saturation compensating input shapers - CiteSeerX

SATURATION COMPENSATING INPUT SHAPERS FOR REDUCING VIBRATION Raynald Eloundou, William Singhose The Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, USA

ABSTRACT Input shaping can be used to eliminate or greatly reduce residual vibration of flexible mechanical systems. The input shaping filter is based on the characteristics of the system, namely the frequencies and damping ratios. When the actuator saturates, the dynamics become nonlinear. As the nonlinearity increases, the effectiveness of standard input shapers tends to degrade. This paper presents a new class of input shapers that account for actuator saturation. 1. INTRODUCTION Unwanted vibration is a major problem that affects the performance of many flexible mechanical systems. For example, robotic arms, construction cranes, and satellite positioning systems are often limited in their speed and accuracy by vibration. Vibration has many possible solutions, each with their advantages and disadvantages. If the system under consideration can be represented by the block diagram shown in Fig. 1, then four possible solutions are obvious. The physical plant, Gp, can be modified to make it less flexible, or the feedback control, G1, can be tuned to damp out vibration. The feedforward block, G2, can be used to inject control effort into the loop, so as to negate vibration. The fourth option is the command generator block, G3. This block has some similarities with the feedforward block, but it does not, in general, inject actuator effort directly onto the physical plant. The focus of this paper is to develop a command generation scheme that can take into account actuator saturation as it attempts to eliminate the vibration. If the system dynamics are known with some confidence, then there are several techniques for generating commands that will negate the system's flexible modes [1-3]. Input shaping is one type of command generation scheme that is implemented by convolving a sequence of impulses with the command signal [4]. This process is depicted in Fig. 2. While this approach is an FIR filter, it

G2 Yd

G3

+ G1 U R - E

GP

Y

Fig.1 Block Diagram of Generic System L

to

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A1

A2 ∆

L

to ∆

Fig.2 Input Shaping Process has been shown to be very effective on mechanical vibration [5]. Input shaping has been implemented on a variety of systems. The performance of long-reach manipulators [6, 7], cranes [8-10], and coordinate measuring machines [11-13] was improved with input shaping. Various other vibration control techniques have been combined with shaping. Hillsley and Yurkovich used input shaping for large-angle movements of a two-link robot, then switched to feedback control when near the desired position [14]. A combined input shaper and feedback controller was successfully implemented by Drapeau and Wang on a five-bar linkage manipulator in [15]. Magee and Book used input shaping in conjunction with feedback control to reduce the vibration of a small articulated robot mounted on the end of a long, slender beam [7]. Kenison and Singhose combined shaping and PD control in an optimal manner [16]. Murphy and Watanabe demonstrated that designing input shapers in the z-domain is easy and effective [17]. Their shaping algorithm made it easy to adapt the shaper parameters to a changing system because only the impulse amplitudes (and not the time spacings) needed to be changed. Magee and Book demonstrated that changing the impulse time locations in real time was

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The benefits provided by input shaping can be easily demonstrated by applying a ZV shaper to a second-order harmonic oscillator. Two types of baseline reference commands are considered, a unit step input and a unit Scurve profile. Figure 4 shows the response to the step input and the shaped step input when there is a 5% error in the modeling frequency. Obviously, the vibration is greatly attenuated by the shaping process. The question arises as to whether this large reduction occurs because a step function is used. As it turns out, the percentage vibration reduction is not a function of the baseline command. Recall the vibration reduction predicted by Fig. 3 is a function of the shaper only. To demonstrate this, Fig. 5 shows the response to an S-curve and a shaped S-curve, again with a 5% error in the modeling frequency. The unshaped vibration is smaller for the S-curve, but the vibration with the shaped S-curve is also smaller. Therefore the percentage vibration

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Fig.4 Response of a 2nd-Order Harmonic Oscillator to a Step and Shaped Step Input with a 5% Error in Modeling Frequency S-curve InputResponses Response to an S-curve Input Shaped S-curve Input Response to a Shaped S-curve Input

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Amplitude

There are many types of input shapers, among them the Zero Vibration (ZV) shaper [1], the Zero VibrationDerivative (ZVD) shaper [4], and the Extra-Insensitive (EI) shaper [23]. Each of the shapers can be best characterized by their robustness. A sensitivity curve, which is a plot of the percentage residual vibration versus the normalized frequency, is a good tool for characterizing these shapers. The sensitivity curves shown in Fig. 3 have the percentage residual vibration on the vertical axis and the normalized frequency, defined as the actual frequency, ωa, divided by the modeling frequency, ωm, on the horizontal axis. To compare robustness quantitatively a tolerance limit of residual vibration is set. For instance, if the tolerable limit of 5% residual vibration is set, the insensitivity is defined as the width of the curve below the specified level of 5%. The larger the insensitivity for a shaper, the more robust that particular shaper is to modeling errors.

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Fig.3 Sensitivity Curves for ZV, ZVD and EI Shapers more problematic [18, 19]. Several others have extended the z-domain shaper design techniques [11, 20, 21], a particularly straightforward approach was described by Tuttle [22].

Step Input Response to a Step Input ShapedResponses Step Input Response to a Shaped Step Input

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Residual Vibration (%)

35 ZV 30 ZVD 25 EI 20 0.40 15 0.28 10 0.06 5 0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ωactual/ω ωmodel) Normalized Frequency (ω

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Fig.5 Response of a 2nd-Order Harmonic Oscillator to an S-Curve and a Shaped S-Curve with a 5% Error in Modeling Frequency reduction is the same, regardless of the baseline reference command. 2. SYSTEMS WITH SATURATION Standard input shapers are designed based on linear system theory. However, the process works well on a large variety of nonlinear systems. This effectiveness arises from its robustness to modeling errors. As the degree of nonlinearity increases, the effectiveness may decrease. Therefore, it is worthwhile to modify the shaping algorithm to compensate for large nonlinearities. Many researchers have developed adaptive shaping algorithms that change the input shaper over time in order to better deal with the nonlinear dynamics [24, 25]. In this paper the nonlinearity caused by actuator saturation is studied. As a baseline system, consider a mass under proportional control described by the transfer function:

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Fig.6 Command Signal (Left) and Saturated Signal (Right)

0.8 0.6 EI-Shaped Step Input Response

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1 ZV-Shaped Step Input Response

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Fig.7 Response to a Standard ZV-Shaped Step Input Under Saturation KP ω n2 y(s) m (1) = = K y d ( s) s 2 + ω n2 s2 + P m Sometimes the force required from the controller to provide the desired results is not physically possible to attain. If the most force the actuator can provide at any time is Umax, then when the controller calls for a force greater than Umax, the system is saturated. Figure 6 illustrates this effect by showing a desired command signal (left) and the saturated signal (right). Traditionally, actuator saturation is a situation that is avoided in practice because it complicates the dynamic response and can make linear controllers perform poorly. As mentioned previously, input shaping is based on the linear characteristics of a non-saturated system, namely modeled frequency, ωm, and damping ratio, ζ. But once the saturation region is encountered, these characteristics are altered. To illustrate this phenomenon, a standard ZV shaper is applied to the system represented by Fig. 6. In this example, the mass, m, is unity, and Kp and Umax are set respectively to 100 N/m and 10 N. In other words, the input shaper is designed to suppress a frequency equal to:

Effective Frequency (rad/s)

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Fig.8 Response to a Standard EI-Shaped Step Input Under Saturation

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= = = = =

1420 N/m 987 N/m 632 N/m 355 N/m 158 N/m

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Fig.9 Effective Frequency vs. Umax for Various Values of Kp 1 Kp (2) = 1.59 Hz. 2π m The response to the ZV shaper is shown in Fig. 7. Clearly, the lack of robustness of this shaper to saturation is detrimental to eliminating residual vibration. The use of a robust shaper is not always a useful alternative for severe nonlinearities. As shown in Fig. 8, an EI shaper designed to limit vibration below 5% produces 18% when subjected to the saturation limits given above. f =

The main challenge of compensating for saturation does not come from predicting how saturation affects the system’s frequency for a given input. As saturation increases, the frequency of the response decreases, as illustrated in Fig. 9. The difficulty is to understand how modifying the input, in turn, changes that frequency. By taking a closer look at Fig. 7, the “effective” frequency can be determined from the remaining oscillations. We define the effective frequency as being the frequency of the residual vibration of the response

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Fig.10 Saturated Response to a Step Command Convolved with a ZV Shaper Based on the Effective Frequency Resulting from a Unit-Step Input Input Parameters

SC Shaper

Yes

Are Constraints Met? No

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Vary Shaper

Fig.11 Saturation Compensation Algorithm when the system is saturated. This frequency is always lower than the natural frequency of the unsaturated system given by the expression (2). The effective frequency, in this example, is 0.67 Hz, which clearly differs from what (2) predicted. If a ZV shaper were to be designed based on a frequency of 0.67 Hz, it would yield the response shown in Fig. 10. This response, although better than the one shown in Fig. 7, is far from being satisfactory as the residual vibration exceeds 30%. This uncertainty on how saturated inputs affect the system’s effective frequency motivates employing optimization to design successful input shapers. 3. SATURATION-COMPENSATING SHAPERS In order to develop Saturation-Compensating (SC) input shapers, the algorithm shown in Fig. 11 is used. This algorithm was implemented using MATLAB’s constraint function “constr.m” and SIMULINK. Once the parameter values for Kp, Umax, and m are set in an m-file, the optimization routine guesses an initial input shaper. This shaper is convolved with a step input to create a staircase reference command. The command is then sent to the block “In1” of the SIMULINK model shown in

Fig.12 Mass Under PD Control with Saturation (Model Implemented in SIMULINK) Fig. 12 in order to simulate the response. If residual vibration remains, the optimization routine modifies the input shaper and sends the new command back to the SIMULINK model to evaluate the response. This process continues until the shortest possible input shaper that yields zero or very little residual vibration is obtained. 3.1 Zero Vibration Saturation-Compensating (ZVSC) Shapers The following development offers a detailed description for obtaining ZVSC shapers. For ZV shapers, the number of impulses, n, is 2. The objective of the optimization is to minimize the duration (time of the last impulse, Tn) of the input shaper under the following set of constraints: 1 - The amplitudes, Ai, of the shaper must sum to the desired setpoint value S: n

∑i =1 A i = S.

(3)

2 - The time location of the first impulse, T1, must be zero: T1 = 0. (4) 3 - The time ordering must be enforced: (5) Ti + 1 − Ti ≥ 0 , i = 1...n - 1. 4 - The amplitudes of the impulses must all be positive: A i > 0 , i = 1...n. (6) 5 - The level of residual vibration, Vtol, must be zero. However, because of software tolerance matters, it is more feasible to require a very small level of residual vibration, ε: (7) Vtol ≤ ε. As an example, a ZVSC shaper is obtained for a unit mass under saturated Proportional-plus-Derivative (PD) control. The derivative and proportional gains are respectively fixed to 2.5 N-s/m and 631 N/m. Without saturation, such values would correspond to a damping ratio of 0.05 and a 4-Hz natural frequency. Also, Umax is 100 N, S equals unity, and ε must remain below 0.5%. After optimization, the following ZVSC shaper is returned:

ZV-Shaped Step Input ZV-Shaped Step Response ZVSC-Shaped Step Input ZVSC-Shaped Step Response

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For undamped systems, the duration of the UMZV shaper is T/3, where T is the period of vibration of the system: T   Ti  0 T 6 3 . (9) A  =   i  1 - 1 1  For the lightly-damped system presented in subsection 3.1, the standard UMZV shaper would be:  Ti  0 0.0454 0.0835 (10) .  A  = 1 -1 1   i  The constraints to develop UMZVSC shapers are similar to those used for ZVSC shapers, except that (6) turns into an equality:

Fig. 13 Saturated Responses to Step Commands Convolved with a ZV Shaper and a ZVSC Shaper

1.75 1.5 1.25 1 0.75 0.5 0.25 0

Compensating (UMZVSC) shapers are compared to standard UMZV shapers and ZVSC shapers.

1

Fig. 14 Saturated Responses to Step Commands Shaped with a UMZV Shaper and a UMZVSC Shaper 0.2024  Ti   0 (8)  A  = 0.5351 0.4649.  i   On the other hand, the standard ZV shaper that does not account for saturation is given by: 0.1252  Ti   0 (9)  A  = 0.5392 0.4608.  i   By comparing (8) and (9), it can be observed that the ZVSC and ZV shapers essentially differ in their duration. In this example, the ZVSC shaper is 61.7% longer than the ZV shaper. Figure 13 shows the simulated responses of the saturated system to both shapers. It corroborates that zero vibration can only be achieved at the cost of an increase in rise time whenever the plant experiences actuator saturation.

(11) A i = (− 1)i +1 , i = 1...n. After optimization, the UMZVSC shaper for this case is given by:  Ti  0 1001 0.2010 (12) .  A  = 1 - 1 1   i  Figure 14 compares the time responses of step commands which are shaped with (10) and (12). It is worth noticing that, under actuator saturation, the UMZVSC shaper is not always shorter than the ZVSC shaper. This is obvious from comparing (8) to (12). In this instance, both shapers have approximately the same duration. This should make sense because the UMZVSC shaper has larger magnitude impulses than the ZVSC shaper. In other words, it tends to saturate actuators even more. 4. CONCLUSION A new class of input shapers has been developed to compensate for severe actuator saturation. The nonlinearity caused by saturation forced the use of optimization to derive these Saturation-Compensating (SC) shapers. Results have shown that SC shapers are longer than the corresponding standard shapers because the system cannot respond as quickly under saturation. It has also been demonstrated that using negative impulses in the SC shapers does not necessarily yield faster rise times than positive-amplitude SC shapers when the system experiences saturation. REFERENCES

3.2 Unity-Magnitude (UM) ZVSC Shapers In an effort to reduce the shaper duration, the design constraints can be changed to allow negative impulse amplitudes. The Unity-Magnitude (UM) ZV shaper [26] requires that the impulse amplitudes switch between 1 and -1. Note that this amounts to releasing the allpositive amplitude constraint given in (6). In this subsection, Unity-Magnitude Zero Vibration Saturation-

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