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