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A Mathematical Introduction to Robotic Manipulation

Richard M. Murray

California Institute of Technology

Zexiang Li

Hong Kong University of Science and Technology

S. Shankar Sastry

University of California, Berkeley

c 1994, CRC Press

All rights reserved

This electronic edition is available from http://www.cds.caltech.edu/~murray/mlswiki. Hardcover editions may be purchased from CRC Press, http://www.crcpress.com/product/isbn/9780849379819.

This manuscript is for personal use only and may not be reproduced, in whole or in part, without written consent from the publisher.

ii

To RuthAnne (RMM)

To Jianghua (ZXL)

In memory of my father (SSS)

vi

Contents

Contents

 

vii

Preface

 

 

xiii

Acknowledgements

xvii

1 Introduction

1

1

Brief History . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Multifingered Hands and Dextrous Manipulation . . . . .

8

3

Outline of the Book . . . . . . . . . . . . . . . . . . . . .

13

 

3.1

Manipulation using single robots . . . . . . . . . .

14

3.2Coordinated manipulation using multifingered robot

 

 

hands . . . . . . . . . . . . . . . . . . . . . . . . .

15

 

3.3

Nonholonomic behavior in robotic systems . . . . .

16

4

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

18

2 Rigid Body Motion

19

1

Rigid Body Transformations . . . . . . . . . . . . . . . . .

20

2

Rotational Motion in R3 . . . . . . . . . . . . . . . . . . .

22

 

2.1

Properties of rotation matrices . . . . . . . . . . .

23

 

2.2

Exponential coordinates for rotation . . . . . . . .

27

 

2.3

Other representations . . . . . . . . . . . . . . . .

31

3

Rigid Motion in R3 . . . . . . . . . . . . . . . . . . . . . .

34

 

3.1

Homogeneous representation . . . . . . . . . . . .

36

 

3.2

Exponential coordinates for rigid motion and twists

39

 

3.3

Screws: a geometric description of twists . . . . . .

45

4

Velocity of a Rigid Body . . . . . . . . . . . . . . . . . . .

51

 

4.1

Rotational velocity . . . . . . . . . . . . . . . . . .

51

 

4.2

Rigid body velocity . . . . . . . . . . . . . . . . .

53

 

4.3

Velocity of a screw motion . . . . . . . . . . . . . .

57

 

4.4

Coordinate transformations . . . . . . . . . . . . .

58

5

Wrenches and Reciprocal Screws . . . . . . . . . . . . . .

61

 

5.1

Wrenches . . . . . . . . . . . . . . . . . . . . . . .

61

vii

 

5.2

Screw coordinates for a wrench . . . . . . . . . . .

64

 

5.3

Reciprocal screws . . . . . . . . . . . . . . . . . . .

66

6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

72

8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3 Manipulator Kinematics

81

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

81

2

Forward Kinematics . . . . . . . . . . . . . . . . . . . . .

83

 

2.1

Problem statement . . . . . . . . . . . . . . . . . .

83

 

2.2

The product of exponentials formula . . . . . . . .

85

 

2.3

Parameterization of manipulators via twists . . . .

91

 

2.4

Manipulator workspace . . . . . . . . . . . . . . .

95

3

Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . .

97

 

3.1

A planar example . . . . . . . . . . . . . . . . . .

97

 

3.2

Paden-Kahan subproblems . . . . . . . . . . . . .

99

3.3Solving inverse kinematics using subproblems . . . 104

3.4General solutions to inverse kinematics problems . 108

4

The Manipulator Jacobian . . . . . . . . . . . . . . . . . .

115

 

4.1

End-e ector velocity . . . . . . . . . . . . . . . . .

115

 

4.2

End-e ector forces . . . . . . . . . . . . . . . . . .

121

 

4.3

Singularities . . . . . . . . . . . . . . . . . . . . . .

123

 

4.4

Manipulability . . . . . . . . . . . . . . . . . . . .

127

5

Redundant and Parallel Manipulators . . . . . . . . . . .

129

 

5.1

Redundant manipulators . . . . . . . . . . . . . . .

129

 

5.2

Parallel manipulators . . . . . . . . . . . . . . . .

132

 

5.3

Four-bar linkage . . . . . . . . . . . . . . . . . . .

135

 

5.4

Stewart platform . . . . . . . . . . . . . . . . . . .

138

6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

144

8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

4 Robot Dynamics and Control

155

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

155

2

Lagrange’s Equations . . . . . . . . . . . . . . . . . . . . .

156

 

2.1

Basic formulation . . . . . . . . . . . . . . . . . . .

157

 

2.2

Inertial properties of rigid bodies . . . . . . . . . .

160

2.3Example: Dynamics of a two-link planar robot . . 164

 

2.4

Newton-Euler equations for a rigid body . . . . . .

165

3

Dynamics of Open-Chain Manipulators . . . . . . . . . .

168

 

3.1

The Lagrangian for an open-chain robot . . . . . .

168

 

3.2

Equations of motion for an open-chain manipulator

169

 

3.3

Robot dynamics and the product of exponentials

 

 

 

formula . . . . . . . . . . . . . . . . . . . . . . . .

175

4

Lyapunov Stability Theory . . . . . . . . . . . . . . . . .

179

viii

 

4.1

Basic definitions . . . . . . . . . . . . . . . . . . .

179

 

4.2

The direct method of Lyapunov . . . . . . . . . . .

181

 

4.3

The indirect method of Lyapunov . . . . . . . . .

184

 

4.4

Examples . . . . . . . . . . . . . . . . . . . . . . .

185

 

4.5

Lasalle’s invariance principle . . . . . . . . . . . .

188

5

Position Control and Trajectory Tracking . . . . . . . . .

189

 

5.1

Problem description . . . . . . . . . . . . . . . . .

190

 

5.2

Computed torque . . . . . . . . . . . . . . . . . . .

190

 

5.3

PD control . . . . . . . . . . . . . . . . . . . . . .

193

 

5.4

Workspace control . . . . . . . . . . . . . . . . . .

195

6

Control of Constrained Manipulators . . . . . . . . . . . .

200

 

6.1

Dynamics of constrained systems . . . . . . . . . .

200

 

6.2

Control of constrained manipulators . . . . . . . .

201

6.3Example: A planar manipulator moving in a slot . 203

7

Summary . . . . . . . . . . . . . . . . . . . . . . . . .

. . 206

8

Bibliography . . . . . . . . . . . . . . . . . . . . . . .

. . 207

9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . 208

5 Multifingered Hand Kinematics

211

1

Introduction to Grasping . . . . . . . . . . . . . . . .

. . 211

2

Grasp Statics . . . . . . . . . . . . . . . . . . . . . . .

. . 214

 

2.1

Contact models . . . . . . . . . . . . . . . . . .

. . 214

 

2.2

The grasp map . . . . . . . . . . . . . . . . . .

. . 218

3

Force-Closure . . . . . . . . . . . . . . . . . . . . . . .

. . 223

 

3.1

Formal definition . . . . . . . . . . . . . . . . .

. . 223

 

3.2

Constructive force-closure conditions . . . . . .

. . 224

4

Grasp Planning . . . . . . . . . . . . . . . . . . . . . .

. . 229

 

4.1

Bounds on number of required contacts . . . .

. . 229

 

4.2

Constructing force-closure grasps . . . . . . . .

. . 232

5

Grasp Constraints . . . . . . . . . . . . . . . . . . . .

. . 234

 

5.1

Finger kinematics . . . . . . . . . . . . . . . .

. . 234

 

5.2

Properties of a multifingered grasp . . . . . . .

. . 237

 

5.3

Example: Two SCARA fingers grasping a box

. . 240

6

Rolling Contact Kinematics . . . . . . . . . . . . . . .

. . 242

 

6.1

Surface models . . . . . . . . . . . . . . . . . .

. . 243

 

6.2

Contact kinematics . . . . . . . . . . . . . . . .

. . 248

 

6.3

Grasp kinematics with rolling . . . . . . . . . .

. . 253

7

Summary . . . . . . . . . . . . . . . . . . . . . . . . .

. . 256

8

Bibliography . . . . . . . . . . . . . . . . . . . . . . .

. . 257

9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . 259

ix

6 Hand Dynamics and Control

265

1

Lagrange’s Equations with Constraints . . . . . .

. . . . . 265

 

1.1

Pfa an constraints . . . . . . . . . . . . .

. . . . . 266

 

1.2

Lagrange multipliers . . . . . . . . . . . .

. . . . . 269

 

1.3

Lagrange-d’Alembert formulation . . . . .

. . . . . 271

 

1.4

The nature of nonholonomic constraints .

. . . . . 274

2

Robot Hand Dynamics . . . . . . . . . . . . . . .

. . . . . 276

 

2.1

Derivation and properties . . . . . . . . .

. . . . . 276

 

2.2

Internal forces . . . . . . . . . . . . . . . .

. . . . 279

 

2.3

Other robot systems . . . . . . . . . . . . .

. . . . 281

3

Redundant and Nonmanipulable Robot Systems

. . . . . 285

 

3.1

Dynamics of redundant manipulators . . . .

. . . . 286

 

3.2

Nonmanipulable grasps . . . . . . . . . . .

. . . . 290

 

3.3

Example: Two-fingered SCARA grasp . . .

. . . . 291

4

Kinematics and Statics of Tendon Actuation . . .

. . . . 293

 

4.1

Inelastic tendons . . . . . . . . . . . . . . .

. . . . 294

 

4.2

Elastic tendons . . . . . . . . . . . . . . . .

. . . . 296

4.3Analysis and control of tendon-driven fingers . . . 298

5

Control of Robot Hands . . . . . . . . . . . . . . . . . . .

300

 

5.1

Extending controllers . . . . . . . . . . . . . . . .

300

 

5.2

Hierarchical control structures . . . . . . . . . . .

302

6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

7

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

313

8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

314

7 Nonholonomic Behavior in Robotic Systems

317

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

317

2

Controllability and Frobenius’ Theorem . . . . . . . . . .

321

 

2.1

Vector fields and flows . . . . . . . . . . . . . . . .

322

 

2.2

Lie brackets and Frobenius’ theorem . . . . . . . .

323

 

2.3

Nonlinear controllability . . . . . . . . . . . . . . .

328

3

Examples of Nonholonomic Systems . . . . . . . . . . . .

332

4

Structure of Nonholonomic Systems . . . . . . . . . . . .

339

4.1Classification of nonholonomic distributions . . . . 340

 

4.2

Examples of nonholonomic systems, continued

. . 341

 

4.3

Philip Hall basis . . . . . . . . . . . . . . . . .

. . 344

5

Summary . . . . . . . . . . . . . . . . . . . . . . . . .

. . 346

6

Bibliography . . . . . . . . . . . . . . . . . . . . . . .

. . 347

7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . 349

8 Nonholonomic Motion Planning

355

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . .

. . 355

2Steering Model Control Systems Using Sinusoids . . . . . 358

2.1First-order controllable systems: Brockett’s system 358

2.2

Second-order controllable systems . . . . . . . . .

361

x

 

 

2.3

Higher-order systems: chained form systems . . . .

363

 

3

General Methods for Steering . . . . . . . . . . . . . . . .

366

 

 

3.1

Fourier techniques . . . . . . . . . . . . . . . . . .

367

 

 

3.2

Conversion to chained form . . . . . . . . . . . . .

369

 

 

3.3

Optimal steering of nonholonomic systems . . . . .

371

 

 

3.4

Steering with piecewise constant inputs . . . . . .

375

 

4

Dynamic Finger Repositioning . . . . . . . . . . . . . . .

382

 

 

4.1

Problem description . . . . . . . . . . . . . . . . .

382

 

 

4.2

Steering using sinusoids . . . . . . . . . . . . . . .

383

 

 

4.3

Geometric phase algorithm . . . . . . . . . . . . .

385

 

5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

389

 

6

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

390

 

7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

391

9

Future Prospects

395

 

1

Robots in Hazardous Environments . . . . . . . . . . . . .

396

 

2

Medical Applications for Multifingered Hands . . . . . . .

398

 

3

Robots on a Small Scale: Microrobotics . . . . . . . . . .

399

A

Lie Groups and Robot Kinematics

403

 

Lie Groups and Robot Kinematics403

 

 

1

Di erentiable Manifolds . . . . . . . . . . . . . . . . . . .

403

 

 

1.1

Manifolds and maps . . . . . . . . . . . . . . . . .

403

 

 

1.2

Tangent spaces and tangent maps . . . . . . . . .

404

 

 

1.3

Cotangent spaces and cotangent maps . . . . . . .

405

 

 

1.4

Vector fields . . . . . . . . . . . . . . . . . . . . . .

406

 

 

1.5

Di erential forms . . . . . . . . . . . . . . . . . . .

408

 

2

Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . .

408

 

 

2.1

Definition and examples . . . . . . . . . . . . . . .

408

 

 

2.2

The Lie algebra associated with a Lie group . . . .

409

 

 

2.3

The exponential map . . . . . . . . . . . . . . . . .

412

 

 

2.4

Canonical coordinates on a Lie group . . . . . . .

414

 

 

2.5

Actions of Lie groups . . . . . . . . . . . . . . . .

415

 

3

The Geometry of the Euclidean Group . . . . . . . . . . .

416

 

 

3.1

Basic properties . . . . . . . . . . . . . . . . . . .

416

 

 

3.2

Metric properties of SE(3) . . . . . . . . . . . . . .

422

 

 

3.3

Volume forms on SE(3) . . . . . . . . . . . . . . .

430

B

A Mathematica Package for Screw Calculus

435

Bibliography

 

441

Index

 

 

449

xi

xii

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