265 generic adaptive control of robots 329 P“D” control with gravity compensa-tion.. 172 PD control with gravity compensation 158 pendulum under PD control with adaptive compensation.. 2
Trang 1418 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,
fL( ˙q) = fL q + c˙ 2 sign( ˙q) (D.18)
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 + (c1+ c2) sign( ˙q) = K R a
a v
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
Trang 2:= and =: 19
L n 2 390
L n ∞ 390
⇐⇒ 19
=⇒ 19
IR 20
IRn 20
IRn ×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
Trang 3420 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
Trang 4PD 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
Trang 5422 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
Trang 6Khosla 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
Trang 7424 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
Trang 8potential 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
Trang 9426 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