Kinematics of general spatial mechanical systems

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 17
Acknowledgments......Page 21
List of Commonly Used Symbols, Abbreviations, and Acronyms......Page 23
About the Companion Website......Page 29
 1.1 General Features of Notation......Page 31
  1.2.2 Equality of Vectors......Page 32
  1.3.2 Cross Product......Page 33
 1.4 Reference Frames......Page 34
 1.5 Representation of a Vector in a Selected Reference Frame......Page 36
  1.6.1 Dot Product......Page 37
  1.6.2 Cross Product and Skew Symmetric Cross Product Matrices......Page 38
 1.7 Mathematical Properties of the Skew Symmetric Matrices......Page 39
  1.8.1 Example 1.1......Page 40
  1.8.3 Example 1.3......Page 41
 2.1 Vector Equation of Rotation and the Rodrigues Formula......Page 43
 2.2 Matrix Equation of Rotation and the Rotation Matrix......Page 45
 2.4 Basic Rotation Matrices......Page 46
 2.5 Successive Rotations......Page 47
 2.6 Orthonormality of the Rotation Matrices......Page 48
  2.7.1 Mathematical Properties of General Rotation Matrices......Page 50
  2.8.1 Example 2.1......Page 52
  2.8.2 Example 2.2......Page 53
  2.8.4 Example 2.4......Page 54
  2.9.1 Scalar Equations of Rotation......Page 55
  2.9.3 Determination of the Axis of Rotation......Page 56
 2.10 Definition and Properties of the Double Argument Arctangent Function......Page 59
 3.1 Matrix Representations of a Vector in Different Reference Frames......Page 61
  3.2.1 Definition and Usage of a Transformation Matrix......Page 62
  3.2.2 Basic Properties of a Transformation Matrix......Page 63
  3.3.2 Row‐by‐Row Expression......Page 64
  3.3.4 Remark 3.2......Page 65
  3.3.6 Example 3.1......Page 66
  3.4.1 Definitions of Direction Angles and Direction Cosines......Page 67
  3.5.1 Correlation Between the Rotation and Transformation Matrices......Page 68
  3.5.2 Distinction Between the Rotation and Transformation Matrices......Page 69
 3.7 Expression of a Transformation Matrix in a Case of Several Successive Rotations......Page 70
  3.7.2 Initial Frame Based (IFB) Formulation......Page 71
  3.8.2 IFB (Initial Frame Based) Euler Angle Sequences......Page 72
  3.8.3 RFB (Rotated Frame Based) Euler Angle Sequences......Page 73
  3.8.5 Remark 3.5......Page 74
  3.8.7 Commonly Used Euler Angle Sequences......Page 75
  3.8.8 Extraction of Euler Angles from a Given Transformation Matrix......Page 76
  3.9.1 Position of a Point Expressed in Different Reference Frames......Page 81
  3.9.2 Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships......Page 82
  3.9.3 Affine Coordinate Transformation Between Two Reference Frames......Page 83
  3.9.4 Homogeneous Coordinate Transformation Between Two Reference Frames......Page 84
  3.9.5 Mathematical Properties of the Homogeneous Transformation Matrices......Page 85
  3.9.6 Example 3.2......Page 88
  4.1.1 Differentiation and Resolution Frames......Page 93
  4.1.2 Components in Different Differentiation and Resolution Frames......Page 94
  4.1.3 Example......Page 95
  4.2.1 First Derivatives and the Relative Angular Velocity......Page 96
  4.2.2 Second Derivatives and the Relative Angular Acceleration......Page 98
  4.3.1 Combination of Relative Angular Velocities......Page 100
  4.4.1 Relative Angular Velocities and Accelerations about Relatively Fixed Axes......Page 101
  4.4.2 Example......Page 102
  4.4.3 Angular Velocities Associated with the Euler Angle Sequences......Page 104
  4.5.1 Velocity of a Point with Respect to Different Reference Frames......Page 107
  4.5.2 Acceleration of a Point with Respect to Different Reference Frames......Page 108
  4.5.3 Velocity and Acceleration Expressions with Simplified Notations......Page 109
 Synopsis......Page 111
  5.1.1 Body Frames and Joint Frames......Page 112
  5.1.3 Joints or Kinematic Pairs......Page 113
 5.2 Position Equations for a Kinematic Chain of Rigid Bodies......Page 114
  5.2.3 Orientation of a Body with Respect to the Base of the Kinematic Chain......Page 115
  5.2.5 Loop Closure Equations for a Kinematic Loop......Page 116
  5.3.1 Relative Angular Velocity between Successive Bodies......Page 117
  5.3.2 Relative Translational Velocity Between Successive Bodies......Page 118
  5.3.4 Translational Velocity of a Body with Respect to the Base......Page 119
 5.4 Acceleration Equations for a Kinematic Chain of Rigid Bodies......Page 120
  5.4.1 Relative Angular Acceleration Between Successive Bodies......Page 121
  5.4.3 Angular Acceleration of a Body with Respect to the Base......Page 122
  5.4.5 Acceleration Equations for a Kinematic Loop......Page 123
  5.5.1 Kinematic Description of the System......Page 124
  5.5.2 Position Analysis......Page 125
  5.5.3 Velocity Analysis......Page 130
  5.5.4 Acceleration Analysis......Page 133
  5.6.1 Kinematic Description of the Mechanism......Page 136
  5.6.2 Loop Closure Equations......Page 138
  5.6.3 Degree of Freedom or Mobility......Page 139
  5.6.4 Position Analysis......Page 140
  5.6.5 Velocity Analysis......Page 149
  5.6.6 Acceleration Analysis......Page 152
  6.1.1 Description of a Joint as a Kinematic Pair......Page 155
  6.1.3 Number of Distinct Joints Between Two Rigid Bodies......Page 156
  6.1.4 Classification of the Joints......Page 157
  6.2.1 Single‐Axis Joints......Page 158
  6.2.2 Universal Joint......Page 160
  6.2.3 Spherical Joint......Page 161
  6.3.1 Line‐on‐Plane Joint......Page 162
  6.3.3 Point‐on‐Surface Joint......Page 163
  6.4.1 Fork‐on‐Surface Joint......Page 164
  6.4.2 Triangle‐on‐Surface Joint......Page 166
  6.5.1 Surface‐on‐Surface Joint......Page 168
  6.5.2 Curve‐on‐Surface Joint......Page 174
  6.5.3 Curve‐on‐Curve Joint......Page 177
  6.6.1 Cone‐on‐Cone Joint......Page 178
  6.6.2 Cone‐on‐Cylinder Joint......Page 185
  6.6.3 Cone‐on‐Plane Joint......Page 187
  6.6.4 Cylinder‐on‐Cylinder Joint......Page 191
  6.6.5 Cylinder‐on‐Plane Joint......Page 194
  6.7.1 Example 6.1: An RRRSP Mechanism......Page 197
  6.7.2 Example 6.2: A Two‐Link Mechanism with Three Point‐on‐Plane Joints......Page 201
  6.7.3 Example 6.3: A Spatial Cam Mechanism......Page 204
  6.7.4 Example 6.4: A Spatial Cam Mechanism That Allows Rolling Without Slipping......Page 207
 7.1 Kinematic Description of a General Serial Manipulator......Page 215
 7.2 Denavit–Hartenberg Convention......Page 216
  7.3.1 Assignment and Description of the Link Frames......Page 217
  7.3.2 D–H Parameters......Page 218
  7.3.4 Alternative Multi‐Index Notation for the D–H Convention......Page 219
 7.4 D–H Convention for the First Joint......Page 220
 7.5 D–H Convention for the Last Joint......Page 223
 7.7 D–H Convention for Successive Joints with Parallel Axes......Page 225
 7.8 D–H Convention for Successive Joints with Coincident Axes......Page 227
 Synopsis......Page 229
 8.1 Forward Kinematics......Page 231
 8.2 Compact Formulation of Forward Kinematics......Page 232
 8.3 Detailed Formulation of Forward Kinematics......Page 233
 8.4 Manipulators with or without Spherical Wrists......Page 235
 8.5 Inverse Kinematics......Page 237
  8.6.1 Regular Manipulator with a Spherical Wrist......Page 238
  8.6.2 Regular Manipulator with a Nonspherical Wrist......Page 241
  8.7.1 Solution by Specifying the Variables of Certain Joints......Page 242
  8.7.2 Solution by Optimization......Page 243
  8.8.1 Compromise in Orientation in Favor of a Completely Specified Location......Page 244
  8.9.1 Forward Kinematics of Velocity Relationships......Page 245
  8.9.2 Forward Kinematics of Acceleration Relationships......Page 246
 8.10 Jacobian Matrices Associated with the Wrist and Tip Points......Page 248
  8.11.1 Orientations of the Links......Page 250
  8.11.4 Angular Velocities of the Links......Page 251
  8.11.8 Accelerations of the Link Frame Origins......Page 252
 8.12 Inverse Motion Analysis of a Manipulator Based on the Jacobian Matrix......Page 253
  8.12.1 Inverse Velocity Analysis of a Regular Manipulator......Page 254
  8.13.1 Inverse Velocity Analysis......Page 255
  8.13.2 Inverse Acceleration Analysis......Page 258
 8.14 Inverse Motion Analysis of a Deficient Manipulator......Page 259
  8.15.1 Inverse Velocity Solution......Page 260
  8.15.2 Inverse Acceleration Solution......Page 261
 9.1 Puma Manipulator......Page 263
  9.1.1 Kinematic Description According to the D–H Convention......Page 264
  9.1.2 Forward Kinematics in the Position Domain......Page 265
  9.1.3 Inverse Kinematics in the Position Domain......Page 267
  9.1.4 Multiplicity Analysis......Page 270
  9.1.5 Singularity Analysis in the Position Domain......Page 272
  9.1.6 Forward Kinematics in the Velocity Domain......Page 274
  9.1.7 Inverse Kinematics in the Velocity Domain......Page 275
  9.1.8 Singularity Analysis in the Velocity Domain......Page 277
  9.2.1 Kinematic Description According to the D–H Convention......Page 280
  9.2.2 Forward Kinematics in the Position Domain......Page 281
  9.2.3 Inverse Kinematics in the Position Domain......Page 283
  9.2.4 Multiplicity Analysis......Page 284
  9.2.6 Forward Kinematics in the Velocity Domain......Page 285
  9.2.7 Inverse Kinematics in the Velocity Domain......Page 286
  9.2.8 Singularity Analysis in the Velocity Domain......Page 287
 9.3 Elbow Manipulator......Page 288
  9.3.1 Kinematic Description According to the D–H Convention......Page 289
  9.3.2 Forward Kinematics in the Position Domain......Page 290
  9.3.3 Inverse Kinematics in the Position Domain......Page 292
  9.3.4 Multiplicity Analysis......Page 294
  9.3.5 Singularity Analysis in the Position Domain......Page 296
  9.3.7 Inverse Kinematics in the Velocity Domain......Page 299
  9.3.8 Singularity Analysis in the Velocity Domain......Page 301
  9.4.1 Kinematic Description According to the D–H Convention......Page 303
  9.4.2 Forward Kinematics in the Position Domain......Page 304
  9.4.3 Inverse Kinematics in the Position Domain......Page 305
  9.4.4 Multiplicity Analysis......Page 307
  9.4.5 Singularity Analysis in the Position Domain......Page 308
  9.4.7 Inverse Kinematics in the Velocity Domain......Page 309
  9.4.8 Singularity Analysis in the Velocity Domain......Page 310
 9.5 An RP2R3 Manipulator without an Analytical Solution......Page 311
  9.5.2 Forward Kinematics in the Position Domain......Page 312
  9.5.3 Inverse Kinematics in the Position Domain......Page 313
  9.5.4 Multiplicity Analysis......Page 315
  9.5.7 Inverse Kinematics in the Velocity Domain......Page 317
  9.5.8 Singularity Analysis in the Velocity Domain......Page 319
  9.6.1 Kinematic Description According to the D–H Convention......Page 320
  9.6.2 Forward Kinematics in the Position Domain......Page 321
  9.6.3 Inverse Kinematics in the Position Domain......Page 323
  9.6.4 Multiplicity Analysis......Page 327
  9.6.6 Forward Kinematics in the Velocity Domain......Page 328
  9.6.7 Inverse Kinematics in the Velocity Domain......Page 329
  9.6.8 Singularity Analysis in the Velocity Domain......Page 331
  9.7.1 Kinematic Description According to the D–H Convention......Page 333
  9.7.2 Forward Kinematics in the Position Domain......Page 334
   9.7.3.1 Solution in the Case of Fully Specified Tip Point Location......Page 335
   9.7.4.1 Analysis in the Case of Fully Specified Tip Point Location......Page 337
   9.7.5.1 Analysis in the Case of Fully Specified Tip Point Location......Page 338
   9.7.5.2 Analysis in the Case of Fully Specified End‐Effector Orientation......Page 339
   9.7.7.1 Solution in the Case of Fully Specified Tip Point Velocity......Page 340
   9.7.7.2 Solution in the Case of Fully Specified End‐Effector Angular Velocity......Page 341
   9.7.8.1 Analysis in the Case of Fully Specified Tip Point Velocity......Page 342
  9.8.1 Kinematic Description According to the D–H Convention......Page 343
  9.8.2 Forward Kinematics in the Position Domain......Page 345
  9.8.3 Inverse Kinematics in the Position Domain......Page 346
  9.8.4 Multiplicity Analysis......Page 353
  9.8.5 Singularity Analysis in the Position Domain......Page 356
  9.8.7 Inverse Kinematics in the Velocity Domain......Page 358
  9.8.8 Singularity Analysis in the Velocity Domain......Page 363
  9.8.9 Consistency of the Inverse Kinematics in the Position and Velocity Domains......Page 365
 Synopsis......Page 371
 10.1 General Kinematic Features of Parallel Manipulators......Page 373
 10.2 Position Equations of a Parallel Manipulator......Page 377
 10.3 Forward Kinematics in the Position Domain......Page 381
 10.4 Inverse Kinematics in the Position Domain......Page 389
 10.5 Velocity Equations of a Parallel Manipulator......Page 398
 10.6 Forward Kinematics in the Velocity Domain......Page 401
 10.7 Inverse Kinematics in the Velocity Domain......Page 407
  10.8.1 Kinematic Description......Page 414
  10.8.2 Position Equations......Page 416
  10.8.3 Inverse Kinematics in the Position Domain......Page 417
  10.8.4 Forward Kinematics in the Position Domain......Page 419
  10.8.5 Velocity Equations......Page 426
  10.8.6 Inverse Kinematics in the Velocity Domain......Page 427
  10.8.7 Forward Kinematics in the Velocity Domain......Page 428
  10.9.1 Kinematic Description......Page 432
  10.9.2 Position Equations......Page 434
  10.9.3 Independent Kinematic Loops and the Associated Equations......Page 437
  10.9.4 Inverse Kinematics in the Position Domain......Page 440
  10.9.5 Forward Kinematics in the Position Domain......Page 442
  10.9.6 Velocity Equations......Page 447
  10.9.7 Inverse Kinematics in the Velocity Domain......Page 448
  10.9.8 Forward Kinematics in the Velocity Domain......Page 450
Bibliography......Page 453
Index......Page 455
EULA......Page 463