Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory (eBook)

Dr. Jaime Gallardo-Alvarado is Professor in the Department of Mechanical Engineering,Instituto Tecnologico de Celaya, TNM, Mexico

Preface 8
Acknowledgments 10
Contents 12
List of Tables 18
List of Figures 20
Part I General Introduction 24
1 An Overview of the Theory of Screws 25
1.1 The Theory of Screws: Historical Contributions 25
1.2 Notable Scientists in Screw Theory 29
References 38
2 An Overview of Parallel Manipulators 41
2.1 Typical Parallel Manipulators 41
References 48
Part II Fundamentals of the Theory of Screws 51
3 Mathematical Background 52
3.1 Introduction 52
3.2 Preamble 52
3.2.1 Velocity State or Twist About a Screw of a Rigid Body 53
3.2.2 The Velocity State as an Equivalence Relationship 56
3.2.3 The Velocity State and the Associated Screw of Rigid Body Motion 58
3.2.4 Plücker Coordinates of Lines 62
3.2.5 Lower Kinematic Pairs and Their Equivalent Infinitesimal Screws 65
3.3 The Lie Algebra se(3) of the Euclidean Group SE(3) 66
3.3.1 The Lie Product and the Euclidean Motion 74
3.3.2 Geometric Interpretation of the Lie Product 76
3.4 Helicoidal Vector Fields 77
3.5 Exercises 82
References 83
4 Velocity Analysis 85
4.1 Introduction 85
4.2 Fundamental Equations of Velocity 85
4.3 Equations of Velocity in Screw Form 100
4.4 Exercises 113
References 116
Part III Higher-Order Kinematic Analyses of Rigid Body 117
5 Acceleration Analysis 118
5.1 Introduction 118
5.2 Fundamental Equations of Acceleration 119
5.3 Equations of Acceleration in Screw Form 122
5.4 Exercises 148
References 149
6 Jerk Analysis 152
6.1 Introduction 152
6.2 Fundamental Jerk Equations 153
6.3 Jerk Equations in Screw Form 160
6.4 Exercises 173
References 175
7 Hyper-Jerk Analysis 178
7.1 Introduction 178
7.2 Fundamental Hyper-Jerk Equations 179
7.3 Hyper-Jerk Equations in Screw Form 186
7.4 Exercises 202
References 204
Part IV Kinematics of Parallel Manipulators by Means of Screw Theory Exemplified 206
8 3R2T Parallel Manipulator 207
8.1 Introduction 207
8.2 Description of the 3R2T Parallel Manipulator 208
8.3 Finite Kinematics of the 3R2T Robot 209
8.3.1 Forward Displacement Analysis of the 3R2T Robot 209
8.3.2 Inverse Displacement Analysis of the 3R2T Robot 211
8.4 Infinitesimal Kinematics of the 3R2T Robot 211
8.4.1 Velocity Analysis of the 3R2T Parallel Manipulator 211
8.4.2 Acceleration Analysis of the 3R2T Parallel Manipulator 214
8.5 Exercises 220
References 220
9 Two-Degree-of-Freedom Parallel Wrist 222
9.1 Introduction 222
9.2 Description of the Parallel Wrist 223
9.3 Finite Kinematics of the Parallel Wrist 224
9.3.1 Forward Displacement Analysis of the Parallel Wrist 224
9.3.2 Inverse Displacement Analysis of the Parallel Wrist 226
9.4 Infinitesimal Kinematics of the Parallel Wrist 226
9.4.1 Velocity Analysis of the Parallel Wrist 227
9.4.2 Acceleration Analysis of the Parallel Wrist 230
9.4.3 Jerk Analysis of the Parallel Wrist 231
9.4.4 Singularity Analysis of the Parallel Wrist 232
9.5 Exercises 233
References 234
10 3-RRPS Parallel Manipulator 235
10.1 Introduction 235
10.2 Description of the 3-RRPS Parallel Manipulator 236
10.3 Finite Kinematics of the 3-RRPS Manipulator 237
10.3.1 Forward Displacement Analysis of the 3-RRPS Manipulator 237
10.3.2 Inverse Displacement Analysis of the 3-RRPS Manipulator 239
10.4 Infinitesimal Kinematics of the 3-RRPS Manipulator 243
10.4.1 Velocity Analysis of the 3-RRPS Manipulator 243
10.4.2 Acceleration Analysis of the 3-RRPS Manipulator 244
10.4.3 Jerk Analysis of the 3-RRPS Manipulator 246
10.5 Exercises 248
References 250
11 3RRRS+3RRPS Parallel Manipulator 252
11.1 Introduction 252
11.2 Description of the 3RRRS+3RRPS Parallel Manipulator 252
11.3 Finite Kinematics of the 3RRRS+3RRPS Parallel Manipulator 254
11.3.1 Forward Displacement Analysis of the 3RRRS+3RRPS Parallel Manipulator 254
11.3.2 Closed-Form Solution of the Forward Position Analysis Using Three Rotary Sensors 256
11.3.3 Inverse Displacement Analysis of the 3RRRS+3RRPS Parallel Manipulator 257
11.4 Infinitesimal Kinematics of the 3RRRS+3RRPS Parallel Manipulator 257
11.4.1 Velocity Analysis of the 3RRRS+3RRPS Parallel Manipulator 258
11.4.2 Acceleration Analysis of the 3RRRS+3RRPS Parallel Manipulator 259
11.4.3 Jerk Analysis of the 3RRRS+3RRPS Parallel Manipulator 261
11.4.4 Hyper-Jerk Analysis of the 3RRRS+3RRPS Parallel Manipulator 263
11.5 Exercises 267
References 267
Part V Emblematic Parallel Manipulators 268
12 Gough's Tyre Testing Machine 269
12.1 Introduction 269
12.2 Description of the General Parallel Manipulator 271
12.3 Finite Kinematics of the General Hexapod 272
12.3.1 Forward Displacement Analysis of the Hexapod 272
12.3.1.1 Typical Formulation of the Forward Displacement Analysis of the Gough Platform 272
12.3.1.2 Forward Displacement Analysis of the Six-Legged Parallel Manipulator with Planar Platforms 274
12.3.1.3 Forward Displacement Analysis of the Six-Legged Parallel Manipulator with Three-Dimensional Platforms: The General Case 279
12.3.2 Inverse Displacement Analysis of the Hexapod 287
12.4 Infinitesimal Kinematics of the Six-Legged Parallel Manipulator 290
12.4.1 Velocity Analysis of the Six-Legged Parallel Manipulator 291
12.4.2 Acceleration Analysis of the Six-Legged Parallel Manipulator 292
12.5 Exercises 293
References 293
13 The Original Stewart Platform 295
13.1 Introduction 295
13.2 Description of the Stewart Platform 297
13.3 Finite Kinematics of the Stewart Platform 299
13.3.1 Forward Displacement Analysis of the Stewart Platform 299
13.3.2 Inverse Displacement Analysis of the Stewart Platform 303
13.4 Infinitesimal Kinematics of the Stewart Platform 304
13.4.1 Velocity Analysis of the Stewart Platform 304
13.4.2 Acceleration Analysis of the Stewart Platform 306
13.4.3 Singularity Analysis of the Stewart Platform 309
13.4.3.1 Direct Singularity of the Stewart Platform 309
13.4.3.2 Inverse Singularity of the Stewart Platform 310
13.4.3.3 Combined Singularity of the Stewart Platform 311
13.5 Exercises 311
References 312
14 Delta Robot 314
14.1 Introduction 314
14.2 Description of the Delta Robot 315
14.3 Finite Kinematics of the Delta Robot 317
14.3.1 Forward Displacement Analysis of the Delta Robot 317
14.3.2 Inverse Displacement Analysis of the Delta Robot 321
14.4 Infinitesimal Kinematics of the Delta Robot 323
14.4.1 Velocity Analysis of the Delta Robot 324
14.4.1.1 Singularity Analysis of the Delta Robot 325
14.4.2 Acceleration Analysis of the Delta Robot 325
14.5 Exercises 328
References 329
Part VI Solved Exercises 330
15 Full Answers to Selected Exercises 331
15.1 Chapter 3: Mathematical Background 331
15.2 Chapter 4: Velocity Analysis 336
15.3 Chapter 5: Acceleration Analysis 344
15.4 Chapter 6: Jerk Analysis 351
15.5 Chapter 7: Hyper-Jerk Analysis 357
15.6 Chapter 12: Gough's Tyre Testing Machine 364
References 366
16 Appendix 1: A Simple Method to Compute the Rotation Matrix 367
17 Appendix 2: Computer Codes 369
17.1 Maple Procedures 369
Index 383