5 2 the state feedback controller for the linearized

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(/5) 2 The state feedback controller for the linearized system is implemented and experimental results are obtained. The control law is given by -01 _ 0 o- 02-0 ° rml] = _K ql rm2J 01 02 41 The feedback gain matrix is chosen to be [304 ~ -0.58 16.50 6.33 0.01 1.01 l K= . 7.07 -7.56 0.14 1.88 -0.64J' which gives the following closed loop eigenvalues for the system as F-0.92 + 53.07i 1 e = | -6.30 4-5.43i 1. k -4.00 4- 3.75i J Figure 8 shows the experimental results of this controller, in which the desired state of the system is defined as 0°=2.65rad, 0°=3.99rad, q°=0, 0°=0, 0°=0, q°=0. Comparing figures 8 and 6, it is clear that slower response is observed for the rigid body modes using the state feedback control based on the linearized state space model in com- parison to that obtained by the ROCT method. However, vibration at the tip of link 5 is suppressed after a very short time. The experimental results show the effect of the state feedback controller designed based on the linearized system obtained by a local lineariza- tion of the nonlinear dynamic model, given in (13), about an equilibrium state. When nonlinearities in the system are not severe, local linearization may be used as an approx- imation to the nonlinear model. However, such manipulator control is not well suited for the manipulator making a large movement. An alternative is to use gain scheduling, in which the controller gain matrix and the equilibrium state of the manipulator change with the manipulator as it moves.
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Control of a 2-DOF manipulator with a flexible forearm 513 3.2 ] 3.1[ Expt. thatal (tad.) 3 k 2.9 ~ 2.8 2. '7 2.6 i i i 2"50 0.5 1 1.5 time (s) 4.E 4.,I 4.3 4.2 4.1 4 3"9 0 015 i I 115 time (s) 20 15 1G Expt. Deflection (ram) -5 -1(~ -15 -2C 0 ; ti5 time (s) Figure 8. Experimental results for feedback control system based on linearized state space model. A hybrid controller is proposed as another altemative to control the flexible-link manipu- lator. The hybrid controller comprises the controller based on the reduced-order computed torque (ROCT) method for the initial large movement of the manipulator, i.e. from the initial state to a state sufficiently close to the desired state. When the manipulator is close to the desired state, the controller based on the ROCT method is switched to the linear state feedback controller. A switching rule is used for such purpose. Figure 9 shows the exper- imental results for the hybrid controller. The initial state and desired state of the system is set identical to that defined in ROCT and state feedback control. The same gains are also used. The switching rule is to switch from ROCT controller to state feedback controller when both 0~ and 02 are within 10 ° from the desired angles. The experimental results show that there is significant improvement in the performance of the controller in terms of speed of response of the angular positions and vibration sup- pression at the tip of the manipulator. The use of the ROCT controller for the initial large movement of the manipulator enables fast response for the angular position. When the ma-
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  • Fall '19
  • Koh Tuck Lye

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