Xianwen Kong and Clement Gosselin

Type Synthesis of Parallel Mechanisms

Spri ringer

Contents

Part I Synthesis Approach

1. Introduction 5 1.1 Parallel Mechanisms and Their Applications 5 1.2 Type Synthesis of Parallel Mechanisms 7 1.3 Representation of Parallel Mechanisms 9 1.4 The State of the Art 13 1.5 Objective and Organization of This Book 17

2. Structural Analysis 19 2.1 Screw Theory 19 2.2 Instantaneous Mobility Analysis of Kinematic Chains 34 2.3 Validity Condition of Actuated Joints in Parallel Mechanisms . . 40 2.4 Summary 42

3. T y P e Synthesis of Single-Loop Kinematic Chains 43 3.1 Introduction 43 3.2 Procedure for the Type Synthesis 43 3.3 Types of Single-Loop Kinematic Chains 49 3.4 Summary 53

4. Classification of Parallel Mechanisms 55 4.1 Motion Patterns of Parallel Mechanisms 55 4.2 The Concept of Virtual Chain 56 4.3 A Preliminary Classification of Motion Patterns 57 4.4 Summary 61

5. Virtual-Chain Approach for the Type Synthesis 63 5.1 Introduction 63 5.2 Conditions for V= parallel kinematic chains 64 5.3 Systematic Type Synthesis of V= Parallel Mechanisms 64 5.4 Step 1: Decomposition of the Wrench System 66

XIV Contents

5.5 Step 2: Type Synthesis of Legs 70 5.6 Step 3: Assembly of Legs 78 5.7 Step 4: Selection of the Actuated Joints 79 5.8 Summary 83

Part II Case Studies

6. Three-DOF P P P = Parallel Mechanisms 89 6.1 Introduction 89 6.2 Wrench System of a P P P = PKC 90 6.3 Conditions for P P P = PKCs 90 6.4 Procedure for the type synthesis 91 6.5 Step 1: Decomposition of the wrench System 91 6.6 Step 2: Type Synthesis of Legs 92 6.7 Step 3: Assembly of Legs 97 6.8 Step 4: Selection of Actuated Joints 99 6.9 Summary 106

7. Three-DOF S = Parallel Mechanisms 109 7.1 Introduction 109 7.2 Wrench System of an S= PKC 110 7.3 Conditions for S= PKCs 110 7.4 Procedure for the type synthesis 110 7.5 Step 1: Decomposition of the wrench system 111 7.6 Step 2: Type Synthesis of Legs 112 7.7 Step 3: Assembly of Legs 117 7.8 Step 4: Selection of Actuated Joints 119 7.9 Summary 124

8. Three-DOF P P R = Parallel Mechanisms 125 8.1 Introduction 125 8.2 Wrench System of a P P R = PKC 126 8.3 Conditions for P P R = PKCs 126 8.4 Procedure for the type synthesis 126 8.5 Step 1: Decomposition of the wrench system 127 8.6 Step 2: Type Synthesis of Legs 127 8.7 Step 3: Assembly of Legs 132 8.8 Step 4: Selection of Actuated Joints 135 8.9 Summary 139

9. Four-DOF P P P R = Parallel Mechanisms 141 9.1 Introduction 141 9.2 Wrench System of a P P P R = PKC 142 9.3 Conditions for P P P R = PKCs 142 9.4 Procedure for the Type Synthesis 143

Contents XV

9.5 Step 1: Decomposition of the Wrench System 143 9.6 Step 2: Type Synthesis of Legs 144 9.7 Step 3: Assembly of Legs 149 9.8 Step 4: Selection of Actuated Joints 150 9.9 Summary 157

10. Four-DOF S P = Parallel Mechanisms 159 10.1 Introduction 159 10.2 Wrench System of an SP= PKC 160 10.3 Conditions for SP= PKCs 160 10.4 Procedure for the type synthesis 161 10.5 Step 1: Decomposition of the Wrench System 161 10.6 Step 2: Type Synthesis of Legs 162 10.7 Step 3: Assembly of Legs 166 10.8 Step 4: Selection of Actuated Joints 167 10.9 Summary 171

11. Five-DOF U S = Parallel Mechanisms 173 11.1 Introduction 173 11.2 Wrench System of a US= PKC 174 11.3 Conditions for US= PKCs 174 11.4 Procedure for the type synthesis 174 11.5 Step 1: Decomposition of the wrench System 175 11.6 Step 2: Type Synthesis of Legs 175 11.7 Step 3: Assembly of Legs 178 11.8 Step 4: Selection of Actuated Joints 180 11.9 Summary 183

12. Five-DOF P P P U = Parallel Mechanisms 185 12.1 Introduction 185 12.2 Wrench System of a PPPU= PKC 186 12.3 Conditions for PPPU= PKCs 186 12.4 Procedure for the type synthesis 186 12.5 Step 1: Decomposition of the wrench System 187 12.6 Step 2: Type Synthesis of Legs 187 12.7 Step 3: Assembly of Legs 193 12.8 Step 4: Selection of Actuated Joints 193 12.9 Summary 197

13. Five-DOF P P S = Parallel Mechanisms 199 13.1 Introduction 199 13.2 Wrench System of a PPS= PKC 199 13.3 Conditions for PPS= PKCs 200 13.4 Procedure for the type synthesis 200 13.5 Step 1: Decomposition of the wrench System 201 13.6 Step 2: Type Synthesis of Legs 201

XVI Contents

13.7 Step 3: Assembly of Legs 204 13.8 Step 4: Selection of Actuated Joints 206 13.9 Summary 210

14. Parallel Mechanisms with a Parallel Virtual Chain 213 14.1 Introduction 213 14.2 Procedure for the Type Synthesis of Parallel Mechanisms 213 14.3 Type Synthesis of 3-PPS= PMs 214 14.4 Type Synthesis of 2-PPPU= PMs 217 14.5 Type Synthesis of US-PPS= PMs 220 14.6 Summary 222

15. Conclusions 223 15.1 Major Contributions 223 15.2 Future Research 224

A. Design of Devices Based on Parallel Mechanisms 227 A.l Common Issues 227 A.2 Specific Issues 229 A.3 Summary 234

B. Mobility Analysis of Parallel Mechanisms 235 B.l Principle of Full-Cycle Mobility Inspection 235 B.2 Procedure for the Mobility Analysis 237 B.3 Examples 239 B.4 Summary 246

C. Method Based on the Displacement Group Theory 247 C.l Displacement Groups and Their Generators 247 C.2 Operations on Displacement Subgroups 249 C.3 Kinematic Bond 252 C.4 Steps for the Type Synthesis of Parallel Kinematic Chains 252

References 255

Index 267