . . . . . 279
8.1.4 Lagrangian Dynamics vs. Newton–Euler Dynamics . . . . 281
8.2 Dynamics of a Single Rigid Body . . . . . . . . . . . . . . . . . . 283
8.2.1 Classical Formulation . . . . . . . . . . . . . . . . . . . . 283
8.2.2 Twist–Wrench Formulation . . . . . . . . . . . . . . . . . 288
. . . . . . . . . . . . . . . . . 290
8.2.3 Dynamics in Other Frames
. . . . . . . . . . . . . . . . . . 291
8.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.3.2 Newton-Euler Inverse Dynamics Algorithm . . . . . . . . 294
8.4 Dynamic Equations in Closed Form . . . . . . . . . . . . . . . . . 294
8.5 Forward Dynamics of Open Chains . . . . . . . . . . . . . . . . . 298
8.6 Dynamics in the Task Space . . . . . . . . . . . . . . . . . . . . . 300
8.7 Constrained Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 301
8.3 Newton–Euler Inverse Dynamics
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iv
Contents
8.8 Robot Dynamics in the URDF . . . . . . . . . . . . . . . . . . . 303
8.9 Actuation, Gearing, and Friction . . . . . . . . . . . . . . . . . . 303
8.9.1 DC Motors and Gearing . . . . . . . . . . . . . . . . . . . 305
8.9.2 Apparent Inertia . . . . . . . . . . . . . . . . . . . . . . . 310
8.9.3 Newton–Euler Inverse Dynamics Algorithm Accounting
for Motor Inertias and Gearing . . . . . . . . . . . . . . . 312
8.9.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.9.5
Joint and Link Flexibility . . . . . . . . . . . . . . . . . . 314
8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.11 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
8.12 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 320
8.13 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9 Trajectory Generation
325
9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
. . . . . . . . . . . . . . . . . . . . . 326
9.2 Point-to-Point Trajectories
9.2.1
Straight-Line Paths
. . . . . . . . . . . . . . . . . . . . . 326
9.2.2 Time Scaling a Straight-Line Path . . . . . . . . . . . . . 328
9.3 Polynomial Via Point Trajectories
. . . . . . . . . . . . . . . . . 334
9.4 Time-Optimal Time Scaling . . . . . . . . . . . . . . . . . . . . . 336
9.4.1 The (s, ˙s) Phase Plane . . . . . . . . . . . . . . . . . . . . 339
9.4.2 The Time-Scaling Algorithm . . . . . . . . . . . . . . . . 341
9.4.3 A Variation on the Time-Scaling Algorithm . . . . . . . . 342
9.4.4 Assumptions and Caveats . . . . . . . . . . . . . . . . . . 344
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
9.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 347
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
9.8 Exercises
10 Motion Planning
353
10.1 Overview of Motion Planning . . . . . . . . . . . . . . . . . . . . 353
10.1.1 Types of Motion Planning Problems . . . . . . . . . . . . 354
10.1.2 Properties of Motion Planners
. . . . . . . . . . . . . . . 355
10.1.3 Motion Planning Methods . . . . . . . . . . . . . . . . . . 356
10.2 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
10.2.1 Configuration Space Obstacles
. . . . . . . . . . . . . . . 358
10.2.2 Distance to Obstacles and Collision Detection . . . . . . . 362
10.2.3 Graphs and Trees . . . . . . . . . . . . . . . . . . . . . . . 364
10.2.4 Graph Search . . . . . . . . . . . . . . . . . . . . . . . . . 365
10.3 Complete Path Planners . . . . . . . . . . . . . . . . . . . . . . . 368
May 2017 preprint of Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org
Contents
v
10.6 Virtual Potential Fields
10.4 Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
10.4.1 Multi-Resolution Grid Representation . . . . . . . . . . . 372
. . . . . . . . . . 373
10.4.2 Grid Methods with Motion Constraints
10.5 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 378
10.5.1 The RRT Algorithm . . . . . . . . . . . . . . . . . . . . . 379
10.5.2 The PRM Algorithm . . . . . . . . . . . . . . . . . . . . . 384
. . . . . . . . . . . . . . . . . . . . . . . 386
10.6.1 A Point in C-space . . . . . . . . . . . . . . . . . . . . . . 386
10.6.2 Navigation Functions . . . . . . . . . . . . . . . . . . . . . 389
10.6.3 Workspace Potential
. . . . . . . . . . . . . . . . . . . . . 390
10.6.4 Wheeled Mobile Robots . . . . . . . . . . . . . . . . . . . 391
10.6.5 Use of Potential Fields in Planners . . . . . . . . . . . . . 392
10.7 Nonlinear Optimization . . . . . . . . . . . . . . . . . . . . . . . 392
10.8 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
10.10Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 397
10.11Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
11 Robot Control
403
11.1 Control System Overview . . . . . . . . . . . . . . . . . . . . . . 404
11.2 Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
11.2.1 Error Response . . . . . . . . . . . . . . . . . . . . . . . . 406
11.2.2 Linear Error Dynamics
. . . . . . . . . . . . . . . . . . . 406
11.3 Motion Control with Velocity Inputs . . . . . . . . . . . . . . . . 413
11.3.1 Motion Control of a Single Joint
. . . . . . . . . . . . . . 414
11.3.2 Motion Control of a Multi-joint Robot . . . . . . . . . . . 418
11.3.3 Task-Space Motion Control
. . . . . . . . . . . . . . . . . 419
11.4 Motion Control with Torque or Force Inputs . . . . . . . . . . . . 420
11.4.1 Motion Control of a Single Joint
. . . . . . . . . . . . . . 421
11.4.2 Motion Control of a Multi-joint Robot . . . . . . . . . . . 429
11.4.3 Task-Space Motion Control
. . . . . . . . . . . . . . . . . 433
11.5 Force Control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
11.6 Hybrid Motion–Force Control . . . . . . . . . . . . . . . . . . . . 437
11.6.1 Natural and Artificial Constraints
. . . . . . . . . . . . . 437
11.6.2 A Hybrid Motion–Force Controller . . . . . . . . . . . . . 439
11.7 Impedance Control . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.7.1 Impedance-Control Algorithm . . . . . . . . . . . . . . . . 443
11.7.2 Admittance-Control Algorithm . . . . . . . . . . . . . . . 444
. . . . . . . . . . . . . . . 445
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
11.8 Low-Level Joint Force/Torque Control
11.9 Other Topics
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Contents
11.10Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
11.11Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
11.12Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 452
11.13Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
12 Grasping and Manipulation
12.1 Contact Kinematics
461
. . . . . . . . . . . . . . . . . . . . . . . . . 463
12.1.1 First-Order Analysis of a Single Contact . . . . . . . . . . 463
12.1.2 Contact Types: Rolling, Sliding, and Breaking Free . . . . 465
12.1.3 Multiple Contacts
. . . . . . . . . . . . . . . . . . . . . . 468
12.1.4 Collections of Bodies . . . . . . . . . . . . . . . . . . . . . 472
12.1.5 Other Types of Contacts . . . . . . . . . . . . . . . . . . . 472
12.1.6 Planar Graphical Methods . . . . . . . . . . . . . . . . . . 473
12.1.7 Form Closure . . . . . . . . . . . . . . . . . . . . . . . . . 478
12.2 Contact Forces and Friction . . . . . . . . . . . . . . . . . . . . . 484
12.2.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
12.2.2 Planar Graphical Methods . . . . . . . . . . . . . . . . . . 487
12.2.3 Force Closure . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.2.4 Duality of Force and Motion Freedoms . . . . . . . . . . . 494
12.3 Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
12.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 503
12.6 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
13 Wheeled Mobile Robots
513
13.1 Types of Wheeled Mobile Robots . . . . . . . . . . . . . . . . . . 514
13.2 Omnidirectional Wheeled Mobile Robots . . . . . . . . . . . . . . 515
13.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
13.2.2 Motion Planning . . . . . . . . . . . . . . . . . . . . . . . 520
13.2.3 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 520
13.3 Nonholonomic Wheeled Mobile Robots . . . . . . . . . . . . . . . 520
13.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
13.3.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . 528
13.3.3 Motion Planning . . . . . . . . . . . . . . . . . . . . . . . 537
13.3.4 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 542
13.4 Odometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
13.5 Mobile Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . 548
13.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
13.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . 554
13.8 Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
May 2017 preprint of Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org
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