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عدد المساهمات : 19001 التقييم : 35505 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
| موضوع: كتاب Robotics and Control - Fundamental Algorithms in MATLAB الثلاثاء 15 فبراير 2022, 2:27 am | |
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أخواني في الله أحضرت لكم كتاب Robotics and Control - Fundamental Algorithms in MATLAB Peter Corke
و المحتوى كما يلي :
Contents 1 Introduction 1 1.1 Robots, Jobs and Ethics . 7 1.2 About the Book 8 1.2.1 MATLAB Software and the Toolboxes . 8 1.2.2 Notation, Conventions and Organization 9 1.2.3 Audience and Prerequisites . 10 1.2.4 Learning with the Book 11 1.2.5 Teaching with the Book 11 1.2.6 Outline 11 Further Reading 12 Part I Foundations . 13 2 Representing Position and Orientation 15 2.1 Working in Two Dimensions (2D) . 20 2.1.1 Orientation in 2-Dimensions 21 2.1.2 Pose in 2-Dimensions 24 2.2 Working in Three Dimensions (3D) . 29 2.2.1 Orientation in 3-Dimensions 30 2.2.2 Pose in 3-Dimensions 44 2.3 Advanced Topics . 47 2.3.1 Normalization 47 2.3.2 Understanding the Exponential Mapping . 48 2.3.3 More About Twists 50 2.3.4 Dual Quaternions . 53 2.3.5 Configuration Space . 53 2.4 Using the Toolbox 54 2.5 Wrapping Up 56 Further Reading 58 Exercises 59 3 Time and Motion . 61 3.1 Time-Varying Pose 61 3.1.1 Derivative of Pose . 61 3.1.2 Transforming Spatial Velocities . 62 3.1.3 Incremental Rotation 64 3.1.4 Incremental Rigid-Body Motion . 65 3.2 Accelerating Bodies and Reference Frames . 66 3.2.1 Dynamics of Moving Bodies 66 3.2.2 Transforming Forces and Torques . 67 3.2.3 Inertial Reference Frame . 67 3.3 Creating Time-Varying Pose 68 3.3.1 Smooth One-Dimensional Trajectories . 68 xviiContents 3.3.2 Multi-Dimensional Trajectories . 71 3.3.3 Multi-Segment Trajectories . 72 3.3.4 Interpolation of Orientation in 3D . 73 3.3.5 Cartesian Motion in 3D 75 3.4 Application: Inertial Navigation . 77 3.4.1 Gyroscopes . 77 3.4.2 Accelerometers . 79 3.4.3 Magnetometers . 83 3.4.4 Sensor Fusion 85 3.5 Wrapping Up 88 Further Reading 88 Exercises 89 Part II Mobile Robots 91 4 Mobile Robot Vehicles 97 4.1 Wheeled Mobile Robots . 97 4.1.1 Car-Like Mobile Robots 97 4.1.2 Differentially-Steered Vehicle 107 4.1.3 Omnidirectional Vehicle 110 4.2 Flying Robots . 112 4.3 Advanced Topics 117 4.3.1 Nonholonomic and Under-Actuated Systems 117 4.4 Wrapping Up . 119 Further Reading . 120 Toolbox and MATLAB Notes 121 Exercises . 121 5 Navigation . 123 5.1 Reactive Navigation 124 5.1.1 Braitenberg Vehicles 124 5.1.2 Simple Automata . 126 5.2 Map-Based Planning . 128 5.2.1 Distance Transform 128 5.2.2 D* . 132 5.2.3 Introduction to Roadmap Methods . 134 5.2.4 Probabilistic Roadmap Method (PRM) 135 5.2.5 Lattice Planner . 138 5.2.6 Rapidly-Exploring Random Tree (RRT) . 142 5.3 Wrapping Up . 144 Further Reading . 145 Resources . 146 MATLAB Notes 146 Exercises . 146 6 Localization 149 6.1 Dead Reckoning . 153 6.1.1 Modeling the Vehicle . 153 6.1.2 Estimating Pose 155 6.2 Localizing with a Map 158 6.3 Creating a Map 163 6.4 Localization and Mapping . 165 6.5 Rao-Blackwellized SLAM . 167 6.6 Pose Graph SLAM . 168 xviiiContents 6.7 Sequential Monte-Carlo Localization . 173 6.8 Application: Scanning Laser Rangefinder 176 Laser Odometry . 177 Laser-Based Map Building 179 Laser-Based Localization 180 6.9 Wrapping Up . 180 Further Reading . 181 Toolbox and MATLAB Notes 183 Exercises . 183 Part III Arm-Type Robots 187 7 Robot Arm Kinematics . 191 7.1 Forward Kinematics 191 7.1.1 2-Dimensional (Planar) Robotic Arms . 192 7.1.2 3-Dimensional Robotic Arms 194 7.2 Inverse Kinematics . 203 7.2.1 2-Dimensional (Planar) Robotic Arms . 203 7.2.2 3-Dimensional Robotic Arms 205 7.3 Trajectories . 209 7.3.1 Joint-Space Motion . 209 7.3.2 Cartesian Motion . 212 7.3.3 Kinematics in Simulink . 212 7.3.4 Motion through a Singularity 213 7.3.5 Configuration Change 214 7.4 Advanced Topics 215 7.4.1 Joint Angle Offsets . 215 7.4.2 Determining Denavit-Hartenberg Parameters 215 7.4.3 Modified Denavit-Hartenberg Parameters . 216 7.5 Applications 218 7.5.1 Writing on a Surface 218 7.5.2 A Simple Walking Robot 219 7.6 Wrapping Up . 223 Further Reading . 224 MATLAB and Toolbox Notes 225 Exercises . 225 8 Manipulator Velocity . 227 8.1 Manipulator Jacobian 227 8.1.1 Jacobian in the World Coordinate Frame 227 8.1.2 Jacobian in the End-Effector Coordinate Frame . 230 8.1.3 Analytical Jacobian . 230 8.2 Jacobian Condition and Manipulability . 232 8.2.1 Jacobian Singularities . 232 8.2.2 Manipulability . 233 8.3 Resolved-Rate Motion Control . 235 8.3.1 Jacobian Singularity 238 8.4 Under- and Over-Actuated Manipulators 238 8.4.1 Jacobian for Under-Actuated Robot . 239 8.4.2 Jacobian for Over-Actuated Robot 240 8.5 Force Relationships 242 8.5.1 Transforming Wrenches to Joint Space 242 8.5.2 Force Ellipsoids 242 8.6 Inverse Kinematics: a General Numerical Approach . 243 8.6.1 Numerical Inverse Kinematics . 243 xixContents 8.7 Advanced Topics 245 8.7.1 Computing the Manipulator Jacobian Using Twists . 245 8.8 Wrapping Up . 245 Further Reading . 246 MATLAB and Toolbox Notes 246 Exercises . 246 9 Dynamics and Control . 249 9.1 Independent Joint Control 249 9.1.1 Actuators . 249 9.1.2 Friction . 250 9.1.3 Effect of the Link Mass 251 9.1.4 Gearbox 252 9.1.5 Modeling the Robot Joint 253 9.1.6 Velocity Control Loop 255 9.1.7 Position Control Loop 259 9.1.8 Independent Joint Control Summary 260 9.2 Rigid-Body Equations of Motion . 261 9.2.1 Gravity Term 262 9.2.2 Inertia Matrix 264 9.2.3 Coriolis Matrix . 265 9.2.4 Friction . 266 9.2.5 Effect of Payload . 266 9.2.6 Base Force 267 9.2.7 Dynamic Manipulability 267 9.3 Forward Dynamics . 269 9.4 Rigid-Body Dynamics Compensation . 270 9.4.1 Feedforward Control . 271 9.4.2 Computed Torque Control 272 9.4.3 Operational Space Control . 273 9.5 Applications 274 9.5.1 Series-Elastic Actuator (SEA) 274 9.6 Wrapping Up . 276 Further Reading . 276 Exercises . 278 Appendices . 281 A Installing the Toolboxes 283 B Linear Algebra Refresher . 287 C Geometry 295 D Lie Groups and Algebras . 307 E Linearization, Jacobians and Hessians 313 F Solving Systems of Equations . 317 G Gaussian Random Variables 327 H Kalman Filter . 331 I Graphs . 337 Bibliography 341 Index 347 Index of People 347 Index of Functions, Classes and Methods 348 General Index . 351 Index A Ackermann, Rudolph 99 Asimov, Issac 1 B Ball, Sir Robert 50, 53 Bayes, Reverend Thomas 155 Black, Harold 4 Bode, Henrik 4 Braitenberg, Valentino 124 Bryan, George 35 C . apek, Karel 1, 3 Cardano, Gerolamo 35 Chasles, Michel 51 Clifford, William 53 Cook, Captain James 150, 165 Coriolis, Gaspard-Gustave de 67 Coulomb, Charles-Augustin de 251 D Delaunay, Boris 135 Denavit, Jacques 196 Descartes, René 17 Devol, George C. Jr. 1, 2 Draper, Charles Stark (Doc) 79, 80, 156 E Einstein, Albert 68 Engelberger, Joseph F. 2 Euclid of Alexandria 16 Euler, Leonhard 34, 66, 263 G Gauss, Carl Friedrich 59 Goetz, Raymond 7 H Hall, Edwin 83 Hamilton, Sir William Rowan 42, 53, 58, 59 Harrison, John 150 Hartenberg, Richard 196 Hershey, Allen V. 218 Hesse, Ludwig Otto 314 I Ilon, Bengt 110 J Jacobi, Carl Gustav Jacob 230 K Kálmán, Rudolf 155 L Lagrange, Joseph-Louis 263 Lazzarini, Mario 172 Leclerc, Georges-Louis 172 Lie, Sophus 307 M Markov, Andrey 135 McCarthy, John 4 McCulloch, Warren 4 Metropolis, Nicholas 172 Minsky, Marvin 4 Moler, Cleve 9 N Newell, Allen 4 Newton, Sir Isaac 65–67, 277 Nyquist, Harold 4 P Pitts, Walter 4 Plücker, Julius 297 Price, Richard 155 R Rodrigues, Olinde 40, 59 S Scheinman, Victor 193 Schmidt, Stanley F. 156 Shannon, Claude 4 Simon, Herbert 4 T Tait, Peter 35, 59 Tesla, Nikola 5 Turing, Alan 4 U Ulam, Stanislaw 172 V von Kármán, Theodore 196 von Neumann, John 172 Voronoy, Georgy Feodosevich 135 W Walter, William Grey 4, 124 Wiener, Norbert 4 Index of Functions, Classes and Methods Classes are shown in bold, Simulink models in italics, and methods are prefixed by a dot. All others are Toolbox functions. A about 75, 116, 143, 202, 211, 262, 316 angdiff 101 angles 46 angvec2r 40 angvec2tr 59 animate 74, 75 AprilTag 162 apriltags 162, 182 atan2 301 B Bicycle 98, 99, 107, 109 Bicycle 161, 164, 166, 174, 183 BinaryOccupancyGrid 146 bug 127 bug.plot 127 bug2 127 –, .path 127, 129 BundleAdjust 337 C ccode 316 ctraj 8, 76, 212, 213 D delta2tr 65 DHFactor 216, 220 –, .dh.command 220 dim 161 Dstar 132 –, .costmap 132 –, .modify_cost 133 –, .niter 133 –, .plan 132, 133 –, .query 133 DXform 129 –, .plan 129, 130 –, .query 129 –, .visualize 129 –, .visualize3d 130 E e2h 27 eig 39 EKF 156, 158, 161, 164–166 eps 48 ETS2 192, 225 –, .fkine 192 –, .plot 193 –, .Rz 192, 194 –, .structure 193, 194 –, .teach 192 –, .Tx 192, 194 ETS3 225 –, .fkine 194 –, .Ry 194 –, .Rz 194 –, .Tx 194 –, .Ty 194 –, .Tz 194 eul2jac 231 eul2r 34, 35 eul2tr 46 eval 216, 220 ExampleHelperRobot Simulator 121 exp 24, 41 expm 24, 41, 45, 50, 52, 59 eye 27, 216 F fcode 316 fkine 212 fminsearch 205349 G gait 223 gaussfunc 327, 328 GeometricJacobian 246 Graph 212 H h2e 27 I icp 178 InitFcn 273 interp 75, 76 InverseKinematics 225 ithin 134 J jacobian 170, 228, 316 Jacobian 235 jsingu 232 jtraj 202, 210, 212, 214, 261, 272, 273 L LandmarkMap 158, 161 Lattice 139, 337 Link 196, 198, 215, 254, 266 –, .A 197 –, .a 197 –, .offset 197 –, .RP 197 log 23, 40 logm 23, 40, 52 lscov 246 lspb 70, 71, 76, 210 M makemap 129, 146 matlabFunction 316 mdl_puma560 199, 205, 261, 263, 272 mdl_quadrotor 116 mdl_twolink 251 meshgrid 263, 264 model 101 models 198 MonteCarloLocalization 183 mplot 236 mstraj 218, 222 mtraj 71, 74, 210 N Navigation 128 null 318 numcols 79, 87, 88, 137, 218, 263, 264 numrows 129, 218 O oa2r 38 P ParticleFilter 175, 183 pcregrigid 181 PGraph 337 –, .add_edge 337, 338 –, .add_node 337 –, .closest 338 –, .cost 338 –, .edges 338 –, .neighbours 338 –, .plot 338 pinv 238, 239, 241 ploop 259, 278 ploop_test 259 plot 177 plot_homline 304 plot_point 26 Plucker 52, 304 –, .L 304 –, .side 297 pol2cart 177 Polygon 147 PoseGraph 170, 171, 177, 337 –, .optimize 171 –, .plot 170, 171 –, .plotoccgrid 179 –, .scan 177 –, .scanmap 179 –, .scanxy 177 –, .time 178 PRM 136, 146, 337 –, .plan 136 –, .query 137 –, .visualize 137 Q q.animate 74 qplot 211 Quaternion 42 R rand 137 randinit 136, 137 randn 137, 327 RandomPath 155, 164, 174 RangeBearingSensor 159, 160, 162, 164–166, 175, 184 –, .h 160 –, .H_w 160 –, .H_x 160 –, .reading 159 Revolute 196 RevoluteMDH 217 RigidBodyTree 225, 246 RNE 272, 278 roblocks 101, 109 Robot 269 Index of Functions, Classes and Methods350 Index rotx 32, 33, 40, 41, 64 roty 32–34 rotz 32, 34 rpy2r 39, 60 rpy2tr 43 RRT 142, 337 –, .path 143 –, .plan 142 –, .visualize 142 running 101 S SE2 55, 170, 199 –, .Rx 209 SE3 55, 201, 202, 206, 208, 209, 212–214, 219, 222, 231 –, .Rx 214 –, .Ry 213 –, .Rz 222 –, .torotvec 231 Sensor 159, 164 –, .H_xf 164 sensorfield 125, 146 SerialLink 198, 204, 210, 222, 225, 247, 263 –, .accel 269 –, .base 201, 202, 263 –, .coriolis 261 –, .edit 198 –, .fdyn 270 –, .fellipse 243 –, .fkine 198, 201, 202, 205, 207, 211, 228, 236 –, .gravity 263 –, .gravload 262, 263, 267 –, .ikcon 225 –, .ikine 206, 208, 213, 214, 222, 225, 244 –, .ikine6s 205–209, 212–214, 219, 225 –, .ikinesym 204 –, .inertia 261, 264, 266, 268 –, .jacob0 229–234, 242, 268 –, .jacobn 230 –, .jtraj 210 –, .links 267 –, .maniplty 213, 234, 269 –, .motordynamics 254 –, .nofriction 270 –, .plot 201, 207, 211, 214, 219, 225, 269 –, .plot3d 225 –, .rne 261, 262, 267 –, .teach 225, 233, 243, 246 –, .tool 201 –, .vellipse 233 shortest 74 sigma 327 simplify 23, 60, 170 skew 309 sl_bicycle 99 sl_braitenberg 124, 125 sl_ctorque 271, 272 sl_driveline 102, 103 sl_drivepoint 101, 102 sl_drivepose 106, 107 sl_fforward 271, 272 sl_jspace 212, 213 sl_lanechange 99, 100 sl_opspace 273 sl_pursuit 104 sl_quadcopter 116 sl_quadrotor 115, 116 sl_rrmc 235, 236 sl_rrmc2 236, 237 sl_sea 275 sl_ztorque 269 SO2 55, 72 SO3 55, 74 spy 325 sqrt 157, 268 sqrtm 302 T T1.torpy 71 t2r 45 T2xyz 212 tags 162 tau-d 257 tau_ff 258 torpy 75 tpoly 69, 70, 76, 89, 90, 210 tr2angvec 39 tr2delta 65 tr2eul 34, 35 tr2rotvec 231 tr2rpy 36 Tracking Controller 109 traj 218 tranimate 32, 33, 59, 60 transl 8, 45, 201, 202, 206, 208, 211, 213, 221, 222, 236 transl2 25, 26 trexp 41, 50, 52, 59 trinterp 76 tripleangle 36, 60 triplepoint 134 trlog 41, 52, 231 trot2 25 trotx 45, 46, 202, 220, 263 troty 213 trotz 220 trplot 25, 33, 45, 59 trplot2 26, 59 Ts –, .t 212 –, .torpy 212 Twist 28, 52, 199 –, .expm 52 –, .line 52 –, .S 52 –, .T 28, 52, 199 U Unicycle 109, 121, 183 UnitQuaternion 43, 44, 48, 66, 74, 79 –, .animate 79 –, .dot 62 –, .dotb 62 –, .omega 79 –, .plot 66 –, .torpy 79351 V Vehicle 154, 156, 158, 183 –, .Fv 156 –, .Fx 156 –, .step 155 vex 23, 24, 40, 41, 309 vloop 255, 278 vloop_test 256 VREP_class 185 X, Y, Z xv 316 General Index General Index Symbols \-operator 69, 317, 319 A A* search 132, 137, 140, 339 absorption shock 275 acceleration 79, 80, 85, 118, 249, 273 –, angular 66 –, centripetal 68 –, Coriolis 68, 89 –, discontinuity 76 –, Euler 68 –, gravitational 68, 81 –, inertial 81 –, proper 81 –, sensor 81, 85 –, specific 81 accelerometer 37, 39, 79–81, 85 –, triaxial 81, 85 Ackermann steering 99 actuation 118 –, electric 254 –, electro-hydraulic 249 actuator 118, 249 –, joint 250 –, saturation 116 –, series-elastic (SEA) 274, 275 addition, vector 287 adjoint –, logarithm of 311 –, matrix 63, 67, 199, 245, 297, 311 adjugate 289 adjustment, bundle 182 AHRS (see attitude and heading reference system) aircraft 117, 119 algebra 307 algebraic group 308 algorithm –, Bresenham 179 –, bug 126–128 –, D* 132 –, FastSLAM (see also Rao-Blackwellized SLAM) 181 –, ICP (iterated closest point) 181 –, Levenberg-Marquardt 244, 320, 321 –, Newton-Euler 261 –, rapidly exploring 143 –, resolved-rate motion control 235 –, RRT (rapidly-exploring random tree) 143 –, skeletonization 134, 135 –, thinning 134, 135 –, velocity loop control 255 ampullae 81 anaglyph, stereo glasses 33 analysis, root-locus 278 analytical Jacobian 231 angle –, Cardan 30, 36 –, declination 83 –, elevation 150 –, Euler 34–36, 38, 57, 73, 194, 230, 231, 245 –, singularity 37 –, heading 85 –, inclination 83 –, joint 5, 12, 196 –, nautical 36 –, representation 34 –, roll-pitch-yaw 35, 36, 76, 210–212, 230 –, rate 74 –, singularity 36 –, rotation 23, 24, 29, 33, 35, 37, 41 –, steering 99, 100, 139, 143 –, Tait-Bryan 36 –, trajectory –, joint 270 –, LSPB (linear segment with parabolic blend) 70, 259, 260 –, XYZ sequence 36 angle-axis representation 39, 43 angular –, acceleration 66 –, momentum 66, 77, 78 –, rate 86 –, uncertainty 157 –, velocity 48, 50, 62, 66, 68, 77, 78, 153, 231, 332 anthropomorphic 145, 200, 201 anti-symmetric matrix 289 Apollo 13 36, 38 –, Lunar Module 37, 79 approach vector 38, 39, 208, 209 April tag 162 architecture, subsumption 125 ArduCopter (software project) 120 artificial intelligence 4 Asimo humanoid robot 6 ASV (see autonomous surface vehicle) attitude and heading reference system (AHRS) 85 automata 126 automated guided vehicle 94 autonomous surface vehicle (ASV) 94 autonomous underwater vehicle (AUV) 94, 118, 119 axis –, instantaneous 62 –, of motion 71 –, optical 38352 Index –, rotation 30, 37, 39, 41, 46, 48, 61, 66 –, Earth 83 –, screw 45, 50 B back –, EMF (electromotive force) 250, 258 –, end 168 ballbot 110 base –, force 267 –, transform 197 Baxter robot 209, 275 behavior-based robot 125 Beidou (satellite navigation system) 151 bi-quaternion (see dual quaternion) bias 86 bicycle model 98, 105, 142, 143 bifilar pendulum 277 blade flapping 113 blend 70 –, parabolic 70 body –, acceleration estimation 81 –, moving 66 body-fixed frame 37, 53, 68, 77, 113 Braitenberg vehicle 124 breaking, stiction 250 Bresenham algorithm 179 Buffon’s needle problem 172 bug algorithm 126 bundle adjustment 182 C C-space 54 calibration –, camera 9 –, sensor 86 camera 168 –, pose 173 car 117–119 Cardan angle sequence 34 Cartesian –, coordinate system 20 –, geometry 17 –, motion 75, 209, 212, 236 –, plane 17 –, point 177 –, trajectory 89, 212, 222 celestial navigation 150 center of mass 62, 66, 113, 251, 262 centripetal –, acceleration 68 –, force 262 Chasles theorem 50 chi-squared distribution 158, 329 Cholesky decomposition 290 circle 74 city block distance 128 closed-form solution 203 clothoid 99 CML (see concurrent mapping and localization) coefficient –, Coulomb 270 –, viscous friction 250, 270 column space 291 compass 39, 83, 106, 149, 151, 153, 162 compensation, gravity 116 computed torque control 272 concurrent mapping and localization (CML) 165 condition number (see matrix condition number) confidence test 162 configuration –, change 214, 215 –, kinematic 196, 206, 207, 213, 214, 236 –, of a system 53 –, space 53, 54, 112, 117, 119, 143, 196, 199, 208, 209 –, zero-angle 195 connected component, graph 137, 338 constraint –, nonholonomic 99, 109 –, rolling 119 control –, feedback 260 –, feedforward 116, 258, 260, 270, 271 –, flexible transmission 12 –, force 273 –, independent joint 249 –, integral –, action 257 –, windup 278 –, joint 249, 260 –, loop, nested 249 –, mobile robot 100–107 –, model-based 190 –, proportional 101, 102, 104, 255 –, derivative 114–116 –, integral 116, 258, 259 –, resolved-rate motion 235, 246 –, shared 7 –, space, operational space 273, 274 –, torque 270 –, computed 272 –, feedforward 271 –, traded 6 –, velocity 100, 255, 259 coordinate –, frame 15, 16, 20 –, 2-dimensional 17 –, 3-dimensional 17 –, end-effector 192 –, global 179 –, moving 66 –, right-handed 29 –, velocity 66 –, generalized 53, 98, 107, 111, 117, 118, 192, 261 –, homogeneous 302 –, joint 196, 216, 227, 261 –, Plücker 50, 52, 296 –, point 20, 24, 45, 49 –, random 337 –, system 17 –, vector 15–17, 287, 295, 302 Coriolis –, acceleration 68, 89 –, force 261, 262, 265, 273353 distortion, iron 85 distribution –, chi-squared 329 –, von Mises 154 DOF (see degree of freedom) down hill 319 drag, aerodynamic 113 dual –, number 53 –, quaternion 53 Dubbins path 99 dynamics 249 –, error 272 –, forward 114, 116, 249, 269, 270 –, inverse 261, 271, 272 –, quadrotor 113, 114 –, rigid-body 261, 270 E Earth –, diameter 79 –, gravity 80 –, shape 79 –, surface 68, 77 east-north-up (ENU) 77 eccentricity 300 effect, Eötvös 89 effective inertia 254 EGNOS (satellite network) 151 eigenvalue 39, 158, 234, 268, 269, 290 eigenvector 39, 290 EISPACK project 9 EKF (see extendet Kalman filter and Kalman filter) EKF SLAM (see Kalman filter, extended, SLAM) elasticity, joint 274 ellipse 157, 299, 329 –, canonical 298, 299 –, confidence 165, 166 –, covariance 158, 164, 329 –, drawing 301 –, equation 329 –, error 158, 161 –, rotated 329 –, size 328 –, velocity 233, 242 ellipsoid 268, 299, 300 –, equation 329 –, force 242, 243 –, shape 234 –, surface 233, 243, 328 –, velocity 242 –, rotational 234 –, volume 234, 301 –, wrench 243 Elsie (robot) 93, 123 encoder 253, 254 end-effector 191 –, coordinate frame 230 –, force 242 –, inertia 273 –, torque 242 –, velocity 227, 228 ENU (see east-north-up) General Index correlation, covariance 152, 328, 334 correspondence, point 178 cost map 132 Coulomb friction 250, 251, 253 covariance –, correlation 152, 328, 334 –, ellipse 158, 164, 329 –, matrix 152, 154, 156, 158, 159, 161, 163, 165, 167, 168, 174, 328 –, extending 163 curvature 139 cybernetics 1, 4, 124, 145 D D* 132 d’Alembert force 67 damped inverse 238 data –, association 162 –, error 151, 162 –, laser scan 177 –, type 55, 56 dead reckoning 95, 149, 153 declination –, angle 83 –, magnetic 83 decomposition –, Cholesky 290 –, spectral 291 Deep Phreatic Thermal Explorer (DEPTHX, AUV) 118, 119 definition –, eigenvalue, eigenvector 39 –, frame 68 –, Mahalanobis distance 329 –, robot 5, 124, 128 degree of freedom (DOF) 37, 54, 71, 112, 118, 119, 189, 191, 193, 206, 208, 229, 232, 234, 238–240 Denavit-Hartenberg –, notation 194, 195, 215, 216, 219, 227 –, modified 216 –, parameter 195, 198, 225 DEPTHX (see Deep Phreatic Thermal Explorer) derivative –, orientation 62, 66, 116 –, pose 61, 62 –, quaternion 62 –, time 61 detector, SURF (speeded up robust feature) 250 determinant 47, 233, 238, 291 differential, kinematics 227 Dijkstra method 130 dimension 15 –, curved 15 direction 307–309 displacement –, rigid body 50, 51 –, spatial 65, 243 distance 162 –, Euclidean 16, 128, 338 –, Mahalanobis 162, 293, 329 –, Manhattan 128, 287 –, threshold 137 –, transform 128, 132, 133, 135354 Index Eötvös, effect 89 ephemeris 150 equation –, differential 49 –, Eulers rotation 66 –, line 295, 303 –, motion 99, 109, 269 –, Euler 114, 261 –, rigid-body 249, 261 –, nonhomogeneous 315 –, solving system 317 equivalence principle 68 error 47, 48, 167, 168 –, cumulative 168 –, edge 170 –, ellipse 157, 164, 166 –, ICP (iterated closest point) 180 –, position 249 –, vector 324 estimation 152 –, Monte-Carlo 155, 173, 181 –, pose 81 ethics 7 Euclidean –, coordinate 27, 302 –, distance 16, 128, 338 –, geometry 16, 17, 20, 295 –, group 19, 25, 44 –, length 287 –, line 295 –, plane 17, 303 –, point 27, 295, 303, 304 –, space 17, 53, 295, 303 Euler –, acceleration 68 –, angle 34–36, 38, 57, 73, 194, 230, 231, 245 –, singularity 37 –, force 68 –, motion equation 66, 114, 261 –, rotation theorem 30, 31, 33–35, 309 explicit complementary filter 86, 87 exponential –, coordinate 41, 231, 322 –, rate 231 –, mapping 48, 50 –, matrix 23, 24, 41, 49 –, product of 194, 198, 199 extended Kalman filter (EKF, see also Kalman filter) 86, 88, 155, 167, 315, 334 exteroceptive sensor 5, 168 F FastSLAM (see also SLAM and Rao-Blackwellized SLAM) 167 feature map 161, 166 feedback control 116, 258–260 feedforward control 116, 258, 260, 270, 271 fibre-optic gyroscope (FOG) 78fi ctitious force 67, 81 field –, magnetic, intensity 84, 85 –, robot 3, 94 file 170 –, MEX 284 filter –, complementary explicit 86, 87 –, Kalman 88, 89, 155, 160–162, 167, 173, 180, 182, 332 –, extended (EKF) 86, 88, 155, 167, 315, 334 –, unscented (UKF) 182 –, Kalman-Bucy 333 –, particle 167, 173–176 flow, current 83fl ux line, magnetic 83 FOG (see fibre-optic gyroscope) font, Hershey 218 force 50, 66, 242, 249 –, apparent 67 –, control 273 –, Coriolis 261, 262, 265, 273 –, d’Alembert 67 –, ellipsoid 242, 243 –, fictitious 67, 81 –, gyroscopic 273 –, inertial 67 –, pseudo 67 –, translational 67 form, homogeneous 27 formula, Rodrigues rotation 35, 40, 41, 50, 51, 59, 64, 309 forward –, dynamics 114, 269 –, kinematics 191, 192, 199, 202, 228 –, instantaneous 229 frame –, body-fixed 53, 68, 77 –, coordinate 15, 16, 20 –, reference 67 –, inertial 66, 67, 77, 81 –, noninertial 68 –, right-handed coordinate 29 –, world coordinate 16, 77 friction 249–251, 260, 261, 266 –, aerodynamic 113 –, Coulomb 250, 251, 253, 266, 270 –, stiction 250 –, viscous 244, 250, 251, 253, 266, 269, 270 front end 168 function –, Gaussian 327 –, Huber loss 321 –, observation 162 –, probability density (PDF) 151, 158, 159, 173, 327, 328 –, Gaussian 173 –, scalar 313 –, Tukey biweight 321 fusion, sensor 85, 86, 161 G gait pattern 223 Galileo (satellite navigation system) 151 gantry robot 189 Gaussian –, distribution 331, 332 –, function 327, 329 –, multivariate 328 –, noise 155, 158, 162, 332, 333 –, probability 158, 162, 329 –, random variable 327, 332, 334 gearbox 252–254355 generalized –, coordinate 53, 98, 107, 111, 117, 118, 192, 261 –, joint 196, 216, 261 –, forces 261 –, joint 242, 244, 261, 262, 264, 266, 267 –, matrix inverse 292 –, Voronoi diagram 134 generator matrix 308, 310 Genghis (robot) 145 geomagnet 83 geometric Jacobian 229 geometry –, algebraic 48 –, analytic 17 –, Cartesian 17 –, Euclidean 16, 17, 20, 295 gimbal 203 –, lock 36, 206, 213, 232 –, low-friction 78 Global Hawk unmanned aerial vehicle (UAV) 4, 112 Global Positioning System (GPS) 5, 115, 149, 151, 163 –, differential 151 –, multi-pathing 151 –, RTK 151 –, selective availability 151 GLONASS (satellite navigation system) 151 goal seeking 126 gradient, descent 319, 320 graph 134, 137, 337 –, A* search 132, 137, 140, 339 –, embedded 337 gravity 68, 82, 113, 249, 251 –, compensation 116 –, disturbance 258 –, load 249, 258, 261–263, 269 –, term 262 –, torque 252, 262 –, vector 82, 261 great circle 74 ground effect 113 group –, algebraic 308 –, Euclidean 19 –, Lie 307 –, orthogonal 22, 32, 290 gyroscope 36, 77, 85, 99, 153 –, fibre-optic (FOG) 78 –, ring-laser (RLG) 78 –, strapdown 78 –, triaxial 78 H Hall effect 83 –, sensor 83 hard-iron distortion 85 heading 83 –, angle 85 –, rate (see yaw rate) helicopter 119 Hershey font 218 Hessian 313, 314, 320 –, approximate 314, 320 –, matrix 314 histogram, cumulative 174 holonomic constraint 54 homogeneous –, equation 318 –, form 25 –, transformation 25, 44, 51, 52, 75, 197, 201, 303 –, normalization 48 homography 162 hovercraft 117–119 Huber loss function 321 humanoid robot 3, 6 hybrid, trajectory 70 hypersurface, quadric 304 I ICP (see iterated closest point) ICR (see instantaneous center of rotation) ideal –, line 303 –, point 303, 304 identity quaternion 43 image –, plane 303 –, processing 128, 134 IMU (see inertial measurement unit) inclination –, angle 83 –, magnetic 83, 84 incremental replanning 132 inertia 251–253 –, effective 254 –, end-effector 273 –, load 253 –, matrix 114, 264 –, motor 253 inertial –, force 67 –, measurement unit (IMU) 37, 85 –, navigation system (INS) 77, 85, 115 –, reference frame 66, 67, 77, 81 –, sensor 85 inflation, obstacle 130 innovation 87, 160, 168, 333 INS (see inertial navigation system) instantaneous center of rotation (ICR) 98, 107 integral –, dynamics 269 –, windup 258 intelligence, artificial 12 intensity –, light 123 –, magnetic field 83, 85 interpolation –, linear 73 –, orientation 73 –, quaternion 58, 74 –, rotational 74 –, scalar 210 –, unit-quaternion 74, 75 inverse –, dynamic control 272 –, dynamics 261, 271, 272 –, left-generalized 317 –, pseudo 238, 240, 292, 317 iterated closest point (ICP) 177, 180, 181 General Index356 Index J Jacobian, Jacobian matrix 213, 216, 227, 228, 245, 313, 315 –, analytical 230, 231 –, condition 232 –, damped inverse 238 –, end-effector coordinate frame 230 –, geometric 229 –, insertion 163, 165 –, manipulability 232, 233 –, manipulator 227, 229, 245, 261 –, matrix 156, 170, 190, 213, 227, 228 –, numerical approximation 315 –, over-actuated robot 240 –, singularity 232, 238 –, transpose 227, 243, 244 –, under-actuated robot 239 jerk 68 Johns Hopkins Beast (robot) 145 joint –, actuator 250 –, angle 5, 12, 196 –, control, independent 249 –, elasticity 274 –, position 273 –, prismatic 191, 193 –, revolute 191 –, sliding 191 –, space 196, 210, 242 –, velocity 227, 228 Joseph form 333 K Kalman filter 88, 89, 155, 160–162, 167, 173, 180, 182, 332 –, extended (EKF) 86, 88, 155, 167, 315, 334 –, SLAM (EKF SLAM) 167 –, gain 333 –, unscented (UKF) 182 kernel, density approach 181 kinematic –, configuration 196, 206, 207, 213, 214, 236 –, model 99, 105, 109, 112, 141, 143, 200 kinematics 191 –, differential 227 –, forward 191, 192, 199, 202, 228 –, instantaneous 229 –, symbolic 204, 228 –, inverse –, closed form 203 –, numerical 204, 207, 243 –, velocity 227 Klein quadric 304 L landmark 150, 162, 167, 180 –, identity 162 –, navigation 149 –, observation 159 laser –, odometry 177 –, rangefinder 176, 177, 179 –, noise 178 –, scanner 168 lateral motion 98 lattice planner 138 law –, Newton –, first 67 –, second 66, 68, 80, 113, 261, 277 –, of robotics 1 least squares problem 238, 239, 244, 317 –, nonlinear 169, 314, 320, 321 –, rotation matrix 318 Levenberg-Marquardt –, algorithm 244, 320, 321 –, optimization 244, 323 lever arm effect 251 Lie –, algebra 51, 52, 307–310 –, group 23, 48, 307, 307–310 light intensity 123 line 304 –, 2D 295 –, 3D 296 –, equation 295, 303 –, Euclidean 295 –, ideal 303 –, of no motion 98 –, Plücker 296–298 –, projection 304 linear segment with parabolic blend (LSPB) trajectory 70, 259, 260 linearization 313 –, general 313 link 250 –, effect 251 –, elasticity 274 –, mass 251, 262 LINPACK project 9 load 275 –, gravity 249, 258, 261–263, 269 –, inertia 253 localization 9, 149, 165, 179 –, CML (concurrent mapping and localization) 165 –, error 151 –, laser-based 180 –, Monte-Carlo 173 –, problem 151, 152 –, SLAM (simultaneous localization and mapping) 165, 167–169, 173 longitude problem 150 longitudinal motion 98 LORAN (radio-based localization system) 151 LORD MicroStrain 77 LSPB (see linear segment with parabolic blend) M magnetic –, declination 83 –, field 84, 85 –, flux 83 –, inclination 83, 84 –, north 83, 85 –, pole 83, 84 magnetometer 83, 85 Mahalanobis distance 162, 293, 329 Manhattan distance 128, 287 manifold 307–309357 manipulability 213, 232–235 –, dynamic 267, 269 manipulator (see also robot) 189 –, Jacobian 229, 242, 261 –, kinematics 227 –, over-actuated 54, 238, 240 –, serial-link, dynamics 249 –, under-actuated 54, 208, 238, 239 manoeuvre 118, 119 manufacturing robot 3 map 158, 162, 167 –, building, laser-based 179 –, feature 161, 166 –, obstacle 129 mapping 165 –, CML (concurrent mapping and localization) 165 –, exponential 48, 50 –, point 54 –, PTAM (parallel tracking and mapping) 173 –, SLAM (simultaneous localization and mapping) 165, 167–169, 173 Mars rover 4, 6 mass 66, 275 –, center of 62, 66, 113, 251, 262 –, distribution 66 –, link 251, 262 –, payload 266 –, proof 80 mathematical morphology 134 MATLAB –, code 9 –, command prompt 9 –, matrix xxv –, MEX-file 284 –, object 8 –, software 8 –, Toolbox conventions xxv matrix 288 –, adjoint 63, 67, 199, 245, 297, 311 –, adjugate 289 –, angular velocity 64 –, anti-symmetric 289 –, condition number 233, 293 –, covariance 152, 154, 156, 158, 159, 161, 163, 165, 167, 168, 174, 328 –, diagonal 159 –, extending 163 –, odometry 158 –, sensor 159 –, definite –, negative 314 –, positive 314, 322 –, diagonalization 291 –, estimation 9 –, exponential 23, 24, 41, 49 –, exponentiation 48 –, generator 308, 310 –, Hessian 314 –, homography 9 –, identity 64 –, indefinite 314 –, inertia 114, 264 –, inverse –, damped 238 –, pseudo 238–240 –, Jacobian 170, 227, 228 –, logarithm 23 –, MATLAB xxv –, normalization 47 –, orthogonal 22 –, orthonormal 32, 47 –, rank 232, 292 –, rotation 22, 33, 40, 48, 64 –, determinant 47 –, normalization 65 –, product 23 –, singular value decomposition 292, 318 –, skew-symmetric 23, 40, 41, 35, 41, 48, 49, 61, 64, 289, 309 –, augmented 310 –, sparse 324 –, transformation, homogeneous 50, 62 MAV (see micro air vehicle) maximum –, torque 257 –, velocity 70 measurement –, odometry 154 –, random 154 –, strapdown inertial 85 –, unit, inertial (IMU) 38, 85 mecanum wheel 110 MEMS (see micro-electro-mechanical system) method –, Newton’s 320 –, Newton-Raphson 319 –, roadmap 134 MEX-file 284 micro-electro-mechanical system (MEMS) 78 micro air vehicle (MAV) 112 Mikrokopter (software project) 120 minimization, nonlinear 319 minimum-norm solution 208, 213, 240 mobile robot 3, 93, 97 mobility 119 model –, bicycle 98, 105, 142, 143 –, kinematic 99, 105, 109, 112, 141, 143, 200 –, motion 97, 107, 110, 112, 113, 138, 142, 153, 269, 331, 332 –, nonlinear 86 –, process 331 –, quadrotor 113 –, screw 46 –, unicycle 105, 109 –, vehicle 105 model-based control 270 moment 50 –, image –, line 296 –, matrix 318 –, of inertia 66, 262 –, torque 66, 113, 114, 242, 267 –, vector 28, 45, 50, 296 momentum, angular 66, 77 Monte-Carlo –, estimation 155, 173, 181 –, localization 173 Moore-Penrose pseudo inverse 292 morphology (see mathematical morphology) General Index358 Index motion 61, 82 –, axis of 71 –, Cartesian 75, 209, 212, 236 –, complex 12 –, control, resolved-rate 232, 236, 237 –, discontinuity 76 –, end-effector 236 –, equation 66, 99, 109, 114, 249, 261, 269 –, inertial frame 82 –, joint-space 209, 214 –, lateral 110 –, longitudinal 98 –, model 97, 107, 110, 112, 113, 138, 142, 153, 269, 331, 332 –, multi-dimensional 71 –, null-space 12 –, omnidirectional 97, 110, 126, 138 –, planner 103 –, resolved-rate 12 –, rigid-body 25, 44, 45, 52, 307, 308 –, incremental 65 –, rotational 49, 50, 66 –, screw 45, 46 –, segment 72 –, sickness 81 –, singularity 213 –, straight-line 212 –, translational 28, 29, 49, 51, 66 motor 253, 254, 275 –, DC 249 –, high-torque 252 –, inertia 253 –, limit 257 –, servo 249 –, stepper 249 –, torque 250 multi-pathing 151 multi-segment trajectory 72 N nautical –, angle 36 –, chronometer 150 –, mile 149 navigation 95, 120, 123, 149 –, aerospace 42 –, algorithm 129 –, Beidou (satellite navigation system) 151 –, chart 151 –, dead reckoning 149 –, Galileo (satellite navigation system) 151 –, GLONASS (satellite navigation system) 151 –, GPS (Global Positioning System) 5, 115, 149, 151, 163 –, inertial 61, 64, 77, 85, 115 –, landmark 149 –, map-based 123 –, marine 165 –, radio 77 –, reactive 123, 124 –, satellite 5, 115, 149, 151, 163 –, spacecraft 36, 78 –, system 77, 85, 115 Navlab project 120 NED (see north-east-down) nested control loop 249 Newton’s –, first law 67 –, method 320 –, second law 66, 68, 80, 113, 261, 277 Newton-Euler method 261, 276, 277 Newton-Raphson method 319 node, graph 18, 137, 139, 142, 168, 337 noise 86, 154, 178 –, Gaussian 155, 158, 162, 332, 333 –, odometry 154, 156, 331 –, random 86, 154, 175 –, scanning laser rangefinder 178 –, sensor 160, 173 nonholonomy, nonholonomic 97 –, constraint 99, 109 –, system 119 normalization –, homogeneous transformation 48 –, rotation matrix 47 normal matrix 290 north –, magnetic 83, 85 –, true 83 north-east-down (NED) 77 null space of matrix 240, 292, 318 number –, denominate 15 –, dual 53 –, random 137, 172, 331 O observation 159 obstacle –, inflation 128 –, map 129 occupancy grid 126, 128, 129, 179 odometer 153 odometry 153, 154, 168 –, differential 153 –, laser 177 –, noise 154, 156, 331 –, wheel 153 omnidirectional –, motion 97, 110, 126, 138 –, vehicle 110 –, wheel 110 OmniSTAR satellite network 151 operational space 53 –, control 273, 274 operator 69 –, associative binary 19 –, asterisc 79 –, backslash 69, 317, 319 –, group 308 –, inverse 65 –, multiplication 52 –, spatial displacement 65 optical axis 38 optimization 171, 173, 180 –, algorithm 244 –, graph 173 –, Levenberg-Marquardt 244, 323 –, pose graph 170–172, 181 –, problem 169, 204359 orientation 15 –, 2-dimensional 21 –, 3-dimensional 30 –, derivative 62, 66, 116 –, end-effector 194 –, error 86 –, estimation 78, 82, 87 –, interpolation 73 –, vector 38 –, vehicle 99, 106 origin 15 orthogonal or orthonormal matrix 32, 289, 292 over-actuated robot 54, 238, 240 over-actuation 119, 238 P parabolic blend 70 parallel tracking and mapping (PTAM) system 173 parallel-link robot 189 parameter –, Denavit-Hartenberg 195, 198, 225 –, ellipse 288 particle filter 167, 173 path 68, 129, 132 payload 12, 249, 260 –, effect 266 –, lift capability 113 –, mass 266 PDF (see probability density function) peak 151 –, velocity 70 pendulum, bifilar 277 phototaxis 124 pitch –, angle 35 –, screw 45, 50 planar –, robot 203 –, surface 95, 117 –, transformation 29 plane 298, 304 –, Cartesian 17 –, Euclidean 17, 303 –, image 303 planning –, algorithm 133 –, map-based 128 –, robot path 128, 132 –, trajectory 145 Plücker –, coordinate 50, 52, 296 –, line 296–298 point 15 –, 3D 29 –, Cartesian 177 –, cloud 179, 182 –, coordinate 24 –, homogeneous 49 –, vector 20, 45 –, corresponding, correspondence 178 –, equation –, ellipsoid surface 233, 243 –, line 303 –, Euclidean 27, 295, 304 –, homogeneous form 27 –, ideal 303, 304 –, instantaneous center of rotation (ICR) 98, 107 –, line equation 303 –, mapping 54 –, moving to 100 –, task space 54 –, tool center (TCP) 201 –, transformation 22 –, vector xxv, 15, 20 –, velocity, angular 62 polar-coordinate robot arm 194 pole –, magnetic 83, 84 –, rotational 28 polynomial –, ellipse 300 –, function of time 69 –, matrix approximation 50 –, trajectory 69 pose 15, 53, 58, 168 –, 2D 55 –, 3D 56 –, camera 173 –, change 61 –, derivative 61, 62 –, end-effector 191, 227 –, error 168, 243 –, estimation 81 –, graph 168, 169 –, optimization 170–172, 181 –, SLAM (simultaneous localization and mapping) 165, 167–169, 173 –, robot (see also manipulator) 177, 179 –, singular 232 –, trajectory 75 position 15 positive definite 290 posterior probability 155 power series 50 probabilistic roadmap (PRM) 135 probability 10, 35, 152, 155, 172 –, conditional 155 –, density function (PDF) 151, 158, 159, 173, 327, 328 –, Gaussian 158, 162, 329 –, posterior 155 –, prior 155 process noise 154, 332 product –, of exponential 198, 199 –, of inertia 66 projection, line 304 Prometheus Project 120 proof mass 80 proprioceptive sensor 5 pseudo –, force 67 –, inverse 238, 240, 292, 317 –, Moore-Penrose 292 –, random numbers 172 PTAM (see parallel tracking and mapping) Puma 560 robot 194, 200, 254, 274 pure –, pursuit 103 –, quaternion 43, 53, 62 General Index360 Index Q quadric 304 –, hypersurface 304 –, Klein 304 quadrotor 54, 95, 97, 112, 118 –, control system 115 –, dynamics 113, 114 –, model 113 quaternion 42 –, computational efficiency 43 –, conjugate 43 –, convert to rotation matrix 43 –, derivative 62 –, double cover 42 –, dual 53 –, identity 43 –, interpolation 58, 74 –, pure 43, 53, 62 –, unit 42, 43, 45, 48, 53, 56, 62, 74 quintic polynomial 69 R radio navigation 77, 151 radius, turning 139 random –, coordinate 337 –, measurement 154 –, noise 86, 154, 175 –, number 137, 172, 331 –, sampling 137, 143 –, variable 327 –, Gaussian 327, 332, 334 rangefinder –, remission 177 –, scanning laser 176–179 rank, matrix 232, 292 Rao-Blackwellized SLAM (see also FastSLAM) 167 rapidly-exploring random tree (RRT) 142, 143 rate –, angular 86 –, exponential coordinate 231 –, roll-pitch-yaw angle 74, 116, 231 –, rotation matrix 62 ratio 266 –, gear 252, 262 recursive Newton-Euler 261 redundant robot 54, 208, 224, 238 Reeds-Shepp path 99 reference –, frame 67 –, inertial 66, 67, 77, 81 –, noninertial 68 –, system, attitude and heading (AHRS) 85 reflectance, reflectivity 177, 178 –, specular 178 remission 177 renormalization 53 replanning, incremental 132 representational singularity 231 resampling 174 resectioning 150 resolved-rate motion control 232, 235 response –, position loop 260 –, velocity loop 256–258 right-hand rule 29 rigid-body –, displacement 44, 50, 51 –, dynamics 261, 270 –, motion 25, 44, 45, 52, 65, 307, 308 ring-laser gyroscope (RLG) 78 roadmap 134 robot (see also manipulator) 189 –, arm 119 –, model 198 –, planar 192, 243 –, polar-coordinate 194 –, PUMA 193 –, serial-link 194 –, SCARA (Selective Compliance Assembly Robot Arm) 189, 193, 208 –, Stanford 193 –, Asimo humanoid 6 –, base transform 201, 216 –, Baxter 209, 275 –, behavior-based 125 –, definition of 5, 124, 128 –, DEPTHX (Deep Phreatic Thermal Explorer, AUV) 118, 119 –, Elsie 94 –, end-effector 190 –, field 3, 94 –, gantry 189 –, high-speed 274 –, humanoid 3, 6 –, joint –, modelling 253 –, structure 193 –, kidnapped 176 –, law 1 –, manipulability 213, 234 –, manufacturing 3 –, maximum payload 266 –, mobile 3, 93, 97 –, over-actuated 54, 240 –, path planning 129, 132 –, parallel-link 189 –, planar 203 –, pose 177, 179 –, Puma 560 194, 200, 254, 274 –, redundant 54, 208, 224, 238 –, Shakey 93 –, service 3 –, singularity 206, 213 –, tele- 5 –, tool transform 201, 202, 216, 220 –, tortoise 93 –, trajectory 167 –, under-actuated 54, 208, 238, 239 –, walking 219 –, wrist 194, 213 Rodrigues –, rotation formula 35, 40, 50, 51, 59, 64, 309 –, vector 40 roll angle 35 roll-pitch-yaw angle 35, 36, 38, 230, 231 –, rate 74, 116, 231361 –, singularity 36 –, XYZ 35, 36, 212, 230 –, ZYX 35 rolling, constraint 119 root, finding 318 Rossum’s Universal Robots (RUR) 3 rotation, rotational 45, 48, 52 –, angle 23, 24, 29, 33, 35, 37, 41 –, axis 30, 37, 39, 41, 46, 48, 61, 66 –, direction 74 –, formula 35, 40, 50, 51, 59, 64, 309 –, incremental 64 –, inertia 66 –, interpolation 74 –, matrix 22, 33, 34, 38, 40, 43, 48, 64, 230 –, determinant 47 –, estimating 318 –, least squares problem 318 –, normalization 65 –, product 23 –, reading 33 –, motion 49, 50, 66 –, pole 28 –, rate 62 –, theorem, Euler’s 30, 31, 33–35, 309 –, torque 67 –, twist 28 –, vector 28 –, velocity 61, 63, 67 row space 291 RRT (see rapidly-exploring random tree) RTK GPS (see Global Positioning System (GPS), RTK) rule, right-hand 29 RUR (see Rossum’s Universal Robots) S saccule 81 sampling –, importance 174 –, probabilistic 145 –, random 137, 143 satellite navigation –, system 5, 115, 149, 151, 163 –, network 151 saturation, actuator 116 scalar 15, 52 –, field 314 –, function 313, 314 –, interpolation 210 –, multiplication 287 scale factor 86 scanning laser rangefinder 176, 177, 179 –, noise 178 SCARA (see Selective Compliance Assembly Robot Arm) Schur complement 324 screw 45, 50 –, axis 45, 50 –, model 46 –, motion 45, 46 –, pitch 45, 50 –, theory 50 SE(2) 25, 32 se(3) 51, 52, 310 SE(3) 44, 46, 51, 52, 71, 75, 310, 311, 322 SEA (see series-elastic actuator) selective availability 151 Selective Compliance Assembly Robot Arm (SCARA) 189, 193, 208 sensor 168 –, acceleration 81, 85 –, bias 86 –, calibration 86 –, drift 86 –, error 168 –, fusion 86, 161 –, Hall effect 83 –, inertial 85 –, noise 160, 173 –, range and bearing 159 serial-link manipulator 191 series-elastic actuator (SEA) 274, 275 Shakey (robot) 93 shape –, change 233, 243 –, Earth 79 –, ellipsoid 234 shared control 7 similarity transform, transformation 291 similar matrix 291 Simulink 10, 270 –, block 99 –, library 109 –, kinematics 212 simultaneous localization and mapping (SLAM) 165 –, back end 168, 172, 173 –, EKF (extended Kalman filter) 167 –, Fast 167 –, front end 168, 172 –, pose graph 165, 167–169, 173 –, Rao-Blackwellized 167 –, system, vision-based 173 singular –, pose 232 –, value 292 –, decomposition 292 –, vector 292 singularity 35, 36, 206, 213 –, angle –, Euler 37 –, roll-pitch-yaw 36 –, Jacobian 232, 238 –, motion 213 –, representational 231 –, three angle representation 36 –, wrist 206, 213 singular value decomposition (SVD) 292, 317, 318 skeleton 135, 201 –, topological 134 skeletonization 134, 135 skew-symmetric matrix 23–25, 40, 41, 48, 49, 61, 64, 88, 289, 304, 309 –, augmented 310 skid steering 109 SLAM (see simultaneous localization and mapping) SO(2) 22, 307, 308 so(3) 52, 231, 309 SO(3) 32, 66, 71, 73, 79, 308, 309 General Index362 Index soft-iron distortion 85 solution –, closed-form 203 –, minimum-norm 240 –, numerical 204 solving system 317 space –, configuration 53, 54, 112, 117, 119, 143, 196, 199, 208, 209 –, control 273, 274 –, Euclidean 17, 53, 295, 303 –, inertial reference equipment (SPIRE) 77 –, joint 196, 210, 242 –, operational 53 –, control 273, 274 –, task 53, 54, 208, 209 –, vector 287 sparse matrix 324 spatial –, displacement 65, 243 –, operator 65 –, velocity 62, 63, 67, 229, 230, 237 –, vector 62 special –, Euclidean group 19, 25, 44 –, orthogonal group 22, 32, 290 spectral decomposition 291 specular reflection 178 speeded up robust feature (SURF) detector 250 spherical –, linear interpolation 74 –, wrist 197, 203, 205 SPIRE (see space inertial reference equipment) spring 80, 275 –, torsional 275 Stanford, robot arm 193 steering –, Ackermann 99, 121 –, angle 99, 100, 139, 143 –, mechanism 97 –, skid 109 stereo glasses 33 stiction 250 straight-line motion 212 strapdown –, configuration 78 –, gyroscope 78 –, inertial measurement 85 subsumption architecture 125 surface –, 3D 130 –, Earth 68, 77 –, ellipsoid 233, 243, 328 –, hypersphere 233 –, planar 95, 117 –, polished 178 –, writing on 218 SVD (see singular value decomposition) Swedish wheel 110 symmetric matrix 264, 289 system –, attitude and heading reference (AHRS) 85 –, configuration 53 –, coordinate 17 –, homogeneous 318 –, inertial navigation (INS) 77, 85, 115 –, nonholonomic 119 –, nonhomogeneous 317 –, nonintegrable 119 –, nonlinear 334 –, under-actuated 118 –, vestibular 78, 81 T tag, April 162 Tait-Bryan angle 36 tangent space 308 task space 53, 54, 208, 209 taxis 124 Taylor series 313 TCP (see tool center point) telerobot 5 temperature drift 86 tensor 287 theorem –, Chasles 50 –, Euler’s rotation 30 theory –, Lie group 23 –, screw 50 thinning (also skeletonization) 134, 135 threshold distance 137 thrust 113 time 61 –, derivative 61 –, series xxv –, varying pose 61, 68 tool –, center point (TCP) 201 –, transform 197, 201, 202, 216, 220 toolbox –, functions 55–57 –, obtaining 283 topological skeleton 134 topology, algebraic 48 torque 249, 251, 252, 273 –, control 270 –, computed 270, 272 –, feedforward 258, 270, 271 –, disturbance 249 –, end-effector 242 –, gravity 252, 262 –, maximum 257 –, moment 66, 113, 114, 242, 267 –, motor 250 –, rotational 67 trace of matrix 291 traded control 6 trajectory 68, 72, 74–76, 88, 137, 167, 207, 209, 221, 223, 249, 261 –, Cartesian 89, 212, 222 –, continuous 72, 218 –, end-effector 249 –, following 103, 138 –, hybrid 70 –, joint-space 210–212, 214 –, lane-changing 100 –, leg 219 –, multi-axis 71363 –, multi-segment 72 –, planning 145 –, polynomial 69 –, pose 75 –, robot 167 transconductance 250 transform –, base 197 –, distance 128, 132, 133, 135 –, planar 29 –, SE(2) 29 –, tool 197, 201, 202, 216, 220 –, homogeneous 25, 44, 51, 52, 75, 197, 201 –, matrix 50, 62 –, point 22 –, SE(2) 25 –, SE(3) 44 –, wrench 242 translation 44, 51, 52 transmission 249, 274 –, flexible 12 –, mechanical 107 transpose, Jacobian 244 trapezoidal trajectory 70 traversability 128, 132 triangulation 150 triaxial –, accelerometer 81, 85 –, gyroscope 78 –, magnetometer 83 triple point 134 true north 83 Tukey biweight function 321 turning radius 98, 139 twist 28, 46, 50, 51, 198, 245, 310 –, axis 45 –, Jacobian computing 245 –, nonunit 29, 46 –, rotational 28 –, transforming 310 –, unit 28, 46, 50, 52 –, vector 28, 29, 45 –, velocity 63, 245 U UAV (see unmanned aerial vehicle) UKF (see unscented Kalman Filter) uncertainty 158, 159, 161 under-actuated 54, 97, 118, 119, 193, 227 –, robot, manipulator 54, 208, 238, 239 –, system 118 unicycle, model 109 Unimation Inc. 2 unit –, inertial measurement (IMU) 38, 85 –, quaternion 42, 43, 45, 53, 56 –, derivative 62 –, interpolation 74 –, normalization 48 –, twist 28, 46, 50, 52 unmanned aerial vehicle (UAV) 112 unscented Kalman Filter (UKF) 182 utricle 81 V VaMoRs system (autonomous van) 120 variable, Gaussian random 332 Vaucanson’s duck 1 vector 15, 287 –, addition 287 –, approach 38 –, bound 15 –, coordinate 15–17, 287, 295, 302 –, error 324 –, field 315 –, gravity 82, 261 –, moment 28, 45, 50, 296 –, normal 38 –, orientation 38 –, point xxv, 15, 20 –, Rodrigues 40 –, rotation 28 –, scalar function of 314 –, singular 292 –, space 287 –, twist 28, 29, 45 –, vector function of 314 –, velocity 62, 228 vehicle –, aerial 119 –, autonomous 6, 94 –, surface (ASV) 94 –, underwater (AUV) 94 –, Braitenberg 124 –, car-like 97, 98 –, configuration 98 –, coordinate system 98 –, differentially-steered 97, 107 –, frame 98 –, micro air (MAV) 112 –, mobile robot 3, 93, 97 –, model 105 –, omnidirectional 110 –, orientation 99, 106 –, path 101, 103, 107 –, underwater 119 –, unmanned aerial (UAV) 94, 112 –, velocity 99 –, wheeled 95, 97 velocity 249, 273 –, angular 48, 50, 62, 66, 68, 77, 78, 153, 231, 332 –, time-varying 66 –, vector 64 –, control 100, 255, 259 –, feedforward 260 –, loop 255, 259 –, coupling torque 262 –, discontinuity 76 –, ellipse, ellipsoid 233, 234, 242 –, end-effector 227, 228 –, joint 227, 228 –, kinematics 227 –, linear 50, 66 –, maximum 70 –, peak 70 –, rotational 61, 63, 67 General Index364 Index –, spatial 62, 63, 67, 229, 230, 237 –, translational 61, 63, 67 –, twist 63, 245 –, vector 62, 228 –, vehicle 99 vestibular system 78, 81 via point 72 viscous friction coefficient 250 visual simultaneous localization and mapping (VSLAM) 182 von Mises distribution 154 Voronoi –, cell 135 –, diagram 134, 135 –, roadmap 135 –, tessellation 135 VSLAM (see visual simultaneous localization and mapping) W WAAS (see wide area augmentation system) walking robot 219 waypoint 155 Wide Area Augmentation System (WAAS) 151 world coordinate frame 16, 77 wrench 63, 67, 242, 243, 261, 267 –, ellipsoid 243 –, end-effector 242, 243 –, transformation 242 wrist 206 –, coordinate frame 201 –, robot 194, 213 –, singularity 206, 213 –, spherical 197, 203, 205 X XYZ, roll-pitch-yaw angle 36, 212, 230 Y yaw angle 35 yaw rate 99, 161 Yoshikawa’s manipulability measure 234 Z zero-angle configuration 195 ZYX roll-pitch-yaw angle 35 ZYZ Euler angles 34
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