Page 10 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

CONTENTS ix

4.3.4 Canonical Solution 166
4.3.5 Near-Canonical Solution 169
4.3.6 The Algorithm 170
4.3.7 Convergence. Computational Complexity 172
4.3.8 Examples 175

5 Motion Planning for Two-Dimensional Arm Manipulators 177
5.1 Introduction 177
5.1.1 Model and Definitions 183
5.2 Planar Revolute–Revolute (RR) Arm 187
5.2.1 Analysis 189
5.2.2 Algorithm 210
5.2.3 Step Planning 211
5.2.4 Example 212
5.2.5 Motion Planning with Vision and Proximity Sensing 213
5.2.6 Concluding Remarks 218

5.3 Distinct Kinematic Configurations of RR Arm 220
5.4 Prismatic–Prismatic (PP, or Cartesian) Arm 226
5.5 Revolute–Prismatic (RP) Arm with Parallel Links 229
5.6 Revolute–Prismatic (RP) Arm with Perpendicular Links 234

5.7 Prismatic–Revolute (PR) Arm 234
5.8 Topology of Arm’s Free Configuration Space 245
5.8.1 Workspace; Configuration Space 249
5.8.2 Interaction Between the Robot and Obstacles 252
5.8.3 Uniform Local Connectedness 255
5.8.4 The General Case of 2-DOF Arm Manipulators 256
5.9 Appendix 258
5.10 Exercises 267

6 Motion Planning for Three-Dimensional Arm Manipulators 271
6.1 Introduction 271
6.2 The Case of the PPP (Cartesian) Arm 276
6.2.1 Model, Definitions, and Terminology 276
6.2.2 The Approach 283
6.2.3 Topology of W-Obstacles and C-Obstacles 285
6.2.4 Connectivity of C 295