References to Additional Literature

1. Angeles J (1982) Spatial kinematic chains, analysis, synthesis, optimisation. Springer, New York 2. Angeles J (1988) Rational kinematics. Springer Tracts in Natural Philosophy 34, New York 3. Angeles J, Hommel G, Kovács P (1993) Computational kinematics. Kluwer, Dordrecht 4. Angeles J (2003) Fundamentals of robotic mechanical systems: Theory, methods and algorithms. 2nd ed. Springer, New York 5. Artobolevski I I (1986) Mechanisms in modern engineering design. v.1–3, Mir, Moscow 6. Beggs J S (1966) Advanced mechanism. Macmillan, New York 7. Beggs J S (1983) Kinematics. Springer, Berlin, Heidelberg, New York 8. Beyer R (1931) Technische Kinematik. Barth, Leipzig 9. Beyer R (1953) Kinematische Getriebesynthese. Springer, Berlin (Engl. trans. Kuenzel H (1963) The kinematic synthesis of linkages. Chapman and Hall, London) 10. Beyer R (1963) Technische Raumkinematik. Springer, Berlin 11. Bogolyubov A N (1967) Razvitie problem mekhaniki mashin (Bibliografiya) [Develop- ment of problems of mechanics in machines (Bibliography)], Naukova Dumka, Kiev 12. Crelier L (1911) Systèmes cinématiques. Gauthier-Villars, Paris 13. de Groot J (Ed.) (1970) Bibliography on kinematics. v.1,2. Eindhoven Univ. of Techn. 14. Dizio˘glu B (1965,1967) Getriebelehre v.1: Grundlagen, v.2: Massbestimmung. Vieweg, Braunschweig 15. Dresig H, Naake S, Rockhausen L (1994) Vollständiger und harmonischer Ausgleich ebener Mechanismen. Fortschritt-Berichte VDI, Reihe 18, 155,71 pp. Düsseldorf 16. Dresig H, Schönherr J, Peisach E E (2000) Typ- und Masssynthese von ebenen Kop- pelgetrieben mit höheren Gliedergruppen. Abschlussbericht DFG-Forschungsvorhaben Dr234/7-2, Bonn 17. Giering O, Hoschek J (eds.) (1994) Geometrie und ihre Anwendungen. Hanser, München 18. Hall A S (1966) Kinematics and linkage design. Prentice-Hall, Englewood Cliffs, New Jersey 19. Heun K (1906) Lehrbuch der Mechanik. Teil 1: Kinematik. Göschen, Leipzig 20. Jones F D, Horton H L, Newell J A (eds.) (1968) Ingenious mechanisms for designers and inventers, vol.1,2,3,4 Industrial Press, New York 21. Julia G (1936) Cours de cinématique (rédigé par J. Dieudonné). 2nd. ed. Gauthier- Villars, Paris 22. Klein F, Müller C (eds.) (1901-1908) Enzyklopädie der Math. Wissenschaften. v.IV: Mechanik. Teubner, Leipzig

23. Koenigs G (1905) Introduction à une théorie nouvelle des méchanismes. Hermann, Paris 24. Lebedev P A (1966) Kinematics of spatial mechanisms (Russ.). Isdat. Maschinostr., Moscow (with references to 115 Russian titles) 25. Lenarˇciˇc J, Ravani B (eds) (1994) Advances in robot kinematics and computational geometry. Kluwer, Dordrecht 26. Lenarˇciˇc J, Parenti-Castelli V (eds.) (1996) Advances in robot kinematics. Kluwer, Dordrecht 27. Lenarˇciˇc J, Husty M L (eds.) (1998) Advances in robot kinematics: Analysis and control, Kluwer, Dordrecht 28. Miura H, Arimoto S (eds.) (1990) Robotics Research. MIT Press, Cambridge, Mass. 29. Müller R (1932) Einführung in die theoretische Kinematik. Springer, Berlin 30. Pandrea N I (2000) Elemente de mecanica solidelor in coordonate Plückeriene. Editura Acad. Romane 31. Pennock G R (ed.) (1994) Mechanism, synthesis and analysis. v.DE-70, ASME 32. Pottmann H (1994) Kinematische Geometrie (with 363 literature ref.). In: [17] 33. Prudhomme R, Lemasson G (1955) Cinématique. Dunod, Paris 34. Reuleaux F (1900) Theoretische Kinematik v.2. Vieweg, Braunschweig. 35. Sandor G N, Erdman A G (1984) v.I: Mechanism Design. Analysis and synthesis. v.II: Advanced Mechanism Design. Analysis and synthesis. Prentice Hall, Englewood Cliffs, New York 36. Schoenflies A (1886) Geometrie der Bewegung in synthetischer Darstellung. Teubner, Leipzig. reprint (2007) VDM-Verlag Dr. Müller, Saarbrücken 37. Schoenflies A, Grübler M (1908) Kinematik. In: [22]:190–278 38. Spillers W R (ed.) (1964) Basic questions of design theory. Northholland, Amsterdam 39. Suh C H, Radcliff W (1978) Kinematics and mechanism design. Wiley, New York Index

(4 × 4) transformation matrix, 85 binary link, 658 Π-points, 435 Bobillier’s theorem, 484 Bresse circle, 473 acceleration, 350, 413, 497 Bryan angles, 9, 38, 40, 333, 660 complex formulation, 470 Burmester point, 632 acceleration distribution, 309 Burmester theory, 629 accelerometer, 318 active height, 536, 545, 550 cam mechanism, 240, 455 addendum, 536, 545 canonical reference frame, 80, 303, 310 addendum circle, 546, 550 Cardan angles, 9 addendum modification, 547 cardinal point, 437, 489 affine transformation, 238 cardioid, 460, 501 angle of pressure, 544, 547, 552 carrier body, 664 angular acceleration, 289, 350 cartesian base, 1 angular momentum, 666 Cayley-Klein parameters, 28, 337 angular velocity, 289, 329, 350 center angular velocity ratio, 509 of acceleration, 309 angular velocity triangle, 360 of curvature, 414 animation of motion, 46 of rotation, 521, 575, 577 antiparallelogram mechanism, 465 center point, 629, 630 Archimedes’ spiral, 508 center point curve, 630, 632, 653 articulation point, 350, 671 centrifugal acceleration, 356 astroid, 501, 520 centrode, 451, 464, 485, 488, 530, 577, 600 Chebychev’s optimality criterion, 509, 619 Ball’s point, 492, 618 Chinese southpointing chariot, 307 ball-in-track joint, 404 circle of singular foci, 598, 606, 648 base circle, 543, 552 circle of vertices, 504 base cylinder, 552, 556 circle point, 425, 629 base helix, 553, 556 circle point curve, 632 base helix angle, 560 circular pitch, 535, 545 Bennett mechanism, 180, 207, 663 closed kinematic chain, 400 Besicovitch, 503 closure condition, 166, 168, 177, 674 bevel differential, 307 co-factor, 3, 5, 35, 41 bevel gear, 306 cognate four-bars, 590, 604, 617, 625 Bezout, 281 cognate slider-crank, 593, 608 bicursal, 633, 653 commutativity conditions, 418, 424

© Springer-Verlag Berlin Heidelberg 2016 679 J. Wittenburg, Kinematics, DOI 10.1007/978-3-662-48487-6 680 Index complex (of lines), 70 Darboux motion, 457 complex line, 71, 141, 296 decomposition complex number, 411, 431, 497, 590 of a rotation, 55, 299 composite system center of mass, 634, 665, of a screw displacement, 116 667 of force screw, 368 conchoid, 459 dedendum, 546 congruence, 70, 77, 206 defect, 138 conjugate screws, 124, 374 degree of freedom, 137, 515, 664, 674 conjugate tooth flanks, 374, 532, 534, 537, Denavit-Hartenberg parameters, 161, 235, 544, 553, 558, 560 271 conservation of kinetic energy, 664 differential equations of motion, 665 constraint force, 665, 671 direct kinematics, 278, 350 contact direct path, 354 force, 544, 554 direction cosine matrix, 1, 13, 17, 33, 34, normal, 536, 542, 561 37, 318, 350, 660 point, 538 direction cosines, 1, 329, 335 ratio, 546, 556, 564 directrix, 77, 125, 132, 379, 487, 489 coordinate matrix, 3 discriminant, 311, 382 Coriolis acceleration, 292, 356 displacement coupler, 567, 639 elementary, 415 centrode, 464 in a plane, 411 coupler curve, 482, 567, 584, 590, 594, 595, sense-preserving, 416, 420 607, 616, 634, 643, 656, 658 sense-reversing, 416, 420 bicursal, 595, 606 displacement group, 443 double points, 599, 648 distribution parameter, 78, 83, 125, 301, straight-line approximation, 617 364, 553, 558 symmetrical, 603, 646 door mechanism, 466 unicursal, 595 double helical gearing, 554 coupler triangle, 590, 643, 656 double-crank, 570, 593, 643 coupling double-rocker, 570, 643 ball-in-track, 405, 407 first kind, 571 bicardanic, 402 second kind, 571, 593 Clemens, 403 draw-bar, 42 Devos, 405 dual angle, 99 Hebson, 401 dual basis, 102 Oldham, 387, 394, 409 dual Bryan angles, 108 Tracta, 401 dual derivative, 98 tripod, 407 dual differentiation, 98, 108, 167, 178, 268 Unitru, 403 dual direction cosine matrix, 102, 108 crank, 570 dual direction cosines, 103 crank-rocker, 570, 593, 623, 632, 643 dual Euler angles, 108 crowning, 531 dual Euler-Rodrigues parameters, 110 cubic of stationary curvature, 487 dual number, 97 curvature, 414, 473, 476, 485, 510 dual quaternion, 110, 112, 278 curvilinear coordinates, 321 dual Rodrigues vector, 110 cusp, 303, 311, 460, 463, 481, 501, 517, 542, dual transformation matrix, 269 548, 654 dual vector, 99 cycloid, 500, 501, 521 dwell mechanism, 509 ordinary, 506 dynamics of mechanisms, 663 cylindroid, 122, 132, 359, 373 principal axes, 123, 124 eigenvalue, 4, 14, 124, 311, 382 principal pitches, 124 eigenvector, 5, 15, 59, 311, 382 Index 681 elliptic trammel, 455, 475, 494, 497, 501, generator, 78, 237 514, 629 gimbal lock, 8, 54 envelope, 517 gimbal suspension system, 8, 51 epicycloid, 500, 508, 535, 541 glide reflection, 419, 421, 441 epitrochoid, 500, 508 Goldberg mechanism, 212 prolate, 542 gorge circle, 363, 558 equilibrium condition, 587 Grübler’s formula, 139, 205, 298, 401 Euler angles, 7, 19, 38, 333, 336 graph theory, 672 Euler vector, 33, 339 Grashof condition, 568, 641 Euler’s equation, 671 group, 4, 21, 30 Euler’s formula, 412 Euler-Rodrigues parameters, 16, 19, 21, 28, half-angle equations, 175 36, 38, 270, 319, 334 helicoid, 556 correction formulas, 345 helix, 367, 374 Euler-Savary equation, 479 herpolhode cone, 305, 332, 342, 390, 395 evolute, 517 Heureka octahedron, 242 external force, 664 homokinetic shaft coupling, 153, 387, 400 homologous points, 425, 629, 630 Fayet’s equation, 481 Hooke’s joint, 154, 205, 305, 388, 663 Fenyi’s joint, 302 angular acceleration, 390 fillet, 536, 546 angular velocity ratio, 389 fixed axode, 301 herpolhode cone, 390 fixed centrode, 452 polhode cone, 390 fixed cone, 305 series-connected, 397 fixed link, 567 hyperbolic paraboloid, 78, 130, 649 focus, 436, 489 hyperboloid, 78, 221, 383, 384 foldable four-bar, 235, 571, 575, 595, 599, hyperboloid of revolution, 82, 262, 363, 558 643 hypocycloid, 500, 535 first kind, 572 hypotrochoid, 500 second kind, 572 foldable spherical four-bar, 651 incidence matrix, 672 force screw, 366, 367 inclination angle of the coupler, 352, 577 intensity, 367 indeterminacy conditions, 54 standard form, 367 inertia matrix, 670 unit screw, 367 inertia tensor, 666 four-bar, 144, 154, 205, 352, 441, 465, 481, inflection circle, 473, 479, 492 663 inflection curve, 309 angular acceleration, 578 inflection point, 460, 473 foldable, 575 input angle, 567, 639 limit angles, 575 input link, 567, 639 transfer function, 573 instantaneous axis of rotation, 452 friction wheel, 530 instantaneous center of acceleration, 472 gear, 374, 457, 497, 529, 593 of rotation, 451, 452, 471, 515, 610 cycloidal, 504, 535 of velocity, 501 gear ratio, 530, 531 instantaneous screw axis (ISA), 293, 395, gearing, 529 452 external, 529 interchange of input link and fixed link, internal, 529 576, 580 generalized acceleration, 197 internal force, 664 generalized coordinates, 588, 664 internal gearing, 542 generalized forces, 588 intersection condition, 66, 101, 133 generalized velocity, 197 inverse motion, 292, 458, 475, 478, 481 682 Index inverse rotation, 12 linkage, 567, 593 inversion circle, 627 inversion in a circle, 627 matrix inverted circle, 628 orthogonal, 3, 336 inverted slider-crank mechanism, 606 unitary, 30 involute mecanum wheel, 324 function, 508 mechanism helical gearing, 552 4R-P, 234 helicoid, 553, 556, 560 5R-C, 185 of a circle, 507, 543, 552 6R-P, 192 spur gear, 543 7R, 188, 194, 198 involutoric, 417 antiparallelogram, 465 Bennett, 180, 207 Jeantaud mechanism, 614 Bricard-Borel, 235 jerk, 292 cam, 240, 455 joint, 137, 349, 664 CRRRC, 181 cylindrical, 159, 352 Dietmaier’s, 229 Fenyi’s, 393 door, 466 helical, 137, 370 Goldberg, 212 homokinetic, 387 hyperboloid, 236 prismatic, 137, 159, 370 Jeantaud, 614 revolute, 137, 159, 351, 370 line-symmetric Bricard, 149, 217 spherical, 137, 162, 352 mobility-one, 140 tripod, 407 multiloop, 138 universal, 352, 387 orthogonal Bricard, 224 joint variables, 161, 349, 355, 665 overconstrained, 140, 205 paraboloid, 236 kinematic chain, 400 Peaucellier inversor, 626 kinematic differential equations, 319, 322, planar, 139 329 planar four-bar, 567 kinematical constraint, 137 planar multiloop, 143 kinematics of joints, 350 plane-symmetric Bricard, 145, 221 kinetic energy, 670 RCCC, 177, 198, 207, 639 Kutzbach’s angular velocity diagram, 469 RCRCR, 181 RRCRP, 233 level-luffing jib crane, 492 RRSRR, 195 lima¸con of Pascal, 460, 466, 497, 514, 520, Sarrus, 205 541 slider-crank, 606, 629 limit position, 569, 641 slider-crank inverted, 606 line contact, 530 spherical, 139 line displacement, 46, 127 spherical four-bar, 639 line geometry, 63 steering, 614 line net, 443 trihedral Bricard, 145, 672 line of contact, 535, 536, 538, 539, 561 mobile polyhedron, 231 line vector, 99, 296 mobility-one mechanism, 140 line-symmetric Bricard mechanism, 149 module, 545, 555 linear complex, 70, 77, 142, 206, 220, 236, moment of momentum, 666 277, 296, 367, 368, 373, 558 momentum, 666 axis, 71 motion on a curved surface, 321 pitch, 71 moving axode, 301 special, 70, 77, 78, 223 moving centrode, 452 linear complex of velocity, 293 moving cone, 305 linear fractional function, 32 multiloop mechanism, 138, 243, 664, 674 Index 683

Newton’s law, 665, 667, 671 opposite, 432 nodal line, 129, 132 pole curve, 432, 433, 488, 629, 630, 632 nonholonomic constraint, 322 pole quadrilateral, 432, 629, 630 normal coordinates, 430 pole triangle, 25, 425, 432, 629 normal poles, 476 polhode cone, 305, 332, 342, 390, 395 normal vector of a plane, 63 position vector, 350 null plane, 70, 73, 297, 367, 373 positive definite, 670 null point, 70, 73, 297, 367, 373 principal pitch, 361, 374 principal screw, 132, 374 offset, 161, 207, 221 principle of transference, 102, 128, 263, 394 Oldham coupling, 458 principle of virtual power, 587, 665 opposite poles, 432 pruning-shears, 589 orthogonality, 18, 112 orthonormality condition, 1, 290 quadratic-involutoric, 431 output angle, 567, 639 quadric, 206 output link, 567, 639 quadrilateral, 27 overconstrained mechanism, 140, 205 quaternion, 20, 38, 55, 110 conjugate, 20 Panhard rod, 42 inverse, 21 parallel curve, 508, 516, 541, 546 norm, 20 parallel robot, 609 of a rotation, 21 path matrix, 672 of resultant rotation, 22 Pauli spin matrix, 30 unit, 20 Peaucellier inversor, 626 quaternion product, 20 pedal curve, 460 pencil of complex lines, 71, 76 raccording axodes, 301, 395, 452 phase curve, 664 raccording hyperboloids, 362, 558 piercing mill, 365 raccording motion, 82, 301, 364 pin gear, 456, 540 rack, 529, 534, 535, 544 pitch, 141, 293, 367, 556 rack cutter, 546, 554 pitch circle, 530, 547, 552 rack shift, 547 pitch cone, 306 radius of curvature, 479, 510 pitch cylinder, 552, 563 rear axle, 42 pitch line, 535, 547 reciprocal cone, 376 pitch plane, 554 reciprocal polar, 74, 220, 298 pitch point, 469, 530 reciprocal screw systems, 372 pitch quadric, 383 reciprocal screws, 368, 369 Plücker coordinates, 64 reciprocal vector, 49 Plücker vector, 64, 69, 99, 125, 141, 206, reciprocity condition, 375, 558 271, 369, 375, 382, 559 recursive equations, 163, 354 normalized, 65 reduction point, 366 planar four-bar, 567 reflected pole, 426 planar mechanism, 139 reflecting line, 417 plane motion, 294, 451 reflecting plane, 58 plane-symmetric Bricard mechanism, 145 reflection, 12, 422, 441 planetary gear, 469 in a line, 12, 39, 65, 90, 94, 415, 425 planetary wheel, 477, 495 in a plane, 58 point contact, 531 reflection point, 425 pointed tooth, 549 reflection tensor, 59 Poisson’s equations, 330 regular labeling, 349, 673 polar coordinates, 413 regulus, 237, 383, 384 pole, 415, 421, 425, 441 relative velocity screw, 362 normal, 476 resultant 684 Index

of elementary displacements, 418 spur gear, 469, 530, 543 of infinitesimal screw displacements, 122, steering mechanism, 614 125 Steiner’s hypocycloid, 501, 503 of reflections, 39, 94, 418 stereographic projection, 31, 651 of rotations, 13, 299 Stewart platform, 277 of screw displacements, 95 triangle-configuration, 284 of velocity screws, 359 straight-line approximation, 568, 617 resultant torque, 367 grashopper linkage, 617 robot arm, 370 Roberts, 618 rocker, 570 Tschebychev, 619 Rodrigues parameters, 34, 38, 338 Watt, 617 Rodrigues vector, 17, 26, 34, 93, 338 straight-line position, 497 rotated velocity, 453, 468 strapdown inertial navigation, 318 rotation, 1, 5, 11, 40, 93, 415, 423, 441 striction line, 79, 83 about a fixed point, 305 striction point, 78, 301, 364, 558 decomposition, 55 Study-quadric, 112 instantaneous center of, 451 sunwheel, 495, 501 resultant, 13, 22, 30, 35, 38, 55 superposition principle, 359, 370, 610 rotation cone, 28, 273, 433 sway bar, 42 rotation tensor, 11, 17, 38, 559 Sylvester’s plagiograph, 593 rotation triangle, 25 system graph, 243 rotation vector, 10, 23 system structure, 664 ruled surface, 70, 78, 125, 206, 237, 301, 553 theorem Bernoulli/de la Hire, 506 sawtooth construction, 483 Bobillier, 484 screw, 72 Bricard, 206 angle, 100, 393 Burmester, 453, 468, 471 axis, 88, 141, 360, 367, 558 Camus, 534 displacement, 85, 88, 89, 91, 94, 100, Cayley, 97 105, 161 Chasles, 87, 105 decomposition, 116 de La Hire, 240 scalar measure, 90, 114, 134 Euler, 5, 104 motion, 235, 244, 294, 394 Grashof, 441, 569, 607 system, 359 Halphen, 94 triangle, 95, 118 Holditch, 493, 503 sense of a triangle, 415, 429, 633 Kennedy/Aronhold, 143, 453, 531 sense-preserving displacement, 416, 420 Painlevé, 301, 452 sense-reversing displacement, 420 Poinsot, 305 sensor calibration, 48 Roberts/Tschebychev, 590 serial chain, 349, 664 Roth, 91 serial robot, 86, 199, 609 Tschebychev, 619 shaft coupling, 153, 387 three-angle equation, 480 shaky, 142 tile, 154, 441 shears, 588 tiling, 154, 441 similarity transformation, 6, 50, 269 time derivative of a vector, 290 simple closed chain, 138, 140, 159, 672 tooth flank, 531 single-loop mechanism, 664 conjugate, see conjugate, 534 slider-crank mechanism, 606, 629 curvature, 532 spherical four-bar, 139, 181, 205, 639, 663 tooth thickness, 548 spherical interpolation, 45 torque, 366 spherical mechanism, 139 torsal line, 125, 132 spherical parallel robot, 658 torse, 83, 553, 560 Index 685 torus, 259 unicursal, 633, 653 trajectory, 456, 462, 493, 495, 534 unitary matrix, 30 c.o.m. of four-bar, 634 transfer function, 567, 573, 640 vector polygon, 162, 177 transformation matrix, 4 velocity, 350, 413, 497 translation, 415, 441 complex formulation, 470 transmission rotated, 453 angle, 577, 604 velocity distribution, 293, 366, 451 characteristics, 611 velocity screw, 206, 293, 299, 366, 367 of forces and torques, 587 intensity, 296 ratio, 567, 578, 581, 613, 663 Plücker vectors, 296 transport acceleration, 292 standard form, 296 transport velocity, 292 unit screw, 296 transversal, 68, 220, 380 velocity triangle, 564 tree structure, 138, 349, 664, 669 vertex, 415 tricircular sextic, 598, 608 condition, 415, 487 trihedral Bricard mechanism, 145, 672 virtual change of velocity, 665 tripod joint, 407 virtual power of a force screw, 368 trochoid, 495 virtual velocity screw, 368 curtate, 500 curvature, 486 Wankel engine, 498 double generation, 499 weighted incidence matrix, 672 first generation, 499 wheel suspension system, 141 prolate, 500 Wiener parameters, 36, 38, 339 two-joint-chain, 257 Wiener vector, 36, 339 Woernle-Lee equations, 166, 222, 226 undercutting, 538, 556 work space, 258