Introduction to Robotics: Mechanics and Control [-]

umnodeled flexibility, index 393 281—282 Lumped models, 282—283 Lyapunov stability analysis, 303—307 trajectory-following control, 275 Lyapunov's method, 290 Linear function with parabolic Lyapunov's second (direct) method, blends, 210—212 304 for a path with via points, Manipulability measure, 241 212—216 Manipulator control, problem of, Linear position control, 11—12 294—295 Linear velocity, 138—139 Manipulator kinematics, 62—100 simultaneous rotational velocity, inverse, 101—134 link transformations: 140—141 concatenating, 76 derivation of, 73—76 Linear-control systems, 262 \"standard\" frames, 89—91 Linearizing and decoupling control Manipulator subspace, 107—109 law, 295 Manipulator-mechanism design, Linearizing control law, 291 Link length, 64 230—261 Link offset, 66 Link parameters, 67 actuation schemes, 244—247 actuator location, 244—245 of a three-link planar reduction/transmission manipulator, 71 systems, 245 —247 Link transformations: articulated manipulator, 235 concatenating, 76 basing design on task derivation of, 73—76 requirements, 231—233 Link twist, 64—65 accuracy, 233 Link-connection description, 65—67 degrees of freedom, number first and last links in the chain, of, 231—232 66—67 load capacity, 233 repeatability, 233 intermediate links in the chain, speed, 233 workspace, 233 66 Cartesian manipulator, 234—235 closed-loop structures, 242—244 link parameters, 67 cylindrical configuration, Link-frame assignment, 72 Links, 5 236—237 force sensing, 253—254 convention for affixing frames kinematic configuration, to, 67—73 234—239 first and last links in the chain, position sensing, 252 68 redundant structures, 241—242 SCARA configuration, 235—236 intermediate links in the spherical configuration, 236 chain, 68 stiffness/deflections, 246—252 link parameter summary, 69 actuators, 250—252 link-frame attachment belts, 249 flexible elements in parallel procedure, 69 and stiffness, 249—250 and in series, 247 Load capacity, 233 Local linearization, 291 Locally degenerate mechanism, 9 Lower pair, 62—63 Low-pass ifiter, 279—280

394 Index principal axes, 169 principal moments of inertia, 169 gears, 248—249 Mass matrix, 177 links, 249—250 Mass moments of inertia, 168 shafts, 247—248 Mass products of inertia, 168—169 Mechanical impedance, 332 well-conditioned workspaces, Mechanical manipulators, See 241 Manipulators Memorization scheme, 192 workspace attributes, Micromanipulators, 242 quantitative measures of, Model-based portion, 273 Moment of inertia, 167 239—241 Motion specification, 345—346 Motor torque constant, 278 workspace generation, efficiency Motor-armature inductance, 279—280 of design in terms of, 240 Mouse, 356 Moving linearization, 291 wrist configuration, 237—239 Multi-input, multi-output (MIMO) Manipulators, 3 control systems, 264, 295 accuracy of, 127 Multiprocess simulation, 358 control problems for, 295—296 design, 10—11 Natural constraints, 319—321 dynamics, 9—10, 165—200 Natural frequency, 268 force control, 13 Newton's equation, 171—172 forward kinematics of, 4—6 Noise, 276 inverse kinematics of, 6—7 Nonautonomous system, 305 kinematics, 62—100 Nonlinear control algorithms, 12—13 linear position control, 11—12 Nonlinear control of manipulators, mechanics and control of, 4—15 nonlinear position control, 290—313 12—13 adaptive control, 311—312 Cartesian-based control systems, off-line programming and simulation, 15 307—311 position and orientation, 4 Cartesian decoupling scheme, programming robots, 13—15 repeatability, 127 310—311 sensors, 10—12 defined, 308 singularities, 7—9 intuitive schemes of Cartesian static forces, 79 control, 309—310 joint-based schemes compared static forces in, 153—156 trajectory generation, 9—10 to, 307—309 velocities, 79 current industrial-robot control workspace, 102 systems, 301—303 Mappings, 7, 24—29 Lyapunov stability analysis, involving general frames, 27—29 303—307 involving rotated frames, 25—27 involving translated frames, manipulators, control problems 24—25 for, 295—296 Mass distribution, 167—171 multi-input, multi-output inertia tensor, 167, 171 (MIMO) control systems, mass moments of inertia, 168 mass products of inertia, 295 168—169 parallel-axis theorem, 170

nonlinear systems, 291 —294 Index 395 practical considerations, language translation to target 296—301 system, 359 dual-rate computed-torque multiprocess simulation, 358 implementation, 298—299 path-planning emulation, 358 Pilot simulator, 360—367 feedforward nonlinear sensors, simulation of, 359 control, 297—298 3-D modeling, 356—357 user interface, 355—356 parameters, lack of knowledge workcell calibration, 359—360 Off-line programming system, 15 of, 299—301 Operating point, 291 Open-loop scheme, 263—264 time required to compute the Operational point, 14 Operational space, 6fn, 76 model, 296—297 Operators, 30—34 time-varying systems, 291 —294 rotational, 31—32 Nonlinear position control, 12—13 transformation, 33—34 translational, 30—31 Nonproper orthonormal matrices, 40 Orientation: Nonrigid body effects, 188—189 angle-set conventions, 46 description of, 20—22 Coulomb friction, 188—189 equivalent angle—axis Coulomb-friction constant, 188 viscous friction, 188—189 representation, 46—50 Notation, 16 Euler parameters, 50—51 Denavit—Hartenberg notation, notation for, 135—138 predefined, 51 67 proper orthonormal matrices, 40 taught, 51 for orientation, 135—138 X—Y—Z fixed angles, 41—43 for time-varying positions, Z—Y—X Euler angles, 43—45 Z—Y—Z Euler angles, 45—46 135—138 Orienting structure, 234 Orthogonal intersecting shafts, 245 vector, 16 Orthonormal matrix, property of the Numerical differentiation, 252 Numerical solutions, 106 derivative of, 141 Numerically controlled (NC) milling Overdamped system, 266 Overload protection, 253 machines, 3 Parallel shafts, 245 Off-line programming (OLP) Parallel-axis theorem, 170 systems, 353—371 Parallelism, 358 Parts-mating tasks, 318 automating subtasks in, 367—369 Pascal, 341 automatic assessment of Passive compliance, 333 errors and tolerances, 369 Path generation at run time, 222—224 automatic planning of coordinated motion, 368 Cartesian-space paths, automatic robot placement, generation of, 223—224 367—368 automatic scheduling, 368 coffision avoidance and path optimization, 368 force-control simulation, 368 central issues in, 355—360 defined, 353 dynamic emulation, 358 kinematic emulation, 357—358

396 Index tray conveyors, geometric algorithms for, 365 joint-space paths, generation of, 2-D vision systems, 366—367 222—223 Pitch, 41 Pneumatic cylinders, 251 Path generator, 216 Points: Path planning, 224—225 computed, 127 coffision-free, 225 operating, 291 Path points, 202 operational, 14 Path-planning emulation, 358 path, 202 Path-update rate, 201 pseudo via, 216—217 Pick and place locations, 233 taught, 127 Pick and place operations, 318 TCP (Tool Center Point), 14 PID control law, 277, 284—285 through, 216 Pilot simulator, 360—367 via, 10, 14, 202, 205—209, adjusting probabilities as a 212—216 function of drop height, wrist, 234 362—363 Poles, 265 Polynomials: alignment of the part during grasp, computation of, and closed-form solutions, 114 cubic, 203—205 364—365 higher-order, 209—210 Position constraints, 320—321 bins, 363 Position control system, 11 bounce, simulation of, 363 Position sensing, 252 computing which part to grasp, Position vector, 20 Position vectors, differentiation of, 364 136—137 connecting tray conveyors/sources and Position-control law, 13 Positioning structure, 234 sinks, 365—366 Position-regulation system, 271—272 Positive definite matrix, 182 default grasp location, Potentiometers, 252 computation of, 364 Predefined orientations, 51 Press-fit joints, and hysteresis, 254 flndspace algorithm, 364 Principal axes, 169 inspector sensors, 367 Principal moments of inertia, 169 part grasping, geometric Prismatic joints, 5, 63 Programming environment, 346 algorithms for, 364 Programming paradigm, 360 part pushing, geometric Programming robots, 13—15 Proper orthonormal matrices, 40 algorithms for, 365 Proprioceptive sensors, 230 part tumbling, geometric Pseudo via points, 216—217 PUMA 560 (Unimation), 235, 284, algorithms for, 361—362 physical modeling and 357 defined, 83—84 interactive systems, 361 proximity sensors, 366 pushbar, 365 pushing of trays, 366 sensors, geometric algorithms for, 366 stable-state estimator algorithm, 363 stable-state probabilities, 362 stacking/tangling, simulation of, 363—364

inverse manipulator kinematics, Index 397 117—121 internal world model vs. external reality, 347—348 kinematics of, 77—83 link parameters, 80 requirements of, 344—347 solutions, 104—105 flow of execution, 346 motion specification, 345—346 Quadratic form, 182 programming environment, 346 RAPID (ABB Robotics), 341 sensor integration, 347 RCC (remote center compliance), world modeling, 344—345 333 robot library for a new general-purpose language, Reachable workspace, 102 Real and equal roots, 267, 269—271 341 Real and unequal roots, 266—267 Redundancies, 241—242 robot library for an existing Redundant degree of freedom, computer language, 341 231—232 sample application, 342—344 specialized manipulation Reference inputs, tracking, 278 Remote center compliance (RCC), languages, 341 task-level programming 333 languages, 342, 354 Repeatability, 127, 233 Robotic manipulation, 19 Repeated roots, 269 Robots: Resolvers, 252 Resonances, 246, 247 gantry, 234 motion of the robot links, 144 structural, 278 programming, 13—15 unmodeled, 281 tool, positionlorientation of, 91 Resonant frequency, estimating, Robust controller, 298 Rodriques's formula, 58, 373 282—283 Roll, 41 Roller chains, 246 Revolute joints, 5, 63 Rotary optical encoder, 252 Rigid-body dynamics, form of, 295 Rotated frames, mappings involving, Robot: 25—27 specialized, 11 universal, 11 Rotation matrix, 21 Robot programming: Rotational operators, 31—32 levels of, 340—342 Rotational velocity, 139—140 teach by showing method, 340 Robot programming languages simultaneous linear velocity, (RPLs), 13—15, 339—350, 140—141 354 categories of, 341 Rotor, 278 defined, 342 RRR mechanism, 69—71 description of paths with, 224 Run time: explicit progranmiing languages, defined, 222 341 —342 path generation at, 222—224 problems peculiar to, 347—350 Sampling rate, 297 context sensitivity, 348—349 SCARA configuration, 235—236 error recovery, 349—350 Second-order linear systems, 264—271

398 Index method of solution, 105—106 multiple solutions, 103—105 characteristic equation, 265 SOLVE function, 126 complex roots, 266, 267—269 Spatial constraints on motion, 202 control of, 271—273 Spatial descriptions, 19—23 initial conditions, 265 defined, 19 Laplace transforms, 265 of a frame, 22—23 poles, 265 of an orientation, 20—22 real and equal roots, 267, of a position, 20 Specialized robot, 11 269—271 Speed, 233 Speed-reduction system, 245 real and unequal roots, 266—267 Spherical configuration, 236 Semiconductor strain gauges, 254 Spline, 10 Sensor integration, 347 Spot welding, 318 Sensors, 10—12 Spray painting, 318 Stable system, 264 proprioceptive, 230 Standard frames, 89—91 simulation of, 359 base frame (B), 89, 125 wrist, 253 goal frame (G), 91, 125 Servo error, 264, 331 location of, 125 Servo portion, 273, 292 station frame (S), 90, 125 Servo rate, 277 tool frame (T), 90, 125—126 Set-point, 285 use in a general robot system, Shafts, 247—248 125—126 Similarity transform, 57 Simple applications, 318 wrist frame (W), 90 Simulation, 9 State-space equation, 180—181 Simulation specific code, 360 Single joint, modeling/controlling, centrifugal force, 181 Coriolis force, 181 278—284 Static forces, 153—156 Cartesian transformation of effective inertia, 280 estimating resonant frequency, velocities and, 157—159 Station frame (5), 90, 125 282—283 Stator, 278 Steady-state analysis, 276 motor-armature inductance, Steady-state error, 276 Stewart mechanism, 243—244 279—280 Stiffness: unmodeled flexibility, 281 —282 actuators, 250—252 Single-input, single-output (SISO) belts, 249 control systems, 264 flexible elements in parallel and Singularities of the mechanism in series, 247 (singularities), 9, 151—153 gears, 248—249 workspace-boundary links, 249—250 singularities, 151 shafts, 247—248 workspace-interior singularities, Strain gauges, 253 Structural length index, 240 152 Sink records, 365—366 Skew shafts, 245 Skew-symmetric matrices, 142 Softening position gains, compliance through, 333—334 Solvability, 101—106 existence of solutions, 102—103

Structural resonances, 278 Index 399 Subspace, 107 Sum-of-angle formulas, 82 path description and generation, Tachometers, 252 201—203 Tangle factor, 363 Task space, 6fn path generation at run time, Task-level programming languages, 222—224 342 Cartesian-space paths, Task-oriented space, 76 generation of, 223—224 Taught orientations, 51 Taught point, 127 joint-space paths, generation TCP (Tool Center Point), 14 Teach and playback manipulators, of, 222—223 path planning, 224—225 127 collision-free, 225 Teach pendant, 285, 340 robot programming languages, Temporal attributes of motion, 202 Three roll wrist, 238 224 3-D modeling, 356—357 Through points, 216 Trajectory-conversion process, Time-varying positions, notation for, 307—309 135—138 Trajectory-following control, defined, Tool frame, 5 Tool frame (T), 90, 125—126 275 Tool, position/orientation of, 91 Torque ripple, 279 Trajectory-following control system, Tracking reference inputs, 278 Trailing subscripts/superscripts, in 272 Transducers, flexibility in, 254 notation, 16 Transform equations, 37—39 Trajectory, defined, 201 Trajectory generation, 10, 201—239 Transform mapping, 34 Transform operator, 34 Cartesian paths, geometric Transformation: problems with, 219—222 of free vectors, 51—52 Cartesian-space schemes, order of, 53 Transformation arithmetic, 34—37 216—219 compound transformations, Cartesian straight-line motion, 34—35 217—219 inverting a transform, 35—37 Transformation operators, 33—34 joint-space schemes, 203—216 Translated frames, mappings cubic polynomials, 203—205 cubic polynomials for a path involving, 24—25 with via points, 205—209 Translational mapping, 24—25 higher-order polynomials, Translational operators, 30—31 Transmission system, 245 209—210 Transpose-Jacobian controller, linear function with parabolic blends, 210—212 309 Trigonometric identities, 372—373 linear function with parabolic Types, 344 blends for a path with via points, 212—216 Underdamped system, 266 Unit quaternion, 50 Universal robot, 11 Universe coordinate system, 19 Unmodeled flexibility, 281—282 Unmodeled resonances, 281 Unstable performance, 264

400 Index Work envelope, 233 Work volume, 233 UPDATE simulation routine, Workcell, 339, 344 287 calibration, 359—360 User interface, 355—356 Workspace, 7, 102—103, 233 VAL language, 285, 341, 345 generation of, efficiency of design in terms of, 240 Vane actuators, 250—251 Vector cross-product, 142 and tool-frame transformation, Vector notation, 16 Vectors: 103 actuator, 77 Workspace attributes, quantitative angular, 137—138 measures of, 239—241 position, differentiation of, Workspace-boundary singularities, 136—137 151 Velocities, Cartesian transformation Workspace-interior singularities, 152 of, 157—159 World modeling, 344—345 Velocity: Wrist configuration, 237—239 Wrist frame (W), 90 angular, 141—144 Wrist point, 234 linear, 138—139 Wrist sensors, 253 Wrist-partitioned class of of a point due to rotating reference frame, 141—142 mechanisms, 234 rotational, 139—140 X—Y—Z fixed angles, 41—43 Velocity transformation, 158—159 Via points, 10, 14, 202 Yasukawa Motoman L-3, 235, 245 defined, 83 cubic polynomials for a path inverse manipulator kinematics, with, 205—209 121—125 linear function with parabolic kinematics of, 83—89 blends for a path with via link frames, assignment of, 87 points, 212—216 link parameters of, 88 Yaw angles, 41 Virtual work, 156 Viscous friction, 188—189 Z—Y—X Euler angles, 43—45 Z—Y—Z Euler angles, 45—46 Welded joints, and hysteresis, 254 Well-conditioned workspaces, designing, 241 WHERE function, 91