Eliminating Multiple Modes of Vibration in a Flexible Manipulator

The flexibility of long reach manipulators presents a difficult control problem when accurate endpoint position is required. To maintain a desired tip trajectory, the residual vibration inherent to flexible systems must be eliminated. Unfortunately, vibration suppression is often difficult to achieve in systems whose parameters are a function of configuration. This paper discusses a modified command shaping 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 significant change in system properties occurs. The modified shaping technique will be compared to previous control methods to demonstrate its effectiveness 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 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 [1]. 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 pre shaping technique is the impulse shaping method developed by Singer and Seering. Their method transforms each sample of the commanded 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 input shaping method have recently been considered. Hillsley and Yurkovich designed a two stage control architecture to achieve accurate endpoint 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, endpoint acceleration feedback accurately positions the tip. This composite control scheme provides vibration suppression during gross motions as well as reducing overshoot at the final endpoint position. Rattan and Feliu presented the design of a feedforward conttoller 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 command shaping 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 shaping technique to eliminate the undesirable motion [7,8]. The modified command shaping technique alters the feedback error in a P.D. control scheme to achieve significant vibration suppression in a time-varying system. The main focus of this paper is the application of modified command shaping to eliminate the first and second modes of vibration ofRALF. The impulse shaping constraint equations are solved separately for each mode of vibration and then combined to eliminate two modes simultaneously. Experimental results verify the modified command shaping's ability to eliminate residual vibration when compared to impulse shaping and an original P.D. feedback control scheme. Simplified Model of System A previous modelling technique to describe the dynamic behavior of RALF utilizes an assumed modes method to approximate the position of the manipulator and then applies Lagrange's equation of motion. This procedure yields a set of nonlinear equations which couples the rigid body and flexible motion of the system. A more elementary approach can be taken to model the two modes of vibration. The flexible motion is assumed to be a linear combination of two second-order systems with the overall transfer function which relates position to input force. Since the two natural frequencies and damping ratios are time varying, these parameters were determined experimentally to achieve accurate results. The manipulator was stimulated with random noise at specified joint configurations throughout the workspace and then the frequency response of the system was taken for each configuration using an HP analyzer. From the frequency response data, the natural frequency and damping ratio can be calculated and then parameterized as a function of joint configuration. Thus, the natural frequency and damping ratio for any joint configuration can be computed when needed by the modified command shaping method. Modified Command Shaping The modified command shaping technique is a time varying filter that accommodates changes in the system parameters. It utilizes the solutions of the impulse shaping method and makes modifications to the impulse output when a change in discrete-time period occurs. Consider the impulse shaping technique that creates three output impulses for each sample of the input. The . continuous-time filter can be written in the form given in Equation (2) where the continuous-time period, delT, is one-half the damped natural period of the system and the coefficients can be found in [9]. The continuous-time period can be transformed into a discrete-time period, deln, for use on a discrete-time control system by utilizing the sampling rate of the control system. The equation to perform this transformation is deln = int (delI'·f.) (3) where f. is the sampling rate of the control system and the int function truncates the argument to an integer. For time-invariant systems, the discrete-time period, deln, is constant even though the damped natural frequency may not be known precisely. No problems arise for the impulse shaping method for this situation. However, if the damped frequency becomes time-varying, the continuous-time period, delT, becomes time-varying as well. A significant change in continuous-time period will generate a change in discrete-time period which produces a vibration in the system. This induced vibration is verified later in the paper. The modified command shaping method accommodates a change in discrete-time period and prevents vibration from being induced into the system. For each discrete-time sample of the desired command, the modified method compares the current discrete-time period to the previous discrete-time period. If the discrete period has increased, it is as if the discrete-time scale has expanded leaving gaps in the filter output. To fill the gaps, the current sample is shaped twice using both discrete-time periods and adding the extra impulses at the appropriate discrete-time locations. If the discrete-time period decreases, it is as if the discrete-time scale is compressed leaving extra impulses in the filter output. These extra impUlses are subtracted from the filter output to accommodate the compressed discrete-time scale. It is worth noting that the extra impulses subtracted for a decrease in discrete-time period correspond to the same impUlses that were added for the increase in discrete-time period [7,8,9]. Two Mode Filter The continuous-time filter given in Equation (2) can be implemented for a second mode of vibration to create an effective filter that eliminates two modes simultaneously. The second filter takes the form where the subscript 'i' signifies the second mode. The effective filter is the multiplication of the two independent