Robot and Multibody Dynamics (eBook)

Robot and Multibody Dynamics: Analysis and Algorithms 1
Preface 7
Contents 11
Part I Serial-Chain Dynamics 21
1 Spatial Vectors 22
1.1 Homogeneous Transforms 22
1.2 Differentiation of Vectors 24
1.2.1 Vector Derivatives in Rotating Frames 24
1.2.2 Rigid Body Vector Derivatives 25
1.3 Spatial Vectors 27
1.3.1 Six-Dimensional Spatial Notation 28
1.3.2 The Cross-Product for Spatial Vectors 28
1.4 The Rigid Body Transformation Matrix (x,y) 30
1.4.1 Spatial Velocity Transformations 31
1.4.2 Properties of f(·) 31
1.5 Spatial Forces 34
2 Single Rigid Body Dynamics 35
2.1 Spatial Inertia and Momentum of a Rigid Body 35
2.1.1 The Spatial Inertia 35
2.1.2 The Parallel-Axis Theorem for Spatial Inertias 38
2.1.3 Spatial Inertia of a Composite Assemblage of Rigid Bodies 39
2.1.4 The Spatial Momentum of a Rigid Body 39
2.2 Motion Coordinates 40
2.2.1 Generalized Coordinates and Velocities 41
2.2.2 Generalized Forces 42
2.2.3 Generalized Accelerations 42
2.3 Equations of Motion with Inertial Frame Derivatives 43
2.3.1 Equations of Motion with ßI = IV(C) 43
2.3.2 Equations of Motion with ßI = IV(z) 45
2.4 Equations of Motion with Body Frame Derivatives 47
2.5 Equations of Motion with an Inertially Fixed Velocity Reference Frame 49
2.6 Comparison of the Different Dynamics Formulations 51
3 Serial-Chain Kinematics 53
3.1 Serial-Chain Model 53
3.2 Hinge Kinematics 54
3.2.1 Hinge Generalized Coordinates 55
3.2.2 Relative and Absolute Coordinates 56
3.2.3 Hinge Generalized Velocities 56
3.2.4 Examples of Hinges 58
3.3 Serial-Chain Kinematics 60
3.3.1 Serial-Chain Configuration Kinematics 60
3.3.2 Serial-Chain Differential Kinematics 61
3.3.3 Differential Kinematics with Bk=Ok 63
3.4 Spatial Operators 65
3.4.1 The Spatial Operator 67
3.4.2 Velocity Operator Expression 68
3.4.3 The Spatial Operator 69
3.5 Recursions Associated with the Operator 69
3.6 The Jacobian Operator 71
4 The Mass Matrix 74
4.1 Mass Matrix of a Serial-Chain System 74
4.1.1 Kinetic Energy of the Serial-Chain 74
4.1.2 Composite Rigid Body Inertias 76
4.1.3 Decomposition of fMf* 78
4.1.4 O(N2) Algorithm for Computing the Mass Matrix 80
4.1.5 Relationship to the Composite Rigid Body Method 82
4.2 Lagrangian Approach to the Equations of Motion 83
4.2.1 Properties of M and C 84
4.2.2 Hamiltonian Form of the Equations of Motion 87
4.2.3 Transformation of Lagrangian Coordinates 87
5 Serial-Chain Dynamics 91
5.1 Equations of Motion for a Typical Link 91
5.1.1 Expression for the Spatial Acceleration (k) 93
5.1.2 Overall Equations of Motion 97
5.1.3 Spatial Operators with Body Frame Derivatives but Bk=Ok 99
5.2 Inclusion of External Forces and Gravity 101
5.2.1 Inclusion of External Forces 101
5.2.2 Compensating for External Forces 102
5.2.3 Inclusion of Gravitational Forces 103
5.3 Inverse Dynamics of Serial-Chains 104
5.3.1 Newton–Euler Inverse Dynamics Algorithm 104
5.3.2 Computing the Mass Matrix Using Inverse Dynamics 106
5.3.3 Composite Rigid Body Inertias Based Inverse Dynamics 107
5.4 Equations of Motion with an Inertially Fixed Velocity Reference Frame 109
5.4.1 Relationship Between I and 112
6 Articulated Body Model for Serial Chains 113
6.1 Alternate Models for Multibody Systems 113
6.1.1 Terminal Body Model 113
6.1.2 Composite Body Model 114
6.1.3 Articulated Body Model 115
6.2 The P(k) Articulated Body Inertia 116
6.2.1 Induction Argument for P(k) 116
6.2.2 The D (k) and G(k) Matrices 117
6.2.3 The (k) and (k) Projection Matrices 118
6.2.4 The P+(k) Matrix 120
6.2.5 Conclusion of the Induction Argument for P(k) 121
6.3 Articulated Body Model Force Decomposition 123
6.3.1 The (k) Vector 125
6.3.2 Acceleration Relationships 126
6.4 Parallels with Estimation Theory 127
6.4.1 Process Covariances 128
6.4.2 Optimal Filtering 128
6.4.3 Optimal Smoothing 129
6.4.4 Extensions 130
7 Mass Matrix Inversion and AB Forward Dynamics 131
7.1 Articulated Body Spatial Operators 131
7.1.1 Some Operator Identities 133
7.1.2 Innovations Operator Factorization of the Mass Matrix 136
7.1.3 Operator Inversion of the Mass Matrix 137
7.2 Forward Dynamics 138
7.2.1 O(N) AB Forward Dynamics Algorithm 139
7.3 Extensions to the Forward Dynamics Algorithm 144
7.3.1 Computing Inter-Link f Spatial Forces 144
7.3.2 Including Gravitational Accelerations 144
7.3.3 Including External Forces 145
7.3.4 Including Implicit Constraint Forces 146
Part II General Multibody Systems 148
8 Graph Theory Connections 149
8.1 Directed Graphs and Trees 149
8.2 Adjacency Matrices for Digraphs 152
8.2.1 Properties of Digraph Adjacency Matrices 152
8.2.2 Properties of Tree Adjacency Matrices 154
8.2.3 Properties of Serial-Chain Adjacency Matrices 154
8.3 Block Weighted Adjacency Matrices 155
8.4 BWA Matrices for Tree Digraphs 156
8.4.1 The 1-Resolvent of Tree BWA Matrices 157
8.5 Similarity Transformations of a Tree BWA Matrix 161
8.5.1 Permutation Similarity Transformations 162
8.5.2 Similarity-Shift Transformations 163
8.6 Multibody System Digraphs 164
8.6.1 BWA Matrices and Serial-Chain, Rigid Body Systems 165
8.6.1.1 E Is a BWA Matrix 167
8.6.2 Non-Canonical Serial-Chains 168
8.7 BWA Matrices and Tree-Topology, Rigid Body Systems 168
8.7.1 Equations of Motion for Tree Topology Systems 169
9 SKO Models 172
9.1 SKO Models 173
9.1.1 Definition of SKO Models 173
9.1.2 Existence of SKO Models 174
9.1.3 Generalizations of SKO Models 174
9.2 SPO Operator/Vector Products for Trees 175
9.2.1 SKO Model O(N) Newton–Euler Inverse Dynamics 178
9.3 Lyapunov Equations for SKO Models 179
9.3.1 Forward Lyapunov Recursions for SKO Models 179
9.3.2 Mass Matrix Computation for an SKO Model 181
9.3.3 Backward Lyapunov Recursions for SKO Models 183
9.4 Riccati Equations for SKO Models 187
9.4.1 The E and SKO and SPO Operators 188
9.4.2 Operator Identities 189
9.5 SKO Model Mass Matrix Factorization and Inversion 192
9.5.1 O(N) AB Forward Dynamics 193
9.6 Generalized SKO Formulation Process 195
9.6.1 Procedure for Developing an SKO Model 196
9.6.2 Potential Non-Tree Topology Generalizations 198
10 Operational Space Dynamics 199
10.1 Operational Space Equations of Motion 199
10.1.1 Physical Interpretation 201
10.1.2 Operational Space Control 201
10.2 Structure of the Operational Space Inertia 202
10.2.1 The Extended Operational Space Compliance Matrix 202
10.2.2 Decomposition of 203
10.2.3 Computing 205
10.2.4 The Operational Space Compliance Kernel 206
10.2.5 Simplifications for Serial-Chain Systems 209
10.2.6 Explicit Computation of the Mass Matrix Inverse M-1 210
10.3 The Operational Space Cos Coriolis/Centrifugal Term 211
10.3.1 The U and U Projection Operators 211
10.3.2 Computing Cos 214
10.4 Divide and Conquer Forward Dynamics 216
11 Closed-Chain Dynamics 221
11.1 Modeling Closed-Chain Dynamics 221
11.1.1 Types of Bilateral Motion Constraints 221
11.1.2 Constrained System Forward Dynamics Strategies 223
11.2 Augmented Approach for Closed-Chain Forward Dynamics 224
11.2.1 Move/Squeeze Decompositions 226
11.2.2 Augmented Dynamics with Loop Constraints 227
11.2.3 Dual-Arm System Example 231
11.3 Projected Closed-Chain Dynamics 233
11.4 Equivalence of Augmented and Projected Dynamics 234
11.5 Unilateral Constraints 236
11.5.1 Complementarity Problems 237
11.5.2 Forward Dynamics 239
12 Systems with Geared Links 240
12.1 Equations of Motion 241
12.1.1 Reformulated Equations of Motion 243
12.1.2 Eliminating the Geared Constraint 244
12.2 SKO Model for Geared Systems 245
12.2.1 Expression for the Mass Matrix 247
12.3 Computation of the Mass Matrix 248
12.3.1 Optimized Composite Body Inertia Algorithm 249
12.4 O(N) AB Forward Dynamics 250
12.4.1 Mass Matrix Factorization and Inversion 250
12.4.2 Recursive AB Forward Dynamics Algorithm 252
12.4.3 Optimization of the Forward Dynamics Algorithm 253
13 Systems with Link Flexibility 255
13.1 Lumped Mass Model for a Single Flexible Body 255
13.1.1 Equations of Motion of the Okj Node 256
13.1.2 Nodal Equations of Motion for the kth Flexible Body 257
13.1.3 Recursive Relationships Across the Flexible Bodies 258
13.2 Modal Formulation for Flexible Bodies 259
13.2.1 Modal Mass Matrix for a Single Body 261
13.2.2 Recursive Relationships Using Modal Coordinates 262
13.2.3 Recursive Propagation of Accelerations 263
13.2.4 Recursive Propagation of Forces 264
13.2.5 Overall Equations of Motion 265
13.3 SKO Models for Flexible Body Systems 265
13.3.1 Operator Expression for the System Mass Matrix 267
13.3.2 Illustration of the SKO Formulation Procedure 267
13.4 Inverse Dynamics Algorithm 269
13.5 Mass Matrix Computation 271
13.6 Factorization and Inversion of the Mass Matrix 272
13.7 AB Forward Dynamics Algorithm 274
13.7.1 Simplified Algorithm for the Articulated Body Quantities 274
13.7.2 Simplified AB Forward Dynamics Algorithm 276
Part III Advanced Topics 280
14 Transforming SKO Models 281
14.1 Partitioning Digraphs 281
14.1.1 Partitioning by Path-Induced Sub-Graphs 282
14.2 Partitioning SKO Models 283
14.2.1 Partitioning SKO Model Operators 283
14.2.2 Partitioning of an SKO Model 285
14.3 SPO Operator Sparsity Structure 286
14.3.1 Decomposition into Serial-Chain Segments 286
14.3.2 Sparsity Structure of the E A SKO Matrix 288
14.3.3 Sparsity Structure of the A Matrix 289
14.3.4 Sparsity Structure of the M Mass Matrix 290
14.4 Aggregating Sub-Graphs 290
14.4.1 Edge and Node Contractions 291
14.4.2 Tree Preservation After Sub-Graph Aggregation 292
14.4.3 The SA Aggregation Sub-Graph 295
14.5 Transforming SKO Models Via Aggregation 295
14.5.1 SKO Operators After Body Aggregation 295
14.5.2 SKO Model for the TS Aggregated Tree 298
14.6 Aggregation Relationships at the Component Level 300
14.6.1 Velocity Relationships 301
14.6.2 Acceleration Relationships 302
14.6.3 Force Relationships 303
15 Constraint Embedding 305
15.1 Constraint Embedding Strategy 305
15.1.1 Embedding Constraint Sub-Graphs 307
15.2 Examples of Constraint Embedding 311
15.2.1 Geared Links 312
15.2.2 Planar Four-Bar Linkage System (Terminal Cut) 313
15.2.3 Planar Four-Bar Linkage System (Internal Cut) 314
15.3 Recursive AB Forward Dynamics 315
15.3.1 Articulated Body Inertias for the Aggregated System 315
15.3.2 Mass Matrix Factorization and Inversion 316
15.3.3 AB Forward Dynamics Algorithm 316
15.4 Computing XS and S 317
15.5 Generalization to Multiple Branches and Cut-Edges 319
16 Under-Actuated Systems 320
16.1 Modeling of Under-Actuated Manipulators 321
16.1.1 Decomposition into Passive and Active Systems 322
16.1.2 Partitioned Equations of Motion 323
16.1.3 Spatial Operator Expression for M-1pp 324
16.1.4 Operator Expressions for S Blocks 326
16.2 O(N) Generalized Dynamics Algorithms 327
16.2.1 Application to Prescribed Motion Dynamics 331
16.3 Jacobians for Under-Actuated Systems 331
16.3.1 The Generalized Jacobian JG 332
16.3.2 Computed-Torque for Under-Actuated Systems 333
16.3.3 The Disturbance Jacobian JD 334
16.4 Free-Flying Systems as Under-Actuated Systems 336
16.4.1 Integrals of Motion for Free-Flying Systems 336
17 Free-Flying Systems 338
17.1 Dynamics of Free-Flying Manipulators 338
17.1.1 Dynamics with Link n as Base-Body 338
17.1.2 Dynamics with Link 1 as Base-Body 339
17.1.3 Direct Computation of Link Spatial Acceleration 343
17.1.4 Dynamics with Link k as Base-Body 343
17.2 The Base-Invariant Forward Dynamics Algorithm 344
17.2.1 Parallels with Smoothing Theory 345
17.2.2 Simplifications Using Non-Minimal Coordinates 346
17.2.3 Computational Issues 347
17.2.4 Extensions to Tree-Topology Manipulators 347
17.3 SKO Model with kth Link as Base-Body 348
17.3.1 Generalized Velocities with kth Link as the Base-Body 348
17.3.2 Link Velocity Recursions with kth Link as the Base-Body 349
17.3.3 Partitioned System 350
17.3.4 Properties of a Serial-Chain SKO Operator 351
17.3.5 Reversing the SKO Operator 352
17.3.6 Transformed SKO Model 354
17.4 Base-Invariant Operational Space Inertias 355
18 Spatial Operator Sensitivities for Rigid-Body Systems 359
18.1 Preliminaries 359
18.1.1 Notation 359
18.1.2 Identities for Vv 360
18.2 Operator Time Derivatives 362
18.2.1 Time Derivatives of (k+1, k), H(k) and M(k) 362
18.2.2 Time Derivatives with Ok=Bk 364
18.2.3 Time Derivative of the Mass Matrix 366
18.3 Operator Sensitivities 367
18.3.1 The bold0mu mumu HHOT1HHHH"0365bold0mu mumu HHOT1HHHHi , bold0mu mumu HHOT1HHHH"0365bold0mu mumu HHOT1HHHHi , and bold0mu mumu HHOT1HHHH"0365bold0mu mumu HHOT1HHHH=i Operators 368
18.4 Mass Matrix Related Quantities 371
18.4.1 Sensitivity of bold0mu mumu MMOT1MMMM* 371
18.4.2 Sensitivity of the Mass Matrix Mi 371
18.4.3 Sensitivity of the Kinetic Energy 372
18.4.4 Equivalence of Lagrangian and Newton–Euler Dynamics 373
18.5 Time Derivatives of Articulated Body Quantities 374
18.6 Sensitivity of Articulated Body Quantities 379
18.7 Sensitivity of Innovations Factors 381
19 Diagonalized Lagrangian Dynamics 384
19.1 Globally Diagonalized Dynamics 384
19.2 Diagonalization in Velocity Space 387
19.2.1 Coriolis Force Does No Work 389
19.2.2 Rate of Change of the Kinetic Energy 389
19.3 The Innovations Factors as Diagonalizing Transformations 390
19.3.1 Transformations Between and 391
19.4 Expression for C(,) for Rigid-Link Systems 392
19.4.1 Closed-Form Expression for 392
19.4.2 Operator Expression for C(,) 394
19.4.3 Decoupled Control 397
19.5 Un-normalized Diagonalized Equations of Motion 398
19.5.1 O(N) Forward Dynamics 399
Useful Mathematical Identities 401
A.1 3-Vector Cross-Product Identities 401
A.2 Matrix and Vector Norms 401
A.3 Schur Complement and Matrix Inverse Identities 402
A.4 Matrix Inversion Identities 404
A.5 Matrix Trace Identities 404
A.6 Derivative and Gradient Identities 405
A.6.1 Function Derivatives 405
A.6.2 Vector Gradients 405
A.6.3 Matrix Derivatives 406
Attitude Representations 407
B.1 Euler Angles 407
B.2 Angle/Axis Parameters 408
B.3 Unit Quaternions/Euler Parameters 410
B.3.1 The E+(q) and E-(q) Matrices 412
B.3.2 Quaternion Transformations 413
B.3.3 Quaternion Differential Kinematics 414
B.4 Gibbs Vector Attitude Representations 416
Solutions 418
References 477
List of Notation 477
Index 494