-+点数片哥 FIGURE 101. 12 Closed-loop control of a robot joint. Source: Adapted from J. Y. S Luh, "Conventional controller design for industrial robots: A tutorial "IEEE Trans. Systems, Man, Cybernetics, voL. SMC-13, no. 3, June 1983.@ 1983 IEEE) the damping of the system. This can be done either using a tachometer or computing the difference in angular displacements of the actuator shaft over a fixed time interval The design of the control system involves fixing the values of K, and K, to achieve the desired response Consider the closed-loop transfer function of the system shown in Fig. 101.12(assuming n=0, ponse 61(s) nk KrKe (101.1) 8, (s) 5 Leff+s(RBeff K,Kb +Ka K,)+ nkrkake where K= gain of the amplifier, K,= torque constant of the motor, K,=back emf constant, Ke= position sensor constant(volts/rad), R= resistance of the motor winding(ohms), and n= gear ratio. A.=link position (rad)and 0m= angular displacement at the actuator side(rad). The effective inertia, J efs, and damping, Beft, are defined as eff=m+ nJL Bn+ nB. where Jm=total inertia on the motor side, Bm=damping coefficient at the motor side, JL= inertia of the robot link, and B.= damping coefficient at the load side This is a second-order system and stable for all values of K and K. The values of Ka and K, are selected to achieve a desired transient response by fixing the damping ratio and the natural frequency of the system and are described below The characteristic equation for the above system is RBo + krk,+ Kk nKrkK (1014) Reff JEff This can be conveniently written as s-+2C0, s +0*=0 (101.5) where the natural frequency O and the damping ratio s of the system are given as c 2000 by CRC Press LLC
© 2000 by CRC Press LLC the damping of the system. This can be done either using a tachometer or computing the difference in angular displacements of the actuator shaft over a fixed time interval. The design of the control system involves fixing the values of Ka and Kv to achieve the desired response. Consider the closed-loop transfer function of the system shown in Fig. 101.12 (assuming nTl = 0), (101.1) where Ka = gain of the amplifier, KT = torque constant of the motor, Kb = back emf constant, Kq = position sensor constant (volts/rad), R = resistance of the motor winding (ohms), and n = gear ratio. qL = link position (rad) and qm = angular displacement at the actuator side (rad). The effective inertia, Jeff, and damping, Beff, are defined as Jeff = Jm + n2JL (101.2) and Beff = Bm + n2BL (101.3) where Jm = total inertia on the motor side, Bm = damping coefficient at the motor side, JL = inertia of the robot link, and BL = damping coefficient at the load side. This is a second-order system and stable for all values of Ka and Kv . The values of Ka and Kv are selected to achieve a desired transient response by fixing the damping ratio and the natural frequency of the system and are described below. The characteristic equation for the above system is (101.4) This can be conveniently written as (101.5) where the natural frequency wn and the damping ratio z of the system are given as FIGURE 101.12 Closed-loop control of a robot joint. (Source: Adapted from J. Y. S. Luh, “Conventional controller design for industrial robots: A tutorial,” IEEE Trans. Systems, Man, Cybernetics, vol. SMC-13, no. 3, June 1983. © 1983 IEEE.) q q q q 1 2 ( ) ( ) ( ) s s nK K K d s RJ s RB K K K K K nK K K a T Tb aT v Ta = eff eff + ++ + s s RB K K K K K RJ nK K K RJ 2 Tb aT v T a + 0 + + + = eff eff eff q s sn n 2 2 + += 2 0 zw w
IAXIMUM VARIATIONS IN TOTAL LINK INERTIAS ARE NDING JOINT, ASSUMED LOAD: 1.8 kg, 422 cm(4 Ib, 27 iny ETRICALLY HELD IN THE HA <H NO LOAD 盘 NO LOAD IN FIGURE 101.13 Variations of link inertias for JPL-Stanford manipulator Source: A K. Bejczy, Jet Propulsion Lab, Pasadena, Calif, American Automatic Control Conference Tutorial Workshop, Washington, D. C, June 18, 1982.) Ik KrKe> o (101.6) <= RBa+K,K,+K,K,K, (1017) 2√nK2KRK6 These systems are designed to operate with critical damping(=1)because an underdamped system(< 1) has fast response but results in an overshoot, whereas an overdamped system(5> 1)is too slow. However, this is not always possible, because the damping ratio given by Eq (101.7)depends on Bef and Jeff which vary during the actual operation of the manipulator. Beff changes with age or repeated use of the manipulator. Jefr varies with the payload. For example, the variation of Je for the Stanford manipulator under various loading conditions is shown in Fig. 101.13. er also varies with the configuration of the manipulator during the actual operation So a compromise solution will be to design the controller such that 52 1 throughout the intended operation The undamped natural frequency @n is selected to be no more than half the resonance frequency of the robot to avoid any structural damage to the robot [Paul, 1981]. These resonances are possible due to the flexibilities associated with the links of the robot and the shafts within the drive system, to name a few.These are called unmodeled resonances because they are not explicitly included in the model In our case, if K and Jer are the effective stiffness and the inertias of the joint, respectively, then the resonance frequency o, is given by Since K r is difficult to estimate but constant for a given joint, we can experimentally determine the resonance frequencies for a known inertia and use this information for fixing the gain. Suppose o is the resonance frequency for a given value of effective inertia h then e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (101.6) (101.7) These systems are designed to operate with critical damping (z = 1) because an underdamped system (z < 1) has fast response but results in an overshoot, whereas an overdamped system (z > 1) is too slow. However, this is not always possible, because the damping ratio given by Eq. (101.7) depends on Beff and Jeff which vary during the actual operation of the manipulator. Beff changes with age or repeated use of the manipulator. Jeff varies with the payload. For example, the variation of Jeff for the Stanford manipulator under various loading conditions is shown in Fig. 101.13. Jeff also varies with the configuration of the manipulator during the actual operation. So a compromise solution will be to design the controller such that z ³ 1 throughout the intended operation. The undamped natural frequency wn is selected to be no more than half the resonance frequency of the robot to avoid any structural damage to the robot [Paul, 1981]. These resonances are possible due to the flexibilities associated with the links of the robot and the shafts within the drive system, to name a few. These are called unmodeled resonances because they are not explicitly included in the model. In our case, if Keff and Jeff are the effective stiffness and the inertias of the joint,respectively, then the resonance frequency wr is given by (101.8) Since Keff is difficult to estimate but constant for a given joint, we can experimentally determine the resonance frequencies for a known inertia and use this information for fixing the gain. Suppose w is the resonance frequency for a given value of effective inertia J, then FIGURE 101.13 Variations of link inertias for JPL-Stanford manipulator.(Source: A.K. Bejczy,Jet Propulsion Lab, Pasadena, Calif., American Automatic Control Conference Tutorial Workshop, Washington, D.C., June 18, 1982.) w q n nKa T K K RJ = > eff 0 z q = RB + K K + K K K nK K RJ K T b a T v a T eff eff 2 wr K J = eff eff