418 D Dynamics of Direct-current Motors Example D.4. Model of the motor with load whose center of mass is located on the axis of rotation (nonlinear friction). Consider the model of nonlinear friction (D.17) for the friction torque between the axis of the rotor and its bearings, and the corresponding load’s friction, f L (˙q)=f L ˙q + c 2 sign( ˙q) (D.18) where f L and c 2 are positive constants. Taking into account the functions (D.17) and (D.18), the motor- with-load model (D.11) becomes (J L + J m )¨q +  f m + K a K b R a + f L  ˙q +(c 1 + c 2 ) sign( ˙q)= K a R a v where for simplicity, we took r =1. ♦ Bibliography Derivation of the dynamic model of DC motors may be found in many texts, among which we suggest the reader to consult the following on control and robotics, respectively: • Ogata K., 1970, “Modern control engineering”, Prentice-Hall. • Spong M., Vidyasagar M., 1989, “Robot dynamics and control”, John Wi- ley and Sons, Inc. Various nonlinear models of friction for DC motors are presented in • Canudas C., ˚ Astr¨om K. J., Braun K., 1987, “ Adaptive friction compen- sation in DC-motor drives”, IEEE Journal of Robotics and Automation, Vol. RA-3, No. 6, December. • Canudas C., 1988, “Adaptive control for partially known systems—Theory and applications”, Elsevier Science Publishers. • Canudas C., Olsson H., ˚ Astr¨om K. J., Lischinsky P., 1995, “A new model for control of systems with friction”, IEEE Transactions on Automatic Control, Vol. 40, No. 3, March, pp. 419–425. An interesting paper dealing with the problem of definition of solutions for mechanical systems with discontinuous friction is • Seung-Jean K., In-Joong Ha, 1999, “On the existence of Carath´eodory solutions in mechanical systems with friction”, IEEE Transactions on Au- tomatic Control, Vol. 44, No. 11, pp. 2086–2089. Index :=and =: 19 L n 2 390 L n ∞ 390 ⇐⇒ 19 =⇒ 19 IR 20 IR n 20 IR n×m 21 IR + 20 ˙x 19 ∃ 19 ∀ 19 ∈ 19 → 19 e.g. xiv i.e. xiv cf. xiv etc xiv ˚ Astr¨om K. J 308, 333, 418 Abdallah C. T. 139, 333, 378 absolute value . . . . 20 actuators 60, 82, 411 electromechanical . . . . 82, 89 hydraulic . . . 82 linear . . . . 82 nonlinear . . . . 87 adaptive gain 329 update law. . 328 adaptive control closed loop. . 330 law 328 parametric convergence . 329 adaptive law . . . . 328 Ailon A. 308 Alvarez R. xv Alvarez–Ramirez J. . . 218 An C 283 Anderson B. D. O. . . 333 Annaswamy A. . . 333, 398 Arimoto S. . xv, 109, 139, 153, 167, 168, 195, 217, 218, 333, 334 armature current 412 armature resistance 83, 411 Arnold V. . 54 Arteaga A. . 332 Arteaga, M. . xv Asada H. 88, 139, 283 asymptotic stability definition. 33, 35 Atkeson C. 283 back emf 83, 412 back emf constant . . 411 bandwidth . . 231 Barb˘alat lemma . 397 Bastin G. . . . 140 Bayard D 238, 331 Bejczy A. K. . . 90, 283 Berghuis H. xv, 308, 333 bifurcation . 187, 196 pitchfork. 188 saddle-node 188 Bitmead R. R. . . 333 Block-diagram computed-torque control . 228 420 Index feedforward control 265 generic adaptive control of robots 329 P“D” control with gravity compensa- tion . . . 293 PD control. 142 PD control plus feedforward 270 PD control with compensation . 245 PD control with desired gravity compensation . 172 PD control with gravity compensation 158 pendulum under PD control with adaptive compensation 370 PID control 202 Proportional control plus velocity feedback 141 robot with its actuators . . 84 Boals M. . 334 Bodson M. . . 333 Borrelli R. . . . . 54 boundedness of solutions . 148, 174 uniform . . 45 Braun K. . . 418 Brogliato B 332 Burkov I. V. . 89 Caccavale F. . . 283 Campa R xv Canudas C. 140, 308, 332, 418 Canudas de Wit C xv Carelli R xv, 238, 332 Cartesian coordinates . . 115 Cartesian positions . . 61 catastrophic jumps . . . 187 Cervantes I. xv, 218 Chaillet A. . . xv chaos 187 Chiacchio P. 283 Chiaverini S. . . . . 89 Choi Y. . . . 218 Christoffel symbol 97 symbols 73 Chua L. O 196 Chung W. K 218 CICESE xiv closed loop . 137, 224 adaptive control . . . . 330 Computed-torque control . . 229 Computed-torque+ control 234 control P“D” with gravity compensa- tion . . . 294 P“D” control with desired gravity compensation . 302 PD control with compensation . 245 PD control with desired gravity compensation . 173 PD control with gravity compensation 159 PD plus feedforward control 271 PD+ control . 249 PID control 205 closed-loop PD control. . . 143 Coleman C. . 54 Computed-torque control closed loop. . 229 control law. . 228 pendulum 231 Computed-torque+ control closed loop. . 234 control law. . 233 CONACyT . . . . xiv control adaptive. . . see adaptive control adaptive Slotine and Li 361 Computed-torque see Computed-torque control Computed-torque+ . . see Computed-torque+ control feedforward . . see feedforward control force . . . 16 fuzzy . . . . 15 hybrid motion/force. 16 impedance 16 law 136, 224 learning . . . 15, 333 motion. 223 neural-networks-based. . . 15 P see PD control P“D” with desired gravity compensa- tion see P“D” control with desired gravity compensation P“D” with gravity compensation see control P“D” with gravity compensation PD see control PD Index 421 PD plus feedforward. see PD plus feedforward control PD with desired gravity compensa- tion . . see PD control with desired compensation of gravity PD with gravity compensation see control PD with gravity compensation PD+ see PD+ control PID see PID control position . . 13 set-point . 136 Slotine and Li see PD controller with compensation specifications. . . 12 variable-structure . . 15 without measurement of velocity . 291 control law . . 136, 224 adaptive . . . . 328 Computed-torque control . . 228 Computed-torque+ control 233 control P“D” with gravity compensa- tion . . . 293 feedforward control 264 P“D” control with desired gravity compensation . 300 PD control. 142 PD control with desired gravity compensation . 171 PD control with gravity compensation 157 PD plus feedforward control 269 PD+ control . 248 PID control 201 Slotine and Li . 244 control P“D” with gravity compensation closed loop. . 294 control law. . . . . 293 controller. . . 136, 224 coordinates Cartesian . . . 61, 115 generalized. 78 joint . . . 60, 115 Coriolis forces . . 72 Craig J. 88, 109, 139, 238, 283, 331, 332 critically damped . . 231 damping coefficient . . . . 84 Dawson D. M. . 139, 140, 333, 378 Dawson, D. M. 308 DC motors 135 de Jager B. . . . . 333 de Queiroz M . 140 degrees of freedom. 4 Denavit–Hartenberg 88 desired trajectory . . 327 Desoer C. . . 398 differential equation autonomous . 28, 49 linear . . . . 28 nonlinear . 28 digital technology 263 direct method of Lyapunov 27 direct-current motor linear model 83 direct-current motors 82, 411 model linear . 416 nonlinear model. . 417 direct-drive robot 77 DOF 4 Dorsey J. . . . . . . 218 dynamic linear 396 dynamics residual . 102 Egeland O. . 89, 259 eigenvalues 24 elasticity . . . 78 electric motors . . . 77 energy kinetic . . . 71, 78 potential . . . 72, 79 equations of motion Lagrange’s . . 62, 72 equilibrium asymptotically stable . 37, 38 bifurcation . . . 187 definition . . 28 exponentially stable . . . . 38 isolated . . . . 55 stable . . . 31, 32 unstable . 39 error position 136, 223 velocity 224 422 Index feedforward control control law. . . . . 264 pendulum 266 fixed point 26 Fixot N . 308, 332 Fomin S. V. . . . 54 forces centrifugal and Coriolis . . . . 72 conservative. 63 dissipative . 76 external 73 friction 76 gravitational . 72 nonconservative . 63 friction coefficient . . . 83 forces . . 76 nonlinear . 417 Fu K 88, 139, 238 function candidate Lyapunov. . 43 continuous . . . 390 decrescent . . . 41 globally positive definite . . 41 locally positive definite . . . . 40 Lyapunov . . . 44 positive definite . . 41, 401 quadratic . . 41 radially unbounded . 41 strict Lyapunov 163, 167, 279 gain adaptive . . . . 329 derivative . . 141 integral 201 position 141, 328 velocity 141, 328 gear . . 83 gears 77, 412 global asymptotic stability . . . . 48 definition. 37, 38 theorem. . . . 47 global exponential stability 48 definition of . 38 theorem. . . . 47 global minimum . . . 181 global uniform asymptotic stability theorem. . . . 47 Godhavn J. M 259 Goldstein H. . . 89 Gonzalez R. . . . . 88, 139, 238 Goodwin G. C. 331 gradient . . 73, 182, 403, 405 adaptive law . see adaptive law Guckenheimer J. . . 196 Hahn W. 54 Hale J. K. . . 54, 196 Hauser W. . . 89 Hessian. . . . . . 182, 387, 403 Hollerbach J 283 Holmes P. . . . . . . 196 Hopfield. . . . . . 55 Horn R. A 397 Horowitz R. . . . . xv, 331, 332, 334 Hsu L 332 Hsu P 331 Ibarra J. M xv In-Joong Ha . . 418 inductance . . . 83, 411 inertia matrix . 72 rotor’s . . 83 input . 82, 84 input–output. 390 inputs . . . 73 instability definition . . 39 integrator . 201 Jackson E. A. . . . . . . 196 Jacobian . . 92, 116 Johansson R 332 Johnson C. R. . . 333, 397 joint elastic . 77 prismatic . . 59 revolute . . . 59 joint positions 60 Kanade T. . 283, 334 Kanellakopoulos I. . . . . 333 Kao W 334 Kawamura S 168, 218 Kelly R. . . . 196, 217, 218, 238, 239, 259, 283, 284, 308, 332, 358, 378 Khalil H. . 54, 333 Khosla P. . . 334 Index 423 Khosla P. K. 283 kinematics direct . . 61 inverse . . . . . 61 Ko¸cak H 196 Koditschek D. E. xv, 259, 260, 333, 334, 359 Kokkinis T. 283 Kokotovi´c P 333 Kolmogorov A. N. . . 54 Kosut R. . . 333 Krasovski˘ı N. N 54 Kristi´c M 333 L¨ohnberg P 308 La Salle . . see theorem La Salle J. . 53 Lagrange’s equations of motion . . . . 62 Lagrangian . . 63, 72, 79 Landau I. D. 332 Landau I. D. 333 Lee C 88, 139, 238 Lefschetz S. . . 53 lemma Barb˘alat’s . . . . 397 Lewis F. L. . . . 139, 333, 378 Li W. . . . . . 54, 259, 331, 333, 377, 378 Li Z 90 linear dynamic system. . . 264 links numeration of . 59 Lipschitz . . 101, 180 Lischinsky P. . . . 418 Lizarralde F. . . 332 Lor´ıa A. 218 Lozano R. 332, 333 Luh J 89 Lyapunov candidate function . 43 direct method . . 27, 44 function 44 second method 27 stability. . . 27 stability in the sense of. . 31 theory . 54 uniform stability in the sense of . . . 32 Lyapunov, A. M. . 53 M’Saad M. 333 manipulator definition . . 4 Mann W. R. . 397 mapping contraction see theorem Marcus M. . 397 Mareels I. M. Y. . 333 Marino R. . . . 89, 333 Marth G. T. 283 Massner W. . . . . . 334 matrix . . . . 21 centrifugal and Coriolis . . . 73, 97 diagonal . 22 Hessian . . . . 182, 387, 403 identity . . . 23 inertia. 72, 95 Jacobian . 92, 116 negative definite . . . 24 negative semidefinite 24 nonsingular . . . 23 partitioned . . 124, 384 positive definite . . 23, 41 positive semidefinite . . . 24 singular 23 skew-symmetric . . . . . . 22, 98 square . 22 symmetric . . . 22 transfer 396 transpose . 21 Mawhin J. . . 54 Mayorga R. V. . 89 Meza J. L. . 217 Middleton R. H. . . 331 Milano N. . 89 Minc H. . . 397 Miyazaki F 168, 195, 217, 218 model direct kinematic 61, 115 dynamic. . . 10, 71 elastic joints. . 77 with actuators. . . . . 82 with elastic joints . 89 with friction . . 75 dynamics . . . . . 88 inverse kinematic. 61, 116 kinematics . . . 88 moment of inertia . . 77 motion equations Lagrange’s . 79 424 Index motor-torque constant 83, 411 multivariable linear system . 230 Murphy S. . 218 Nagarkatti S. P. 140 Naniwa T. . . 334 Narendra K . . 333, 398 Nicklasson P. J. . 139 Nicosia S. . . 89, 308 Nijmeijer H. . xv, 308 norm Euclidean 20 spectral 25 numerical approximation . 291 observers. . 291 Ogata K . 418 Olsson H. 418 open loop 265 operator delay 299 differential . . . . 292 optical encoder . . . . 291 optimization 181 Ortega R. . . xv, 109, 139, 218, 238, 259, 308, 332, 378 oscillator harmonic . 32 van der Pol 39 output . . 82 outputs. 73, 84 P´amanes A. xv P“D” control with desired gravity compensation closed loop. . 302 control law. . . . . 300 Paden B. . . 168, 260, 283 Panja R. . . . . . . 168, 260 parameters adaptive . . . . 328 of interest 317 parametric convergence. . . . 329, 330 parametric errors. . . 330 Parker T. S 196 Parra–Vega V. . . 334 passivity . . 111 Paul R 88, 139, 217 PD control . . . . . 141 closed-loop 143 control law. . 142 pendulum . 146, 150 PD control with adaptive compensation 363 closed loop. . 364 PD control with adaptive desired gravity compensation adaptive law . 339 closed loop. . 342 control law. . 339 PD control with compensation closed loop. . 245 PD control with desired gravity compensation closed loop. . 173 control law. . 171 pendulum . 174, 187 PD control with gravity compensation 157 closed loop. . 159 control law. . 157 pendulum 168 robustness 168 PD control with gravity precompensa- tion . . see PD control with desired gravity compensation PD plus feedforward control closed loop. . 271 control law. . 269 experiments. . 283 pendulum 273 tuning . . 273 PD+ control closed loop. . 249 control law. . 248 pendulum 252 pendulum . 30 Computed-torque control . . 231 feedforward control 266 kinetic energy . . 45 PD control 146, 150 PD control with desired gravity compensation . 174 PD control with gravity compensation 168 PD plus feedforward control 273 PD+ control . 252 PID control 218 Index 425 potential energy. . . 45 with friction . . 57 permanent-magnet . . . . 411 PID control closed loop. . 205 control law. . . . . 201 modified 213 robustness 217 tuning . . . . 213 pitchfork. . see bifurcation potentiometer . . . . . 291 Praly L . . . 333 properties gravity vector . . 101 of residual dynamics . . . . 102 of the Centrifugal and Coriolis matrix 97 of the inertia matrix . . 95 Qu Z 139, 218 Queiroz M. S. de . . . 308 Ramadarai A. K . 283 Rayleigh–Ritz . . . see theorem Reyes F 284 Riedle B. D 283, 333 Rizzi A. . . . . 260, 333, 334, 359 robot Cartesian . 69 definition . . 4 direct-drive . . . . 77 dynamic model. 71 mobile . 3 robots navigation . . 13 stability of . 75 Rocco P 218 rotors 79 Rouche N . . 54 saddle-node . . . see bifurcation Sadegh N. 331, 332 Salgado R. 283 sampling period 299 Samson C. . . . xv, 217 Santib´a˜nez V 110, 217, 283 Sastry S. 54, 331, 333 Schwartz inequality . 21 Schwartz inequality . . 21 Sciavicco L. . 139 sensors . . 78, 136, 224 Seung-Jean K. . 418 Siciliano B. . 89, 139, 140 singular configuration 118 Sira-Ram´ırez H. . . xv, 139 Slotine and Li . see control control law. . 244 Slotine J. J . .xv, 54, 88, 139, 259, 331, 333, 377, 378 space L n 2 390 L n p 390, 397 L n ∞ 391 Spong M. xv, 88, 89, 109, 139, 153, 167, 217, 238, 259, 332–334, 378, 418 stability definition. 31, 32 of robots 75 semiglobal . 207 theorem. . . 44 Stoten D. P. . 333 Stoughton R 283 Sylvester . . see theorem symbols of Christoffel 73 system dynamic lineal 202 tachometer . . . . 291 Takegaki M. . . 153, 167, 195 Takeyama I 283 Tarn T. J 90, 283 Taylor A. E. . . 397 theorem contraction mapping . . 26 application . . 147 contraction mapping theorem application . . 180 global asymptotic stability . . . . . . 47 global exponential stability. . . . . . 47 global uniform asymptotic stability47 La Salle application . . 145 La Salle’s . . . 49, 51 application . 184, 211 use of 160 mean value . . 392 mean value for integrals. 388 of Rayleigh–Ritz . . . 24 426 Index of Sylvester . . 23 of Taylor 387 stability. . . 44 uniform stability . 44 Tomei P . 195, 308, 333, 359 torsional fictitious springs . . . . . . 79 Tourassis V. . 89 tuning. 201, 213 PD plus feedforward control 273 uncertainties parametric . . . . 265 uncertainty . . . 313 van der Pol see oscillator vector . 20 gravity 101 of external forces 73 of gravitational forces . 72 parametric errors . . 330 Vidyasagar M . .88, 109, 139, 153, 167, 217, 238, 333, 334, 378, 397, 398, 418 voltage . . . 83, 412 Wen J. T. . . . xv, 140, 218, 238, 283, 331 Whitcomb L. L .xv, 260, 333, 334, 359 Wiggins S. . . . . 196 Wittenmark B 333 Wong A. K 89 Yoshikawa T. . 88, 89, 139, 153, 167, 238 Yu T 332 Yun X 90 Zhang F. . 140 . feedforward control 273 uncertainties parametric . . . . 265 uncertainty . . . 313 van der Pol see oscillator vector . 20 gravity 101 of external forces 73 of gravitational forces . 72 parametric. 87 adaptive gain 329 update law. . 328 adaptive control closed loop. . 330 law 328 parametric convergence . 329 adaptive law . . . . 328 Ailon A. 308 Alvarez R. xv Alvarez–Ramirez J. . . 218 An. Ogata K., 1970, “Modern control engineering”, Prentice-Hall. • Spong M., Vidyasagar M., 1989, Robot dynamics and control , John Wi- ley and Sons, Inc. Various nonlinear models of friction for