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Copyrighted Material Index 2R planar manipulator mixed Coriolis, 859 dynamics, 1207, 1258 mixed double, 823, 859 equations of motion, 1209 multibody, 823 forward acceleration, 873 particle, 830 general dynamics, 1258 Razi, 860, 861, 886 ideal, 1207 tangential, 726, 812, 824, 832, 859 inverse acceleration, 876 tidal, 190 inverse kinematics, 664 transformation, 792, 818 inverse velocity, 778, 780 Active transformation, 409 Jacobian matrix, 761, 763 Actuator kinetic energy, 1208 force and torque, 1159, 1177 Lagrangian, 1208 torque equation, 1169 mass moments, 1051 Air resistance, 163 Newton–Euler dynamics, 1168 Angle potential energy, 1208 attitude, 829 recursive dynamics, 1172 cruise, 829 4-Bar linkages Euler, 379 dynamics, 1163 heading, 829 spatial, 560, 602 nutation, 379 precession, 379 A sideslip, 829 Abel, Niels Henrik, 149 spin, 379 Acatastatic constraint, 966 zenith, 509 Acceleration Angular acceleration, 788, 789, 805, 823, angular, 789, 796, 809, 824 824 applied, 851 decomposition, 795 bias vector, 875 Euler parameters, 796, 809 body point, 725, 824 matrix, 788, 808 centripetal, 726, 824, 832 natural frame, 809 Coriolis, 812, 832, 859 quaternion, 809 definition,COPYRIGHTED 51 MATERIALrelative, 792, 799 double mixed, 861 Rodriguez formula, 799 energy, 1009, 1011 rotational transformation, 788, 791 forward kinematics, 872, 873 vector, 788 gravitational, 1254 Angular jerk, 808 inverse kinematics, 874 Cartesian, 880 kinematics, 788 matrix, 808 local, 832 Angular velocity, 385, 387, 388, 432, 695, matrix, 788, 864, 865 697, 699–702, 706, 789, 794 mixed, 816, 817 alternative definition, 715 1305 1306 Index Angular velocity (continued) Canonical equation, 1003 combination, 699 Cardan coordinate transformation, 709 angles, 390 decomposition, 705 frequencies, 390 elements of matrix, 714 Cardioid, 258 Euler parameters, 713 Cardioidal coordinate system, 342 instantaneous, 697, 698 Cardioidcylindrical coordinate system, 342 instantaneous axis, 699 Cartesian matrix, 695 angular velocity, 387 principal matrix, 702 coordinate system, 32, 243 quaternion, 711 end-effector position, 777 rotation matrix, 707 end-effector velocity, 778 Appell function, 1012 orthogonality condition, 32 Appell, Paul Emile, 1011 unit vectors, 34 Applied force, 933 Casscylindrical coordinate system, 342 Arc length, 40 Caster theory, 649 Articulated arm, 666, 671, 769 Catastatic constraint, 966 Assembling kinematics, 615 Center point, 903 Atan2 function, 665 Central force, 993 Atwood, George, 1001 Central frame, 1038 Atwood machine, 1000 Central principle, 1230, 1231, 1239 Australia, 109, 885 Centrifugal moment, 1033 Automorphism, 457 Chasles, Michel, 121 Axis–angle of rotation, 422, 425, 426, Chasles theorem, 517, 526 440, 442, 446, 461 Christoffel, Elwin Bruno, 328 Christoffel operator, 334 B Christoffel symbol, 325, 327, 333, 334, bac –cab rule, 22, 29 1203, 1211 Basic lemma, 1217 first kind, 327 Bernoulli equation, 931 second kind, 327 Bernoulli, Johann, 942 third kind, 335 Bernoulli, Johann and Jacob, 1221 Circular integrals, 153 Bipolar coordinate system, 341 Collision, 177, 181 Bipolarcylindrical coordinate system, 341 elastic, 177, 178 Bispherical coordinate system, 340, 342 inelastic, 181, 182 Bong, 53 oblique, 187 Book-stacking problem, 193 plastic, 182 Boom, 1126 restitution coefficient, 182 Brachistochrone, 1220 Concave surface, 79 Bragg condition, 296 Condition Broucke, Roger, 30 Bragg, 296 Bryant angles, 390 orthogonality, 31, 32 Bushehr, 858 reciprocality, 271 Configuration C coordinate, 924 Camber theory, 658 degree of freedom, 923, 930 Canada, 109, 885 path, 47 Index 1307 space, 890 Convex surface, 79 trajectory, 890 Coordinate Confocalellip coordinate system, 342 cyclic, 1243 Confocalparab coordinate system, 343 cylindrical, 506 Conical coordinate system, 343 ignorable, 1243 Conservation non-Cartesian, 1201 energy, 1241 nonorthogonal, 32 Jacobi Integral, 1242 parabolic, 1201 laws, 1240 spherical, 744 momentum, 1243 Coordinate frame, 3, 5 Conservative force, 174 Cartesian, 282 Constant of motion, 977, 1241 curvilinear, 300, 309 Constraint reciprocal base vector, 302 acatastatic, 966 reciprocal unit vector, 302 acceleration, 922 space scale factor, 302 catastatic, 966 natural, 54, 55 constraint-free space, 929 neshin, 616 force, 932, 933, 937, 943 nonorthogonal, 24, 274, 282, 284, 285, frozen, 915 300 generalized, 975 vector product, 285, 289, 291 holonomic, 913 oblique, 294, 298, 300 inequality, 962 acceleration, 299 jerk, 923 velocity, 299 just-, 928, 929 origin, 5 least, 999 orthogonal, 5, 31 limit, 962 orthogonality condition, 31 motion, 999 principal, 284, 1058 multiple, 919 transformation, 284 nonholonomic, 952 reciprocal, 278, 284 nonintegrable, 952 transformation, 284 over-, 928, 929 rim, 586 Pfaffian forms, 966, 967, 1238 takht, 616 plane, 916 tire, 608 rheonomic, 913 vehicle, 608 rolling, 956–958 wheel, 608 scleronomic, 913 wheel–body, 608 slip, 962 Coordinate system, 4, 243 total, 952 bipolar, 341 under-, 928, 929 bipolarcylindrical, 341 unicycle, 955 bispherical, 340, 342 Contravariant cardioidal, 342 metric, 279 cardioidcylindrical, 342 vector components, 278 Cartesian, 243 Control casscylindrical, 342 bang-bang, 1225 confocalellip, 342 directional control system, 594 confocalparab, 343 minimum time, 1225 conical, 343 1308 Index Coordinate system (continued) radius, 65 cylindrical, 250, 253, 257, 258 surface, 84 gradient, 257 vector, 63, 64 orthogonality, 253 vectorial expression, 62 ellcylindrical, 343 Curvilinear ellipsoidal, 343 acceleration, 331 elliptic-hyperbolic cylindrical, 267 base vector, 301 hypercylindrical, 343 coordinate frame, 309 invcasscylindrical, 344 coordinate system, 293, 295, 300–302, invellcylindrical, 344 309, 320, 1245 invoblspheroidal, 344 reason, 334 logcoshcylindrical, 344 vector product, 319 logcylindrical, 344 work, 331 maxwellcylindrical, 344 differential, 339 natural, 334 geometry, 320 nonorthogonal, 269 kinematics, 325, 335 oblatespheroidal, 345 natural coordinate, 330, 331 oblique, 277 parallelepiped, 323, 326, 329 orthogonal, 243, 341 velocity, 331 parabolic cylindrical, 261 Cyclic coordinate, 1243 paraboloidal1, 345 Cycloid, 247, 586, 1220 paraboloidal2, 345 curtate, 247, 586 paracylindrical, 345 ordinary, 247 principal, 278 prolate, 247, 586 rosecylindrical, 345 sixsphere, 345 D spherical, 263, 265, 266, 339 D’Alembert, Jean Le Rond, 942 orthogonality, 265 D’Alembert principle, 943 tangentcylindrical, 345 Lagrange’s form, 942 tangentsphere, 345 Da Vinci, Leonardo, 938 toroidal, 346 Darboux, Jean Gaston, 70 Coriolis Darboux vector, 70 acceleration, 802, 832, 833 Degree of freedom, 923 force, 833, 836 configuration, 891, 919, 923, 928, 952, Coriolis, Gaspard Gustave de, 835 963 Costate variable, 1225 event, 896 Coulomb, Charles Augustin de, 938 generalized, 1250 Covariant joint, 555 metric, 279 rigid body, 1034 vector components, 278 state, 901, 923, 980 Crackle, 53 state–time, 909 Curl, 93 Del, 86 potential force, 173 Denavit–Hartenberg Curvature, 70 method, 563, 571 center, 77 notation, 563 plane curve, 61 parameters, 563, 750, 754 principal, 84 rule, 563 Index 1309 transformation, 566–568, 574, 575, Dynamic 643, 1293–1296, 1298, 1299 coupling, 1211 Derivative decoupling, 1211 mixed double, 822, 858–860 Dynamics, 114, 1157 mixed second, 819 2R planar manipulator, 1168, 1172 transformation 4 bar linkage, 1163 mixed, 724, 725 actuator’s force and torque, 1177 simple, 724 application, 188 transformation formula, 724, 819 backward Newton–Euler, 1170 Deviation moment, 1033 definition, 130 Dido problem, 1221 forward Newton–Euler, 1172 Differential fundamentals, 114 manifold, 408 global Newton–Euler, 1157 Differential geometry, 37 isolated system, 116 immersed surface, 80 modeling, 189 parametric line, 41 multibody, 1251 quadratic surface, 45 Newton–Euler, 1157 space curve, 38, 40, 41 Newtonian, 118 surface and plane, 43, 44 one-link manipulator, 1159 surface expression, 81 recursive Newton–Euler, 1157, 1170 tangent line, 41 special solutions, 131 Differentiating, 718 B-derivative, 718, 722, 814 E G-derivative, 718, 725 Earth second, 726 free fall, 840, 846, 848, 849 second derivative, 810 kinetic energy, 1200 transformation formula, 724 moving vehicle, 801 Directional revolution energy, 1200 angle, 572 rotation effect, 802 control system, 594, 595, 629 rotation energy, 1200 cosine, 6–8, 18, 38, 55, 57, 58, 81, 92, shrinking, 185 398–400, 422, 457, 460, 482, 581 spherical, 185 curvature, 85 Eddington, Arthur Stanley, 115 derivative, 89, 91, 92, 257 e-delta identity, 26, 29 line, 5, 120, 126 Eigenvalue, 433 Displacement Eigenvector, 433 actual, 916, 935, 936 Einstein summation convention, 28, impossible, 916, 935, 936 301 possible, 916, 935, 936 Ellcylindrical coordinate system, 343 virtual, 935–937 Ellipsoid Distance energy, 1092 topocentric, 508 momentum, 1092 Divergence, 93 Ellipsoidal coordinate system, 343 Double factorial, 153 Elliptic function, 148, 149, 1129, 1130, Double pendulum, 232, 926, 1016, 1018 1132 Dubai, 109 limiting, 153 Duffing equation, 1136, 1137 period, 153 1310 Index Elliptic integral, 148, 1132 integrability, 388 arc length, 155 body frame equation, 1076, 1095 complete first kind, 148 equation, 126 complete second kind, 149 coordinate frame, 388 complete third kind, 149 equation, 1076, 1093, 1095, 1159, first kind, 148 1171 second kind, 148 equation of motion, 1072 third kind, 149 frequencies, 385, 387, 706, 805 End-effector general equation, 1098 acceleration, 872 global rotation matrix, 403 angular velocity, 776 inverse matrix, 403 configuration vector, 872 –Lexell-Rodriguez formula, 424 configuration velocity,
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