Maximally Robust Input Preconditioning for Residual Vibration ...

D. Economou Research Assistant, Mechanical Design and Control Systems Division, Department of Mechanical Engineering, National Technical University of Athens, Athens, Greece e-mail: [email protected]

C. Mavroidis Mem. ASME Associate Professor, Department of Mechanical and Aerospace Engineering, Rutgers University, The State University of New Jersey, 98 Brett Rd., Piscataway, NJ 08854-8058 e-mail: [email protected]

I. Antoniadis Mem. ASME Assistant Professor, Mechanical Design and Control Systems Division, Department of Mechanical Engineering, National Technical University of Athens, Athens, Greece e-mail: [email protected]

C. Lee Graduate Student, Department of Mechanical and Aerospace Engineering, Rutgers University; Currently, Senior Research Engineer at General Motors e-mail: [email protected]

1

Maximally Robust Input Preconditioning for Residual Vibration Suppression Using Low-Pass FIR Digital Filters A method for suppressing residual vibrations in flexible systems is presented and experimentally demonstrated. The proposed method is based on the preconditioning of the inputs to the system using low-pass Finite Impulse Response (FIR) digital filters. Provided that the cutoff frequency of FIR filters is selected lower than the lowest expected natural frequency of the system and their stop-band is maximized, we show that these filters can be designed to exhibit maximally robust behavior with respect to changes of the system natural frequencies. To perform the proper design of FIR filters for robust vibration suppression, this paper introduces a series of dimensionless performance indexes and the Delay-Error-Order (DEO) curves that represent graphically the delay time introduced by the filter as a function of the remaining residual vibrations, and the filter order. Several classes of FIR filters such as: a) Parks-McClellan; b) Window-based methods (using Chebyshev window); and c) Constrained Least Squares method, are shown to present maximally robust behavior, almost identical to the theoretically predicted. Parallel, they demonstrate excellent vibration suppression while they introduce the minimum possible delay. Further advantages offered by the proposed method, is that no modeling of the flexible system is required, the method can be used in a variety of systems exhibiting vibrations, it is independent of the guidance function and it is simple to implement in practical applications. The results are experimentally verified on a flexible aluminum beam with a significantly varying mass, attached to the end-effector of a robot manipulator. The beam is rotated, using one joint of the manipulator, from an initial to a final position. It is shown that the preconditioned inputs to the flexible system induce very little amount of residual vibrations compared to the inputs with no preconditioning. 关DOI: 10.1115/1.1434272兴 Keywords: Vibration Suppression, Input Preconditioning, FIR Digital Filters

Introduction

Residual vibration suppression is important in a broad range of mechanical engineering applications such as in the deployment of space structures and cranes or in the operation of machine tools and flexible robots. The traditional approaches to minimize the effect of residual vibrations are focused on either increasing the structural stiffness which increases the system’s size and weight or using closed loop control methods, which require advanced instrumentation and control equipment, that increases the system cost and complexity. An alternative approach for supressing residual vibrations is the proper conditioning of the pre-specified excitation pattern 共‘‘Guidance’’兲, so that the system moves exactly to the desired end position without any residual vibrations. This concept is very attractive, since it can drastically simplify the system requirements and complexity. Significant research effort has been devoted towards this direction. One of the first approaches used a partitioning of the desired excitation pattern into two distinct steps, the second one of which is delayed 关1兴. Other approaches included the introduction of multi-switch Bang-Bang inputs, or the convolution of the desired guidance function with a series of impulses 关2,3兴. Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division July 27, 2000. Associate Editor: J. Tu.

Methods have been proposed that approximate the guidance function by a series of typical mathematical functions such as splines, sinusoids, ramped sinusoids, or polynomials 关4 –7兴. All these methods require the exact knowledge of the system dynamic parameters. Small parameter variation during operation such as additional machine payload or modeling errors may cause deviations from the desired response and result in significant residual vibrations. Recently, several methods have been proposed so that the desired guidance function exhibits a robust performance. The splines method 关4兴 provided satisfactory results for a big range of ‘‘alternative dynamic models’’ of the system, which are obtained from variations of its nominal parameters. However, this method can only be used in systems where the desired guidance function can be approximated by splines, while the choice of the ‘‘alternative models’’ is based mainly on heuristic criteria. Preconditioning methods and closed-loop control methods are in principle two different and independently applied procedures for vibration suppression, each one presenting its own advantages and disadvantages. Preconditioning approaches show their vibration suppression capabilities independently of the system closedloop controller, which is used to perform the position task and not the vibration suppression task. The advantage of the preconditioning methods is that by planning appropriately the input trajectories to the system, the level of the system residual vibrations is very small, almost eliminated, so that actually the system actuators do not have to be used for vibration suppression and they can be used

Journal of Dynamic Systems, Measurement, and Control Copyright © 2002 by ASME

MARCH 2002, Vol. 124 Õ 85

all the time for the position task. In active feedback control for vibration suppression either the system has to be redundant 共i.e. have more actuators than needed to perform the position task兲 which increases the cost or the system needs to interrupt the positioning task to perform the vibration suppression and this increases operation time. Furthermore, preconditioning methods can even be efficient in suppressing vibrations in open loop systems, where no closed loop controller of any kind is used. A limitation of the preconditioning methods compared to active vibration control is the treatment of external disturbances to the system, since preconditioning methods alone are not able to suppress vibrations excited by external disturbances. However, even in the case where a closed loop control method must be used, its proper combination with a preconditioning method can lead to additional benefits for both methods. Of special interest in the present work, are methods to suppress residual vibrations, based on the convolution of the guidance function with a series of impulses. In single-degree-of-freedom systems, a limited number of impulses, such as 3 to 5, in several forms were shown to present a robust behavior 关3,8兴. In multidegree-of-freedom systems, the above basic impulse pattern is further convoluted or delayed in time 关3,9,10兴. This impulse sequence, which can be considered as a specific case of a Finite Impulse Response 共FIR兲 digital filter, is known as ‘‘input shapers’’ and the methodology as ‘‘input shaping’’. A lot of research has been performed to propose design techniques for the input shapers, so that they can suppress multiple modes of vibration and exhibit increased robustness 关11–16兴. Input shaping has been used successfully in many applications for vibration suppression such as long reach manipulators, cranes and coordinate measuring machines 关17–19兴. While input shaping methods have a very good performance in a variety of systems and applications, their robustness is limited in local areas around the system natural frequencies and can be increased only, by increasing the total duration of the pulse sequence. This results in unnecessary delays in the application of the total guidance and consequently, in the total duration of the system motion. In a first attempt to extend the robustness of vibration suppression methods based on the convolution of the guidance function with a series of impulses, a general approach for the impulse sequence convolution procedure has been proposed 关20兴, leading to a set of three different methods for the proper design of the impulse sequence. The impulse sequence approach is significantly extended 关21兴, by establishing a framework, according to which the design of an appropriate guidance function can be transformed to the problem of a proper design of a conventional FIR digital filter. In contrast to the extensive research that has been performed for the input shaping, which is in fact a special form of FIR filters, very little research has been performed in applying conventional FIR digital filters in vibration suppression. Although some relevant results have been reported 关22兴 the design specifications for the FIR digital filters were not properly selected and this contributed to their nonsuccessful implementation. In this paper, a preconditioning approach is proposed, based on the proper design of conventional low-pass FIR digital filters. Based on five requirements that ensure robust vibration suppression with minimum residual error, short delays, increased robustness, and rigid body motion at zero frequency a general method for determining the FIR Filter Design Parameters is developed. The FIR filter design method is based on the use of novel dimensionless performance indexes and the Delay-Error-Order 共DEO兲 curves. Then, the application of the method to ten types of low pass FIR digital filters is considered: These filters are: 共a兲 ParksMcClellan; 共b兲 Window-based method, using Hamming, Hanning, Boxcar, Kaiser, Bartlett, Blackman and Chebyshev windows; 共c兲 Least Squares method; 共d兲 Constrained Least Squares method. For these filters, that are the state of art best performing conventional FIR filters, we show, that while all types present good vibration suppression properties, only three of the filters 共i.e., Parks86 Õ Vol. 124, MARCH 2002

McClellan; Window-based method using Chebyshev window; and Constrained Least Squares method兲 introduce very small time delays and present maximal robustness to changes in the system’s dynamic parameters. The method is experimentally verified using a flexible aluminum beam attached at the end-effector of a robot manipulator. The beam is rotated, using one joint of the manipulator, from an initial to a final position. During this motion, vibrations of the beam are excited due to inertial forces. It is clearly shown that the preconditioned inputs to the flexible system induce a very small amount of residual vibrations, compared to the inputs with no preconditioning. Robustness tests were conducted by attaching a variable mass at the end of the beam in order to change the natural frequency of the flexible system. Even if the system mass changed as much as 287 percent of the original mass, the amount of vibrations obtained from the filtered input function was less than 8.32 percent of the residual vibrations obtained from the original input function.

2

Theoretical Development

2.1 Requirements for Residual Vibration Suppression Using Input Preconditioning. The motion of a typical degree of freedom or natural mode of a dynamic system with natural frequency f 0 , damping ratio ␨, excitation function 共also called command or guidance function兲 g(t) and response vector x(t), can be described in the following state space form:

A⫽

冋

x˙共 t 兲 ⫽Ax共 t 兲 ⫹bg 共 t 兲 0

1

⫺ ␻ 20

⫺2 ␨␻ 0

册 冋册 ,

0 , 1

b⫽

(1a)

␻ 0 ⫽2 ␲ f 0

(1b)

Using the Duhamel Integral, the system response is equal to:

冕 冋 冕 t

x共 t 兲 ⫽e At x共 0 兲 ⫹

e A 共 t⫺ ␶ 兲 bg 共 ␶ 兲 d ␶

0

t

⫽e At x共 0 兲 ⫹

e ⫺A ␶ bg 共 ␶ 兲 d ␶

0

册

(2)

The application of the Sylvester expansion method to the transition matrix e At leads to: 2

e ⫽ At

兺e i⫽1

qit

Hi

(3a)

q 1,2⫽⫺ ␨␻ 0 ⫾ ␻ 0 冑1⫺ ␨ 20 Hi ⫽

1 关 A⫺q i I兴 q i ⫺q j

(3b)

for i, j⫽1,2 i⫽ j

(3c)

Thus, substituting Eq. 共3兲 in Eq. 共2兲, the expression for the response can be written as:

冋 冋

⫽e

At

兺 H b冕 e 2

x共 t 兲 ⫽e At x共 0 兲 ⫹

i⫽1

t

i

⫺q i ␶

g共 ␶ 兲d␶

0

2

x共 0 兲 ⫹

兺 H bG 共 s;s⫽q 兲 i⫽1

i

i

册

册

(4a)

(4b)

where the integral in Eq. 共4a兲 is the Laplace transform G(s) of the guidance function, evaluated at the two system poles q i from Eq. 共3b兲. In the case of a rigid body motion 共i.e. ␻ 0 ⫽0兲, Eq. 共4b兲 can take the form: xR 共 t 兲 ⫽HR 共 t 兲关 x共 0 兲 ⫹bG 共 s;s⫽0 兲兴 HR 共 t 兲 ⫽

冋 册 1

t

0

1

(5a) (5b)

Transactions of the ASME

According to the proposed preconditioning approach, instead of the direct implementation of the original guidance function g(t) in Eq. 共1a兲, a conditioned guidance function u(t) can be alternatively used, obtained by the original guidance function g(t) by introducing a sequence of the form shown in Eq. 共6兲: N

u共 t 兲⫽

兺 c g 共 t⫺nT 兲

(6)

n

n⫽0

where 兵 c n 其 is a series of coefficients of length N⫹1. The corresponding Laplace transform of u(t) is: U 共 s 兲 ⫽F 共 s 兲 G 共 s 兲

(7a)

N

F共 s 兲⫽

兺ce n

n⫽0

⫺snT

(7b)

The purpose of introducing the sequence in Eq. 共6兲, is that the conditioned guidance function u(t) is able to move the system in essentially the same way as the original guidance function g(t), without the effect of the residual vibrations. This requirement can be stated as: F 共 s;s⫽q i 兲 ⫽0

(8a)

F 共 s;s⫽0 兲 ⫽1

(8b)

In view of Eqs. 共4b兲, 共7a兲, Eq. 共8a兲 implies that the residual vibration effect can be completely canceled just by the proper selection of F(s), quite independently from the properties of the original guidance function g(t). In the same way, Eq. 共8b兲 implies that u(t) maintains all properties of g(t) at steady state, so that the system reaches its desired end-position with no vibrations. 2.2 Input Preconditioning Using FIR Digital Filters. FIR filters are a series of constants 兵 c n 其 关23兴. The mathematical foundation of filtering is a convolution procedure, resulting to a discrete filtered signal u(kT S ), which is obtained from an original discrete input signal g(kT S ) according to Eq. 共9兲, N

u 共 kT S 兲 ⫽

兺 c g 共 kT ⫺nT

n⫽0

n

S

SF 兲

(9)

where k is an integer, T S is the sampling period of the discrete signals u(t k ) and g(t k ), T SF is the sampling period of the filter and 兵 c n 其 is a series of constants of length N⫹1. The z-transform of the filtered function U(z) is related to the z-transform of the original input function G(z) by: N

U 共 z 兲 ⫽F 共 z 兲 G 共 z 兲 ⫽

兺cz

n⫽0

n

⫺n

G共 z 兲

(10)

where: N

F共 z 兲⫽

兺cz n

n⫽0

⫺n

(11a)

with a corresponding frequency response function of the form: N

F 共 j ␻ 兲 ⫽F 共 j2 ␲ f 兲 ⫽

兺ce

n⫽0

n

⫺ jn2 ␲ f T SF

(11b)

Provided that the filter sampling period T SF is chosen equal to the period T of the preconditioning sequence, Eq. 共10兲 represents also the z transform of Eq. 共7b兲. This fact clearly implies that the design of a proper function F(s), is completely equivalent to the design of an FIR filter in the form of Eq. 共9兲. Provided that the frequency response function F( j ␻ ) of the filter is zero at frequencies coinciding with the expected natural frequencies of the dynamic system according to Eq. 共8a兲, this filter is capable of completely eliminating the residual vibrations effect. In addition, according to Eq. 共8b兲, the response for zero frequency of this filter should be kept equal to one, in Journal of Dynamic Systems, Measurement, and Control

Fig. 1 Typical frequency response of a low-pass filter

order to ensure the proper motion of the mechanical system as a rigid body. In view of Eq. 共11b兲, this last requirement becomes: N

F共 0 兲⫽

兺 c ⫽1

n⫽0

n

(12)

Considering robust residual vibration suppression, the robustness properties for the preconditioning procedure can be directly met, by extending the requirement for zero frequency response of the filter not only for individual frequencies coinciding with the expected natural frequencies of the system, but also for extended areas 共stop-band areas兲 of the filter frequency response function F( j ␻ ), in order to cover now additionally the possible variations of the system natural frequencies. Provided that the stop-band area of the filter becomes the maximum possible, the maximum possible degree of robustness of the guidance function can be attained.

3

FIR Filter Design Procedure

In this section we present a novel methodology for properly designing FIR filters for robust vibration suppression in mechanical systems. The method is general because it can be applied to any FIR digital filter and to any mechanical system. 3.1 Low-Pass Filter Design Requirements for Robust Residual Vibration Suppression. Low-pass filters have a passband at the lower frequencies determined by the pass-band frequency limit f B , and a stop-band at the higher frequencies determined by a cutoff frequency f C . The ideal response of a low-pass filter is shown in Fig. 1 关23兴. However, the actual implementation of a low-pass filter in the form of any FIR filter is quite different, as shown in Fig. 1, due to several reasons. First, according to the ‘‘sampling theorem,’’ if f SF is the sampling frequency of the filter, the frequency response is symmetric about the frequency f SF /2 and the frequency response is repeated over the sampling frequency in every frequency range 关 k f SF ,(k ⫹1) f SF 兴 , where k is any integer. The frequency range of the stop-band can be easily concluded from Fig. 1 to be 关 f C , f SF ⫺ f C 兴 . Thus, the width of the stop-band w SB is: w SB ⫽ 共 f SF ⫺ f C 兲 ⫺ f C ⫽ f SF ⫺2 f C

(13)

Additionally, due to a number of factors affecting their design 共e.g. Gibbs effects, finite duration requirements, etc兲, a nonsharp transition band is necessary between the pass-band and the stopband and their actual frequency response presents a number of ripples 关23兴. The maximum value e P , e S of these ripples determine the maximum allowable error in the pass-band and the stopband respectively. Provided that Eq. 共12兲 is satisfied, no ripples MARCH 2002, Vol. 124 Õ 87

are present at zero frequency. The ripples e S in the stop band prevent the requirement expressed by Eq. 共8a兲 to be exactly satisfied. However, if a very small amount of vibration is accepted instead of zero vibrations, then the requirements for vibration suppression for actual FIR filters can be stated as: F 共 s;s⫽q i 兲 ⭐ 共 e S 兲 per

兺 c ⫽1

n⫽0

n

(14b)

where (e S ) per is the permissible size of ripples 共permissible vibration error兲. In view of Section 2.1, the design of a low pass FIR digital filter for vibration suppression needs to satisfy the following requirements: R共a兲 The cut-off frequency f C must be equal or lower than the system’s lowest damped natural frequency f 0 , to ensure vibration suppression of all modes. R共b兲 The stop-band width w SB should be designed as large as possible in order to cover the possible expected variations of all natural frequencies of the system, ensuring the maximum possible robust behavior of the guidance function 共Eq. 共13兲兲. R共c兲 The response for zero frequency must be equal to one, in order to achieve the appropriate rigid body motion 共Eq. 共14b兲兲. R共d兲 The ripples on the stop-band should be less than a prespecified acceptable residual vibration error (e S ) per 共Eq. 共14a兲兲. R共e兲 The delay introduced by the filtering process should be minimal. 3.2 Basic Filter Design Parameters. A large number of FIR filter types has been developed, each one being designed with a specific procedure 共analytical or even numerical兲 and using a specific set of parameters 关23兴. Therefore, as a first step for their common use for robust vibration suppression, a set of four general parameters has been identified, specifying the behavior of all filters. These important and basic ‘‘FIR Filter Design Parameters’’ are: 共a兲 the filter order N, 共b兲 the cutoff frequency f C , 共c兲 the pass-band edge f B , d兲 the sampling frequency f SF . The selection of the proper FIR Filter Design Parameters is not a trivial problem, especially for residual vibration suppression. The problem is that, due to features inherent in all filter design methods 关24兴, the values of the FIR Filter Design Parameters, as they are re-evaluated from the actual frequency response of the filter after it has been designed, may differ from their corresponding values used as inputs for the filter design. This implies that the actual performance of the designed filter may be different than the expected performance, as expected by the input design parameters. This can result in significantly decreased vibration suppression capabilities for the FIR filter. 3.3 Nondimensional Performance Indices. First, in an attempt to reduce the number of parameters needed for the design, two nondimensional ratios are used: 共a兲 The pass-band width ratio r B : fB rB⫽ f SF/2 共b兲 The stop-band width ratio r C : fC rC⫽ f SF/2

(15)

(17)

共a兲 The relative robustness r R to variations of the dynamic system parameters. Since every change to these parameters reflects to changes of the natural frequencies of the system, the relative robustness can be expressed using the stop-band width w SB , according to Eq. 共13兲, as: wSB f SF⫺2f C rR⫽ ⫽ (18) f0 f0 共b兲 The relative robustness width r W covered by the robustness region 共stop-band兲 of the filter in proportion to the overall effective frequency range of the filter: wSB f SF⫺2f C rW⫽ ⫽ (19) f SF f SF 共c兲 The relative delay d introduced by the filter: TD NTSF Nf 0 d⫽ ⫽ ⫽ (20) T0 T0 f SF In the above Eqs. 共18兲, 共19兲, 共20兲, f 0 is the system’s lowest damped natural frequency, T 0 the corresponding period, T SF the sampling period of the filter and T D the duration of the filter, which is equal to the total time delay introduced for linear phase symmetric FIR filters. 3.4 Limits of the Performance Indices for Theoretically Maximum Robustness. According to Eq. 共20兲, the sampling frequency f SF can be written as: f SF ⫽

Nf0 d

(16)

(21)

Substituting Eq. 共16兲 in Eq. 共20兲 and using requirement R共a兲 according to which the filter cutoff frequency f C must be almost equal to the system’s lowest damped natural frequency f 0 , then the following relationships are obtained between the performance indices and the stop-band width ratio r C : r R ⫽2

冉 冊

1 N ⫺1 ⫽ ⫺2 rC d

r W ⫽1⫺r C ⫽1⫺2

d N

(22)

(23)

Equations 共22兲 and 共23兲 indicate that a filter exhibiting theoretically maximum robustness, can be completely characterized by the following limit values of the performance indices: 共 r R 兲 TH →⬁

(24a)

共 r W 兲 TH ⫽1

(24b)

Both of them can be simultaneously met, just by the following limit value of the stop-band width ratio r C : 共 r C 兲 TH ⫽0

An additional purpose for the introduction of the above two ratios is the fact that they are independent from the value of the sampling frequency f SF . Thus, using now the ratios r B and r C , filters suppressing vibrations can be ‘‘predesigned’’ in a general non-dimensional form, independently from the actual values of the frequencies f B , f C and f SF and then straightforwardly applied 88 Õ Vol. 124, MARCH 2002

0⬍r B ⬍r C ⬍1 Additionally, three filter performance indices are defined:

(14a)

N

F 共 s;s⫽0 兲 ⫽

to any dynamic system, according to the specific values of these frequencies. In view of Fig. 1, the following relation between r B and r C exists:

(24c)

As a result of Eqs. 共17兲 and 共24c兲 the following limit value for the pass-band width ratio r B is obtained: 共 r B 兲 TH ⫽0

(24d)

Therefore, a filter presenting theoretically maximal robustness is completely characterized by a stop-band width ratio r C equal to zero or, equivalently, by a relative robustness width r W equal to one. Transactions of the ASME

3.5 Effects of Maximal Robustness Requirements to Filter Performance Indices. According to the results of Section 3.4, the relative delay d, can be expressed as: d⫽

Nr C 2

(25)

In view of Eqs. 共24c兲 and 共25兲, the following equation holds for theoretically maximum robustness:

冉冊 d N

⫽0

(26)

TH

Equation 共25兲 indicates that, for a specific value of the filter order N the minimization of the relative delay d can be simultaneously achieved with the minimization of the stop-band width ratio r C and consequently by the maximization of its robustness. Furthermore, in view of Eq. 共26兲, an additional effective mechanism for increasing an FIR filter robustness is by increasing the filter order N. The theoretically maximal robust behavior cannot be met by any practical realization of any FIR filter. First, Eq. 共24d兲 and consequently 共24c兲 are in contradiction to Requirement R共c兲. Additionally the filter duration and consequently the filter order must remain finite. These constraints, together with constraints inherent in FIR design procedures 关24兴, generate a number of implications for the design of actual FIR filters, meeting the robust residual suppression requirements. First, values for r B below a certain limit may result in a frequency response for zero frequency smaller than the desired value of one. Additionally, values for r C very close to r B may cause increased ripples in the stop-band, larger than the acceptable vibration error 共requirement R共d兲兲. Furthermore, in certain cases, the resulting values for r C and r B , as they finally appear in the actual frequency response of the designed filter, cannot be decreased below certain limits, regardless of any further reduction of their corresponding values, used as input parameters for the filter design. 3.6 Delay-Error-Order „DEO… Curves and the Filter Design Procedure. The minimum relative delay d min , the filter order N and the acceptable vibration error (e S ) per are the important parameters for the robust design of FIR filters for vibration suppression 共see Section 3.5兲. To properly select these parameters, not only for filters designed analytically but also for filters designed numerically, we are introducing a new graphical tool called the Delay-Error-Order 共DEO兲 curves. Once these parameters are selected using the DEO curves then the filter robustness indices are calculated using Eqs. 共22兲 and 共23兲. The DEO curves are the plots of the minimum value of d versus the filter order N. For different values of the maximum permissible error (e S ) per different curves are obtained. Although each point on these curves represents an optimum filter, the ratios r C and r B used for the design do not explicitly appear on the DEO curves. For this reason, look-up tables are complementary created, including the values of the corresponding design parameters which lead to each minimum delay d. It has to be emphasized, that the DEO curves are characteristics of the FIR filter itself, that is used and they do not depend on the physical system that is being studied. The DEO curves are generated using an iterative algorithm. A generic form of this algorithm is shown in Fig. 2. For each FIR filter, a multi-loop, iterative process is implemented. This algorithm varies all the filter design parameters, resulting in all filters obtainable in the available filter design parameters space. Then, we select the filters that for each filter order and each maximum permissible error (e S ) per , present the minimum time delay, and meet all the five design requirements R共a兲 to R共e兲, described in Section 3.1. The algorithm’s outer loop specifies the range for the permissible filter ripples 共residual vibration errors兲 (e S ) per . RepresentaJournal of Dynamic Systems, Measurement, and Control

Fig. 2 Algorithm to generate the delay-error-order „DEO… curves for FIR filters

tive values for (e S ) per are within 1 percent to 10 percent. This loop makes sure that requirement R共d兲 is satisfied when a set of optimal FIR Filter Design Parameters have been selected using the other requirements. The second loop spans the space of possible filter orders N from 1 to N max . Ideally, in view of Eq. 共26兲, N and N max should be selected as high as possible. However, the maximum value of N and N max is limited in practice by the following constraints: C共a兲 The guidance function is in many cases a discrete signal. In order to be convolved with the discrete filter coefficients, the sampling frequency of the guidance function must be an integer multiple of the sampling frequency of the filter f SF or at least equal to f SF . Since the sampling frequency of the guidance function usually depends on the physical system and its computer hardware and cannot be selected freely, this constraint defines the upper limit for f SF . According to Eq. 共21兲, the upper limit for f SF sets an upper limit for N and N max , provided that d and f 0 are known. Also, due to the requirement for f SF to be a sub-multiple of the sampling frequency of the guidance function, only specific values of N between 1 and N max can be used. C共b兲 In the case where the filtering process is performed online, high values of N cause increased computational effort that may create problems in real-time applications. The third and the fourth loops span the ranges of values for ratios r B and r C respectively. The range for r C , according to Eqs. 共17兲 and 共24c兲, is set from 1 down to a value close to zero and the corresponding range for r B , according to Eqs. 共17兲 and 共24d兲, is varying from r C down to a value very close to zero. For each MARCH 2002, Vol. 124 Õ 89

Fig. 3 Delay-error-order „DEO… curves for FIR filters designed using the following methods. „a… Constrained least squares method, „b… window-based method „Chebyshev Window…, „c… Parks-McClellan method, and „d… window-based method „Hamming Window….

value of N, r C , and r B , the coefficients of the FIR filter are calculated using the specific filter design method 关24兴. In order to satisfy requirement R共c兲 and Eq. 共14b兲, the sum of the calculated filter coefficients should be equal to 1. If this is not true, then no further consideration is given to these coefficients and the algorithm proceeds to the next iteration. The next step is to compute the filter frequency response and from it to calculate the actual resulting values for the maximum vibration error e S , the stopband area width ratio r C and the relative delay d. If the maximum vibration error e S , is larger than the maximum permissible error (e S ) per then, according to requirement R共d兲, the algorithm disregards this filter and proceeds to the next iteration. To satisfy requirements R共b兲 and R共e兲, the index d should be minimized. For each value of the filter order N and for all the values of d that have been calculated while spanning the ranges of ratios r C and r B for the same N, the minimum value of d is found. 3.7 Optimal Conventional FIR Filters. The general procedure of Section 3.6 is then applied to ten FIR filter types, covering in practice almost all conventional FIR filters, considered as best performing 关24兴. These filters have been designed according to the following methods, according to which they are also named: 共A兲 Parks-McClellan FIR design method; 90 Õ Vol. 124, MARCH 2002

共B兲 Window-based design method, using Hamming, Hanning, Boxcar, Kaiser, Bartlett, Blackman and Chebyshev windows; 共C兲 Least Squares method; 共D兲 Constrained Least Squares method. They have been implemented in the algorithm of Section 3.6 by the use of MATLAB ‘‘Signal Processing Toolbox’’ 关25兴. DEO curves have been developed for all ten above FIR filters. Figure 3 shows the DEO curves for four of these filters. The DEO curves of Fig. 3 were created for N max being equal to 256 and (e S ) per values selected between 1 to 10 percent. For better readability of the figure, only three curves have been plotted for each filter design method. Additionally, the corresponding look-up tables were created. Part of a look-up table is shown in Table 1. The robustness indices, corresponding to the filters presented in Fig. 3, are shown respectively in Figs. 4 and 5. The major impact from this analysis, is that all the filters designed, exhibit a maximally robust behavior, which, for relatively high filter orders, is almost identical to the one theoretically expected. The difference between the theoretical limit of the relative robustness width r W , as predicted in Eq. 共24b兲, with the ones Transactions of the ASME

Fig. 4 Relative robustness r R for FIR filters designed using the following methods. „a… Constrained least squares method, „b… window-based method „Chebyshev Window…, „c… Parks-McClellan method, and „d… window-based method „Hamming Window….

resulting in Fig. 5 for all filters tested, is less than 1 percent, for a filter order of 256. This implies that the robust region of the filters in this case occupies 99 percent of the entire filter band. Quite robust behavior is obtained also for significantly lower filter orders. For example, filters with an order of 64 offer a relative robustness width of more than 95 percent and filters with an order of 16 offer a relative robustness width of more than 80 percent– 85 percent, dependent on the filter type. Moreover, the relative robustness width is practically insensitive to the residual vibration error. For filters with an order greater than 64, it is additionally independent from the filter type. This fact implies that although the algorithm of Section 3.6 has been applied to specific filter types, in principle it is inherently capable to ‘‘tune’’ any ‘‘proper’’ FIR filter to be maximally robust. From the relative robustness values shown in Fig. 4, it is concluded that the robust region of the three best performing filters for a filter order above 128, can extend up to multiples of orders of 102 of the lowest expected natural mode of the system. For example, according to the results in Table 1, the relative robustness of a Parks-McClellan filter type of order 128 is 105, implying that the robust region extends up to 105 times the lowest expected natural frequency of the system. Robustness of such orders of magnitude, is far beyond any one offered by existing preconditioning techniques and more than satisfactory for practical applications, especially taking into account Journal of Dynamic Systems, Measurement, and Control

that the higher modes of a flexible system have a significantly less contribution than the lowest ones in the overall vibration response. Moreover, perhaps more important from the practical point of view, is that this maximal robustness is achieved, using quite satisfactory delay times and residual vibration errors. In the specific example of the Parks-McClellan filter mentioned above, the filter induces a residual vibration error at most of 5 percent and presents a delay time of 1.18 times the maximum expected natural period of the system. The DEO curves of Fig. 3 show that the delay introduced by each filter is strongly influenced by the maximum permissible error (e S ) per . It becomes larger when (e S ) per becomes small and vice versa. For very small values of the filter order N, the minimum time delays are small and they increase as N increases. Above a certain critical value of N, the minimal time delays practically converge to a constant value independent of N. This result, in view of Eq. 共22兲, has also a major impact. It implies that the relative filter robustness for the filters considered can be increased just by increasing the filter order N, while retaining practically the same total filter duration time and resulting to the same residual vibration error. For example, in the ParksMcClellan filter mentioned above, the robustness of the filter can be doubled to 210 times the lowest expected natural frequency of the system, just by doubling the filter order to 256, without any other practical consequence to the delay time and vibration error. MARCH 2002, Vol. 124 Õ 91

Fig. 5 Relative robustness width r W for FIR filters designed using the following methods. „a… Constrained least squares method, „b… window-based method „Chebyshev Window…, „c… Parks-McClellan method, and „d… window-based method „Hamming Window….

Table 1 Part of a look-up table „Parks-McClellan design method, permissible vibration error „ e s …perÄ5%…

Three of the filters 共i.e., 共a兲 Constrained Least Square Method—Fig 3共a兲; 共b兲 Window-Based Method 共Chebyshev Window兲—Fig 3共b兲; 共c兲 Parks-McClellan Method—Fig. 3共c兲 reveal simultaneously the smallest relative delay d the largest relative robustness r R . All other filters, such as the Window-Based Method with the Hamming Window shown in Figs. 3共d兲 and 4共d兲, introduced much larger delays, and less relative robustness. 3.8 Implementation Procedure to Specific Dynamic Systems. Once the DEO curves have been obtained for a filter type, the four FIR Filter Design Parameters can be easily calculated for every specific dynamic system, using the following procedure. First, the system’s lowest expected damped natural frequency f 0 is calculated. This is the only information necessary from the flexible system and can be readily estimated, either from experimental data or from quite simplified system models. In case that experimental data are used to identify f 0 the method described in this paper for suppressing residual vibrations requires no model at all. According to requirement R共a兲 in Section 3.1, the filter cutoff frequency f C is set equal to the system’s lowest damped natural frequency f 0 . Then, according to the specific task specifications and con-

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Transactions of the ASME

Table 2 Physical properties of the aluminum beam used in the experiments

of the filter f SF results directly from Eq. 共21兲. The actual values of f B and f C can be calculated from the frequency response of the designed filter.

4

straints for the filter time delay T D 共and hence the relative time delay d兲 and/or for the maximum permissible error (e S ) per , a specific DEO curve is selected. The value N of the filter order is selected as high as possible, provided that constraints C共a兲 and C共b兲 are satisfied. This leads to a point on the selected DEO curve. Finally, using as input parameters the values of r B and r C taken from the look-up table corresponding to this specific point on the DEO curve, the optimal filter is designed. The sampling frequency

Experimental Demonstration

4.1 Experimental System and Procedures. The proposed method is experimentally verified using a thin, long, rectangular, flexible aluminum beam with the physical properties presented in Table 2. The beam is attached at the end-effector of a five-degreeof-freedom Mitsubishi RV-M2 robot manipulator. Figure 6 shows a schematic and a picture of all mechanical and electrical components of this system. In the experiments performed in this work, the beam is rotated at an angle of 30 degrees from the vertical position, using one joint of the manipulator wrist as it is shown in Fig. 7. The plane of rotation is perpendicular to the plane formed by the beam’s body. During this motion, the inertial forces developed excite vibrations of the beam. Sets of experiments are performed with a different mass M attached at the beam’s free endpoint 共Fig. 6兲. In order to test the robustness of the proposed method, the mass M is allowed to vary from zero up to 287 percent of the total mass of the beam causing a significant variation of the natural frequency of the beam. Five cases of additional

Fig. 6 The Rutgers University experimental system

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Fig. 7 „a… Rotation of the flexible beam using one joint of the manipulator; „b… original and preconditioned guidance functions

mass at the end of the beam are considered and the corresponding measured natural frequencies of the first mode for the flexible beam/mass system are shown in Table 3. The input function to the flexible beam is the angular displacement ␪ (t) introduced by one of the manipulator wrist joints. The joint is under closed-loop PID control to ensure accurate implementation of the desired input function. Prior to the execution of the experiment, the input function is preconditioned off-line, using one of the three optimal low-pass FIR filters described in Section 3.7. The original 共nonfiltered兲 input function used in these experiments is the fastest rotation that the manipulator’s wrist was able to follow. The original input function, shown in Fig. 7, has a duration of 0.30 s. It is approximately a ramp with smoothened edges, has a very low content of high frequencies and thus no excitation of the higher mode vibrations occurred, at least with an amplitude able to be measured. The Dell® OptiPlex Gxa™ PC system with INTEL® Pentium II™ 333 MHz CPU and 128 Mbyte RAM is used to provide the data acquisition and control needs for the experiments. It is augmented with two US Digital® PC7166™ PC to incremental encoder interface cards and two Datel® PC-412C™ Analog I/O boards. The PC collects the sensor readings either through the data acquisition boards or the encoder interface cards, performs the feedback control calculation for the proper implementation of the desired input function, and then sends out the signal to the actuators of the robotic systems through the D/A converter and laboratory built amplifiers. An Entran Accelerometer 共Model EGE732B-2000D-/RS兲, which is a strain gauge type sensor, is attached at the free-end of the flexible beam to record the beam’s oscillations. Noise from the accelerometer signal was eliminated using a simple Butterworth filter with cut-off frequency at 30 Hz. Since no higher mode vibrations were present at the recorded data, the filter did not cut any useful frequencies needed in these experiments.

Table 3 Measured system natural frequencies „Hz… of the first mode for 5 different added masses

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WinRec v.1, a software developed at the Robotics and Mechatronics Laboratory at Rutgers University, provides deterministic fast timers based on MSDN library under Windows NT platforms and can be used in both real-time control and data acquisition. The timer in this experiment is set at 200 Hz that is fast enough for the cut-off frequency to cover the first few modes of a flexible specimen 关26兴. 4.2 Low-Pass FIR Filters Used in the Experiments. Experimental results are presented only for the filters designed with the three methods that introduced minimum delay. As shown in Section 3.7, these methods are the Parks-McClellan method, the window-based method using Chebychev window and the Constrained Least Square method. The procedure of Section 3.8 was followed to implement the filters. The lowest expected natural frequency f 0 of the system, used as basis for the calculations, is 2.5 Hz, corresponding to Case A of Table 3 with the highest mass. The maximum permissible residual vibration error (e S ) per was set equal to 5 percent. The order N of the filter was selected to be 19. Although the filter order is relatively small, the relative robustness r R of all three filters is approximately 14, according to the results of Fig. 4. This value results in a filter stop-band between 2.5 Hz and 37.5 Hz. This band, where the filters exhibit their robustness properties, occupies almost 88 percent of the entire filter band according to Fig. 5. In view of the expected variations of the lowest natural frequency of the system, as it is presented in Table 3, is in fact 9 times wider than actually needed for the specific experiments. From the DEO curves shown in Figs. 共3共a,b,c兲兲, the minimum value of the relative delay d was found to be around 1.18 for each one of the three methods considered. According to Eq. 共20兲 and these values for the natural frequency and relative delay, the total delay to be expected by the filter, was calculated to be 0.47 s. The sampling frequency f SF according to Eq. 共21兲 is approximately 40 Hz. Using the look-up tables described in Section 3.6 the parameters r B and r C were retrieved. Then using the corresponding commands from the Signal Processing Toolbox of MATLAB 关25兴 the coefficients of the 3 filters were calculated 共see Table 4兲. The frequency responses of the three filters used in these experiments are presented in Fig. 8. They are almost identical, except some slight differences concerning the size and the distribution of the ripples in the stop-band. The similarity of the filter coefficients in Table 4, as well as of the frequency responses and the equal distribution of the ripples verify the optimality of the procedure, used for their generation. Transactions of the ASME

Table 4 Coefficients of the filters used in the experiments

Table 5 Residual vibrations induced by the original and the filtered inputs

Fig. 8 Filter frequency responses with respect to the normalized frequency f Õ f 0 for the three filters used in the experiments: „a… Parks-McClellanFilter, „b… Chebyshev Window Method, and „c… constrained least squares filter „ f 0 Ä2.5 Hz….

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Fig. 9 Case A: beam vibration with 200 gr. additional mass

Fig. 12 Case D: beam vibration with 20 gr. additional mass

4.3 Results. The filter was then applied off-line to the original guidance function. Figure 7 presents the filtered guidance function as obtained by the Parks-McClellan filter. Table 5 contains a summary of the experimental results. By preconditioning the inputs with anyone of the three filters the level of vibrations of

Fig. 13 Case E: beam vibration with no additional mass

Fig. 10 Case B: beam vibration with 100 gr. additional mass

Fig. 11 Case C: beam vibration with 40 gr. additional mass

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the flexible beam was dramatically reduced to only 7– 8 percent of the level of vibrations obtained with no preconditioning. Even if the filters were designed for the natural frequency of Case A, they performed equally well in all five cases, thus, verifying the robustness of the proposed method. Figures 9–13 present the vibrations of the tip of the beam for the five cases when the preconditioning of the input is done with a filter designed with the Constrained Least Squares method. These vibrations are compared with the residual vibrations induced by the unfiltered input function. Similar results are obtained when the inputs are preconditioned with anyone of the other two filters. In Figs. 9–13 the continuous lines represent the response of the tip for a filtered input while the dashed lines represent the response with the corresponding unfiltered input. The first part of each curve in Figs. 9–13, represented with a light colored line, shows the transient vibration 共vibration during the motion兲 while the largest second part, shown with a dark line, represents the residual vibrations. It is clearly seen that the preconditioned input functions resulted in a significantly reduced amount of residual vibrations compared with the amount of vibrations obtained from the original input function. Even if the system mass changed as much as 287 percent of the original mass, the amount of vibrations obtained from the filtered input function was less than 8.32 percent of the residual vibrations obtained from original input function. Transactions of the ASME

5

Conclusions

Preconditioning any guidance function by filtering it with a properly designed low-pass FIR digital filter drastically reduces residual as well as transient vibrations in mechanical systems. Maximal robustness can be achieved, capable to cover not only extended variations of the multiple modes of the system, but also the major part of the frequency axis. Moreover, this maximal robustness results in quite satisfactory delay times and residual vibration errors for the filters tested. To further increase this robustness for filter orders above a minimum value, the filter order should be increased proportionally to the robustness increase. For example, the robust region of a Parks-McClellan filter type of order 128 extends up to approximately 105 times the lowest expected natural frequency of the system, and occupies 98 percent of the entire filter band. Furthermore, it induces residual vibration errors up to 5 percent and presents a delay time of 1.18 times the maximum expected natural period of the system. This robustness can be doubled to approximately 210 times the lowest expected natural frequency of the system, just by doubling the filter order to 256, without any other practical consequence to the rest of the filter performance characteristics mentioned above. In addition to the above excellent performance characteristics, the application of the method to actual mechanical systems is quite simple and versatile, since its practical implementation requires just the application of an already predesigned digital filter. The corresponding filtering operation can be performed either online or offline, quite independently from the type of the original guidance. Thus, the method can be easily applied in practice to any mechanical system, with any form of original guidance, either derived through mathematical approaches 共e.g., path planning methods兲, or input directly to the system 共e.g., through direct operator commands兲. The filtering approach shows its vibration suppression capabilities at any stage of the motion 共both transient and end of trajectory兲. The only adverse effect 共cost paid兲 for vibration suppression is the delay introduced by the filter. However, there are many other additional benefits that can be claimed using the filtering approach. Due to the maximally robust effect of the filter, practically the flexible system behaves as a rigid body during the motion. Thus, almost all the energy from the actuator is transformed to kinetic energy, without any practical loss to elastic energy. This can result in energy minimization and optimal actuator utilization.

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Journal of Dynamic Systems, Measurement, and Control

关4兴 Antoniadis, I., and Kanarachos, A., 1996, ‘‘A Spline-Based Guidance With Enhanced Frequency Robustness for the Motion of Flexible Part Handling Manipulators,’’ Mechatronics, 6, pp. 761–777. 关5兴 Aspinwall, D. M., 1980, ‘‘Acceleration Profiles for Minimizing Residual Response,’’ ASME J. Dyn. Syst., Meas., Control, 102, pp. 3– 6. 关6兴 Bhat, S. P., Tanaka, M., and Miu, D. K., 1991, ‘‘Experiments on Point-to-Point Position Control of a Flexible Beam Using Laplace Transform Technique— Part I: Open-Loop,’’ ASME J. Dyn. Syst., Meas., Control, 113, pp. 432– 437. 关7兴 Meckl, P. H., and Seering, W. P., 1987, ‘‘Reducing Residual Vibration in Systems with Time Varying Resonances,’’ Proceedings of the 1987 IEEE International Conference on Robotics and Automation, 1690–1695. 关8兴 Singhose, W. E., Porter, L. J., Tuttle, T. D., and Singer, N. C., 1997, ‘‘Vibration Reduction Using Multi-Hump Input Shapers,’’ ASME J. Dyn. Syst., Meas., Control, 119, pp. 320–326. 关9兴 Singh, T., and Heppler, G. R., 1993, ‘‘Shaped Input Control of a System with Multiple Modes,’’ ASME J. Dyn. Syst., Meas., Control, 115, pp. 341–347. 关10兴 Singh, T., and Vadali, S. R., 1995, ‘‘Robust Time-Delay Control of MultiMode Systems,’’ Int. J. Control, 62, pp. 1319–1339. 关11兴 Murphy, B. R., and Watanabe, I., 1992, ‘‘Digital Shaping Filters for Reducing Machine Vibration,’’ IEEE Trans. Rob. Autom., 8, pp. 285–289. 关12兴 Tuttle, T. D., and Seering, W. P., 1994, ‘‘A Zero-Placement Technique for Designing Shaped Inputs to Suppress Multiple-Mode Vibration,’’ Proceedings of the 1994 American Control Conference, Baltimore, MD, pp. 2533–2537. 关13兴 Pao, L. Y., 1996, ‘‘Input Shaping Design for Flexible Systems with Multiple Actuators,’’ Proceedings of the 1996 IFAC World Congress, San Francisco, CA. 关14兴 Singhose, W. E., Seering, W., and Singer, M. C., 1996, ‘‘Input Shaping for Vibration Reduction With Specified Insensitivity to Modeling Errors,’’ Proceedings of the 1996 Japan-USA Symposium on Flexible Automation, Boston MA. 关15兴 Lim, S., Stevens, H. D., and How, J. P., 1999, ‘‘Input Shaping Design for Multi-Input Flexible Systems,’’ ASME J. Dyn. Syst., Meas., Control, 121, pp. 443– 447. 关16兴 Pao, L. Y., and Lau, M. A., 2000, ‘‘Robust Input Shaper Control Design for Parameter Variations in Flexible Structures,’’ ASME J. Dyn. Syst., Meas., Control, 122, pp. 63–70. 关17兴 Magee, D. P., and Book, W., 1995, ‘‘Filtering Micro-Manipulator Wrist Commands to Prevent Flexible Base Motion,’’ Proceedings of the 1995 American Control Conference, Seattle, WA, pp. 924 –928. 关18兴 Singer, N. C., Singhose, W., and Kriikku, E., 1997, ‘‘An Input Shaping Controller Enabling Cranes to Move Without Sway,’’ Proceedings of the 1997 ANS Topical Meeting on Robotics and Remote Systems, Augusta, GA. 关19兴 Jones, S. D., and Ulsoy, A. G., 1999, ‘‘Approach to Control Input Shaping with Application to Coordinate Measuring Machines,’’ ASME J. Dyn. Syst., Meas., Control, 121, pp. 242–247. 关20兴 Antoniadis, I., 1999, ‘‘Guidance Preconditioning by an Impulse Sequence for Robust Residual Vibration Suppression,’’ Shock and Vibration, 6, pp. 133– 145. 关21兴 Antoniadis, I., and Economou, D., 1999, ‘‘Robust Residual Vibration Suppression by Digital Filtering the Guidance Function,’’ Proceedings of 3rd Greek National Congress on Computational Mechanics. 关22兴 Singer, N., Singhose, W., and Seering, W. P., 1999, ‘‘Comparison of Filtering Methods for Reducing Residual Vibration,’’ European Journal of Control, 5, pp. 208 –218. 关23兴 Proakis, J. G., and Manolakis, D. G., 1988, Introduction to Digital Signal Processing, Mcmillan Publishing Company. 关24兴 Parks, T. W., and Burrus, C. S., 1987, Digital Filter Design, Wiley, NY. 关25兴 The Mathworks Inc., 1998, Signal Processing Toolbox, for Use with MATLABUser’s Guide, v.5.2, Natick, MA. 关26兴 Lee, C., and Mavroidis, C., 2000, ‘‘WinReC v.1: Real-Time Control Software for Windows NT and its Applications,’’ Proceedings of the 2000 American Control Conference, June 28 –30, 2000, Chicago, IL, pp. 651– 655.

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