Kinematic Differential Geometry and Saddle Synthesis of Linkages

DELUN WANG, Dalian University of Technology, China WEI WANG, Dalian University of Technology, China

"After reading this book, the reader will be convinced that the intended audience for it consists of researchers in differential geometry and discrete kinematic geometry, particularly in multi- dimensional space and in the kinematics of manipulators with multiple degrees of freedom. The book is useful also for scientists and engineers as well as graduate and Ph.d students interested in the theory and applications of multibody systems." (Zentralblatt MATH 2016)

• Preface ixAcknowledgments xi1 Planar Kinematic Differential Geometry 11.1 Plane Curves 21.1.1 Vector Curve 21.1.2 Frenet Frame 61.1.3 Adjoint Approach 101.2 Planar Differential Kinematics 141.2.1 Displacement 141.2.2 Centrodes 181.2.3 Euler–Savary Equation 261.2.4 Curvatures in Higher Order 331.2.5 Line Path 421.3 Plane Coupler Curves 491.3.1 Local Characteristics 491.3.2 Double Points 511.3.3 Four-bar Linkage I 551.3.4 Four-bar Linkage II 611.3.5 Oval Coupler Curves 671.3.6 Symmetrical Coupler Curves 731.3.7 Distribution of Coupler Curves 751.4 Discussion 78References 802 Discrete Kinematic Geometry and Saddle Synthesis of Planar Linkages 832.1 Matrix Representation 842.2 Saddle Point Programming 852.3 Saddle Circle Point 882.3.1 Saddle Circle Fitting 892.3.2 Saddle Circle 922.3.3 Four Positions 952.3.4 Five Positions 972.3.5 Multiple Positions 1002.3.6 Saddle Circle Point 1012.4 Saddle Sliding Point 1062.4.1 Saddle Line Fitting 1082.4.2 Saddle Line 1092.4.3 Three Positions 1112.4.4 Four Positions 1142.4.5 Multiple Positions 1162.4.6 Saddle Sliding Point 1162.5 The Saddle Kinematic Synthesis of Planar Four-bar Linkages 1202.5.1 Kinematic Synthesis 1222.5.2 Crank-rocker Linkage 1292.5.3 Crank-slider Linkage 1392.6 The Saddle Kinematic Synthesis of Planar Six-bar Linkages with Dwell Function 1452.6.1 Six-bar Linkages 1462.6.2 Local Saddle Curve Fitting 1492.6.3 Dwell Function Synthesis 1502.7 Discussion 163References 1673 Differential Geometry of the Constraint Curves and Surfaces 1713.1 Space Curves 1713.1.1 Vector Representations 1713.1.2 Frenet Trihedron 1753.2 Surfaces 1773.2.1 Elements of Surfaces 1773.2.2 Ruled Surfaces 1833.2.3 Adjoint Approach 1863.3 Constraint Curves and Surfaces 1923.4 Spherical and Cylindrical Curves 1953.4.1 Spherical Curves (S–S) 1953.4.2 Cylindrical Curves (C–S) 1973.5 Constraint Ruled Surfaces 2013.5.1 Constant Inclination Ruled Surfaces (C′–P′–C) 2013.5.2 Constant Axis Ruled Surfaces (C′–C) 2043.5.3 Constant Parameter Ruled Surfaces (H–C, R–C) 2083.5.4 Constant Distance Ruled Surfaces (S′–C) 2123.6 Generalized Curvature of Curves 2143.6.1 Generalized Curvature of Space Curves 2153.6.2 Spherical Curvature and Cylindrical Curvature 2183.7 Generalized Curvature of Ruled Surfaces 2243.7.1 Tangent Conditions 2243.7.2 Generalized Curvature 2253.7.3 Constant Inclination Curvature 2273.7.4 Constant Axis Curvature 2283.8 Discussion 228References 2304 Spherical Kinematic Differential Geometry 2334.1 Spherical Displacement 2334.1.1 General Expression 2334.1.2 Adjoint Expression 2354.2 Spherical Differential Kinematics 2404.2.1 Spherical Centrodes (Axodes) 2404.2.2 Curvature and Euler–Savary Formula 2454.3 Spherical Coupler Curves 2574.3.1 Basic Equation 2574.3.2 Double Point 2574.3.3 Distribution 2624.4 Discussion 263References 2665 Discrete Kinematic Geometry and Saddle Synthesis of Spherical Linkages 2675.1 Matrix Representation 2675.2 Saddle Spherical Circle Point 2695.2.1 Saddle Spherical Circle Fitting 2695.2.2 Saddle Spherical Circle 2725.2.3 Four Positions 2745.2.4 Five Positions 2755.2.5 Multiple Positions 2785.2.6 Saddle Spherical Circle Point 2795.3 The Saddle Kinematic Synthesis of Spherical Four-bar Linkages 2825.3.1 Kinematic Synthesis 2835.3.2 Saddle Kinematic Synthesis of Spherical Four-bar Linkages 2895.4 Discussion 298References 3006 Spatial Kinematic Differential Geometry 3036.1 Displacement Equation 3036.1.1 General Description 3046.1.2 Adjoint Description 3066.2 Axodes 3106.2.1 Fixed Axode 3106.2.2 Moving Axode 3126.3 Differential Kinematics of Points 3146.3.1 Point Trajectory 3156.3.2 Darboux Frame 3196.3.3 Euler–Savary Analogue 3206.3.4 Generalized Curvature 3236.4 Differential Kinematics of Lines 3266.4.1 Frenet Frame 3266.4.2 Striction Curve 3306.4.3 Spherical Image Curve 3326.4.4 Connecting Kinematic Pairs 3346.4.5 Constant Axis Curvature 3386.4.6 Constant Parameter Curvature 3496.5 Differential Kinematics of Spatial Four-Bar Linkage RCCC 3556.5.1 Adjoint Expression 3556.5.2 Axodes 3586.5.3 Point Trajectory 3616.5.4 Line Trajectory 3686.6 Discussion 378References 3807 Discrete Kinematic Geometry and Saddle Synthesis of Spatial Linkages 3837.1 The Displacement Matrix 3847.2 Saddle Sphere Point PSS 3867.2.1 Spherical Surface Fitting 3867.2.2 Saddle Spherical Surface 3907.2.3 Five Positions 3917.2.4 Six Positions 3937.2.5 Multiple Positions 3967.2.6 Saddle Sphere Point 3967.3 Saddle Cylinder Point PCS 4017.3.1 Cylindrical Surface Fitting 4027.3.2 Saddle Cylindrical Surface 4047.3.3 Six Positions 4067.3.4 Seven Positions 4077.3.5 Multiple Positions 4107.3.6 Saddle Cylinder Point 4107.3.7 The Degeneration of the Saddle Cylinder Point (R–S, H–S) 4127.4 Saddle Constant Axis Line LCC 4177.4.1 Ruled Surface Fitting 4177.4.2 Saddle Spherical Image Circle Point 4187.4.3 Saddle Striction Cylinder Point 4207.4.4 Saddle Constant Axis Line 4257.5 Degenerate Constant Axis Lines LRC and LHC 4267.5.1 Saddle Characteristic Line LRC (R–C, R–R) 4267.5.2 Saddle Characteristic Line LHC (H–C, H–R, H–H) 4287.6 The Saddle Kinematic Synthesis of Spatial Four-Bar Linkages 4447.6.1 A Brief Introduction 4457.6.2 The Spatial Linkage RCCC 4507.6.3 The Spatial Linkage RRSS 4547.6.4 The Spatial Linkage RRSC 4587.7 Discussion 461References 464Appendix A Displacement Solutions of Spatial Linkages RCCC 467Appendix B Displacement Solutions of the Spatial RRSS Linkage 473Index 477