Effects of Input Shaping on Two-Dimensional Trajectory ... - CiteSeerX

IEEE TRA.NSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 12, NO. 6, DECEMBER 1996

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Effects of Input Shaping on Two-Dimensional Trajectory Following William E. Singhose and Neil C. Singer

Abstract- Input shaping is a method of reducing residual vibrations in computer-controlled machines. Input shaping is implemiented by convolving the desired command signal with a sequence of impulses. The result of the convolution is then used to drive the system. Because input shaping alters the commanded trajectory, it has had questionable utility for trajectory following applications such as painting, cutting, and scanning. The effects of input shaping on trajectory following were investigated by simulating the response of a fourth-order system with orthogonal modes ;and conducting experiments on an S I 7 positioning stage. For nearly all values of experimental parameters, input shaping improvted trajectory following.

I. INTRODUCTION

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NPIJT shaping improves settling time and positioning accuracy by reducing residual vibrations in computer-controlled machines. The system model required with input shaping consists only 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. Because input shaping alters the commanded trajectory, it has had questionable utility for trajectory following applications such as painting, cutting, and scanning. The effectiveness of input shaping for twodimensional (2-D) trajectory following will be discussed in this palper. Robust input shaping has been investigated and extended by many researchers since its original presentation [17]. A technique for improving input shaping's insensitivity to 'modeling errors ,and parameter variations was developed [20]-[22]. The effectiveness of input shaping on multiple-mode systems was demonstrated 171. Input shapers containing negative impulses were shown to improve response. time [23]. Shaping was 'combined with a postmaneuver damping controller to improve large angle slewing [6]. Several researchers have used the ideas presented in [3], [16], and [17] to develop methods for designing input shapers in the x-plane [91, [151, [181, 1241. Input shaping reduced residual vibration and maximum deflections during the simulation of a large nonlinear spacebased antenna [ l ] , [2]. Two-mode input shapers were used to Manuscript received August 1, 1994; revised March 2, 1995. This work was supported by the Office of Naval Research Fellowship Program and Convolve, Inc. under NASA Contract NAS5-32034 and NSF Grant 9 101 441. This paper was recommended for publication by Associate Editor M. Peshkin and Editor A. Desrochers upon evaluation of reviewers' comments. W. E. Singhose is with the Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02141 USA. N.C. Singer is with Convolve, Inc., New York, NY 10038 USA. Publisher Item Identifier S 1042-296X(96)07251-5.

increase the throughput of a wafer-handling robot [13]. The performance of long-reach manipulators was improved with input shaping [SI, [ l l ] . Input shaping has been modified for use on systems with constant-force actuators [2],[lo], [12], [181, [191. The input shaper used to alter the command signal is determined from a set of constraint equations that limits the residual vibration. The constraint on vibration amplitude can be expressed as the ratio of residual vibration amplitude with shaping to that without shaping. If we assume the system is a second-order harmonic oscillator, then the vibration ratio can be determined from the expression for vibration amplitude given in [4]. The vibration from an input shaper is divided by the vibration from a single impulse to get the desired ratio, as shown in (1) at the bottom of the following page, where n is the number of impulses in the input shaper, A, and t , are the amplitudes and time locations of the impulses, w is the vibration frequency, and C is the damping ratio. Specifications for an input shaper usually require some amount of insensitivity to errors in the system model. The insensitivity constraint first proposed requires the derivative of the percentage vibration equation (1) (shown at the bottom of the next page) to be zero at the modeling frequency [ 161, [ 171. These constraints yield a zero vibration and zero derivative (ZVD) input shaper. An alternate constraint achieves significantly more insensitivity by relaxing the requirement of zero vibration at the damped modeling frequency 1201-[22]. By limiting the residual vibration at the modeling frequency to some small value, V, instead of zero, the zero vibration constraint can be enforced at two frequencies close to the modeling frequency. This set of constraints leads to extra-insensitive (EI) input shapers that are essentially the same duration in time as the ZVD shapers, but have significantly more insensitivity. Fig. 1 compares the sensitivity curves (plots of residual vibration versus normalized system frequency) for ZVD and E1 shapers. The curves in Fig. 1 show that the residual vibration will be small even if the system model is off by &20%. The effectiveness of input shaping for reducing residual vibration in point-to-point motions has been well established. However, very little work has been done to determine how input shaping affects trajectory following. Experiments in [5] show a five-bar-linkage manipulator follows a clover pattern better with shaping than without shaping. Other forms of feedforward compensation have shown promise for trajectory following applications [ 141. The shaping process alters the desired trajectory, so it is often assumed that input shaping

1042-296X/96$05.00 0 1996 IEEE

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 12, NO. 6, DECEMBER 1996

882

0.7

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Normalized Frequency ( u&t,,al /U ,,,,,'Icl )

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Fig. 1. ZVD and E1 sensitivity curves.

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Fig. 2.

Two-mode model of a flexible system under PD control

will degrade trajectory following. While this may be true for temporal trajectories (trajectories where the location as a function of time is important), this paper will show it is untrue for spatial trajectories, where only the shape of the trajectory is important. For this investigation we studied a simple two-mode system and gave it simple yet representative trajectories to follow. Our model, shown in Fig. 2, represents a system with two orthogonal modes under proportional-plus-derivative (PD) control. The flexibility and damping of the controller and structure have been lumped together into a single spring and damper for each mode. The inputs to the system are z and y position commands. This model is representative of gantry robots, coordinate measuring machines, and X Y stages. Hardware experiments on an X Y stage were used to verify the simulation results. The next section of this paper describes how input shaping affects the response to circular and square trajectory inputs. The following section presents two simple methods for altering the unshaped command to better utilize input shaping. In the final section, experimental results are presented and compared to simulation results.

11. CIRCULARTRAJECTORIES The response of our model to unshaped and shaped constantvelocity unit-circle inputs was simulated for a large range of

Fig. 3 . Comparison of ZVD-shaped and unshaped responses to a unit-circle input.

modeling parameters. The response without input shaping is a function of the commanded speed around the circle, the ratio of the two vibration modes, the damping, and the initial departure angle relative to the lowest mode (for this paper the low mode will always be in the x direction). The response to shaped circular inputs depends on the above variables and the type of input shaper selected. Fig. 3 compares the unsh,aped and ZVD-shaped responses for the case where the frequencies are fz = fy = 1 Hz, the damping ratios are Cz == Cy = 0.05, and the unit-circle trajectory command has a duration of 10 s. The circle is initiated in the -n: direction at location (0, 0). By examining Fig. 3 we can say, qualitatively, that the shaped response is closer to the desired trajectory than the unshaped response. We can compare the responses quantitatively by examining the maximum and minimum values of the response radius, the envelope enclosing the radius, and the mean and standard deviation of the error. This comparison is shown in Table I along with the performance nieasures for the desired unit-circle response. The shaped respoinse is substantially closer to the desired performance measures in every category except mean value. The mean value results are understandable because the unshaped response oscillates about the desired radius, while the shaped response tracks almost the entire circle with a nearly constant, but slightly smaller than desired radius. Input shaping leads to a smaller than commanded radius because the shaped inputs lag the unshaped inputs. We will address this issue in a subsequent section. Input shaping improves circular trajectory following over a large range of I , T (frequency ratio), and command speeds. To display these data, we combine f3:and the command speed into one unit called vibration cycles/circle. This measure tells us how fast the system i,s commanded relative to its lowest natural frequency. For the responses shown in Fig. 3, the command speed was 10 cycles/circle because the frequency was 1 Hz

SINGHOSE AYD SINGER: TWO-DIMENSIONAL TRAJECTORY FOLLOWING

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TABLE I PERFORMANCE MEASURES FOR A N UNSHAPED, AND DESIRED UNIT CIRCLE RESPONSE ZVD-SHAPED,

- m -ZVD Shaped, r=1.5 -ZVD Shaped, ~ 1 . 3

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-80

Unshaped,