Singular Configurations of Mechanisms and Manipulators (eBook)

Preface 6
Contents 9
Generalized Singularity Analysis 10
1 Introduction 10
2 The Standard View of Singularity 13
2.1 The Simplest Case: Serial Chain Singularity 13
2.2 The Serial–Parallel Duality 15
2.3 A Generalization: The Input–Output Equation 16
2.4 The Counterexamples 17
3 Singularity via Differentiable Maps 20
3.1 Serial Chains 20
3.2 Chains with Closed Loops 20
3.3 Regular and Singular Configurations 22
4 Singularity via Instantaneous Kinematics 23
4.1 Instantaneous Motions 24
4.2 Instantaneous Definition of Singularity 25
4.3 Singularity Types and Classes 26
4.4 Classes of Singularities 28
5 Common and Uncommon Singularities 29
5.1 Generic and Nongeneric Singularity Types 29
5.2 Parallel Mechanisms with Extendable Legs 30
5.3 The Isotropic Singularity of the Orthoglide 31
6 Constraint Singularities of Parallel Manipulators 35
6.1 Another ``Isotropic'' Configuration 35
6.2 Constraints, Freedoms, and Constrained Singularities 36
6.3 Singularities of 3-RPS Manipulators 37
6.4 The Simplest Constraint Singularity 39
6.5 Constraint Singularities as Branching Points 40
7 Conclusions 45
References 46
Singularity Set Computation: A Hands-On Session with the CUIK Suite 48
1 Introduction 48
1.1 Basic Concepts 49
1.2 An Illustrative Example 50
2 Singularity Set Computation with the CUIK Suite 53
2.1 Preparation 53
2.2 Dextar's Geometry 54
2.3 Assembly Constraints 55
2.4 Inverse Singularity Conditions 57
2.5 Forward Singularity Conditions 58
2.6 Computation and Output Plots 59
2.7 Visualizing the Output Portrait 69
2.8 Visualizing the Input Portrait 70
2.9 Workspace Boundaries and Interior Barriers 72
References 74
Cuspidal Robots 75
1 Introduction 75
1.1 Preliminaries 75
1.2 Brief History of Cuspidal Robots 76
1.3 Questions of Interest 77
1.4 Chapter Outline 78
2 Postures 78
2.1 Definition 78
2.2 Postures and Aspects Case 1: Non-cuspidal Robots 78
2.3 Postures and Aspects Case 2: Cuspidal Robots 80
2.4 Non-singular Change of Posture Versus Singular Change of Posture 82
3 Path Feasibility 85
3.1 Showing an Example 85
3.2 Regions of Feasible Paths for Non-cuspidal Robots 86
3.3 Characteristic Surfaces 88
3.4 Uniqueness Domains and Regions of Feasible Paths for Cuspidal Robots 91
4 Identification, Enumeration and Classification of Cuspidal and Non-cuspidal Robots 92
4.1 Simplifying Geometric Conditions 92
4.2 Identification of Cuspidal Robots and Classification 95
5 Higher-Degree-of-Freedom Robots and Parallel Robots 101
5.1 6-dof Serial Robots 101
5.2 Parallel Robots 102
6 Conclusions 105
References 106
Mechanism Constraints and Singularities—The Algebraic Formulation 108
1 Introduction 109
2 Parameterization of SE(3) 109
3 Constraint Equations 113
3.1 Geometric Constraint Equations 114
3.2 Elimination Method 128
3.3 Linear Implicitization 130
3.4 Singularities in Serial Chains 139
4 Algebraic Basics 141
4.1 Ideals and Affine Varieties 141
4.2 Standard Bases 153
4.3 Elimination 156
4.4 Dimension, Primary Decomposition 159
5 Complete Kinematic Analysis of the TSAI Parallel Manipulator 162
5.1 Introduction 162
5.2 Manipulator Design 163
5.3 Deduction of Constraint Equations 164
5.4 Solving the System 167
6 The Manipulator's Operation Modes 171
7 Conditions for Singular Poses 174
7.1 Changing Operation Modes 180
8 Conclusion 185
References 185
Local Investigation of Mobility and Singularities of Linkages 188
1 Introduction 188
2 Kinematic Model of Single-Loop Linkages 190
2.1 Geometric Loop Closure Constraints and the Configuration Space 190
2.2 Velocity Loop Closure Constraints 192
3 Kinematic Model of Multi-loop Linkages 194
4 Phenomenology and Definitions 196
5 Local Analysis of Smooth Finite Motions 204
5.1 Kinematic Tangent Cone to the C-Space 205
5.2 Higher-Order Loop Closure Constraints 206
5.3 Computational Procedure 208
5.4 Examples 209
6 Local Analysis of Smooth Finite Motions with Certain Rank 215
6.1 Kinematic Tangent Cone 215
6.2 Higher-Order Time Derivatives of Jacobian Minors 217
6.3 Examples 218
7 Interlude: The Tangent Cone to a Variety 221
8 Local Analysis of the Configuration Space 225
8.1 Higher-Order local Approximation 225
8.2 Higher-Order Series Expansion of the Constraint Mapping 225
8.3 Examples 227
9 Local Analysis of Subvarieties of the C-Space of Points with Certain Rank 231
9.1 Higher-Order Local Approximation 232
9.2 Higher-Order Series Expansion of Jacobian Minors 232
9.3 Examples 232
10 Discussion 233
References 234