Kinematic Differential Geometry and Saddle Synthesis of Linkages

Preface ix Acknowledgments xi 1 Planar Kinematic Differential Geometry 1 1.1 Plane Curves 2 1.1.1 Vector Curve 2 1.1.2 Frenet Frame 6 1.1.3 Adjoint Approach 10 1.2 Planar Continuous Kinematics 14 1.2.1 Displacement 14 1.2.2 Centrodes 18 1.2.3 Euler-Savary Equation 26 1.2.4 Curvatures in Higher Order 33 1.2.5 Line Path 42 1.3 Plane Coupler Curves 49 1.3.1 Local Characteristics 49 1.3.2 Double Points 51 1.3.3 Four-bar Linkage I 55 1.3.4 Four-bar Linkage II 61 1.3.5 Oval Coupler Curves 67 1.3.6 Symmetrical Coupler Curves 73 1.3.7 Distribution of Coupler Curves 75 1.4 Discussion 78 References 80 2 Discrete Kinematic Geometry and Saddle Synthesis of Planar Linkages 83 2.1 Matrix Representation 84 2.2 Saddle Point Programming 85 2.3 Saddle Circle Point 88 2.3.1 Saddle Circle Fitting 89 2.3.2 Saddle Circle 92 2.3.3 Four Positions 95 2.3.4 Five Positions 97 2.3.5 Multiple Positions 100 2.3.6 Saddle Circle Point 101 2.4 Saddle Sliding Point 106 2.4.1 Saddle Line Fitting 108 2.4.2 Saddle Line 109 2.4.3 Three Positions 111 2.4.4 Four Positions 114 2.4.5 Multiple Positions 116 2.4.6 Saddle Sliding Point 116 2.5 The Saddle Kinematic Synthesis of Planar Four-bar Linkages 120 2.5.1 Kinematic Synthesis 122 2.5.2 Crank-rocker Linkage 129 2.5.3 Crank-slider Linkage 139 2.6 The Saddle Kinematic Synthesis of Planar Six-bar Linkages with Dwell Function 145 2.6.1 Six-bar Linkages 146 2.6.2 Local Saddle Curve Fitting 149 2.6.3 Dwell Function Synthesis 150 2.7 Discussion 163 References 167 3 Differential Geometry of the Constraint Curves and Surfaces 171 3.1 Space Curves 171 3.1.1 Vector Representations 171 3.1.2 Frenet Trihedron 175 3.2 Surfaces 177 3.2.1 Elements of Surfaces 177 3.2.2 Ruled Surfaces 183 3.2.3 Adjoint Approach 186 3.3 Constraint Curves and Surfaces 192 3.4 Spherical and Cylindrical Curves 195 3.4.1 Spherical Curves (S-S) 195 3.4.2 Cylindrical Curves (C-S) 197 3.5 Constraint Ruled Surfaces 201 3.5.1 Constant Inclination Ruled Surfaces (C'-P'-C) 201 3.5.2 Constant Axis Ruled Surfaces (C'-C) 204 3.5.3 Constant Parameter Ruled Surfaces (H-C, R-C) 208 3.5.4 Constant Distance Ruled Surfaces (S'-C) 212 3.6 Generalized Curvature of Curves 214 3.6.1 Generalized Curvature of Space Curves 215 3.6.2 Spherical Curvature and Cylindrical Curvature 218 3.7 Generalized Curvature of Ruled Surfaces 224 3.7.1 Tangent Conditions 224 3.7.2 Generalized Curvature 225 3.7.3 Constant Inclination Curvature 227 3.7.4 Constant Axis Curvature 228 3.8 Discussion 228 References 230 4 Spherical Kinematic Differential Geometry 233 4.1 Spherical Displacement 233 4.1.1 General Expression 233 4.1.2 Adjoint Expression 235 4.2 Spherical Differential Kinematics 240 4.2.1 Spherical Centrodes (Axodes) 240 4.2.2 Curvature and Euler-Savary Formula 245 4.3 Spherical Coupler Curves 257 4.3.1 Basic Equation 257 4.3.2 Double Point 257 4.3.3 Distribution 262 4.4 Discussion 263 References 266 5 Discrete Kinematic Geometry and Saddle Synthesis of Spherical Linkages 267 5.1 Matrix Representation 267 5.2 Saddle Spherical Circle Point 269 5.2.1 Saddle Spherical Circle Fitting 269 5.2.2 Saddle Spherical Circle 272 5.2.3 Four Positions 274 5.2.4 Five Positions 275 5.2.5 Multiple Positions 278 5.2.6 Saddle Spherical Circle Point 279 5.3 The Saddle Kinematic Synthesis of Spherical Four-bar Linkages 282 5.3.1 Kinematic Synthesis 283 5.3.2 Saddle Kinematic Synthesis of Spherical Four-bar Linkages 289 5.4 Discussion 298 References 300 6 Spatial Kinematic Differential Geometry 303 6.1 Displacement Equation 303 6.1.1 General Description 304 6.1.2 Adjoint Description 306 6.2 Axodes 310 6.2.1 Fixed Axode 310 6.2.2 Moving Axode 312 6.3 Differential Kinematics of Points 314 6.3.1 Point Trajectory 315 6.3.2 Darboux Frame 319 6.3.3 Euler-Savary Analogue 320 6.3.4 Generalized Curvature 323 6.4 Differential Kinematics of Lines 326 6.4.1 Frenet Frame 326 6.4.2 Striction Curve 330 6.4.3 Spherical Image Curve 332 6.4.4 Connecting Kinematic Pairs 334 6.4.5 Constant Axis Curvature 338 6.4.6 Constant Parameter Curvature 349 6.5 Differential Kinematics of Spatial Four-Bar Linkage RCCC 355 6.5.1 Adjoint Expression 355 6.5.2 Axodes 358 6.5.3 Point Trajectory 361 6.5.4 Line Trajectory 368 6.6 Discussion 378 References 380 7 Discrete Kinematic Geometry and Saddle Synthesis of Spatial Linkages 383 7.1 The Displacement Matrix 384 7.2 Sphere Point PSS 386 7.2.1 Spherical Surface Fitting 386 7.2.2 Saddle Spherical Surface 390 7.2.3 Five Positions 391 7.2.4 Six Positions 393 7.2.5 Multiple Positions 396 7.2.6 Saddle Sphere Point 396 7.3 Cylinder Point PCS 401 7.3.1 Cylindrical Surface Fitting 402 7.3.2 Saddle Cylindrical Surface 404 7.3.3 Six Positions 406 7.3.4 Seven Positions 407 7.3.5 Multiple Positions 410 7.3.6 Saddle Cylinder Point 410 7.3.7 The Degeneration of the Saddle Cylinder Point (R-S, H-S) 412 7.4 Constant Axis Line LCC 417 7.4.1 Ruled Surface Fitting 417 7.4.2 Saddle Spherical Image Circle Point 418 7.4.3 Saddle Striction Cylinder Point 420 7.4.4 Saddle Constant Axis Line 425 7.5 Degenerate Constant Axis Lines LRC and LHC 426 7.5.1 Saddle Characteristic Line LRC (R-C, R-R) 426 7.5.2 Saddle Characteristic Line LHC (H-C, H-R, H-H) 428 7.6 The Saddle Kinematic Synthesis of Spatial Four-Bar Linkages 444 7.6.1 A Brief Introduction 445 7.6.2 The Spatial Linkage RCCC 450 7.6.3 The Spatial Linkage RRSS 454 7.6.4 The Spatial Linkage RRSC 458 7.7 Discussion 461 References 464 Appendix A Displacement Solutions of Spatial Linkages RCCC 467 Appendix B Displacement Solutions of the Spatial RRSS Linkage 473 Index 477