Singularities of Robot Mechanisms (eBook)

Preface 8
Motivation and Purpose 8
Highlights 9
Intended Audience 10
Acknowledgments 10
Contents 12
Notation 15
Scalars, Vectors, and Tuples 15
Vector Components and Compound Vectors 15
Screws 16
Matrices 16
Sets 16
Mechanism Symbols 17
Maps 17
Other Symbols 18
List of Figures 19
1 Introduction 22
1.1 Historical Context 24
1.2 Assumptions and Scope 26
1.3 Reader's Guide 29
References 30
2 Singularity Types 34
2.1 Forward and Inverse Singularities 34
2.2 A Geometric Interpretation of Singularities 38
2.2.1 Singularities Yield Shaky Mechanisms 38
2.2.2 C-Space, Input, and Output Singularities 41
2.2.3 Singularities as Silhouette Points 43
2.2.4 Indeterminacies in the Mechanism Trajectory 47
2.3 Lower-Level Singularity Types 48
2.4 A Simple Mechanism with All Singularities 54
References 56
3 Numerical Computation of Singularity Sets 57
3.1 A Suitable Approach 57
3.2 Formulating the Equations of the Singularity Set 59
3.2.1 The Assembly Constraints 59
3.2.2 The Velocity Equation 62
3.3 Isolating the Singularity Set 65
3.3.1 Reduction to a Simple Quadratic Form 65
3.3.2 Initial Bounding Box 66
3.3.3 A Branch-and-Prune Method 66
3.3.4 Computational Cost and Parallelization 70
3.4 Visualising the Singularity Sets 70
3.5 Case Studies 72
3.5.1 The 3-RRR Mechanism 72
3.5.2 The Gough-Stewart Platform 81
3.5.3 A Double-Loop Mechanism 86
References 92
4 Workspace Determination 94
4.1 The Need of a General Method 94
4.2 The Workspace and Its Boundaries 95
4.2.1 Jacobian Rank Deficiency Conditions 96
4.2.2 Barrier Analysis 98
4.3 Issues of Continuation Methods 100
4.4 Exploiting the Branch-and-Prune Machinery 103
4.4.1 Joint Limit Constraints 103
4.4.2 Equations of the Generalised Singularity Set 105
4.4.3 Numerical Solution and Boundary Identification 108
4.5 Case Studies 110
4.5.1 Multicomponent Workspaces 110
4.5.2 Hidden Barriers 111
4.5.3 Degenerate Barriers 117
4.5.4 Very Complex Mechanisms 126
References 127
5 Singularity-Free Path Planning 130
5.1 Related Work 131
5.2 Formulating the Singularity-Free C-Space 132
5.3 Exploring the Singularity-Free C-Space 135
5.3.1 Constructing a Chart 135
5.3.2 Constructing an Atlas 136
5.3.3 Focusing on the Path to the Goal 140
5.3.4 The Planner Algorithm 141
5.4 Case Studies 143
5.4.1 A Simple Example 144
5.4.2 A 3-underlineRRR Parallel Robot 145
References 153
6 Planning with Further Constraints 156
6.1 Wrench-Feasibility Constraints 157
6.2 The Planning Problem 161
6.2.1 A Characterisation of the Wrench-Feasible C-Space 161
6.2.2 Conversion into Equality Form 163
6.2.3 The Navigation Manifold 164
6.2.4 Addition of Pose Constraints 165
6.3 Proofs of the Properties 165
6.3.1 Nonsingularity of the Screw Jacobian 166
6.3.2 Smoothness of the Navigation Manifold 166
6.3.3 Smoothness of Lower-Dimensional Subsets 167
6.4 Case Studies 169
6.4.1 Planning in Illustrative Slices 169
6.4.2 Planning in the IRI Hexacrane 174
6.4.3 Planning in Rigid-Limbed Hexapods 177
6.4.4 Problem Sizes and Computation Times 178
6.5 Details About the Wrench Ellipsoid 179
6.6 Extensions 182
References 183
7 Conclusions 185
7.1 Summary of Results 185
7.2 Future Research Directions 187
References 190
Author Index 192
Subject Index 196