Proceedings of the 1994 American Control Conference
Initial Investigations into the Effects of Input Shaping on Trajectory Following William Singhose Neil Singer, Ph.D. Convolve, Inc., Armonk, NY 914-273-4042
[email protected] 20
Vibration Percentage
Abstract 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 XY positioning stage. For nearly all values of damping and frequency ratio, the shaped inputs result in significantly better trajectory following than unshaped inputs. When the trajectory improvement techniques in this paper are used, the shaped inputs perform considerably better for all parameter values.
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Introduction Input shaping improves settling time and positioning accuracy by reducing residual vibrations in computer controlled machines. The method requires only 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. The input shaper is determined from a set of constraint equations that limit the residual vibration of the system. The constraint on vibration amplitude can be expressed as the ratio of residual vibration amplitude with shaping to that without shaping. This percentage vibration is given by:
%Vibration = e − ζωtn {(ΣAi eζωti cos(ω 1− ζ 2 ti ))2 1
+ (ΣAi eζωti sin(ω 1− ζ 2 ti ))2 } 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. Specifications for an input shaper usually require some amount of insensitivity to errors in the system model. The insensitivity constraint proposed by Singer and Seering requires the derivative of the percentage vibration equation (Eq. 1) to be zero at the modeling frequency [9,10]. These constraints yield a zero vibration and zero derivative (ZVD) input shaper. An alternate constraint achieves significantly more insensitivity by relaxing the constraint of zero vibration at the damped modeling frequency [11,12]. 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 length in time as the ZVD shapers presented in [10], but have significantly more insensitivity. Figure 1 compares the sensitivity curves, plots of residual vibration vs. system frequency for ZVD and EI shapers. The EI shaper results in low levels of vibration over a wider range of frequencies than the ZVD shaper, hence the name, extra-insensitive. A great variety of restrictions can be added to the constraint equations to customize input shaping to a specific system. For multiple mode systems, a set of equations consisting of one
Figure 1: ZVD and EI Sensitivity Curves vibration constraint (Eq. 1) for each mode and one insensitivity constraint for each mode yields very effective input shapers [4]. The response time can be improved by allowing the input shaper to contain negative impulses [8,14]. Input shaping can be used successfully on systems with only constant force actuators, such as on-off reaction jets, by specifying the amplitudes of the impulses in the input shaper [6,13,16]. Harmful effects from course digitization of the input can be eliminated [7]. The effectiveness of input shaping for reducing residual vibration in point-to-point motions has been well established [1-16]. However, very little work has been done to determine how input shaping effects trajectory following. Experiments in [3] show a five-bar-linkage manipulator follows a clover pattern better with shaping than without. The shaping process alters the desired trajectory, so it is often assumed that input shaping 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 move is important. Spatial trajectories compose a large percentage of the trajectory following applications including cutting, scanning, and measuring. One major reason the effects of input shaping on trajectory following have not been rigorously investigated is due to the difficulty in measuring the deviation from a desired trajectory. For this reason we studied a simple two-mode system and gave it uncomplicated, yet representative trajectories to follow. Our model, shown in Figure 2, represents a system with two orthogonal modes under 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 x and y position commands. This model is very representative of gantry robots, coordinate measuring machines, and XY stages with flexible structures mounted to them. 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.
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Table 1: Performance Measures for an Unshaped, ZVD Shaped, and Desired Unit Circle Response. Radius of ZVD Response Unshaped Shaped Desired 0.5000 Maximum 0.5313 0.5000 0.5000 Minimum 0.4804 0.4927 0 Envelope 0.0509 0.0073 0.5000 Mean 0.5052 0.4935 0 Std 0.0091 0.0020
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Figure 4: Unshaped and ZVD Shaped Command.
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Circular Trajectories The response of the model shown in Figure 2 to unshaped and shaped unit-circle inputs was simulated in MATLAB. The unshaped response is a function of commanded speed around the circle, frequency ratio (r=high frequency/low frequency), 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 unit-circle inputs is a function of the above variables and the type of input shaper selected. Figure 3 compares the unshaped and ZVD shaped responses for the case where the frequencies are f1 =f2 =1 Hz, the damping ratios are ζ1=ζ 2 =0.05, and the unit-circle command has a duration of 10 seconds. The circle is initiated in the +x direction at location (0,0). By examining Figure 3 we can say, qualitatively, that the shaped response is closer to the desired trajectory than the unshaped. We can also compare the responses quantitatively by examining the maximum and minimum values of the response radius, the envelope enclosing the radius, as well as, calculating the mean and standard deviation of the radius. This comparison is shown in Table 1 along with the performance measures for the desired unit-circle response. The shaped response 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
Radius Envelope
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smaller than commanded radius because the convolution process yields inputs that lag the unshaped inputs. This can be seen in Figure 4 where shaped and unshaped command signals are compared. Input shaping improves circular trajectory following over a large range of ζ, r (f2 /f1), and command speed. To display this data, we combine f1 and the command speed into one unit called vibration cycles/circle. This measure tells us how fast the system is commanded relative to its natural frequency. In the above example, the command speed was 10 cycles/circle because the frequency was 1 Hz and the desired input lasted for 10 seconds. As this measure is decreased, the system is required to move more rapidly, so performance is poor for low values of cycles/circle. When our two-mode model has different frequencies in the x and y directions, the cycles/circle will be measured relative to the low mode. Figure 5 compares the ZVD shaped and unshaped radius envelope (maximum radius - minimum radius) as a function of cycles/circle and ζ for the case of r=1. For every command speed and damping ratio, the envelope on the shaped response is smaller than that for the unshaped response. Furthermore, the performance is no longer a function of ζ when input shaping is used because there is no vibration to damp out. The results shown in Figure 5 are only for the specific case where r=1. For all values of r, the benefit from shaping is
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Figure 6: Radius Envelope as a Function of Departure Angle for Cycles/Circle=11.
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Figure 8: Comparison of ZVD Shaped and Unshaped Responses to a Unit-Square Input (r =1.5, ζ1=ζ2=0.05, Cycles/Square=15).
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Frequency (Hz) Figure 7: Response Radius Envelope as System Frequency Varies (Inputs Were Designed for f1 =1 Hz, r=1, ζ1 =ζ 2 =0, and Cycles/Circle=11). greatest in cases with low damping. As damping is increased the performance of the unshaped response approaches the shaped performance. When r differs from 1, the input shaper needed to eliminate the two modes will be longer than the shaper for r=1 (provided the same type of input shaper is used). The longer the shaper, the more the shaped command lags the unshaped and, therefore, the greater the deviation from the desired radius. The increase in shaper length is significant for 1.3≤ r ≤ 2.5. When r ≤1.3, a single mode EI shaper can be used effectively to cancel both modes. For r ≥ 2.5, the shaper length will be only slightly longer than for r=1. For all of the results presented so far, the unit circle command has been initiated in the +x direction, which is parallel to the low mode of our model. If the angle of departure is varied relative to the +x direction, the trajectory following performance will also vary. The performance measures repeat every 180 o of departure angle, so we will present results that span from -90o to +90o relative to the +x direction, that is, from the -y direction to the +y direction. Figure 6 shows the unshaped and ZVD shaped radius envelope (max radius - min radius) for the cases of 11 cycles/circle, ζ1=ζ 2 =0, and r=1.5, 1.3. For every value of the departure angle, the envelope with shaping is at least four times smaller than without shaping. Figure 7 also reveals that it is a poor idea to start the circle in the direction of the low mode when shaping is not used. This makes sense, as the start-up transient excites the low mode instead of the high mode. When shaping is used, it makes little difference what departure angle is used to commence the circular trajectory. It is interesting to note that for the case of r=1.3, the best departure angle is not parallel to the high mode, rather at an angle of +70 o relative to the x direction. This phenomenon only occurs with low (r
