1 Proceedings of the 1996 IFAC World Congress UNITY-MAGNITUDE INPUT SHAPERS AND THEIR RELATION TO TIME-OPTIMAL CONTROL Lucy Y. Pao University of Colorado Boulder, CO William E. Singhose Massachusetts Institute of Technology Cambridge, MA 2139 Abstract: Input shaping reduces residual vibrations by convolving a sequence of impulses, an input shaper, with the desired system command. Using negative impulses in the shaper leads to faster maneuvers. Unfortunately, when negative input shapers are used, there is no guarantee that the shaped command will satisfy actuator limitations. A new type of negative input shaper is presented that satisfies actuator limits for a large class of unshaped commands. The performance of these input shapers is compared to the time-optimal control and other types of input shapers. Keywords: Shaping Filters, Feedforward Compensation, Point-to-Point Control, Modeling Errors 1. INTRODUCTION Input shaping reduces residual vibrations in computer controlled machines. A sequence of impulses, an input shaper, is convolved with the desired system command to produce a shaped input. This process is demonstrated in Figure 1. The amplitudes and time locations of the impulses are determined by solving a set of equations. Many papers have been published on robust input shaping since its original presentation (Singer and Seering, 199). Methods for increasing the insensitivity to modeling errors have been presented (Singhose et al., 1994a; Singhose et al., 1995). Input shaping was shown to be effective for multiple-mode systems (Hyde and Seering, 1991), as well as for systems equipped with constant-force actuators (Liu and Wie, 1992; Singh and Vadali, 1994; Singhose et al., 1996). Input shaping was used to improve the throughput of a silicon wafer handling robot (Rappole et al., 1994) and it has been proposed as a means of reducing residual vibrations of long reach manipulators (Magee and Book, 1995). Input shaping was also a major component of an experiment in flexible system control which flew on the Space Shuttle in March 1995 (Tuttle and Seering, 1995). Move time can be significantly reduced by allowing the input shaper to contain negative impulses (Singhose et al., 1994b). The equivalence of time-optimal control and input shaping using special negative input shapers has been demonstrated (Pao and Singhose, 1995b). Others have developed negative input shapers using zero placement algorithms (Seth et al., 1993; Jones and Ulsoy, 1994; Tuttle 1..5 Unshaped Input * A 1 A 2 Input Shaper 1. Shaped Input A.5 1 Component Shaped A 2 Component Input Figure 1: The Input Shaping Process. and Seering, 1994; Magee and Book, 1995). Unfortunately, the negative input shapers presented thus far can lead to shaped commands that overcurrent the actuators. That is, the shaped commands have small periods when they require more current (torque) than the unshaped commands. This paper will first present a new type of negative shaper that can be used to shape a large class of inputs without causing overcurrenting. The new shapers are compared to other shaping methods. The properties that will be compared are move speed, robustness to modeling errors, and transient vibration amplitude. 2. UNITY-MAGNITUDE INPUT SHAPERS The constraint equations used to design an input shaper always limit the amplitude of residual vibration. The constraints are based on a superposition of simple systems like the one shown in Figure 2. The constraint on vibration amplitude can be expressed as the ratio of residual vibration
2 u m 1 x 1 b k m 2 x 2 Figure 2: System with Flexible and Rigid Body Modes. amplitude with shaping to that without shaping. This percentage vibration can be determined by using the expression for residual vibration of a second-order harmonic oscillator of frequency ω radians/sec and damping ratio ζ, which is given in (Bolz and Tuve, 1973). The vibration from a series of impulses is divided by the vibration from a single unity-magnitude impulse to get the percentage vibration: V(ω) = e ζωt n C 2 + S 2 (1) where, C = n i=1a i e ζωt i cos(ω d t i ) and S = n i=1a i e ζωt isin(ω d t i ). Ai and ti are the amplitudes and time locations of the impulses, n is the number of impulses in the input shaper, and ω d = ω 1 ζ 2. In addition to limiting vibration amplitude, robust shaping formulations require some amount of insensitivity to modeling errors. Insensitivity refers to the shaper s ability to reduce residual vibration in the presence of modeling errors. A shaper's insensitivity is displayed graphically by a sensitivity curve: a plot of vibration versus frequency, (Eq. 1 plotted as a function of ω). A sensitivity curve reveals how much residual vibration will exist when there is an error in the estimation of the system frequency. This paper describes three types of input shapers: Zero Vibration (ZV) shapers. These shapers satisfy Eq. 1 with V set equal to zero at the modeling frequency (Smith, 1958; Singer and Seering, 199). Zero Vibration and Derivative (ZVD) shapers. These satisfy the ZV constraints and the constraint that the derivatives of Eq. 1 with respect to ω and ζ be zero at the modeling frequency (Singer and Seering, 199). Extra-Insensitive (EI) shapers. These shapers satisfy Eq. 1 with V set equal to a small non-zero value. The insensitivity to modeling errors is then maximized (Singhose et al., 1994a; Singhose et al., 1995). The vibration reducing characteristics of the input shapers can be compared using sensitivity curves as shown in Figure 3. The ZV shaper is very sensitive to modeling errors; small errors in the modeling frequency lead to significant residual vibration. The ZVD shaper has considerably more insensitivity to modeling errors, which is evident by noting that the width of the ZVD curve is much larger than the width of the ZV curve. The additional insensitivity of the ZVD incurs a time penalty; the ZVD shaper is longer than the ZV shaper by one half period of the vibration. This means that a ZVD shaped command will be one half period of vibration longer than a ZV shaped Vibration Percentage ZV Shaper ZVD Shaper EI Shaper Normalized Frequency (ω actual /ω model ) Figure 3: Sensitivity Curves for the ZV, ZVD, and EI Input Shapers. command. The EI shaper is essentially the same length as the ZVD shaper, but it is considerably more insensitive. Robustness can be compared quantitatively by measuring the curve width at a specific level of vibration. For example, the 5% insensitivity is obtained by measuring the width at the level indicated by the dashed line in Figure 3. In addition to constraints on residual vibration and sensitivity to modeling errors, some type of constraint must be placed on the amplitudes of the impulses in the shaper. Requiring that the sum of the amplitudes equal one: n A i = 1 (2) i=1 ensures that the shaped command will reach the same final setpoint as the unshaped command. If no additional constraints are placed on the amplitudes, then when the constraint equations are solved while minimizing the length of the shaper, the impulse amplitudes will go to positive and negative infinity (Singer, 1989). The most common solution to this problem is to restrict the amplitudes to only positive values. This is an attractive solution because when an all-positive input shaper is convolved with any unshaped input, the actuator limitations are preserved. That is, if the unshaped command does not cause actuator limits to be exceeded, then neither will the shaped command. Although positive input shapers are well-behaved, they move the system slower than shapers containing negative impulses. If the positive constraint is abandoned, another amplitude constraint must be used to limit the impulses to finite positive and negative values. A previously proposed constraint limits the partial sums of the impulse sequence to be less than one (Singhose et al., 1994b): p A j j=1 1, p = 1, 2,..., n. (3) When solving (1)-(3) while minimizing the shaper length, the impulse amplitudes are such that the equality sign in (3) holds, giving impulse amplitudes of: A = [ ]. (4) When convolving this shaper with a step command whose amplitude is equal to the maximum acceleration level, the result of the convolution is a series of alternating-sign pulses which is equivalent to the time-optimal control (Pao and Singhose, 1995b).
3 1-2 2 * Over-Currenting -2 Over-Currenting Figure 4: Shaping with Negative Shapers Can Lead to Overcurrenting. Unfortunately, most real inputs are not a single step in acceleration. Figure 4 shows that when a shaper of the form given in (4) is convolved with the acceleration command corresponding to a trapezoidal velocity profile there will be short periods of overcurrenting. The presence of the overcurrenting is tolerable for many applications because most systems have peak current capabilities that greatly exceed the steady-state levels. However, elimination of the overcurrenting altogether is desirable and has motivated the derivation of a new amplitude constraint. Velocity commands that consist of ramps and constants are very common. These correspond to acceleration commands that consist of step changes: u = u i (t) T i 1 < t < T i, i = 1, 2,...,r (5) where u i 1 (where we assume the actuator limits are at -1 and +1 for simplicity), T i is the time where the command transitions from u i to u i+1, and r is the number of step changes in the command. When commanding a flexible system, each transition causes vibration. If the magnitude change and timing of the transitions are carefully selected, there may be very little or no vibration at the end of the command. However, commands are usually specified independently of the flexibility in the system and input shapers can be used to reduce residual vibration. An input shaper shapes each of the transitions in (5) to reduce or eliminate residual vibration at the end of the command. Evaluating the shaping of the i th transition yield constraints that ensure that the shaped command does not violate actuator limits. Before the transition, u = u i where -1 u i 1; after the transition, u = u i+1, where -1 u i+1 1. If the shaper impulse amplitudes satisfy p 1 u i M i A j 1 u i, p = 1, 2,...,n (6) j=1 where M i = u i+1 - u i, then the actuator limits will not be violated. When convolved with the shaper, the shaped command u s equals u i for t < t 1 (the time of the first impulse). At t 1, the shaped command u s becomes M i A 1 + u i and it is necessary to have 1 M i A 1 + u i 1 Command Figure 5: Shaping a Bang-Bang Command with a Unity-Magnitude Shaper. for actuator limits not to be exceeded. Similarly, at t 2, it is necessary to have 1 M i ( A 1 + A 2 )+ u i 1 etc. which leads to constraint (6). Constraint (6) assumes that the i th transition must be completely shaped before the (i+1) th transition is reached. This requires that the minimum length of time between transitions satisfy min( T i T i 1 ) > t n (7) i.e., the minimum time between transitions must be larger than the shaper length. If constraint (7) is not met, actuator limits may or may not be exceeded depending on the unshaped command and the shaper design. If (2), (6), and (7) are satisfied, then actuator limits will not be exceeded. For speed critical applications, bang-bang acceleration commands are very common. These commands are a subset of the class of commands characterized in (5) with u i taking on only the values +1 or -1. M i is either +2 or -2 and constraint (6) simplifies to p A j 1, p = 1, 2,...,n. (8) j=1 When solving for the minimum-time input shapers satisfying (2) and (8), the impulse amplitudes are: A i = ( 1) i+1, i = 1,..,n (9) where n is odd. For example, if n = 5, then A = [ ]. (1) Note that a shaper meeting the amplitude constraint of (9) automatically satisfies (2). For bang-bang acceleration commands, satisfaction of (2), (7), and (8) guarantee that actuator limits are not exceeded. Figure 5 shows the shaped command resulting from the convolution of a shaper satisfying (9) with a bang-bang command. Condition (7) is easily satisfied for most real bang-bang commands because shapers are generally about one period of vibration in length, but typical moves last longer than several periods of vibration. 2.1 Unity-Magnitude ZV and ZVD Shapers Combining the amplitude constraint of (9) with the vibration constraint of (1), the impulse time locations of the
4 new negative unity-magnitude ZV shapers were solved for as a function of damping ratio for a one-mode system as in Figure 2. To satisfy the ZV constraints there must be three impulses in the shaper. The solutions were obtained using GAMS (Brooke et al., 1988) a linear and non-linear programming package. To eliminate the need for using an optimizer to calculate these shapers, curve fits to impulse time locations are provided in Table 1. To obtain a shaper for a system under consideration, substitute in the approximate frequency and damping ratio into the tabulated equations. ZV shapers do not work well for most applications because they are sensitive to modeling errors, as shown in Figure 3. To generate shapers that work on most real systems, sensitivity constraints like those outlined above must be added. To satisfy the ZVD constraints, the shaper must contain five impulses, i.e., its amplitudes will be given by (1). Table 1 gives the impulse time locations as a function of system frequency and damping ratio. Additional derivatives of (1) with respect to ω and ζ can be constrained to zero to further increase the insensitivity to modeling errors (Singer and Seering, 199). 2.2 Unity-Magnitude EI Shapers Extra-insensitive constraints achieve significantly more insensitivity by relaxing the constraint of zero vibration at the damped modeling frequency. If the residual vibration at the modeling frequency, ω, is limited to some small value, V, instead of zero, and the zero vibration constraint is enforced at two frequencies, one higher than ω and the other lower than ω, then this set of constraints leads to input shapers that are essentially the same length in time as the ZVD shapers, but have more insensitivity. For a detailed discussion of these constraints see the references. Equations describing the EI shapers as a function of system frequency and damping are given in Table 1. The EI algorithm can be extended by requiring more than one hump in the sensitivity curve (Singhose et al., 1995). 3. COMPARISON WITH OTHER SHAPERS The time-optimal control for rest-to-rest motion of a rigid body is bang-bang with the switch occurring at midmaneuver, as shown in Figure 6. The bang-bang input can be interpreted as the convolution of a 3-impulse sequence and a step input. Applying these bang-bang control inputs to flexible structures can cause large amounts of both residual and move-time vibration. In this section, the negative shapers of the previous section are used to shape the time-optimal rigid-body command to obtain shaped commands for flexible systems. Since the bang-bang commands that are being shaped are the time-optimal commands for a rigid body (RB), the commands generated by the negative shapers are denoted as, NEG ZVD RB, and. t 1 t 2 t 3 * Figure 6: Using Input Shaping to Form the - Optimal Control of a Rigid Body. t 2 t 3 When time-optimal rigid-body commands are shaped with a positive shaper, the resulting commands are referred to as,, and POS EI RB (Pao and Singhose, 1995a). A major advantage of using positive input shapers is that they can be specified analytically in terms of the system parameters ω, ζ, and L, where L is the desired move distance. However, using the negative unitymagnitude shapers of the previous section will lead to faster maneuvers. And, given the information in Table 1, they are almost as easy to use. In this section, the tradeoffs between move duration, robustness to modeling errors, and vibration reduction are examined. The negative unity-magnitude shapers are compared to positive shapers, as well as to the time-optimal commands subject to actuator constraints. The command that results from shaping the rigid-body bang-bang command is not itself a bang-bang signal. The shaped command has a variable amplitude that is not always equal to the maximum actuator limit; see Figure 5. For rest-to-rest motion, Pontryagin's maximum principle states that the time-optimal control consists of a series of alternating positive and negative pulses. That is, commands exist that are shorter in duration than the shaped commands, yet still meet the requirement of zero residual vibration. It has been shown that the time-optimal commands satisfying the ZV, ZVD, and EI constraints are a series of bang-bang pulses (Pao and Singhose, 1995b). The time-optimal control can be generated with input shaping if the unshaped input is a step input whose magnitude is equal to the maximum actuator limit and the input shaper has the form: A [ i t i ] = [ 1 t t 3 t 4... t n 1 t n ]. (11) Because the time-optimal shapers lead to constant amplitude pulse (CAP) commands, the time-optimal shapers are denoted as ZV CAP, etc. There are no positive and negative versions of the time-optimal shapers; they must contain negative impulses. Figure 7 shows the relative time-optimality of the various shapers compared to the ZV CAP shaper, which is the timeoptimal command for rest-to-rest motion of the system given only actuator constraints. The results shown are for shapers that guarantee no overcurrenting. That is, the results for the,, and NEG EI
5 % Increase in Move Duration % Increase in Move Duration Saved with Negative Shapers Damping Ratio, ζ Saved with Negative Shapers a) Move Distance, L b) Figure 7: Percentage Increase in Move Duration Over the ZV CAP Shaper. a) As a Function of Damping (L=1), b) As a Function of Move Distance (ζ=.4). RB shapers are only given for parameter values where the constraints (2), (7), and (8) are met. For the same robustness constraints (ZV or ZVD), the negative shapers lead to faster maneuvers than the corresponding positive shapers. The and the yield approximately the same maneuver times. Figure 8 presents 5% insensitivities of the various shapers relative to that for the ZV CAP. The 5% insensitivity is the width of the sensitivity curve at V=.5 and represents the frequency range over which the vibration would remain below 5%. Although the and cost somewhat (2 to 45%) in maneuver time, they offer a significant increase in robustness. Compared with the ZV CAP shaper, the and shapers exhibit increases of 3 to 45 percent in 5% insensitivity. For most parameter values, the 5% insensitivities for the are 1 to 3% greater than the. Thus, while the ZV CAP shaper leads to the fastest maneuvers given actuator limits, the and NEG ZVD RB lead to moderately longer maneuvers that are significantly more robust to modeling errors. Another quality to compare among the shapers is the amount of vibration during the move rather than just at the end of a maneuver. Decreasing vibration during the move can increase the lifetime of many systems and improve the trajectory following of the endpoint. The vibration during the move is defined as the mean of the magnitude of deviation of the flexible structure position from the position of a rigid body experiencing the same actuator force inputs; that is, the amount of additional motion the flexible structure has over a rigid body subject to the same forces. % Increase in 5% Insensitivity % Increase in 5% Insensitivity Damping Ratio, ζ a) Positive ZV RB Positive ZVD RB Negative ZV RB Negative ZVD RB Negative EI RB Move Distance, L b) Figure 8: Percentage Increase in 5% Insensitivity Over the ZV CAP Shaper. a) As a Function of Damping (L=1), b) As a Function of Move Distance (ζ=.4). For the one-bending-mode model of a flexible structure as shown in Figure 2, the vibration during a maneuver is V m = 1 t n x t 2 (t) x 1 (t) dt (12) n where x 1 and x 2 are the positions of the masses in Figure 2. Figure 9 shows the amount of move vibration, Vm, for the various shapers. The negative ZV and ZVD shapers yield about 5 to 1% more move vibration than their positive shaper counterparts, and the performances of the NEG ZVD RB and shapers are again very close in terms of move vibration. In general, the ZVD RB is seen to be slower, more robust to parameter variations, and causes less vibration during moves than the other shapers. However, given the tradeoff of gains in robustness over loss in speed, the shaper gives the greatest increase in insensitivity and greatest decrease in move vibration for only a moderate increase in maneuver time. The and NEG ZVD RB shapers are 1 to 3% faster than the and shapers, respectively, while only degrading the insensitivity and move vibration by 2 to 1%. 4. CONCLUSIONS A new method of designing negative shapers has been proposed which will not cause actuator limits to be exceeded for a large class of commands. Various negative shapers using this method have been tabulated, evaluated and compared with other shaping strategies. Using the performance metrics of speed, insensitivity, and transient vibration, the new negative shaper designs give better overall performance than corresponding positive shapers.
6 t2 t3 t4 t5 t1 % Decrease in Move Vibration % Decrease in Move Vibration Damping Ratio, ζ a) Move Distance, L b) Figure 9: Percentage Decrease in Move Vibration Over the ZV CAP Shaper. a) As a Function of Damping (L=1), b) As a Function of Move Distance (ζ=.4). Table 1: Unity-Magnitude Shapers ti = ( M + M 1 ζ + M 2 ζ 2 + M 3 ζ 3 )T, T = 2π ω Shaper A i t i M M 1 M 2 M 3 Unity- 1 t 1 Magnitude -1 t ZV 1 t Unity- 1 t 1 Magnitude -1 t ZVD 1 t t t Unity- 1 t 1 Magnitude -1 t EI 1 t V=5% -1 t t Unity- 1 t 1 Magnitude -1 t Hump EI 1 t V=5% -1 t t t t 7 5. REFERENCES Bolz, R. E. and Tuve, G. L., (1973). CRC Handbook of Tables for Applied Engineering Science, Boca Raton, FL: CRC Press, Inc., pp Brooke, A., Kendrick, D. and Meeraus, A., (1988). GAMS: A User s Guide, Redwood City, CA: The Scientific Press. Hyde, J. M. and Seering, W. P., (1991). Inhibiting Multiple Mode Vibration in Controlled Flexible Systems, Proceedings of the American Control Conference, Boston, MA. Jones, S. D. and Ulsoy, A. G., (1994). Control Input Shaping for Coordinate Measuring Machines, Proceedings of the American Control Conference, Baltimore, MD, Vol. 3, pp Liu, Q. and Wie, B., (1992). Robust -Optimal Control of Uncertain Flexible Spacecraft, Journal of Guidance, Control, and Dynamics, Vol. 15(3), pp Magee, D. P. and Book, W. J., (1995). Filtering Micro- Manipulator Wrist Commands to Prevent Flexible Base Motion, American Control Conference, Seattle, WA. Pao, L. Y. and Singhose, W. E., (1995a). A Comparison of Constant and Variable Amplitude Command Shaping Techniques for Vibration Reduction, IEEE Conference on Control Applications, Albany, New York. Pao, L. Y. and Singhose, W. E., (1995b). On the Equivalence of Minimum Input Shaping with Traditional -Optimal Control, IEEE Conference on Control Applications, Albany, NY, pp Rappole, B. W., Singer, N. C. and Seering, W. P., (1994). Multiple-Mode Impulse Shaping Sequences for Reducing Residual Vibrations, Proceedings of the ASME Mechanisms Conference, Minneapolis, MN. Seth, N., Rattan, K. S. and Brandstetter, R. W., (1993). Vibration Control of a Coordinate Measuring Machine, IEEE Conference on Control Applications, Dayton, OH, pp Singer, N. C., 1989, Residual Vibration Reduction in Computer Controlled Machines, MIT Artificial Intelligence Lab. Singer, N. C. and Seering, W. P., (199). Preshaping Command Inputs to Reduce System Vibration, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 112(March), pp Singh, T. and Vadali, S. R., (1994). Robust -Optimal Control: A Frequency Domain Approach, AIAA Journal of Guidance, Control and Dynamics, Vol. 17(2), pp Singhose, W., Derezinski, S. and Singer, N., (1996). Extra- Insensitive Input Shapers for Controlling Flexible Spacecraft, Accepted to the AIAA Journal of Guidance, Control, and Dynamics. Singhose, W., Porter, L. and Singer, N., (1995). Vibration Reduction Using Multi-Hump Extra-Insensitive Input Shapers, American Control Conference, Seattle, WA, Vol. 5, pp Singhose, W., Seering, W. and Singer, N., (1994a). Residual Vibration Reduction Using Vector Diagrams to Generate Shaped Inputs, ASME Journal of Mechanical Design, Vol. 116(June), pp Singhose, W., Singer, N. and Seering, W., (1994b). Design and Implementation of -Optimal Negative Input Shapers, International Mechanical Engineering Congress and Exposition, DSC 55-1, Chicago, IL, pp Smith, O. J. M., (1958). Feedback Control Systems, New York: McGraw-Hill Book Company, Inc., pp Tuttle, T. D. and Seering, W. P., (1994). A Zero-placement Technique for Designing Shaped Inputs to Suppress Multiplemode Vibration, Proceedings of the American Control Conference, Baltimore, MD, Vol. 3, pp Tuttle, T. D. and Seering, W. P., (1995). Vibration Reduction in -g Using Input Shaping on the MIT Middeck Active Control Experiment, Proceedings of the American Control Conference, Seattle, WA, Vol. 2, pp