Index

A six-dimensional vector, 102, 108, 118, Acceleration analysis 122, 123, 125, 127 acceleration, definition, 97 trucks in relative motion, 119–125 angular acceleration vector, 98–101 two-degree-of-freedom serial Coriolis acceleration, 100 manipulator, 107–110 decomposition of, 164, 176 six-legged parallel manipulator, 278–279 Delta robot, 312–314 Stewart platform, 292–293 four-bar linkage mechanism, 128, 332–334 trucks in relative motion, 100–101 linear acceleration vector, 98–100 Active revolute joint, 206, 220, 222, 223, 232, parallel wrist, 213 238, 239, 303, 310 planar timing mechanism, 128, 336–339 Agile Eye, 24, 205 relative motion of three rigid bodies, An Application of Screw Algebra to the 99–100 Acceleration Analysis of Serial 3-RRPS manipulator, 228–229 Chains, 106 3RRRS+3RRPS parallel manipulator, Argos robot, 24 244–246 3R2T parallel manipulator, 196–198 screw form B angular acceleration vector, 101, 102, Ball, Sir Robert Stawell, 4, 7, 31, 80, 97, 101, 106, 109, 110, 112, 117, 127 158 Geneva wheel, 110–112, 128, 334–335 helicoidal vector fields, 102–103, 122, 124 C hexaglide-type robot, 124–127 Carpal Wrist, 24 linear acceleration vector, 102, 105 Cayley, Arthur, 6, 10, 12 multibody mechanical system, Chasles, Michel, 3, 4, 6, 9 105–106 Chasles’ theorem, 38, 68, 69 reduced acceleration state, 104–107, Clifford, William Kingdon, 4–6, 12, 53 109, 110, 118, 122, 124, 127 Computer codes, 357–370 2RRR+PPR planar parallel manipulator Coriolis hyper-jerk, 162 (see 2RRR+PPR planar parallel Coriolis jerk, 139 manipulator) Crank-slider linkage mechanism, 88–90, 94, serial chains, by helical pairs, 107 327

© Springer International Publishing Switzerland 2016 371 J. Gallardo-Alvarado, Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory, DOI 10.1007/978-3-319-31126-5 372 Index

D two-degree-of-freedom parallel wrist Delta robot, 22, 24 (2-DOF) advantage of, 301 forward position analysis, 207–209 applications, 302 inverse position analysis, 209 characteristics, 297 Flat kinematic pair, 45 description, 302–303 Forward displacement analysis finite kinematics closed-form solution of, 241 forward displacement analysis, 304–308 Delta robot, 304–308 inverse displacement analysis, 308–310 parallel wrist, 207–209 infinitesimal kinematics 3-RRPS manipulator, 221–223 acceleration analysis, 312–314 3RRRS+3RRPS manipulator, 239–240 infinitesimal screws, 310 3R2T robot, 191–192 singularity analysis, 312 six-legged parallel manipulator temporal behavior, moving platform, Cayley’s formula, 259 314–315 with control points, 260–261 velocity analysis, 311–312 input data, 261–262 Dimentberg, F.M., 5, 14, 53 o2 point position, 263–264 Discorso matematico sopra il rotamento orthogonal matrix, 259 momentaneo dei corpi (Mozzi), 4, with planar platforms, 261–262 7–8 three-dimensional platform, 265–266, Distinct kinematic chains, 206 269–273 2-DOF parallel wrist. See Two-degree-of- triangular-prism moving platform, freedom parallel wrist (2-DOF) 266–269 vector position, 258–259 Stewart platform, 285–289 E Four-bar linkage mechanism, 85–88, 303, 326, Euclidean group SE (3). See Lie Algebra se (3) 330, 332, 333 Freudenstein, Ferdinand, 6, 15, 219

F Finite kinematics G Delta robot Geneva wheel/Maltese cross forward displacement analysis, 304–308 acceleration analysis, 110–112, 128, inverse displacement analysis, 308–310 334–335 3-RRPS parallel manipulators velocity analysis, 90–93 coordinates of, moving platform, 223, Gough–Stewart platforms, 22, 219, 237, 255, 226 282 coordinates of, spherical joints, Gough’s tire testing machine 223–225 description, 257–258 forward position analysis, 221–223 finite kinematics, 256 3RRRS+3RRPS parallel manipulators hexapod (see Six-legged parallel algebraic equations, 240 manipulator) forward displacement analysis, 239–240 infinitesimal kinematics, 256–257 hexagonal moving platform, 239 acceleration analysis, 278–279 inverse displacement analysis, 242 velocity analysis, 277 rotary sensor, 241 octahedral hexapod, 255–256 3R2T parallel manipulators singularity analysis, 256, 279, 353 forward position analysis, 191–192 Grassmann, Hermann Günther, 11, 12 inverse position analysis, 193 Stewart platform forward displacement analysis, H 285–289 Hamilton, Sir William Rowan, 6, 9, 12 inverse displacement analysis, 289 Hammer blow force, 131 Index 373

Helicoidal vector fields I angular velocity tensor, 60–61 Illness regions, parallel manipulators, 88, 297 definition, 57 Infinitesimal kinematics equivalence relationship Delta robot reflexive, 59 input–output acceleration equation, 314 symmetric, 59 input–output velocity equation, 311 transitive, 60 joint-velocity rates, 313 hyper-jerk analysis, 166 Lie screw of acceleration, 313 jerk analysis, 141 reduced active matrix, 311 reduced acceleration state, 102, 103 3-RUU translational parallel two helicoidal fields, 57–59 manipulator, 310 velocity state and, 66–67, 85 second-order driver matrix, 314 Hexaglide-type robot, 124–127 singularity analysis, 312 History temporal behavior, moving platform, Discorso matematico sopra il rotamento 314–315 momentaneo dei corpi (Mozzi), 4 3-RRPS parallel manipulators “dynamo” (Plücker), 4 acceleration analysis, 228–229 finite kinematics of rigid body, 6 connector chains, 227 Kinematic Geometry of Mechanisms, 6 connector limbs, 230 Motorrechnung, ein neues Hilfsmittel in input-output equation, 227–228 Mechanik (von Mises), 5 Jacobian matrix, 228 Poinsot’s theorem, 4 jerk analysis, 231–232 principle of transference, 3 Lie screw of, jerk, 230 rigid body motion, 3, 8 periodic functions, 232 The Screw Calculus and Its Applications in screw theory, 232, 233 Mechanics (Dimentberg), 6, 14 velocity analysis, 227 Screw Systems in Spatial Kinematics 3RRRS+3RRPS parallel manipulators (Hunt), 6 acceleration analysis, 244–246 Vector and Tensor Analysis (Brand), 5 hyper-jerkor, 248–251 H4 robot, 22 jerk analysis, 246–247 Hunt, Kenneth Henderson, 6, 15, 16, 189, Klein form, 243 282 Lie screws of, acceleration, 245 Hybrid manipulator, 19 screws of, 242 Hyper-jerk analysis velocity analysis, 243–244 angular hyper-jerk vector, 158–161 3R2T parallel manipulators Coriolis hyper-jerk, 162 angular and linear acceleration, input–output equation, 182–183, 200–201 350–352 angular and linear velocity, 200–201 linear hyper-jerk vector, 159–161 forward position analysis, 199–200 rotating disk, 162–166, 182, 345–347 Lie screw, acceleration, 197 3RRRS+3RRPS parallel manipulator, periodic functions, 198–199 248–249 reduced acceleration state, 196–197 screw form spherical joints, 199–200 helical pair, 171–173 velocity analysis, 193–196, 198 helicoidal vector, 166, 167 Stewart platform Lie screw, 172–173 acceleration analysis, 292–295 paint-spraying robot (see Paint-spraying combined singularity analysis, 297 robot) direct singularity analysis, 295–296 reduced hyper-jerk state, 166, 168, 169, inverse singularity analysis, 296–297 180–181 velocity analysis, 290–292 telescoping antenna, 182–183, 347–350 two-degree-of-freedom parallel wrist time derivatives, 161–162 (2-DOF) Hyper-redundant manipulators, 19, 65, 189 acceleration analysis, 213 374 Index

Infinitesimal kinematics (cont.) time derivatives, 133–134, 139, 147 angular velocity, 210 trucks in relative motion, 139–140 input–output equation, 211–213 water velocity, nozzle, 154, 339–340 jerk analysis, 214 Klein form, 212 Plücker coordinates, 210 K screws of, 211 Kinematic Geometry of Mechanisms, 6 singularity analysis, 215–216 Klein, Christian Felix, 10–12 Instantaneous screw axis (ISA), 40, 41, 68, 69, Kotelnikov, Aleksandr Petrovich, 13–14 323 Inverse displacement analysis Delta robot, 308–310 L hexapod, 273–276 Lie, Marius Sophus, 13 parallel wrist, 209 Lie Algebra se (3) 3-RRPS manipulator, 223–227 addition, 46, 47 3RRRS+3RRPS parallel manipulator, 242 equivalence classes, 45, 48 3R2T robot, 193 generic kinematic state, 46 Stewart platform, 289 Killing and Klein forms, 51–52 Lie product, 46–48 distributive, 49 J and Euclidean motion, 53–55 Jerk analysis geometric interpretation of, angular jerk vector, 133, 138 55–56 applications of, 132 Jacobi identity, 48–50, 55 Cam mechanism, 155, 341–342 nilpotent, 49 Coriolis jerk, 139 noncommutative, 50 development, 131–132 multiplication by a scalar, 46, 47 hammer blow force, 131 reciprocal screw, 52–53 input–output equation, 150, 152, 153, 156 Lie product joint-jerk rate, 133, 231, 247 distributive, 49 linear jerks, 133, 138 and Euclidean motion, 53–55 parallel wrist, 214 geometric interpretation of, 55–56 rocket fired vertically, 134–137 Jacobi identity, 48–50, 55 rotating disk with circular slot, 155–156, nilpotent, 49 342–344 noncommutative, 50 3-RPR parallel manipulator, 156, 344–345 Lie screw of acceleration, 107, 111, 112, 123, 3-RRPS manipulator, 230–231 127, 153, 197, 213, 229, 278, 293, 3RRRS+3RRPS parallel manipulator, 313, 339, 351 246–247 Lie screw of jerk, 150, 154, 214, 230, 352 screw form Lower kinematic pairs, 44–45, 61, 321 helical pair, 146–148 Lower-mobility parallel manipulators, helicoidal vector field, 141 20, 22 lie screw of jerk, 150, 154 linear jerk, 141, 144 multibody mechanical system, 145–146 M reduced jerk state, 141, 142, 145, 146, Maple procedures, 357–370 148, 153 Motor acceleration. See Acceleration analysis robot arm, 143–145 Motor algebra, 5, 6, 14, 31, 45 serially connected kinematic chain, Motor product/dual vector product. See Lie 148–150 product six-dimensional vector, 142 Motorrechnung, ein neues Hilfsmittel in slider spring mechanism, 154–155, Mechanik (von Mises), 5 340–341 Mozzi del Garbo, Giuseppe Giulio, 7–8 Index 375

N Poinsot, Louis, 4, 8 Newton–Raphson method, 200, 232, 258, 294, Poinsot’s theorem, 4 314 Pollard’s spray painting machine, 20–21 Nonredundant parallel manipulators, 20, 95, Principle of transference, 3 115, 220, 221, 296 6-PSU parallel manipulator. See Hexaglide- Normalized screw, 41–42 type robot

O Q Oxymoron, The, 20 Quattro robot, 22, 301

P R Paint-spraying robot Reduced active matrix, 311 acceleration analysis, 175–176 Redundant parallel manipulators, 20 Coriolis acceleration, 176–178 Rossum’s Universal Robots (Capek),ˇ 19 infinitesimal screws, 178 Rotating disk, 162–163, 182, 343, 345–347 model of, 178–179 Rotation matrix, 54, 92, 207–209, 222, 256, reduced acceleration state, 179–180 259, 260, 273, 276, 288, 355–356 reduced hyper-jerk state, 181 3-RRPS parallel manipulators reduced jerk state, 180 description, 220–221 velocity analysis, 174–175 finite kinematics Parallel manipulators coordinates of, moving platform, 223, Gough’s hexapod, 21 226 Gough–Stewart platforms, 22 coordinates of, spherical joints, illness regions, 88, 297 223–225 linear, 23 forward position analysis, 221–223 lower-mobility mechanisms, 20, 22 infinitesimal kinematics nonredundant, 20, 115, 220, 221, 296 acceleration analysis, 228–229 octahedral hexapod, 22 connector chains, 227 parallelograms, 22 connector limbs, 230 redundant, 20 input-output equation, 227–228 3-RRPS (see 3-RRPS parallel Jacobian matrix, 228 manipulators) jerk analysis, 231–232 3RRRS+3RRPS (see 3RRRS+3RRPS Lie screw of, jerk, 230 parallel manipulators) periodic functions, 232 3R2T (See 3R2T parallel manipulators) screw theory, 232, 233 Schönflies motion generator robot, 22 velocity analysis, 227 serial manipulators vs., 19–20 2RRR+PPR planar parallel manipulator spherical, 20, 24 acceleration analysis spray painting machine, 20, 21 characteristic equation of manipulator, translational, 20, 22, 24 115 universal tyre testing machine, 21 displacement analysis, 114–115 videos, 25–26 first-order coefficient matrix, 117 Phillips, Jack Raymond, 6, 16, 56 first-order driver matrix, 117 Plücker, Julius, 10–11 gradual improvement, 112–114 Plücker coordinates infinitesimal screws, 115–116 angular velocity, 41 input-output equation of velocity, 117, infinitesimal screw, 43 119 moment-par, 41 Lie screw of acceleration, 118 normalized/unit screw, 41–42 passive joint-velocity rate, 118, 119 primal and dual part, 42 right mechanism, 113 triple vectorial product, 43 R*RRS-type kinematic chains, 116, 117 376 Index

2RRR+PPR planar parallel manipulator (cont.) Six-legged parallel manipulator second-order driver matrix, 119 forward displacement analysis singularity analysis, 128, 335–336 Cayley’s formula, 259 virtual joint rates, 116 with control points, 260–261 3RRRS+3RRPS parallel manipulators input data, 261–262 description, 237–238 o2 point position, 263–264 finite kinematics orthogonal matrix, 259 algebraic equations, 240 with planar platforms, 261–262 forward displacement analysis, 239–240 three-dimensional platform, 265–266, hexagonal moving platform, 239 269–273 inverse displacement analysis, 242 triangular-prism moving platform, rotary sensor, 241 266–269, 279, 352–353 infinitesimal kinematics vector position, 258–259 acceleration analysis, 244–246 infinitesimal kinematics hyper-jerk analysis, 248–251 acceleration analysis, 278–279 infinitesimal screws of, 242 infinitesimal screws, 276 jerk analysis, 246–247 velocity analysis, 277 velocity analysis, 243–244 inverse displacement analysis 3R2T parallel manipulators B1 point positions, 273–274 description, 190–191 coordinates of, point B1, 273, 275 finite kinematics real solutions, 273, 275 forward position analysis, 191–192 rotation matrix, 275, 276 inverse position analysis, 193 linear and rotary actuators, 238 infinitesimal kinematics Snap or jounce. See Hyper-jerk analysis angular and linear acceleration, Stewart platform 200–201 Cappel’s flight simulator, 282–283 angular and linear velocity, 200–201 description, 283–284 forward position analysis, 199–200 finite kinematics Lie screw, acceleration, 197 forward displacement analysis, 285–289 periodic functions, 198–199 inverse displacement analysis, 289 reduced acceleration state, 196–197 flight simulator, 281–282 spherical joints, 199–200 infinitesimal kinematics velocity analysis, 193–196, 198 acceleration analysis, 292–295 limited-DOF, 189–190 combined singularity analysis, 297 direct singularity analysis, 295–296 inverse singularity analysis, 296–297 S velocity analysis, 290–292 SCARA robot. See Selective-compliance tyre test machine, 282 assembly robot arm (SCARA) robot Study, Christian Hugo Eduard, 5, 6, 13 Schönflies motion generator robot, 22 Sylvester dialytic method, 192, 199, 222 Screw Systems in Spatial Kinematics (Hunt), 6 Selective-compliance assembly robot arm (SCARA) robot, 22, 94, 325–326 T Serial manipulators, 19, 20, 25, 65, 94, 107, The Screw Calculus and Its Applications in 108, 158, 178, 189, 206 Mechanics (Dimentberg), 5 Singularity analysis Three-dimensional platform, 258, 260, combined singularity, 297 265–266, 269–273 Delta robot, 312 Triple vectorial product, 39, 41, 43, 48, 55, 56, direct singularity, 295–296 320 inverse singularity, 296–297 Twist, screw. See Velocity state parallel wrist, 215–216 Two-degree-of-freedom parallel manipulator Stewart platform, 295–297 complementary jerk matrix, 154 Six-degree-of-freedom (DOF), 189, 233. See first-order driver matrix, 152 also Six-legged parallel manipulator input-output equation of velocity, 152 Index 377

Plücker coordinates, 151 linear velocity vector, 66, 69 second-order driver matrix, 153 multibody system, 74–77 third-order driver matrix, 154 parallel wrist, 210–213 Two-degree-of-freedom parallel wrist (2-DOF) right-hand orthonormal basis, description, 206 70–73 finite kinematics 3-RRPS manipulator, 227–228 forward displacement analysis, 207–209 3RRRS+3RRPS parallel manipulator, inverse displacement analysis, 209 243–244 infinitesimal kinematics 3R2T parallel manipulator, 193–196 acceleration analysis, 213 SCARA, 94, 325–326 first-order coefficient matrix, 212 screw form infinitesimal screws, 211 actuating mechanism for a telescoping input–output equation, 211–213 antenna on a spacecraft, 83–85 Jacobian matrix, 212 crank-slider linkage mechanism, 88–90, jerk analysis, 214 94, 327 screws of, 211 four-bar linkage mechanism, 85–88 singularity analysis, 215–216 Geneva wheel/Maltese cross geometry velocity analysis, 210–211 scheme, 90–93 Two-degree-of-freedom serial manipulator, helical pair, 80–82 107–110 serial kinematic chain, 82–83 singularity analysis, 94, 326 six-bar dwell mechanism, 95, 330 V six-dimensional vector, 67, 68, 79–80 Vector and Tensor Analysis (Brand), 5 six-legged parallel manipulator, 277 Velocity analysis stewart platform, 290–292 actuating mechanism of telescoping time derivatives, 70, 71, 73–75, 93, 94, antenna on spacecraft, 78–80 324–325, 328 angular velocity vector, 93, 324, 325 vector, defined, 65 application of, 65 Velocity state Chasles’ theorem, 37, 68 angular velocity vector, 32, 33 coupler platform, 94, 327 associated screw of rigid body motion, decomposition of, 164, 175 37–41 Delta robot, 311–312 as equivalence relationship free vector, 73, 75 equivalence class, 37 helical pair, 69–70 reflexive, 35–36 helicoidal vector fields, 67 symmetric, 36 high-torque mechanism, 95–96, 331 transitive, 36 input–output equation, 87–89, 94, 95, linear velocity vector, 32, 36 328–330 six-dimensional vector, 32 ISA, 68–69 von Mises, Richard, 14