Jain Robot and Multibody Dynamics

Analysis and Algorithms
1. Auflage 2010
ISBN: 978-1-4419-7267-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 510 Seiten

Robot and Multibody Dynamics: Analysis and Algorithms provides a comprehensive and detailed exposition of a new mathematical approach, referred to as the Spatial Operator Algebra (SOA), for studying the dynamics of articulated multibody systems. The approach is useful in a wide range of applications including robotics, aerospace systems, articulated mechanisms, bio-mechanics and molecular dynamics simulation. The book also: treats algorithms for simulation, including an analysis of complexity of the algorithms, describes one universal, robust, and analytically sound approach to formulating the equations that govern the motion of complex multi-body systems, covers a range of more advanced topics including under-actuated systems, flexible systems, linearization, diagonalized dynamics and space manipulators. Robot and Multibody Dynamics: Analysis and Algorithms will be a valuable resource for researchers and engineers looking for new mathematical approaches to finding engineering solutions in robotics and dynamics.

Autoren/Hrsg.

Jain, Abhinandan

Inhaltsverzeichnis

1;Robot and Multibody Dynamics: Analysis and Algorithms;1
1.1;Preface;7
1.2;Contents;11
1.3;Part I Serial-Chain Dynamics;21
1.3.1;1 Spatial Vectors;22
1.3.1.1;1.1 Homogeneous Transforms;22
1.3.1.2;1.2 Differentiation of Vectors;24
1.3.1.2.1;1.2.1 Vector Derivatives in Rotating Frames;24
1.3.1.2.2;1.2.2 Rigid Body Vector Derivatives;25
1.3.1.3;1.3 Spatial Vectors;27
1.3.1.3.1;1.3.1 Six-Dimensional Spatial Notation;28
1.3.1.3.2;1.3.2 The Cross-Product for Spatial Vectors;28
1.3.1.4;1.4 The Rigid Body Transformation Matrix (x,y);30
1.3.1.4.1;1.4.1 Spatial Velocity Transformations;31
1.3.1.4.2;1.4.2 Properties of f(·);31
1.3.1.5;1.5 Spatial Forces;34
1.3.2;2 Single Rigid Body Dynamics;35
1.3.2.1;2.1 Spatial Inertia and Momentum of a Rigid Body;35
1.3.2.1.1;2.1.1 The Spatial Inertia;35
1.3.2.1.2;2.1.2 The Parallel-Axis Theorem for Spatial Inertias;38
1.3.2.1.3;2.1.3 Spatial Inertia of a Composite Assemblage of Rigid Bodies;39
1.3.2.1.4;2.1.4 The Spatial Momentum of a Rigid Body;39
1.3.2.2;2.2 Motion Coordinates;40
1.3.2.2.1;2.2.1 Generalized Coordinates and Velocities;41
1.3.2.2.2;2.2.2 Generalized Forces;42
1.3.2.2.3;2.2.3 Generalized Accelerations;42
1.3.2.3;2.3 Equations of Motion with Inertial Frame Derivatives;43
1.3.2.3.1;2.3.1 Equations of Motion with ßI = IV(C);43
1.3.2.3.2;2.3.2 Equations of Motion with ßI = IV(z);45
1.3.2.4;2.4 Equations of Motion with Body Frame Derivatives;47
1.3.2.5;2.5 Equations of Motion with an Inertially Fixed Velocity Reference Frame;49
1.3.2.6;2.6 Comparison of the Different Dynamics Formulations;51
1.3.3;3 Serial-Chain Kinematics;53
1.3.3.1;3.1 Serial-Chain Model;53
1.3.3.2;3.2 Hinge Kinematics;54
1.3.3.2.1;3.2.1 Hinge Generalized Coordinates;55
1.3.3.2.2;3.2.2 Relative and Absolute Coordinates;56
1.3.3.2.3;3.2.3 Hinge Generalized Velocities;56
1.3.3.2.4;3.2.4 Examples of Hinges;58
1.3.3.3;3.3 Serial-Chain Kinematics;60
1.3.3.3.1;3.3.1 Serial-Chain Configuration Kinematics;60
1.3.3.3.2;3.3.2 Serial-Chain Differential Kinematics;61
1.3.3.3.3;3.3.3 Differential Kinematics with Bk=Ok;63
1.3.3.4;3.4 Spatial Operators;65
1.3.3.4.1;3.4.1 The Spatial Operator;67
1.3.3.4.2;3.4.2 Velocity Operator Expression;68
1.3.3.4.3;3.4.3 The Spatial Operator;69
1.3.3.5;3.5 Recursions Associated with the Operator ;69
1.3.3.6;3.6 The Jacobian Operator;71
1.3.4;4 The Mass Matrix;74
1.3.4.1;4.1 Mass Matrix of a Serial-Chain System;74
1.3.4.1.1;4.1.1 Kinetic Energy of the Serial-Chain;74
1.3.4.1.2;4.1.2 Composite Rigid Body Inertias;76
1.3.4.1.3;4.1.3 Decomposition of fMf*;78
1.3.4.1.4;4.1.4 O(N2) Algorithm for Computing the Mass Matrix;80
1.3.4.1.5;4.1.5 Relationship to the Composite Rigid Body Method;82
1.3.4.2;4.2 Lagrangian Approach to the Equations of Motion;83
1.3.4.2.1;4.2.1 Properties of M and C;84
1.3.4.2.2;4.2.2 Hamiltonian Form of the Equations of Motion;87
1.3.4.2.3;4.2.3 Transformation of Lagrangian Coordinates;87
1.3.5;5 Serial-Chain Dynamics;91
1.3.5.1;5.1 Equations of Motion for a Typical Link;91
1.3.5.1.1;5.1.1 Expression for the Spatial Acceleration (k);93
1.3.5.1.2;5.1.2 Overall Equations of Motion;97
1.3.5.1.3;5.1.3 Spatial Operators with Body Frame Derivatives but Bk=Ok;99
1.3.5.2;5.2 Inclusion of External Forces and Gravity;101
1.3.5.2.1;5.2.1 Inclusion of External Forces;101
1.3.5.2.2;5.2.2 Compensating for External Forces;102
1.3.5.2.3;5.2.3 Inclusion of Gravitational Forces;103
1.3.5.3;5.3 Inverse Dynamics of Serial-Chains;104
1.3.5.3.1;5.3.1 Newton–Euler Inverse Dynamics Algorithm;104
1.3.5.3.2;5.3.2 Computing the Mass Matrix Using Inverse Dynamics;106
1.3.5.3.3;5.3.3 Composite Rigid Body Inertias Based Inverse Dynamics;107
1.3.5.4;5.4 Equations of Motion with an Inertially Fixed Velocity Reference Frame;109
1.3.5.4.1;5.4.1 Relationship Between I and ;112
1.3.6;6 Articulated Body Model for Serial Chains;113
1.3.6.1;6.1 Alternate Models for Multibody Systems;113
1.3.6.1.1;6.1.1 Terminal Body Model;113
1.3.6.1.2;6.1.2 Composite Body Model;114
1.3.6.1.3;6.1.3 Articulated Body Model;115
1.3.6.2;6.2 The P(k) Articulated Body Inertia;116
1.3.6.2.1;6.2.1 Induction Argument for P(k);116
1.3.6.2.2;6.2.2 The D (k) and G(k) Matrices;117
1.3.6.2.3;6.2.3 The (k) and (k) Projection Matrices;118
1.3.6.2.4;6.2.4 The P+(k) Matrix;120
1.3.6.2.5;6.2.5 Conclusion of the Induction Argument for P(k);121
1.3.6.3;6.3 Articulated Body Model Force Decomposition;123
1.3.6.3.1;6.3.1 The (k) Vector;125
1.3.6.3.2;6.3.2 Acceleration Relationships;126
1.3.6.4;6.4 Parallels with Estimation Theory;127
1.3.6.4.1;6.4.1 Process Covariances;128
1.3.6.4.2;6.4.2 Optimal Filtering;128
1.3.6.4.3;6.4.3 Optimal Smoothing;129
1.3.6.4.4;6.4.4 Extensions;130
1.3.7;7 Mass Matrix Inversion and AB Forward Dynamics;131
1.3.7.1;7.1 Articulated Body Spatial Operators;131
1.3.7.1.1;7.1.1 Some Operator Identities;133
1.3.7.1.2;7.1.2 Innovations Operator Factorization of the Mass Matrix;136
1.3.7.1.3;7.1.3 Operator Inversion of the Mass Matrix;137
1.3.7.2;7.2 Forward Dynamics;138
1.3.7.2.1;7.2.1 O(N) AB Forward Dynamics Algorithm;139
1.3.7.3;7.3 Extensions to the Forward Dynamics Algorithm;144
1.3.7.3.1;7.3.1 Computing Inter-Link f Spatial Forces;144
1.3.7.3.2;7.3.2 Including Gravitational Accelerations;144
1.3.7.3.3;7.3.3 Including External Forces;145
1.3.7.3.4;7.3.4 Including Implicit Constraint Forces;146
1.4;Part II General Multibody Systems;148
1.4.1;8 Graph Theory Connections;149
1.4.1.1;8.1 Directed Graphs and Trees;149
1.4.1.2;8.2 Adjacency Matrices for Digraphs;152
1.4.1.2.1;8.2.1 Properties of Digraph Adjacency Matrices;152
1.4.1.2.2;8.2.2 Properties of Tree Adjacency Matrices;154
1.4.1.2.3;8.2.3 Properties of Serial-Chain Adjacency Matrices;154
1.4.1.3;8.3 Block Weighted Adjacency Matrices;155
1.4.1.4;8.4 BWA Matrices for Tree Digraphs;156
1.4.1.4.1;8.4.1 The 1-Resolvent of Tree BWA Matrices;157
1.4.1.5;8.5 Similarity Transformations of a Tree BWA Matrix;161
1.4.1.5.1;8.5.1 Permutation Similarity Transformations;162
1.4.1.5.2;8.5.2 Similarity-Shift Transformations;163
1.4.1.6;8.6 Multibody System Digraphs;164
1.4.1.6.1;8.6.1 BWA Matrices and Serial-Chain, Rigid Body Systems;165
1.4.1.6.1.1;8.6.1.1 E Is a BWA Matrix;167
1.4.1.6.2;8.6.2 Non-Canonical Serial-Chains;168
1.4.1.7;8.7 BWA Matrices and Tree-Topology, Rigid Body Systems;168
1.4.1.7.1;8.7.1 Equations of Motion for Tree Topology Systems;169
1.4.2;9 SKO Models;172
1.4.2.1;9.1 SKO Models;173
1.4.2.1.1;9.1.1 Definition of SKO Models;173
1.4.2.1.2;9.1.2 Existence of SKO Models;174
1.4.2.1.3;9.1.3 Generalizations of SKO Models;174
1.4.2.2;9.2 SPO Operator/Vector Products for Trees;175
1.4.2.2.1;9.2.1 SKO Model O(N) Newton–Euler Inverse Dynamics;178
1.4.2.3;9.3 Lyapunov Equations for SKO Models;179
1.4.2.3.1;9.3.1 Forward Lyapunov Recursions for SKO Models;179
1.4.2.3.2;9.3.2 Mass Matrix Computation for an SKO Model;181
1.4.2.3.3;9.3.3 Backward Lyapunov Recursions for SKO Models;183
1.4.2.4;9.4 Riccati Equations for SKO Models;187
1.4.2.4.1;9.4.1 The E and SKO and SPO Operators;188
1.4.2.4.2;9.4.2 Operator Identities;189
1.4.2.5;9.5 SKO Model Mass Matrix Factorization and Inversion;192
1.4.2.5.1;9.5.1 O(N) AB Forward Dynamics;193
1.4.2.6;9.6 Generalized SKO Formulation Process;195
1.4.2.6.1;9.6.1 Procedure for Developing an SKO Model;196
1.4.2.6.2;9.6.2 Potential Non-Tree Topology Generalizations;198
1.4.3;10 Operational Space Dynamics;199
1.4.3.1;10.1 Operational Space Equations of Motion;199
1.4.3.1.1;10.1.1 Physical Interpretation;201
1.4.3.1.2;10.1.2 Operational Space Control;201
1.4.3.2;10.2 Structure of the Operational Space Inertia;202
1.4.3.2.1;10.2.1 The Extended Operational Space Compliance Matrix;202
1.4.3.2.2;10.2.2 Decomposition of ;203
1.4.3.2.3;10.2.3 Computing ;205
1.4.3.2.4;10.2.4 The Operational Space Compliance Kernel;206
1.4.3.2.5;10.2.5 Simplifications for Serial-Chain Systems;209
1.4.3.2.6;10.2.6 Explicit Computation of the Mass Matrix Inverse M-1;210
1.4.3.3;10.3 The Operational Space Cos Coriolis/Centrifugal Term;211
1.4.3.3.1;10.3.1 The U and U Projection Operators;211
1.4.3.3.2;10.3.2 Computing Cos;214
1.4.3.4;10.4 Divide and Conquer Forward Dynamics;216
1.4.4;11 Closed-Chain Dynamics;221
1.4.4.1;11.1 Modeling Closed-Chain Dynamics;221
1.4.4.1.1;11.1.1 Types of Bilateral Motion Constraints;221
1.4.4.1.2;11.1.2 Constrained System Forward Dynamics Strategies;223
1.4.4.2;11.2 Augmented Approach for Closed-Chain Forward Dynamics;224
1.4.4.2.1;11.2.1 Move/Squeeze Decompositions;226
1.4.4.2.2;11.2.2 Augmented Dynamics with Loop Constraints;227
1.4.4.2.3;11.2.3 Dual-Arm System Example;231
1.4.4.3;11.3 Projected Closed-Chain Dynamics;233
1.4.4.4;11.4 Equivalence of Augmented and Projected Dynamics;234
1.4.4.5;11.5 Unilateral Constraints;236
1.4.4.5.1;11.5.1 Complementarity Problems;237
1.4.4.5.2;11.5.2 Forward Dynamics;239
1.4.5;12 Systems with Geared Links;240
1.4.5.1;12.1 Equations of Motion;241
1.4.5.1.1;12.1.1 Reformulated Equations of Motion;243
1.4.5.1.2;12.1.2 Eliminating the Geared Constraint;244
1.4.5.2;12.2 SKO Model for Geared Systems;245
1.4.5.2.1;12.2.1 Expression for the Mass Matrix;247
1.4.5.3;12.3 Computation of the Mass Matrix;248
1.4.5.3.1;12.3.1 Optimized Composite Body Inertia Algorithm;249
1.4.5.4;12.4 O(N) AB Forward Dynamics;250
1.4.5.4.1;12.4.1 Mass Matrix Factorization and Inversion;250
1.4.5.4.2;12.4.2 Recursive AB Forward Dynamics Algorithm;252
1.4.5.4.3;12.4.3 Optimization of the Forward Dynamics Algorithm;253
1.4.6;13 Systems with Link Flexibility;255
1.4.6.1;13.1 Lumped Mass Model for a Single Flexible Body;255
1.4.6.1.1;13.1.1 Equations of Motion of the Okj Node;256
1.4.6.1.2;13.1.2 Nodal Equations of Motion for the kth Flexible Body;257
1.4.6.1.3;13.1.3 Recursive Relationships Across the Flexible Bodies;258
1.4.6.2;13.2 Modal Formulation for Flexible Bodies;259
1.4.6.2.1;13.2.1 Modal Mass Matrix for a Single Body;261
1.4.6.2.2;13.2.2 Recursive Relationships Using Modal Coordinates;262
1.4.6.2.3;13.2.3 Recursive Propagation of Accelerations;263
1.4.6.2.4;13.2.4 Recursive Propagation of Forces;264
1.4.6.2.5;13.2.5 Overall Equations of Motion;265
1.4.6.3;13.3 SKO Models for Flexible Body Systems;265
1.4.6.3.1;13.3.1 Operator Expression for the System Mass Matrix;267
1.4.6.3.2;13.3.2 Illustration of the SKO Formulation Procedure;267
1.4.6.4;13.4 Inverse Dynamics Algorithm;269
1.4.6.5;13.5 Mass Matrix Computation;271
1.4.6.6;13.6 Factorization and Inversion of the Mass Matrix;272
1.4.6.7;13.7 AB Forward Dynamics Algorithm;274
1.4.6.7.1;13.7.1 Simplified Algorithm for the Articulated Body Quantities;274
1.4.6.7.2;13.7.2 Simplified AB Forward Dynamics Algorithm;276
1.5;Part III Advanced Topics;280
1.5.1;14 Transforming SKO Models;281
1.5.1.1;14.1 Partitioning Digraphs;281
1.5.1.1.1;14.1.1 Partitioning by Path-Induced Sub-Graphs;282
1.5.1.2;14.2 Partitioning SKO Models;283
1.5.1.2.1;14.2.1 Partitioning SKO Model Operators;283
1.5.1.2.2;14.2.2 Partitioning of an SKO Model;285
1.5.1.3;14.3 SPO Operator Sparsity Structure;286
1.5.1.3.1;14.3.1 Decomposition into Serial-Chain Segments;286
1.5.1.3.2;14.3.2 Sparsity Structure of the E A SKO Matrix;288
1.5.1.3.3;14.3.3 Sparsity Structure of the A Matrix;289
1.5.1.3.4;14.3.4 Sparsity Structure of the M Mass Matrix;290
1.5.1.4;14.4 Aggregating Sub-Graphs;290
1.5.1.4.1;14.4.1 Edge and Node Contractions;291
1.5.1.4.2;14.4.2 Tree Preservation After Sub-Graph Aggregation;292
1.5.1.4.3;14.4.3 The SA Aggregation Sub-Graph;295
1.5.1.5;14.5 Transforming SKO Models Via Aggregation;295
1.5.1.5.1;14.5.1 SKO Operators After Body Aggregation;295
1.5.1.5.2;14.5.2 SKO Model for the TS Aggregated Tree;298
1.5.1.6;14.6 Aggregation Relationships at the Component Level;300
1.5.1.6.1;14.6.1 Velocity Relationships;301
1.5.1.6.2;14.6.2 Acceleration Relationships;302
1.5.1.6.3;14.6.3 Force Relationships;303
1.5.2;15 Constraint Embedding;305
1.5.2.1;15.1 Constraint Embedding Strategy;305
1.5.2.1.1;15.1.1 Embedding Constraint Sub-Graphs;307
1.5.2.2;15.2 Examples of Constraint Embedding;311
1.5.2.2.1;15.2.1 Geared Links;312
1.5.2.2.2;15.2.2 Planar Four-Bar Linkage System (Terminal Cut);313
1.5.2.2.3;15.2.3 Planar Four-Bar Linkage System (Internal Cut);314
1.5.2.3;15.3 Recursive AB Forward Dynamics;315
1.5.2.3.1;15.3.1 Articulated Body Inertias for the Aggregated System;315
1.5.2.3.2;15.3.2 Mass Matrix Factorization and Inversion;316
1.5.2.3.3;15.3.3 AB Forward Dynamics Algorithm;316
1.5.2.4;15.4 Computing XS and S;317
1.5.2.5;15.5 Generalization to Multiple Branches and Cut-Edges;319
1.5.3;16 Under-Actuated Systems;320
1.5.3.1;16.1 Modeling of Under-Actuated Manipulators;321
1.5.3.1.1;16.1.1 Decomposition into Passive and Active Systems;322
1.5.3.1.2;16.1.2 Partitioned Equations of Motion;323
1.5.3.1.3;16.1.3 Spatial Operator Expression for M-1pp;324
1.5.3.1.4;16.1.4 Operator Expressions for S Blocks;326
1.5.3.2;16.2 O(N) Generalized Dynamics Algorithms;327
1.5.3.2.1;16.2.1 Application to Prescribed Motion Dynamics;331
1.5.3.3;16.3 Jacobians for Under-Actuated Systems;331
1.5.3.3.1;16.3.1 The Generalized Jacobian JG;332
1.5.3.3.2;16.3.2 Computed-Torque for Under-Actuated Systems;333
1.5.3.3.3;16.3.3 The Disturbance Jacobian JD;334
1.5.3.4;16.4 Free-Flying Systems as Under-Actuated Systems;336
1.5.3.4.1;16.4.1 Integrals of Motion for Free-Flying Systems;336
1.5.4;17 Free-Flying Systems;338
1.5.4.1;17.1 Dynamics of Free-Flying Manipulators;338
1.5.4.1.1;17.1.1 Dynamics with Link n as Base-Body;338
1.5.4.1.2;17.1.2 Dynamics with Link 1 as Base-Body;339
1.5.4.1.3;17.1.3 Direct Computation of Link Spatial Acceleration;343
1.5.4.1.4;17.1.4 Dynamics with Link k as Base-Body;343
1.5.4.2;17.2 The Base-Invariant Forward Dynamics Algorithm;344
1.5.4.2.1;17.2.1 Parallels with Smoothing Theory;345
1.5.4.2.2;17.2.2 Simplifications Using Non-Minimal Coordinates;346
1.5.4.2.3;17.2.3 Computational Issues;347
1.5.4.2.4;17.2.4 Extensions to Tree-Topology Manipulators;347
1.5.4.3;17.3 SKO Model with kth Link as Base-Body;348
1.5.4.3.1;17.3.1 Generalized Velocities with kth Link as the Base-Body;348
1.5.4.3.2;17.3.2 Link Velocity Recursions with kth Link as the Base-Body;349
1.5.4.3.3;17.3.3 Partitioned System;350
1.5.4.3.4;17.3.4 Properties of a Serial-Chain SKO Operator;351
1.5.4.3.5;17.3.5 Reversing the SKO Operator;352
1.5.4.3.6;17.3.6 Transformed SKO Model;354
1.5.4.4;17.4 Base-Invariant Operational Space Inertias;355
1.5.5;18 Spatial Operator Sensitivities for Rigid-Body Systems;359
1.5.5.1;18.1 Preliminaries;359
1.5.5.1.1;18.1.1 Notation;359
1.5.5.1.2;18.1.2 Identities for Vv;360
1.5.5.2;18.2 Operator Time Derivatives;362
1.5.5.2.1;18.2.1 Time Derivatives of (k+1, k), H(k) and M(k);362
1.5.5.2.2;18.2.2 Time Derivatives with Ok=Bk;364
1.5.5.2.3;18.2.3 Time Derivative of the Mass Matrix;366
1.5.5.3;18.3 Operator Sensitivities;367
1.5.5.3.1;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
1.5.5.4;18.4 Mass Matrix Related Quantities;371
1.5.5.4.1;18.4.1 Sensitivity of bold0mu mumu MMOT1MMMM*;371
1.5.5.4.2;18.4.2 Sensitivity of the Mass Matrix Mi;371
1.5.5.4.3;18.4.3 Sensitivity of the Kinetic Energy;372
1.5.5.4.4;18.4.4 Equivalence of Lagrangian and Newton–Euler Dynamics;373
1.5.5.5;18.5 Time Derivatives of Articulated Body Quantities;374
1.5.5.6;18.6 Sensitivity of Articulated Body Quantities;379
1.5.5.7;18.7 Sensitivity of Innovations Factors;381
1.5.6;19 Diagonalized Lagrangian Dynamics;384
1.5.6.1;19.1 Globally Diagonalized Dynamics;384
1.5.6.2;19.2 Diagonalization in Velocity Space;387
1.5.6.2.1;19.2.1 Coriolis Force Does No Work;389
1.5.6.2.2;19.2.2 Rate of Change of the Kinetic Energy;389
1.5.6.3;19.3 The Innovations Factors as Diagonalizing Transformations;390
1.5.6.3.1;19.3.1 Transformations Between and ;391
1.5.6.4;19.4 Expression for C(,) for Rigid-Link Systems;392
1.5.6.4.1;19.4.1 Closed-Form Expression for ;392
1.5.6.4.2;19.4.2 Operator Expression for C(,);394
1.5.6.4.3;19.4.3 Decoupled Control;397
1.5.6.5;19.5 Un-normalized Diagonalized Equations of Motion;398
1.5.6.5.1;19.5.1 O(N) Forward Dynamics;399
1.5.7;Useful Mathematical Identities;401
1.5.7.1;A.1 3-Vector Cross-Product Identities;401
1.5.7.2;A.2 Matrix and Vector Norms;401
1.5.7.3;A.3 Schur Complement and Matrix Inverse Identities;402
1.5.7.4;A.4 Matrix Inversion Identities;404
1.5.7.5;A.5 Matrix Trace Identities;404
1.5.7.6;A.6 Derivative and Gradient Identities;405
1.5.7.6.1;A.6.1 Function Derivatives;405
1.5.7.6.2;A.6.2 Vector Gradients;405
1.5.7.6.3;A.6.3 Matrix Derivatives;406
1.5.8;Attitude Representations;407
1.5.8.1;B.1 Euler Angles;407
1.5.8.2;B.2 Angle/Axis Parameters;408
1.5.8.3;B.3 Unit Quaternions/Euler Parameters;410
1.5.8.3.1;B.3.1 The E+(q) and E-(q) Matrices;412
1.5.8.3.2;B.3.2 Quaternion Transformations;413
1.5.8.3.3;B.3.3 Quaternion Differential Kinematics;414
1.5.8.4;B.4 Gibbs Vector Attitude Representations;416
1.5.9;Solutions;418
1.5.10;References;477
1.5.11;List of Notation;477
1.5.12;Index;494