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عدد المساهمات : 19002 التقييم : 35506 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
| موضوع: كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis السبت 14 يناير 2023, 10:07 pm | |
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أخواني في الله أحضرت لكم كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis Jorge Angeles, Shaoping Bai
و المحتوى كما يلي :
Contents 1 Introduction to Kinematic Synthesis . 1 1.1 The Role of Kinematic Synthesis in Mechanical Design 1 1.2 Glossary 5 1.3 Kinematic Analysis Versus Kinematic Synthesis 8 1.3.1 A Summary of Systems of Algebraic Equations 10 1.4 Algebraic and Computational Tools . 10 1.4.1 The Two-Dimensional Representation of the Cross Product . 11 1.4.2 Algebra of 2 × 2 Matrices . 13 1.4.3 Algebra of 3 × 3 Matrices . 14 1.4.4 Linear-Equation Solving: Determined Systems . 14 1.4.5 Linear-Equation Solving: Overdetermined Systems . 18 1.5 Nonlinear-Equation Solving: The Determined Case 29 1.5.1 The Newton–Raphson Method . 31 1.6 Overdetermined Nonlinear Systems of Equations . 32 1.6.1 The Newton–Gauss Method . 33 1.7 Packages Relevant to Linkage Synthesis . 36 References . 38 2 The Qualitative Synthesis of Kinematic Chains 41 2.1 Notation 41 2.2 Background . 42 2.3 Kinematic Pairs 47 2.3.1 The Kinematic Pair 49 2.4 Graph Representation of Kinematic Chains . 50 2.5 Groups of Displacements 53 2.5.1 Displacement Subgroups 56 2.6 Kinematic Bonds . 60 viiviii Contents 2.7 The Chebyshev–Grübler–Kutzbach–Hervé Formula . 63 2.7.1 Trivial Chains . 63 2.7.2 Exceptional Chains . 66 2.7.3 Paradoxical Chains . 69 2.8 Applications to Robotics 69 2.8.1 The Synthesis of Robotic Architectures and Their Drives 69 References . 73 3 Linkage Synthesis for Function Generation . 75 3.1 Introduction . 75 3.2 Input-Output (IO) Functions . 76 3.2.1 Planar Four-Bar Linkages . 76 3.2.2 The Denavit-Hartenberg Notation 79 3.2.3 Spherical Four-Bar-Linkages 80 3.2.4 Spatial Four-Bar-Linkages . 86 3.3 Exact Synthesis 91 3.3.1 Planar Linkages 91 3.3.2 Spherical Linkages . 96 3.3.3 Spatial Linkages . 98 3.4 Analysis of the Synthesized Linkage 99 3.4.1 Planar Linkages 99 3.4.2 Spherical Four-Bar Linkages 110 3.4.3 Spatial Four-Bar Linkages . 113 3.5 Approximate Synthesis 120 3.5.1 The Approximate Synthesis of Planar Four-Bar Linkages 123 3.5.2 The Approximate Synthesis of Spherical Linkages 125 3.5.3 The Approximate Synthesis of Spatial Linkages 127 3.6 Linkage Performance Evaluation . 131 3.6.1 Planar Linkages: Transmission Angle and Transmission Quality . 131 3.6.2 Spherical Linkages: Transmission Angle and Transmission Quality . 136 3.6.3 Spatial Linkages: Transmission Angle and Transmission Quality . 137 3.7 Design Error Versus Structural Error 139 3.7.1 Minimizing the Structural Error 142 3.7.2 Branch-Switching Detection . 144 3.7.3 Introducing a Massive Number of Data Points 145 3.8 Synthesis Under Mobility Constraints . 145 References . 148Contents ix 4 Motion Generation . 151 4.1 Introduction . 151 4.2 Planar Four-Bar Linkages 151 4.2.1 Dyad Synthesis for Three Poses 153 4.2.2 Dyad Synthesis for Four Poses . 154 4.2.3 Dyad Synthesis for Five Poses . 156 4.2.4 Case Study: Synthesis of a Landing Gear Mechanism . 158 4.2.5 The Presence of a P Joint in dyad Synthesis 165 4.2.6 Approximate Synthesis . 170 4.3 Spherical Four-Bar Linkages . 174 4.3.1 Dyad Synthesis for Three Attitudes . 177 4.3.2 Dyad Synthesis for Four Attitudes 178 4.3.3 Dyad Synthesis for Five Attitudes 179 4.3.4 Spherical dyads with a P Joint . 181 4.3.5 Approximate dyad Synthesis . 181 4.3.6 Examples . 189 4.4 Spatial Four-Bar Linkages . 195 4.4.1 Geometric Constraints of CC and RC dyads 196 4.4.2 The Synthesis of the CC dyad 198 4.4.3 The Synthesis of the RC dyad 200 4.4.4 Synthesis of Four-Bar Linkages 202 4.4.5 A Semigraphical Solution of the Direction Equations 202 4.4.6 Solving the Linear Equations of the Moment Variables 204 4.4.7 Congruences of the Fixed and the Moving Axes 204 4.4.8 Examples . 205 4.4.9 Summary . 208 References . 210 5 Trajectory Generation 213 5.1 Planar Linkages 214 5.1.1 Planar Path Generation with Prescribed Timing . 214 5.1.2 Coupler-Curve Synthesis of Planar Four-Bar Linkages 221 5.2 Trajectory Generation for Spherical Four-Bar Linkages 234 5.2.1 Path Synthesis with Discrete Positions 235 5.2.2 Path Synthesis with Prescribed Timing 236 5.2.3 Examples . 237 5.2.4 Summary . 240 5.3 Path Generation for RCCC Linkages 240 5.3.1 A Generic RCCC Linkage . 240 5.3.2 An Alternative Coordinate Frame 242 5.3.3 Constraint Equations of the CC and RC dyads 243 5.3.4 N-point Path Synthesis 245 5.3.5 Example 247 5.3.6 Summary . 249 References . 249x Contents 6 Synthesis of Planar Complex Linkages . 251 6.1 Six-Bar Linkages . 251 6.2 Synthesis of the Stephenson-III Linkage . 253 6.2.1 Elimination of Motion Variables . 254 6.3 Extension to Stephenson-I and II Linkages . 255 6.3.1 Stephenson-II Linkage 256 6.3.2 Parameterized Coordinates of Point D 256 6.3.3 Synthesis Equations for the Stephenson-II Linkage . 258 6.3.4 Stephenson-I Linkage 259 6.4 Synthesis in the Presence of Incomplete Data Sets . 260 6.4.1 A Design Example . 262 6.5 A 10-Bar Linkage 262 6.6 Synthesis for m Prescribed Poses . 265 6.7 Solution of the 10-Posture Synthesis Equations . 266 6.8 A Synthesis Example . 268 References . 272 Appendix A: A Summary of Dual Algebra . 275 Appendix B: Coupler Curves . 313 Index . 32 Index A Algebraic system degree, 10 eliminant, 10 resolvent, 10 Alternative coordinate frame, 242 Alternative coordinates, 242 Analysis, 8 Angular-velocity matrix, 292 Approximate synthesis function generation, 120 planar four-bar linkage, 123 spatial linkage, 127 spherical linkage, 125 Assemblability, 70 Augmented synthesis equations, 125 Axial vector, 285 B Bezout number, 10, 171, 173, 185, 206, 270 Bicycle, 75 Bimodal linkage, 101 Bivariate equations, 9 Branching defect, 191 Branch switching, 144 Burmester problem planar, 151 spatial, 195 spherical, 176 Burmester-Roth Theorem, 180 Burmester theory, 314 C Canonical equation, 116 Cartesian decomposition, 44 Cayley–Hamilton Theorem, 285 CC dyad synthesis five poses, 198, 205 four poses, 199, 207 Centerpoint, 154, 175 Centerpoint curve, 156, 161, 179 Chain exceptional, 63, 66 paradoxical, 63, 69 trivial, 63 Chain rule, 21 Chebyshev-Grübler-Kutzbach formula, 64 Chebyshev-Grübler-Kutzbach-Hervé Formula, 63 Chebyshev norm, 15 Chirality, 42 Cholesky decomposition, 18 Circle of sliders, 167 Circlepoint, 154, 175 Circlepoint curve, 155, 161, 173, 179, 204 Circular guideway, 181, 193 Cognate linkages, 229, 233 Conceptual design, 3 Condition number, 15 Cone axis, 174 Cone element, 174 Congruence, 204, 207 Conical cubic surface, 180 Conical surface, 174 Conjugate branches, 101 Conjugate postures, 102 Constrained minimization, 182 Constrained optimization problem, 182 Constraint Jacobian, 183 Contour-intersection, 85 Convex combination, 127 algebraic equation, 221 bivariate polynomial, 223 exact synthesis, 221, 228 four-bar linkage, 221 RCCC linkage, 248 spherical , 235 synthesis, 222 approximate, 222 exact, 225, 229 univariate polynomial, 225 Coupler-curve equation, 320 planar four-bar linkage, 316 spatial four-bar linkage, 247 spherical four-bar linkage, 322 Crank, 105 Cross product, 11 Cross-product matrix, 43 Cubic of stationary curvature, 214 Cylindrical pair, 49 D Damping, 34 Damping factor, 34 Data-conditioning, 158 Deadpoint, 102 Delta robot, 69 Denavit-Hartenberg frames, 80 notation, 79 Denavit-Hartenberg parameters, 5 Descartes’ rules, 231 Design functions, 3 specifications, 3 variants, 3 Design error, 120, 139, 145 least-square, 121 Design-error vector, 120 Design parameter, 266 Detailed design, 3 Determined system, 29 Dialytic elimination, 186 Dimensionally homogeneous, 170 Dimensional synthesis, 8 Dimensioning, 5 functional, 5 mechanical, 5 Direction equation, 197 Displacement subgroups, 56 Dual angle, 89, 139, 277 Dual angular velocity, 304 matrix, 304 vector, 305 Dual Euler–Rodrigues parameters, 299 Dual exponential, 276 Dual Freudenstein parameters, 89 Dualization, 88 Dualization operation, 88 Dual matrix, 278 determinant, 279 eigenvalue, 279 eigenvector, 279 inverse, 278 pure, 278 square, 278 Dual-matrix exponential, 275 Dual number, 275 Dual quaternion, 276 Dual rotation matrix, 292, 294 Dual trigonometric function, 277 Dual unit, 88 Dual vector, 196 dual part, 282 Dwell, 213 Dyad CC, 196, 244 definition, 151 PR, 165 RC, 196, 243 Dyad synthesis approximate, 181 CC dyad, 198 planar five poses, 156, 163, 169 four poses, 154, 161, 168 three poses, 153, 160, 167 RC dyad, 200 E Embodiment design, 3 Engineering design, 1 process, 3 Error vector, 19 Euclidean norm, 15, 77 Euler–Rodrigues parameters, 304 Euler’s Theorem, 284 Exact synthesis, 266 function generation, 91 planar linkage, 91 spherical four-bar linkage, 96 F First-order normality conditions, 170Index 327 Floating-point operation, 22 Flop, see floating-point operation Four-bar linkage path generation with timing, 214 path synthesis, 214 planar, 76 spatial, 86, 113 spherical, 80 spherical linkage, 110 4C linkage, 202 4C linkage synthesis five poses, 205 Fractal, 32 Franz Reuleaux, 3 Freudenstein equation, 84 planar linkages, 79 Freudenstein parameters, 76, 78 spherical linkage, 83 Frobenius norm, 15 Function generation, 8, 75 Function-generation problem, 76 G Generalized CGK formula, 64 Global minimum, 174 Gnomonic projection, 323 Gram-Schmidt orthogonalization, 24, 122 Graph, 50 Groups of displacement, 41, 53, 63 H Householder reflection, 26, 122 Hyperbolic paraboloid, 116 I kinematic pair, 49 Incomplete data set, 260 Infinity norm, 15 Inflection circle, 214 Input-output equation spherical linkage, 84 Input-output functions, 76 Instant-screw parameters, 305 IO equation, 77 Isotropic matrix, 17, 217 J Jacobian matrix, 31 K Kinematic bond, 60 Kinematic chain architecture, 80 multiloop, 41 Kinematic synthesis, 4, 8 Kinetostatics, 11 L Lagrange multipliers, 182 Lamé curve, 158 Landing gear, 158 Latitude, 186 Least-square approximation, 158, 219 Least-square error, 21, 33 normality condition, 19 Least-square solution, 33 Lemniscate, 232 Linear invariants, 294 Linkage, 7 Linkage discriminant, 106, 136 Linkage feasibility, 105 Local maximum, 173 Local minimum, 173 Longitude, 186 Lower Kinematic Pairs (LKP), 41 LU-decomposition, 17 M Machine, 1 function, 1 Manifold, 269 Marcus Vitruvius Polio, 3 Mechanical system, 1 Mechanical-system gesture, 8 Mechanical-system posture, 8 Mechanism, 7 MEL microfinger, 71 Minimum-size linkage, 128 Mobility constraints, 145 Moment equation, 197 Motion generation, 8, 151 Motion representation, 41 Motion variable, 266 Mozzi–Chasles axis, 287, 292, 295, 296, 299 Mozzi–Chasles Theorem, 287, 293 N Natural invariants, 205, 284, 320 Negative-definite, 173 Newton-Gauss method, 33, 185328 Index Newton-Raphson method, 31, 35, 185 Normality condition, 34 Number synthesis, 8 O Optimization multiobjective, 4 Optimum design, 170 Orthogonal complement, 130, 183 Overdetermined system, 18, 32 P Parallelism condition, 165 Parameterized coordinates, 216, 256, 318 Parametric model, 4 Path generation, 8, 213 Path synthesis RCCC linkage, 240 five points, 247 nine points, 249 N points, 245 spherical, 234, 238 discrete points, 234, 235 discrete points with timing, 236 great-arc, 234 Pencil of lines, 59 Performance evaluation, 131 Planar four-bar linkage feasibility condition, 92 PRRR linkage, 92, 108 PRRP linkage, 108 RRRP linkage, 92 Planar linkage Bloch synthesis, 94 crank-rocker, 105 double crank, 105, 109 double rocker, 105, 110 drag-link, 106 Grashof, 106 non-Grashof, 106 rocker-crank, 105 transmission angle, quality, 131 Planar pair, 49 Plücker coordinates, 195, 287 normalized, 277 Polynomial bivariate sextic, 313 cubic, 156 dodecic, 206 multivariate, 10 nonic, 172, 174 octic, 270 quadratic, 180, 218 quartic, 183 root finding, 206 sextic, 180, 221, 229 trivariate quartic, 322 univariate, 10, 225, 232 Pose error, 170 Positive-definite, 173 Principle of Transference, 87, 275 Prismatic pair, 48 Projection theorem, 22 Proper dual number, 276 Pure dual number, 276 Putative least-square approximation, 156 Q QR decomposition, 122, 203 Qualitative synthesis, 69 R Rank, 18 Rank-nullity theorem, 43 RCCC linkage, 196 Reciprocal bases, 14 Reliability, 213 Repeatability, 213 Revolute pair, 48 Rigid body, 7, 42 Rigid-body guidance, 151 Rigid-body kinematics, 283 Rigid-body pose, 7 Rigid-body twist, 8 Ring, 154 Roberts-Chebyshev Theorem, 233 Robotic gripper, 93 Robustness, 158 Robust selection, 261 Rocker, 101, 105 Roundoff error, 219 Ruled surface, 213 S Saddle point, 173 Sarrus mechanism, 66 SCARA systems, 57 Schönflies-motion generator, 68 Schwarz’s Theorem, 173 Scissor, 75 Screw pair, 48 Semigraphical method, 85, 202, 207, 208Index 329 Sign-indefinite, 173 Signum function, 28, 100 Singularity, 103 Stephenson-II, 258 Six-bar linkages, 251 Skew-symmetric matrix, 282, 285 Slack variable, 147 Slider-crank mechanism, 67 Spatial four-bar linkage, 195 Spatial linkages, 98 function generation, 118 mobility, 119 Spatial motion synthesis RR dyad, 201 Spherical centerpoint curve, 204 Spherical dyad, 181 Spherical dyad synthesis approximate, 195 five attitudes, 179, 191 four attitudes, 178, 190 three attitudes, 177, 189 with P joint, 181, 192 Spherical pair, 49 Stationary point, 34, 173 Stephenson linkage, 216 Stephenson-I, 259 Stephenson-II, 256 Stephenson-III, 253 Structural error, 139, 142, 145 Sylvester’s Theorem, 18, 22 Synthesis, 8 approximate, 9, 79, 170 equations, 9 exact, 9 planr kinkage 11 poses, 174 qualitative, 4, 41 semigraphical methods, 9 Synthesized linkage analysis, 99 mobility analysis, 113 Synthesized planar linkage mobility analysis, 106 T 10-bar linkage, 262 10-pose synthesis, 266 3-RRR planr linkage, 262 Trajectory generation, 234 Transcendental, 10 Transmission angle, 136 Transmission defect, 133 Transmission quality, 132, 137 spatial linkage, 137 Triad, 214 RRR, RPR, 214 Trial and error, 127 Type synthesis, 8 U Unconstrained minimization, 186 Unconstrained nonlinear least squares, 186 Underdeterminacy, 261 W Watt curve, 232 Weighted error-norm, 126 Weighting matrix, 33 Working region, 158 Y Yang-and-Freudenstein linkage, 118
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