Eliminating Multiple Modes of Vibration in a Flexible Manipulator

The flexibility of long reach manipulators presents a dificult control problem when accurate endpoint position is required. To maintain a desired tip trajectov, the residual vibration inherent to flexible systems must be eliminated. Unfortunately, vibration suppression is ofren difficult to achieve in systems whose parameters are a function of configuration. This paper discusses a modiJied command filtering technique that eliminates the first two modes of vibration in a large, flexible manipulator with position dependent parameters. The elimination of residual vibration is demonstrated using a circular trajectory located in the workspace of the manipulator so that a change in system properties occurs. The modi$ed command filtering technique will be compared to previous control methods to demonstrate its efectiveness in eliminating residual vibration. Introduction A recent development in the control of flexible manipulators is preshaping commanded inputs to reduce the residual vibration of the system. Work conducted by Meckl and Seering demonstrated that force profiles can be developed to control flexible dynamic systems with minimal vibration. By defining an appropriate cost function, a force profile can be derived that efficiently allocates kinetic energy so that excitation is minimized at the system resonances and maximum energy is used for system motion [ 11. Recently, Meckl and Kudo applied the minimum energy force profile technique to control the position of disk drive heads [2]. Using a finite series of ramped sinusoids, simulation results showed significant reduction in vibration at the head positioner natural frequency. However, they concluded that the feedforward nature of the control would likely produce final positioning errors that would require the use of an existing closed-loop feedback system. Another feedforward preshaping technique is the impulse shaping method developed by Singer and Seering. Their method transforms each sample of the desired input into a new set of impulses that do not excite the system resonances [3]. The idea involves delaying a portion of the input by half the damped natural period of the system to cancel the vibration induced by the original input. The method was implemented strictly in a feedforward manner and depended heavily on accurate system information to be effective. Some interesting extensions of the impulse shaping method have recently been considered. Hillsley and Yurkovich designed a two stage control architecture to achieve accurate end-point position control for point-topoint movements [4]. The first stage implements Singer and Seering's shaping algorithm to achieve vibrationless motion for large angle slewing and then shaft position, end-point acceleration feedback accurately positions the tip. This composite control scheme provides vibration suppression during gross motions as well as reducing overshoot at the final end-point position. Rattan and Feliu presented the design of a feedforward controller using a dynamic model inversion technique [5 ] . Using a simplified discrete-time controller that is independent of the reference input, a continuous controller is derived that contains a finite sum of delay terms. In fact, if the delay period is chosen to equal one-half the damped natural period, a shaping controller similar to Singer and Seering's impulse shaping method results. The shaping method can also be applied directly in a feedback control architecture. Noakes, Petterson and Werner applied an acceleration profile to damp oscillations in objects transported by overhead cranes [6] . By delaying a constant acceleration input by one-half the period of the swinging object, the object is able to move with constant velocity through the workspace. The same strategy is applied to decelerate the object and bring it to a stop without overshoot. Magee and Book have also used a modified command filtering scheme in a feedback manner to eliminate the first mode of vibration in a two-link, flexible manipulator named RALF (Robotic Arm, Large and Flexible). They demonstrated that the original impulse shaping scheme developed by Singer and Seering induces vibration in time-varying systems and then created a modified command filtering technique to eliminate the 474 1050-4729/93 $3.00