Design and Implementation of Time-optimal Negative Input Shapers

Input shaping is a method for reducing residual vibrations in computer controlled machines. Vibration is eliminated by convolving a sequence of impulses, an input shaper, with a desired system command to produce a shaped input. The shaped input is then used to drive the system. The input shaper has traditionally contained only positively valued impulses. However, when the impulses are allowed to have negative amplitudes, the length of the shaper can be greatly reduced. Decreasing shaper length is desirable because rise time is limited by shaper length. A simple look-up method is presented that allows the design of negative shapers without the usual requirement of solving a set of simultaneous transcendental equations. Solutions to the problem of high-mode excitation that can occur with negative shapers are also presented. The performance of negative shapers is evaluated experimentally. INTRODUCTION Input shaping improves response time and positioning accuracy by reducing residual vibrations in computer controlled machines. The method requires only a simple system model consisting of estimates of the natural frequencies and damping ratios. Input shaping is implemented by convolving a sequence of impulses, an input shaper, with a desired system command to produce a shaped input that is then used to drive the system. Many papers have been published on input shaping since its original presentation in (Singer and Seering, 1990). A method for increasing the insensitivity to modeling errors was presented in (Singhose et al., 1990). Input shaping was shown to be effective for multiple mode systems (Hyde and Seering, 1991). Flexible systems equipped with constant-force actuators were shown to be compatible with input shaping (Liu and Wie, 1992; Wie et al., 1993; Rogers, 1994; Singhose et al., 1994a). Harmful effects from course digitization of the input signal can be eliminated (Murphy and Watanabe, 1992). Input shaping was used to reduce residual vibration and maximum deflections during the simulation of a large spacebased antenna (Banerjee, 1993). Residual vibrations of a long reach manipulator were also decreased with input shaping (Jansen, 1992). Input shaping was used in conjunction with an adaptive controller (Tzes and Yurkovich, 1993). Input shaping was shown to be beneficial for trajectory following (Drapeau and Wang, 1993; Singhose and Singer, 1994). Input shaping was acknowledged as an established technique for controlling flexible structures in (Book, 1993). The constraint equations used to determine the input shaper usually require positive values for the impulse amplitudes. However, move time can be significantly reduced by allowing the shaper to contain negative impulses. The work of several authors (Smith, 1958; Liu and Wie, 1992; Wie et al., 1993; Singhose et al., 1994a) can be viewed as special purpose trajectories precomputed with negative shapers. However, these negative shapers cannot be used with arbitrary commands. Others have used the results of (Singer and Seering, 1990) in zero placement algorithms that can give rise to sub-optimal negative shapers (Jones, 1993; Seth et al., 1993; Tuttle and Seering, 1994). The first paper dedicated to the subject of time-optimal negative input shapers (Rappole et al., 1993) required the numerical solution of a set of simultaneous transcendental equations. This paper presents a look-up method that allows the design of negative input shapers without solving a set of complicated equations. The constraint equations used to design an input shaper can vary greatly depending on the application, but they always include limitations on the amplitude of vibration at problematic frequencies. The constraint on vibration amplitude can be expressed as the ratio of residual vibration amplitude with shaping to that without shaping. This percentage vibration ratio is given by: %Vibration = en{(ΣAiei cos(ω 1−ζ 2 ti ))2 + (ΣAiei sin(ω 1−ζ 2 ti ))2} 1 2 (1) where Ai and ti are the amplitudes and time locations of the impulses, tn is the time of the last impulse, ω is the vibration frequency, and ζ is the damping ratio. . -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0.0 0.40 0.80 1.2 1.6 2.0 2.4 2.8 Unshaped Command A cc el er at io n C om m an d