Used to control a flexible link manipulator with

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used to control a flexible-link manipulator with minimal deterioration in performance, if the discarded terms in the model-based control law have only a negligible effect on the dynamics of the system. For example, friction, viscous damping or flexibility in the system may be ignored if the system is well lubricated or is fairly rigid. The objective of this experiment is to study the deterioration in performance, if any, of the controller designed based on the ROCT method. The physical system is described by the matrix equation given by (8). Assuming link 5 is rigid, ql, q2 ..... q~ and 01, q2 ..... qee are set equal to zero in (8) so that the resulting equation is the dynamic model of an equivalent 'rigid' manipulator. This equation is given by Z'm2.] COS(02 -- 01) a2 + Ym2 ] L d3 sin(02 - 01)~) 2 J + [B~ 1 902] [~12] "{" Fb"sgn(0.")l Lbm2sgn(02)] . (9) Based on the reduced-order dynamic model given by (9) and ignoring the coulomb friction and viscous damping terms which are not known exactly, a control law is obtained based on the computed torque method which is given by l rm2l LO2a q- kp2e2 -t- kv2e2 + ki2 fo e2dtJ + h(Ol, 0z, 01, 02) (10) where [ dl + Jml d3 c0s(02 - 01)] M(OI' 02) =- [.d3 cos(02 - 01) d2 + Jm2 J F-d sin 0 - h(O1,02, 0,, 02)= L d3 sin(02 - 01>~} 2 J el = Old -- O1 e2 = 02d -- 02 The control law given by (10) is implemented to study position and vibration control of the end effector of the manipulator. Figure 5 shows the controller based on ROCT method which has been implemented on the system. Figure 6 shows the experimental results of
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Control of a 2-DOF manipulator with a flexible forearm 509 t~ d Physical system . . . . . M(.0, 0 z ) D(~+C~+/~([email protected] +l.l(O i ,02) So<,, IOadt I I +.x~- 6 ~ + ,- (t 0 O~ Figure 5. Controller based on reduced- order computed torque method. 3.2 xpt. thetal 2. 2.8 2.7 2.6 2.5 015 (rad.) time (s) 20 , , f 10 5 Expt. Deflection (mm) 10 15 20 t I i 0.5 1 1.5 time (s) 4.5 4.4 Expt. theta2 (rad.) 4.2 4.1 4 3.g o'.s 1 l'.s 2 0 time (s) 2 Figure 6. Experimental results for ROCT.
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510 Koh Tuck Lye et al 5 4 Simulated result. Effect of 1st and 2nd modes (mm). 3 0 -1 -2 -3 -4 -5 0 0.5 1 1.5 2 time (s) Figure 7. Relative magnitude of~bl (Is)q1 (t) and (02(15)q2(t). the ROCT method. Note that the desired state of the system is defined as 01 = 2.65rad, 02 = 3.99rad. The gains are setto Kpl = 100, KOl = 1 and Kp2 = 50, Kv2 = 4.25. Figure 6 shows that there is deterioration in the performance of the controller based on the ROCT method. Although the controller is able to control the rigid body modes of the manipulator, it is unable to control the vibratory modes of flexible link 5. Thus it is necessary to use a different control law for position and vibration control of the manipulator. Rayleigh's method for approximation of the fundamental frequency of vibration is used to estimate the first fundamental frequency of vibration. Rayleigh's quotient is given by 1/2 ogi = {£'SEl(p~'(u)2du/£lSpgpi(u)2du+mL~i(15)} (11) Since f~5 Elck~,(u)2du = o92pl5, fd5/9~bi(u)2du = pl 5 and all the other parameters are known, the first fundamental frequency of vibration can be estimated. The parameter values were substituted into (11) and Wl was found to be 8.09Hz. From figure 6, it is observed that the natural frequency of vibration is about 7-8 Hz, which is very close to o91.
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  • Fall '19
  • Koh Tuck Lye

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