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, 872 local rotation matrix, 403 force, 1172 parameters, 439, 442, 443, 448, 449, link, 563 451, 463, 711, 713 orientation, 777 rotation matrix, 382, 403 position kinematics, 584 theorem, 378, 437–439 position vector, 771 Euler-Chasles theorem, 438 SCARA position, 505 Eulerian viewpoint, 737 space station manipulator, 580 Euler-Lagrange time-optimal control, 1225 equation of motion, 1215, 1217 velocity, 757, 766, 778 Euler, Leonhard, 126, 463 velocity vector, 757 Event space, 896 Energy Event trajectory, 896 classical form equation, 950 Earth kinetic, 1200 F first-form equation, 950 Fargo, ND, 885 kinetic, 165 Ferris wheel, 249 kinetic rigid body, 1036, 1037, 1077 Field, 77 kinetic rotational, 1073 curl, 93 mechanical, 166, 1200 cylindrical, 258 multibody, 1252 derivative, 89, 94, 95, 98 potential, 166, 1203 directional derivative, 91 primitive-form equation, 950 divergence, 93 second-form equation, 950 gradient, 88 Envelope, 136 index notation, 99 Epitrochoid, 247 isosurface, 86, 92 Equation isovalue, 86 Bernoulli, 931 Laplacian, 94 inhomogeneous, 931 scalar, 77, 85, 99 linear ODE, 931, 962 second derivative, 97 Equation of motion, 119, 197 spherical, 267 integral form, 119 stationary, 77 Equivalent Lagrangian, 1213 tensor, 99 Euclidean space, 887, 888 timeinvariant, 77 Euler total derivative, 89 angles, 379, 382, 384, 462 vector, 77, 88, 92, 97–99 Index 1311

Final rotation formula, 435, 436 Grubler,¨ 559, 560 First integral, 977 inversion, 530 First variation, 1216, 1224 Kutzbach, 560 Fixed frame, 4 parallel axes, 1047 Flash space, 910 relative acceleration, 792, 795, 799 Flash–time space, 910 relative angular velocity, 700, 705 Flicker space, 911 Rodriguez, 447, 456, 463, 466, Flicker–time space, 911 475, 477, 520, 699, 754, Focus point, 903 785, 799 Forbidden umbrella, 136, 137 trigonometric, 1300 Force Forward kinematics, 584 action, 1158 Foucault pendulum, 853, 855 actuator, 1177 Frame body, 120 central, 1102 central, 993 principal, 1073 conservative, 166, 1203 Free-body diagram, 156 constraint, 932, 937, 943 Freezing process, 915 contact, 120 Frenet Coriolis, 833, 836 frame, 58 derivative, 131 trihedron, 58 driven, 1158 Frenet, Jean Fred´ eric,´ 65 driving, 1158 Frenet–Serret formula, 64 effective, 839 Friction external, 120 force, 938 friction, 937 law, 938 function, 117 Fundamental equation, 941, 942, 950, general, 941 996, 997, 1014, 1233, 1240, 1242 generalized, 1192, 1254 first form, 942, 1014 gravitational, 994 second form, 950 gravitational vector, 1255 third form, 950 internal, 120 Magnus, 1110 G potential, 166, 1203 Galilean transformation, 890 potential field, 1200 Galilei, Galileo, 1221 reaction, 1158 Gauss resultant, 120 equation, 999 system, 120 function, 999 total, 120 Gauss dynamics, 999 Formula Gauss, Johann Carl Friedrich, 464, 999 axis–angle, 520 Generalized derivative transformation, 724, 725, applied force, 1013–1015 819, 820, 822, 859, 864 constraint, 973, 975 derivative transport, 797 coordinate, 923, 926–929, 975, 1189, final rotation, 435, 436 1192, 1194, 1205 Frenet–Serret, 64, 65 force, 986, 1002, 1191, 1192, 1194, geometric transformation, 693 1202, 1203, 1206, 1209, 1246, Gibbs, 447 1252 1312 Index

Generalized (continued) Hamilton principle, 1228, 1230 inertia force, 1013–1015 general form, 1228 kinetic energy, 971 Hamilton, William, 464 mechanics, 970 Harmonic oscillator, 142 momenta, 1248 Helix, 51, 517 momentum, 986, 1002, 1004, 1192, Hertz, Heinrich Rudolf, 954 1248, 1268 Hodograph, 47, 49 potential force, 972 Homogeneity, 887, 888 space, 928, 929, 974 position, 889 velocity, 1249 time, 888 work, 1247 Homogeneous Geodesics problem, 1218 compound transformation, 500 Gibbs coordinate, 482, 489 matrix, 447 direction, 489 vector, 447 general transformation, 490, 498 Gibbs–Appell inverse transformation, 494, 496, 498, dynamics, 1009 501 equation, 1009, 1011, 1012 position vector, 483 function, 1009, 1011 scale factor, 482 Gibbs, Josiah Willard, 1011 transformation, 482, 487–489, 494, 497 Gorz, 131 Huygens–Steiner theorem, 1047 Grad, 86 Hyperbolic integrals, 153 Gradient, 84, 86 Hypercylindrical coordinate system, 343 definition, 89 Hypergeometric function, 152, 153 operator, 86 Hypertrochoid, 247 Gram determinant, 295 Hypotrochoid, 247 Grassmanian, 537 Gravitation I Newton law, 121 Ignorable coordinate, 1243 universal constant, 121 Impenetrability, 889 Gravity center, 123 Impossible displacement, 936 Group properties, 408 Impulse, 119, 176, 177 Grubler¨ formula, 560 Impulsive point, 893 Guldin, Paul, 226, 227 Ince, Edward Lindsay, 970 Guldin theorem, 226, 227 Index notation, 26 India, 858 H Inequality constraint, 962 Hamilton Inertial frame, 889 dynamics, 1002 Inhomogeneous equation, 931 equation, 1002, 1003 Integrability, 388 function, 1002 Integral of motion, 976, 977, 1241 Hamiltonian, 1225 Integrating factor, 960 dynamics, 1002 Interpolant, 14 equation, 1002 Interpolation, 14 function, 1002 linear, 14 method, 1002 problem, 14 system, 1006 vector, 14 Index 1313

Invariant vector, 244 prismatic, 555 Invcasscylindrical coordinate system, 344 revolute, 555 Invellcylindrical coordinate system, 344 rotary, 555 Inverse kinematics translatory, 555 Pieper technique, 676 universal, 560 transformation technique, 675 variable vector, 757 Inverse Lagrangian, 1214 velocity vector, 757, 767 Invoblspheroidal coordinate system, 344 Jolt, 53 Iran, 109, 858, 885 Jounce, 53 Isotropy, 887, 888 Just constraint, 928, 929 position, 889 Just rigid, 1033 time, 888 K J Kane Jacobi, Carl Gustav Jacob, 149 dynamics, 1013 Jacobi function, 148, 149 equation, 1013, 1014 Jacobian, 315 Kane equation, 1016 analytical, 778 Kennedy theorem, 734 displacement matrix, 757 Kinematic elements, 776 length, 565 generating vector, 765, 767 space, 910 geometrical, 778 Kinematics link, 1252 acceleration, 789 matrix, 757, 758, 763, 767, 769, 773, assembling, 615 774, 778, 781, 872, 873, 877 foot–leg, 513 polar manipulator, 764, 874 forward, 584 rotational matrix, 757 forward acceleration, 872 Jeeq, 53 forward velocity, 757 Jerk, 53, 807, 808 inverse, 675 angular, 808 inverse acceleration, 874 body point, 808 inverse velocity, 778 definition, 51 velocity, 754 global, 807 Kinetic energy, 165 matrix, 865 Earth, 1200 rotational transformation, 807 parabolic coordinate, 1202 transformation, 866, 867 rigid body, 1036, 1037, 1077 Jerkmeter, 53 rotational body, 1073 Joint, 555 Kinetic potential, 984 acceleration vector, 872 Kronecker delta, 26, 681, 1042 active, 555 angle, 565 L distance, 565 Lagrange equivalent spherical, 562 dynamics, 996, 1251 inactive, 555 equation, 1217 multiple, 561 equation of motion, 1189, 1203, 1215, parameters, 565 1234, 1238, 1239 passive, 555 explicit equation, 1202, 1210 1314 Index

Lagrange (continued) classification, 573 mechanics, 1203 compound, 561 multiplier, 933, 1221 end effector, 563 Lagrange, Joseph-Louis, 213 Euler equation, 1171 Lagrangian, 984, 1203, 1241 kinetic energy, 1252 dynamics, 1189 length, 565 equivalent, 1213 Newton–Euler dynamics, 1157 form invariant, 1214 offset, 565 fundamental theorem, 1234 parameters, 565 generality, 1238 recursive dynamics, 1170 inverse, 1214 translational velocity, 756 viewpoint, 737 twist, 565 Laplacian, 94, 98 velocity, 754 Larz, 53 Lissajous curves, 144 Law Lituus, 260 conservation, 176, 1240 Load, 120 cosine, 105 Lobachevsky space, 207 force, 114 Location vector, 519, 521, 650 gravitational, 121, 122 Logcoshcylindrical coordinate inertia, 890 system, 344 inverse square, 994 Logcylindrical coordinate system, 344 Kepler’s second, 994 Kepler’s third, 419 M motion, 114, 126 Manipulator motion second, 1101 2R planar, 1207 Newton’s first, 114 one-link dynamics, 1159 Newton’s second, 114 planar polar, 1257 Newton’s third, 114 SCARA, 505 rotational motion, 1077 Mass, 115 sine, 508 Mass center, 122, 126, 1039, 1040, 1102 Laws of motion, 114 Mass moment, 1033 Least constraint, 999 about a line, 1042 Legendre, Adrien-Marie, 150 about a plane, 1042 Levi-Civita symbol, 26, 27, 448 about a point, 1042 Levi-Civita, Tullio, 900 characteristic equation, 1062 Libration, 142 diagonal elements, 1041 Lie group, 408 matrix, 1035 Limit constraint, 962 parallel-axes theorem, 1044 Linear space, 15 polar, 1035 Link principal, 1058 angular velocity, 756 principal directions, 1059 class 1 and 2, 1293 principal frame, 1058 class 3 and 4, 1294 principal invariants, 1062 class 5 and 6, 1295 principal planes, 1065 class 7 and 8, 1296 product, 1035 class 9 and 10, 1298 pseudomatrix, 1037 class 11 and 12, 1299 rigid body, 1072 Index 1315

rotated-axes theorem, 1044 polar, 1035 transformation, 1044 principal, 1058 Matrix principal axes, 1073 orthogonality condition, 399 principal invariants, 1062 skew symmetric, 423, 425, 439 product, 1035 Maxwellcylindrical coordinate system, 344 pseudomatrix, 1037 Measure number, 5 rigid body, 1072 Mechanical energy, 166 rotated-axes theorem, 1044 Mechanics Moment of momentum, 126 Lagrangian, 1189 Momentum, 125, 176, 177 Newtonian, 118 angular, 126 Mechanism definition, 115 3D slider–crank, 598 ellipsoid, 1092 4-bar, 560, 1163 linear, 125 inverted slider–crank, 731 translational, 125 slider–crank, 597 Motion Melbourne, 109, 885 central force, 993 Methods of dynamics, 996 impossible, 913 Gauss method, 999 possible, 913 Gibbs–Appell method, 1009 spiral, 245 Hamilton method, 1002 Moving frame, 4, 58 Kane method, 1013 Multibody Lagrange method, 996, 998 2R manipulator, 676, 1168, 1169, 1172 Mangerone–Deleanu method, 1018 3D slider–crank mechanism, 598 Nielsen method, 1017, 1018, 1020 articulated manipulator, 640, 666, 671 Tzenoff´ method, 1020, 1021 assembling, 615, 616, 621 Methods of solution, 200 best spherical wrist, 623 perturbation methods, 200 closed-loop mechanism, 597 series methods, 200 connection, 555 Metric tensor, 279 Denavit–Hartenberg rule, 563 Moment directional control system, 594 action, 1158 direct kinematics, 584 applied, 126 dynamics, 1157, 1251, 1254 driven, 1158 forward kinematics, 584, 643 driving, 1158 four-bar linkage, 1163 of a force, 120 gravitational vector, 1255 reaction, 1158 industrial links, 573 resultant, 120 inertia matrix, 1253 total, 120 inverse kinematics, 662 Moment of inertia, 1033 inverse transformation, 674 about a line, 1042 kinematics, 555 about a plane, 1042 kinetic energy, 1252, 1253 about a point, 1042 Lagrange dynamics, 1251 characteristic equation, 1062 Lagrange equation, 1254 diagonal elements, 1041 Lagrangian, 1254 Huygens–Steiner theorem, 1047 order-free rotation, 628 parallel-axes theorem, 1044 order-free transformation, 635 1316 Index

Multibody (continued) Node point, 903 potential energy, 1253 Nonholonomic Constraint, 952 recursive dynamics, 1170–1172 Nonpotential force, 170 rest position, 565 shuttle manipulator, 577, 588 O spherical arm, 619 Oblatespheroidal coordinate system, 345 spherical robot, 576, 677 Oklahoma City, 109 spherical wrist, 616 Optimal control, 1225 takht and neshin, 616 Hamiltonian, 1225 tire–wheel–vehicle, 606, 611–614 Lagrange equation, 1215 trebuchet, 581 linear system, 1225 universal joint, 602 objective function, 1225 velocity coupling vector, 1255 switching point, 1226 Orientable surface, 79 N Orlando, 858 Nabla, 86, 100 Orthogonality condition, 31, 399 identities, 100 Orthogonal mesh, 79 Natural coordinate frame, 54, 66, 71 Osculating binormal, 57 plane, 54, 56, 78, 86 bivector, 56 sphere, 66 curvature, 57 Overconstraint, 928, 929 orthogonality, 62 Overrigid, 1034 principal normal line, 56 n-body problem, 194, 992 P Neshin, 616 Paraboloidal1 coordinate system, 345 Neutron star, 1127 Paraboloidal2 coordinate system, 345 New Delhi, 858 Paracylindrical coordinate system, 345 Newton equation, 126 Pars, Leopold Alexander, 942 body frame, 1102 Partial definition, 822 angular velocity, 1015 global frame, 1101 derivative, 78, 79, 86, 1245 Lagrange form, 1191 velocity, 1013, 1014, 1016, 1017 rotating frame, 839 Particle, 889 Newton–Euler definition, 115 backward equations, 1170 impenetrability, 115 equation of motion, 1171 Passive transformation, 409 equations of motion, 1157 Path forward equations, 1171, 1172 admissible, 1215 global equations, 1157 configuration, 47 recursive equations, 1170 cycloid, 247 Newtonian frame, 58 dynamics, 922 kinematics, 46 Newton, Isaac, 126, 992 minimizing, 1215, 1216 New York, 885 motion, 98 Nielsen optimal, 1217, 1224 dynamics, 1017 projectile, 135, 144, 221 equation, 1017, 1020 shortest, 1218 Index 1317

spiral, 260 Plucker¨ variable, 1215 angle, 541 Pendulum, 145, 149, 921 classification coordinate, 538 acceleration, 800 distance, 541 compound, 1271 line coordinate, 534–537, 541, connected, 1269 544–546, 648 constraint, 981 moment, 540 double, 1016, 1018 ray coordinate, 535, 537 elastic, 1005, 1012, 1206 reciprocal product, 541 first integral, 981 screw, 545 flexible support, 1268 virtual product, 541 Foucault, 853, 855 Poinsot, Louis, 121 heavy, 1272 Point integral of motion, 988 at infinity, 489, 490 inverted, 1264 singular, 902 in wind, 1264 Point mass, 889 moving support, 921, 925, 1266, 1269 Pole, 493 oscillating support, 1193 Pop, 53 planar, 925, 1156 Position vector, 5 simple, 145, 152, 800, 1192 Potential sliding, 1193 energy, 166 spherical, 229, 803, 804, 933, 988, 1204 force, 166, 1203 turning, 1181, 1267, 1275 Principal variable mass, 1271 angular acceleration, 794, 795 velocity, 800 angular velocity, 702, 794 Permutation symbol, 26, 27, 448 axes, 461, 486, 1065, 1073, 1133, 1138 Perpendicular plane, 54 base vector, 327 Persia, 584, 860 base vectors, 301, 1245 Persian, 616 body frame, 990 Persian Gulf, 109, 858, 885 central screw, 526, 532, 533 Pfaffian forms, 966, 967, 1238 components, 283 Pfaff, Johann Friedrich, 969 coordinate frame, 278, 1060, 1073 Pfaff problem, 970 coordinates, 310, 311 Phase curvature, 84 plane, 901, 902, 906, 1006, 1227 derivative, 327, 334 Hamilton, 1006, 1007 differential rotation, 752 modified, 1006, 1007 direction, 85 portrait, 901, 903, 905, 1121 frame, 278, 279, 287, 304, 309 velocity invariants, 1062 modified, 1007 mass moment, 1058–1060, 1062, 1063, Physical 1065, 1139 force, 933 metric, 326 quantity nonorthogonal frame, 284 scalaric, 12 normal line, 56 vectorial, 10 plane, 243 Pieper technique, 676 planes, 1065 Planar curve, 65 radius of gyration, 1044 1318 Index

Principal (continued) free fall, 847 rotation matrix, 423, 455, 707, 1061 geodesic, 1218 unit vectors, 243, 271, 278, 282, 298, interpolation, 14 299, 301, 304, 1245 inverse acceleration, 875 Principle inverse kinematics, 662 angular impulse, 177 inverse Lagrangian, 1214 angular impulse and moment of inverse velocity, 778 momentum, 177 minimization, 1217 central, 1230, 1231, 1239 minimum time, 1225 conservation of energy, 165, 166, 985, n-body, 30, 194, 992 1242 open, 367, 379 conservation of moment of momentum, Pfaff, 970 177 projectile, 134 conservation of momentum, 176 regularization, 898 D’Alembert, 942, 943, 1014 resolved rates, 778 decoupling, 663 rigid body, 1112 determinacy, 130 rotation, 460 Galileo relativity, 889 shortest path, 1218 Gauss, 197 three-body, 29, 213–216, 219 Hamilton, 1228, 1229, 1231, 1232 time-optimal, 1225 impulse and momentum, 176, 183 two-body, 194, 210 least constraint, 999 vector interpolation, 14 minimum constraint, 197 walking, 513 Newton–Laplace, 130 Projectile, 134–141, 156 Pontryagin, 1225 antiprojectile gun, 220 projection, 289 height, 157 relative velocity, 742 height time, 157 superposition, 118 in air, 156, 160 variational, 1228, 1230 limit velocity, 159 virtual work, 937, 942, 945 path, 158 work and energy, 165 range, 157 Problem range time, 157 antiprojectile, 220 umberella, 137 book-stacking, 193 variable gravitational acceleration, 198 brachistochrone, 1220 bug, 51 Q central force, 993 Quaternions, 449, 463 central-force motion, 1250 addition, 450 collision, 898 composition rotation, 454 composition, 465 flag form, 450 decomposition, 465 inverse rotation, 453 Dido, 1221 multiplication, 450 dynamic, 116 rotation, 451 foot–leg kinematic, 513 forward acceleration, 872 R forward kinematics, 584 Radius of gyration, 1043 forward velocity, 758 principal, 1044 Index 1319

Razi acceleration, 860 kinetic energy, 1036, 1037, 1077 Razi, Zakariya, 860 mass moment, 1072 Reciprocal moment of inertia, 1072 base vectors, 272, 273, 275, 283, 301, motion, 477 340 motion classification, 527 components, 281, 283 motion composition, 481 coordinate frame, 278, 282 motion condition, 430 coordinates, 310, 311 motion theorem, 1035 derivative, 334 overrigid, 1034 frame, 284 Poinsot interpretation, 1135 geometric interpretation, 277 principal rotation matrix, 1061 metric, 335 rolling disc, 1143 principal vector, 276 rotational kinetics, 1072 reciprocality condition, 271 rotation condition, 430 unit vectors, 269, 298, 299, 301 rotation theorem, 433 vectors, 269, 271, 272 spherical, 1039, 1112, 1115 scalar triple product, 271, 276 stability, 1138–1141 Rectifying plane, 54, 57, 75 steady rotation, 1083 Rectilinear motion, 906 torque-free, 1112 escape, 908 translational dynamics, 1101 libration, 907 translational kinetics, 1101 limitation, 907 velocity, 727, 730 lost, 908 Robot rest, 906 articulated, 666, 769, 774 Reference frame, 4 recursive dynamics, 1170 Rest rest position, 572 point, 893 SCARA, 505 position, 902 Right-handed convention, 3 spherical, 677, 767 Right-hand Rule, 4 Rocket motion, 183 Rigid body final velocity, 184 acceleration, 823 multistage, 185 angular momentum, 1074, 1078 Rodriguez angular velocity, 432 rotation formula, 424, 427, 441, asymmetric, 1112, 1128 445–447, 456, 461, 467, 477, 520, axially symmetric, 1116 532, 699, 754 axisymmetric, 1112, 1116 vector, 446, 447, 466 body frame, 1033 Rodriguez, Benjamin, 463 centrosymmetric, 1115 Roll angle, 1107 classical problems, 1112 Roller coaster, 42, 72, 77 definition, 1033, 1034 Rolling constraint, 956–958 Duffing equation, 1136 Roll–pitch–yaw equimomental, 1112 frequency, 394 Euler equation, 1076, 1093 global angles, 455 general motion, 1141 global rotation matrix, 455 just-rigid, 1033 Rosecylindrical coordinate kinematics, 477 system, 345 1320 Index

Rotation, 425 S acceleration transformation, 788, 791 Saddle point, 903 axis–angle, 422, 425, 426, 440, 442, Scalar, 12 446, 461 equal, 12 composition, 465 equivalent, 12 decomposition, 465 triple product, 23, 33 eigenvalue, 433 Scale, 5 eigenvector, 433 SCARA robot, 505 exponential form, 445 Screw, 51, 121, 517, 522, 527 general, 397 axis, 517 global axes, 357 central, 518, 519, 521, 523, 533, 546, infinitesimal, 444 566, 643, 648 kinematics, 357 combination, 531, 532 local axes, 373, 376, 378 coordinate, 517 local versus global, 395 decomposition, 533 matrix, 460 exponential, 532 order free, 628 forward kinematics, 643 pole, 734 instantaneous, 546 quaternion, 451 intersection, 648 reverse, 425 inverse, 529, 530, 532 Stanley method, 449 left-handed, 517 successive global axes, 360, location vector, 519 363–365 motion, 565, 738 successive local axes, 376, 377 parameters, 518, 525 X-matrix, 358 pitch, 517 x-matrix, 360 Plucker¨ coordinate, 545 Y -matrix, 358 principal, 526, 532, 533 y-matrix, 373 reverse central, 519 Z -matrix, 357 right-handed, 3, 517 z-matrix, 373 special case, 525 Rotational jerk, 807 transformation, 520, 523 Rotator, 425, 457 twist, 517 Routhian, 1008 Second derivative, 726 Rule Second variation, 1216, 1224 bac –cab, 22, 29 Series solution, 29, 210, 213 chain, 746 Serret, Joseph Alfred, 65 Denavit–Hartenberg, 563, 566–568 Shake, 131 homogeneous transformation, 610, Sharang, 131 741 Shiraz, 885 Napier, 369 Sina, Abu Ali, 584 parallel-axes, 1044 Sixsphere coordinate system, 345 relative angular acceleration, 792 Slip constraint, 962 relative angular velocity, 792, 860 Snap, 53 right-hand, 4, 565 Snatch, 131 rotated-axes, 1044 Sooz, 53 Rush space, 910 Space Rush–time space, 910 configuration, 890 Index 1321

constraint-free, 929 Stanley method, 449 curve, 38, 40, 41, 48 Stark effect, 1201 describing, 890 State Euclidean, 888 space, 900 event, 896 trajectory, 900 flash, 910 State–time flash–time, 910 space, 908 flicker, 911 trajectory, 908, 909 flicker–time, 911 Steering axis kinematic, 910 caster angle, 650 Lobachevsky, 207 caster plane, 650 rush, 910 forward location, 651 rush–time, 910 lateral location, 651 spark, 911 lean angle, 650 sparkle, 911 lean plane, 650 sparkle–time, 911 Sundman, Karl Frithiof, 900 spark–time, 911 Suspension state, 900 caster angle, 650 state–time, 908 caster plane, 650 Sparkle space, 911 forward location, 651 Sparkle–time space, 911 lateral location, 651 Spark space, 911 lean angle, 650 Spark–time space, 911 lean plane, 650 Spatial integral, 165 location vector, 651 Special solutions, 131 steering axis, 650 force of position, 141 Symbols, xv force of time, 132 System of particles Force of velocity, 156 angular momentum, 128 Spherical kinetic energy, 174 arm, 619 motion equation, 127 coordinate system, 263 rotational motion, 129 Earth, 185 work, 175 field, 267 Szebehely, Victor, 419, 900 kinematics, 265 pendulum, 803, 804 T rigid body, 1039 Takht, 616 wrist, 616 Tangentcylindrical coordinate system, 345 Spinor, 425, 457 Tangent plane, 78, 79, 86 Spiral Tangentsphere coordinate system, 345 Archimedes, 260 Tavaan, 131 Fermat, 260 Tehran, 109 hyperbolic, 260 Temporal integral, 165, 176 logarithmic, 260 Tetrad, 4 motion, 245, 260 Theorem Stable Chasles, 121, 517, 526 focus point, 903 Euler, 378, 435, 437, 438, 461, 1142 node point, 904 Euler–Chasles, 438 1322 Index

Theorem (continued) wheel–body to vehicle frame, 614 fundamental Lagrangian, 1234 Transformation matrix Guldin, 226, 227 derivative, 749 Huygens–Steiner, 1047 differential, 752, 753 Kennedy, 734 elements, 400 order-free rotations, 628 velocity, 739 order-free transformations, 635 Trebuchet, 581 parallel-axes, 1044, 1047 Triad, 3 Poinsot, 121 coordinate frame, 3 rigid-body motion, 1035 natural, 55 rigid-body rotation, 433 negative, 3 rotated-axes, 1044 nonorthogonal, 269, 270 Three-body problem, 29, 213 opposite, 3 series solution, 29–31 orthogonal, 3 Tides, 189 positive, 3 neap, 193 right-handed, 3 ocean, 189 standard, 3 spring, 193 Trigonometric equation, 668 Time derivative, 718 Trochoid, 247, 586 Time history, 896 Tug, 131 Tire Two-body problem, 210 coordinate frame, 608, 609 Tzenoff´ Tireprint, 607 dynamics, 1021 Top, 384, 990, 1151 equation, 1020, 1021 dynamics, 1150 Topocentric distance, 508 U Toroidal coordinate system, 346 Underconstraint, 928, 929 Toronto, 109, 885 Unicycle constraint, 955 Torque, 120 Unit system, xv Torsion, 65, 70 Unitvector,5,34–36 Trajectory Cartesian frames, 34 actual, 916 definition, 36 configuration, 890 spherical frames, 34 event, 896 Universal joint, 560, 602, 604, 605 possible, 916 history, 604 state, 900 speed ratio, 605 state–time, 908 Unstable Transformation focus point, 903 active and passive, 409 node point, 904 general, 397 USA, 109 homogeneous, 482 order free, 635 V tire to vehicle frame, 613 Variable tire to wheel–body frame, 611, 612 scalar, 46 tire to wheel frame, 610, 611 Variation, 940 wheel to tire frame, 609, 611 Vecface, 10 wheel to wheel–body frame, 612 Vecfree, 10 Index 1323

Veclane, 10 outer product, 17, 285, 289, 291 Vecline, 10 plane, 10 Vecpoface, 10 point, 10 Vecpoint, 10 point-free, 11 Vecpolane, 10 point-line, 10 Vecpoline, 10 point-plane, 10 Vecporee, 11 position, 5 Vector principal base, 301 absolute value, 5 principal unit, 278, 301 addition, 12, 15 quaternion product, 17 algebra, 12 reciprocal, 269–271, 273, 275–277 angular acceleration, 788 reciprocal base, 278 anticommutative property, 17 reciprocality condition, 271 associative property, 12, 15 requirements, 10 axis, 10 reverse, 271 bounded, 10 Rodriguez, 447 Cartesian, 245 rotation, 432 characteristics, 10 scalar triple product, 23, 294 commutative property, 12, 15, 17 sliding, 10 components, 5 space, 24 contravariant components, 278 start point, 10 covariant components, 278 surface, 10 decomposed expression, 6, 9 tangent base, 312 decomposition, 6, 9 triple product, 24 definition, 10 types, 10 derivative, 46, 48 unit, 34 direction, 10 variable, 8 end point, 10 variable direction, 8 equation, 20 variable length, 8 free, 10 vecface, 10 function, 46, 51 vecfree, 10 Gibbs, 447 vecline, 10 gravitational force, 1255 vecpoface, 10 inner product, 17, 285, 289, 291 vecpoint, 10 interpolation, 14 vecpolane, 10 invariant, 244 vecpoline, 10 inverse-element property, 16 vecporee, 11 length, 5, 10 velocity coupling, 1255 length-invariant property, 402 Vehicle dynamics line, 10 Lagrange method, 1091 line of action, 10 Newton–Euler equations, 1109 modulus, 5 pitch angle, 1107 multiplication, 17 pitch rate, 1107 natural expression, 6, 9 roll angle, 1107 normal, 79 roll dynamics, 1107 normal base, 312 roll equation, 1109 null-element property, 16 roll rate, 1107 1324 Index

Vehicle dynamics (continued) W yaw angle, 1107 Wheel yaw rate, 1107 camber angle, 608 Velocity coordinate frame, 608, 609 coefficient matrix, 751 degrees of freedom, 608 definition, 51 spin, 608 end effector, 757 Wheel–body hodograph, 49 coordinate frame, 608, 609 inverse transformation, 739 kinematics, 606 limit, 159 Work, 165 matrix, 865 actual, 935 operator matrix, 749 virtual, 935, 1192 prismatic transformation matrix, Wrench, 121 751 Wrist revolute transformation matrix, spherical, 576, 773 751 transformation matrix, 618 transformation matrix, 739, 743, 749 Y Virtual Yank, 131 displacement, 936, 937 product, 541 Z velocity, 938 Zero-velocity point, 733 work, 935, 937, 941 Zoor, 131