ROBUST RESIDUAL VIBRATION SUPPRESSIONUSING IIR DIGITAL FILTERS

D. Economoua, C. Mavroidisb and I. Antoniadisa

a National Technical University of AthensDepartment of Mechanical Engineering,

9 Heroon Polytechniou Str., 15773 Athens, Greece,Email: dim_economou@yahoo.com, antogian@central.ntua.gr

b Rutgers University, The State University of New Jersey,Department of Mechanical and Aerospace Engineering,

98 Brett Rd., Piscataway, NJ, USA 08854-8058,Email: mavro@jove.rutgers.edu

Abstract: A method for suppressing residual vibrations in flexible systems is presented andexperimentally demonstrated. This method is based on the preconditioning of the inputs to thesystem using Infinite Impulse Response (IIR) digital filters. The proposed method is robust tovariations of the system dynamic parameters and simple to implement in practice. The method isexperimentally verified using a flexible aluminum beam. Copyright © 2001 IFAC.

Keywords: Vibration, Digital Filters, Robustness.

1. INTRODUCTION

Residual vibration suppression is important in abroad range of mechanical engineering applicationssuch as the deployment of space structures andcranes or the operation of machine tools and flexiblerobots. The traditional approaches to minimize theeffect of residual vibrations are focused on eitherincreasing the structural stiffness, which increases thesystem's size and weight or using closed loop controlmethods, which require accurate on-board electronicsand increase the system’s cost and complexity.

An alternative approach for suppressing residualvibrations is the proper conditioning of the pre-specified excitation pattern, the so-called Guidance,so that the system moves exactly to the desired endposition without any residual vibrations. This conceptis very attractive, since it can significantly simplify

the system requirements and complexity. Significantresearch effort has been done on the properconditioning of a guidance function when applied onflexible systems, for more than 40 years (Smith,1958). Since methods in this category aretraditionally considered to be quite sensitive tovariations of the system dynamic parameters,significant research effort has been devoted toincrease their robustness features. A brief survey ofrelevant methods has been performed by Antoniadis(1999).

A typical well known example of such methods is theinput shaping approach, based on the convolution ofan arbitrary guidance function with a series ofimpulses (Singer and Seering, 1990; Singh andVadali, 1995; Singhose et al., 1997). A lot ofresearch has been done to propose design techniquesfor the input shapers so that they suppress multiple

modes of vibration and have increased robustness(Tuttle and Seering, 1994; Pao, 1996; Lim et al.,1999; Pao and Lau, 2000). While input shapingmethods present good performance in a variety ofapplications, their robustness is limited in local areasaround the system natural frequencies and can beincreased only by increasing the total duration of thepulse sequence. This results in unnecessary delays inthe total duration of the system motion.

In a first attempt to extend the robustness of thevibration suppression method based on theconvolution with a series of impulses, a generalapproach has been proposed by Antoniadis, (1999),leading to a set of three different methods for thedesign of the impulse sequence. The impulsesequence approach has been extended (Antoniadisand Economou, 2001), by establishing a framework,according to which the design of an appropriateguidance function can be transformed to the properdesign of a FIR digital filter. The effectiveness ofFIR filters for vibration suppression has beenexperimentally verified (Economou et al., 2000).

In this paper, a preconditioning approach is proposed,based on the proper design of conventional low-passIIR digital filters. First, requirements are establishedfor a preconditioned guidance function that moves aflexible system to a desired end position withoutexciting residual vibration effects, having a robustperformance with respect to variations of the flexiblesystem dynamic parameters. Then the aboverequirements are shown to be met by the properdesign of a digital IIR filter which is subsequentlyused to filter the original guidance function. Theresults are experimentally verified using a flexiblealuminum beam. The goal of the experiments is tomove the flexible beam, using an industrial robotmanipulator, from an initial location to a final one,without residual vibrations. A low-pass IIR filterdesigned using the method proposed by Thiran(1971) is chosen to filter the input to the system.Robustness tests are conducted attaching a differentmass M at the beam's free endpoint.

2. THEORETICAL DEVELOPMENT

2.1. Requirements for Residual VibrationSuppression Using Input Preconditioning

The motion of a typical mode of a dynamic systemwith natural frequency f0, damping ratio ζ, excitationfunction (guidance) g(t) and response vector x(t), canbe described in the following state space form:

( ) ( ) ( ) t t g t= +x Ax b (1.a)

0 02

0 0

0 1 0= , = , 2

2 1fω π

ω ζω=

− −

A b (1.b)

Using the Duhamel Integral, and the Sylvester

expansion method to the transition matrix teA , theexpression for the response can be written as:

2

i ii=1

= (0) + ( ; )( ) te G s s qt = ∑A x H bx (2.a)

2

1 2 0 0 0= - 1-,q ζω ω ζ± (2.b)

[ ]i ii j

1 = for i, j=1,2 i j q

q q− ≠

−H A I (2.c)

where G(s) is the Laplace transform of the guidancefunction, evaluated at the two system poles qi.

According to the proposed preconditioning approach,instead of the direct implementation of the originalguidance function g(t) in Equation (1.a), aconditioned guidance function u(t) can bealternatively used, obtained by the original guidancefunction g(t) by introducing a sequence of the formshown in Eq. (3):

N M

n nn=0 n=1

( ) ( n ) ( n ) SF SFu t c g t - T a g t - T= −∑ ∑ (3)

where {cn} and {an} are series of coefficients oflength N+1 and M respectively and TSF is a constantperiod, characterizing the time instants where thepast values of g(t) and u(t) are evaluated.

The corresponding Laplace transform of u(t) is:

( ) = ( ) ( )U s F s G s (4.a)

N- n

nn=0

M- n

nn=1

( ) =1

SF

SF

s T

s T

c eF s

a e+

∑

∑ (4.b)

The purpose of introducing the sequence in Eq. (3), isthat the conditioned guidance function u(t) is able tomove the system in essentially the same way as theoriginal guidance function g(t), without the effect ofthe residual vibrations. This requirement can bestated as:

i i( ; ) 0 ( ) = 0U s s q F s; s=q= = ⇔ (5.a)

( ; 0) ( ; 0) ( 0) = 1U s s G s s F s; s == = = ⇔ (5.b)

In view of Eqs. (2.a) and (4.a), Eq. (5.a) implies thatthe residual vibration effect can be completelycanceled just by the proper selection of F(s), quiteindependently from the properties of the originalguidance function g(t). In the same way, Eq. (5.b)implies that u(t) maintains all the properties of g(t) atsteady state, so that the system reaches its desiredend-position with no vibrations.

2.2. Input Preconditioning Using IIR Digital Filters

Infinite Impulse Response (IIR) filters are series ofconstants {cn} and {an} (Proakis and Manolakis1988). The mathematical foundation of filtering is aconvolution procedure, resulting to a discrete filteredsignal u(kTS), which is obtained from an originaldiscrete input signal g(kTS) according to Eq. (6),

N M

n nn=0 n=1

( ) ( n ) ( n )S S SF S SFu kT c g kT - T a g kT - T= −∑ ∑ (6)

where k is an integer, TS is the sampling period of thediscrete signals u(tk) and g(tk), TSF is the samplingperiod of the filter, {cn} and {an} are series ofconstants of length N+1 and M respectively. Themaximum of N and M is the filter order. The z-transform of the filtered function U(z) is related tothe z-transform of the original input function G(z) by:

N-n

nn=0

M-n

nn=1

( ) ( ) ( ) = ( )1

c zU z F z G z G z

a z=

+

∑

∑ (7)

where:N

-nn

n=0M

-mm

m=1

( ) = 1

c zF z

a z+

∑

∑ (8.a)

with a frequency response function of the form:

N- n 2

nn=0

M- n 2

nn=0

( ) ( 2 )1

SF

SF

j fT

j fT

c eF j F j f

a e

π

πω π= =

+

∑

∑ (8.b)

Eqs. (6), (7) and (8.a,b) clearly imply that the designof a proper function F(s), is equivalent to the designof an IIR filter. Provided that the frequency responsefunction F(jω) of the filter is zero at frequenciescoinciding with the natural frequencies of thedynamic system, this filter is capable to completelyeliminate the residual vibration effect. In addition,according to Eq. (5.b), the response for zerofrequency of this filter must be equal to one. In viewof Eq. (8.b), this last requirement becomes:

N

nn=0M

nn=0

(0) 1c

Fa

= =∑

∑ (9)

The robustness properties for the preconditioningprocedure can be directly met, by extending therequirement for zero frequency response of the filternot only for individual frequencies coinciding withthe expected natural frequencies of the system, but

also for extended areas of the frequency responsefunction F(jω), in order to cover now additionally thepossible variations of the system natural frequencies.

3. FILTER DESIGN CRITERIA

If at least one of the coefficients {an} has non-zerovalue, then the filter is an Infinite Impulse Response(IIR) filter. IIR filters are capable to meet a given setof specifications with a much lower filter order thanthat of a corresponding FIR filter, but they introducelarger total delay.

According to Eq. (5.a,b), an appropriate filter musthave a pass-band (response equal to 1) for the lowestfrequencies and a stop-band (response equal to 0) forfrequencies equal or higher of the natural frequencyof the flexible system. This type of filter is calledlow-pass filter.

An extra insensitive filter capable to suppress theresidual vibrations of a flexible system, must coverthe following requirements:R.a) Cut-off frequency (lower limit of the stop-band)

smaller than the system's smallest naturalfrequency.

R.b) Stop-band quite wide in frequency in order tocover variations of the system frequencies andalso to cover higher modes of the system.

R.c) Response for zero frequency equal to one.R.d) Ripples on the stop-band smaller than a pre-

specified acceptable residual vibration error.R.e) Minimum possible filter delay

Several methods are used to design IIR filters (tocalculate the coefficients {cn} and {an}) that canapproximate the desired frequency response. Most ofthese methods make use of analog prototype filters,such as the Butterworth or the elliptic prototype.Filters designed using these methods have non-linearphase and they introduce large delays.

In this work an IIR filter, proposed by Thiran (1971),was used. This filter introduces smaller delays thanthe other IIR filters.

4. EXPERIMENTAL DEMONSTRATION

4.1 Experimental System and procedures

The proposed method is experimentally verifiedusing a thin, long, rectangular, flexible, aluminumbeam that has the physical characteristics shown inTable 1. The beam is attached at the end-effector of afive-degree-of-freedom Mitsubishi RV-M2 robotmanipulator. Figure 1 shows a schematic and apicture of all mechanical and electrical componentsof this system. The input function to the flexiblebeam is the angular displacement θ(t) introduced byone of the manipulator wrist joints, as it is shown inFigure 2. The rotation is by an angle of 30 degrees ofthe vertical position and the plane of the rotation is

Fig. 1. The Picture and Schematic of the RutgersUniversity Experimental System

Table 1: Physical Properties of the Aluminum Beam

Characteristic ValueDensity 2.7x103 Kg/ m3

Weight 0.0697 KgLength 0.386 mThickness 0.62x10-3 mWidth 0.108 m

perpendicular to the beam's body. The joint is underclosed-loop PID control to ensure accurateimplementation of the desired input function.

Prior to the execution of the experiment, the inputfunction is off-line preconditioned using a low-passIIR filter as it is described in Section 4.2. Theoriginal (non-filtered) input function used in theseexperiments is the fastest rotation that themanipulator’s wrist is able to follow. The duration ofthe original input function is 0.3 sec. The duration ofthe preconditioned inputs is 0.9 sec, introducing adelay of 0.6 sec. Figure 3 shows, the original and thepreconditioned input.

Sets of experiments are performed with differentmass M attached at the beam's free endpoint (seeFigure 1). To test the robustness of the proposedmethod, the mass M is allowed to vary from zero upto 287% of the total mass of the beam causing avariation of the natural frequency of the beam. Fivecases of additional mass at the end of the beam areconsidered and the corresponding natural frequenciesof the first mode for the flexible beam / mass systemare shown in Table 2.

An Entran Accelerometer (Model EGE-732B-2000D-/RS) is attached at the free-end of the flexiblebeam to record the beam's oscillations. WinRec v.1, asoftware developed at the Robotics and MechatronicsLaboratory at Rutgers University, is used in bothreal-time control and data acquisition (Lee and

Fig. 2: a) Rotation of the Flexible Beam Using theWrist Joint, b) Original and Filtered GuidanceFunctions

Table 2: Measured System frequencies (Hz) of FirstMode for 5 Different Added Masses

Case AM=200 g

Case BM=100 g

Case CM=40 g

Case DM=20 g

Case EM=0 g

2.5 3.2 4.13 4.8 6.4

Mavroidis, 2000). The timer in this experiment is setat 200Hz.

4.2 Design of the Filter

A filter designed using the method proposed byThiran (1971) was used. This filter must cover therequirements (R.a) to (R.e) of section 3. In order toensure the proper design of the filter, the stop-bandwidth ratio, a non-dimensional index is used:

/ 2C

C

SF

fr

f= (10)

The purpose of the introduction of the above ratio isthe fact that, for an already designed filter, this ratiohas constant value regardless of the value of thesampling frequency fSF. Using the ratio rC, the filtercan be designed in a general non-dimensional form,independently from the actual values of thefrequencies fC and fSF and therefore independentlyfrom the actual dynamic system.

Additionally, a filter performance index, the relativedelay d introduced by the filter, is defined as:

0

0 0

L LD SF

SF

T T fd

T T f= = = (11)

where TD is the total time delay and L is the totaldelay expressed as number of samples. Given theinfinite nature of IIR filters, total delay is the timeneeded for a filtered signal to settle within a smallrange around its steady state value. Taking in accountthat 0Cf f≥ , according to requirement (R.a), andusing Eq. (10), Eq. (11) can be written as:

minL L

( )2 2

C Cr rd d≥ =⇒ (12)

where (d)min the minimum value of d. From Eq. (12)it is clearly seen that the relative delay is independentfrom the dynamic system. Minimization of the totaldelay TD introduced by the filtering process,according to requirement (R.e), is equivalent tominimization of the relative delay d, which can beperformed independently from the dynamic system.

In these experiments the acceptable residual vibrationerror (requirement (R.d)) was arbitrarily set at 5% ofthe vibrations induced by the original input. For thisvalue of the acceptable vibration error, an iterativealgorithm was used to vary the design parameters, todesign the filter for each set of the design parametersand then to calculate the relative delay d introducedby the filter. The software package Matlab version5.2 and its Signal Processing Toolbox (TheMathworks Inc., 1998) were used for the executionof the iterative algorithm. The minimum relativedelay was found to be about 1.5 for filter order M=7.

Based on these results, a low-pass filter of order 7was designed for these experiments, using thesoftware package Matlab version 5.2. The frequencyresponse in the stop-band was less than 5% of theamplitude of the pass-band. The total duration of thefilter was 0.6 sec and also the total delay TD was 0.6sec, which is 1.5 times of the system's largesteigenperiod and the corresponding samplingfrequency of the filter was equal to 13.33 Hz. Thecoefficients of the filter are shown in Table 3 and thefrequency response is shown in Figure 3.

Table 3: Coefficients of the filter used.

n cn an0 0.02062 -1 - -3.485502 - 5.516623 - -5.092944 - 2.941075 - -1.056456 - 0.217577 - -0.01975

Figure 3: Frequency Function of the Filter Used

4.3. Results

Table 4 contains a summary of the experimentalresults. By preconditioning the inputs with theselected filter the level of vibrations of the flexiblebeam was dramatically reduced to only 5-6% of thelevel of vibrations obtained with no preconditioning.

Fig. 4: Beam Vibration With: A) 200 g. AttachedMass, B) 100 g. Attached Mass, C) 40 g.Attached Mass, D) 20 g. Attached Mass and E)Without Attached Mass.

Although the filter was designed for the naturalfrequency of Case A, it performed equally well in all

Table 4: Residual Vibrations Introduced from theOriginal and the Filtered Inputs

Case Addedmass (g)

Peak-to-peak

amplitude(original )

(mm)

Peak-to-peak

amplitude(filtered)

(mm)

Filteredvibration as

percentage ofthe originalvibration

A 200 85.7 4.7 5.46%B 100 78.2 3.2 4.06%C 40 46.9 2.6 5.56%D 20 57.0 3.0 5.32%E 0 64.9 1.9 2.90%

five cases, thus, showing the robustness of theproposed method.

Figure 4 presents the vibrations of the tip of the beamfor the five cases. These vibrations are comparedwith the residual vibrations induced by the unfilteredinput function. It is clearly seen that thepreconditioned input functions resulted in asignificantly reduced amount of residual vibrationscompared with the amount of vibrations obtainedfrom the original input function. Even if the systemmass changed as much as 287% of the original mass,the amount of vibrations obtained from the filteredinput function was less than 5.56% of the residualvibrations obtained from original input function.

It must be noticed that, although the vibrations of thepreconditioned input are slightly larger than thetheoretically expected 5%, the cancellation ofresidual vibrations is quite satisfactory. The mainreason for this difference is that the actual filteredinput, provided by the manipulator’s wrist, is only anapproximation of the desired preconditioned input.

5. CONCLUSION

The preconditioning of any guidance function byfiltering with a properly designed low-pass IIRdigital filter, drastically reduces residual vibrations inmechanical systems. The cost paid for thissuppression is a delay equal to 1.5 times themaximum expected natural period. The residualvibration can be practically suppressed over anextended frequency range. Satisfactory robustness isachieved, capable to cover extended variations of thedynamic characteristics of the flexible system. Thisrobustness makes the method applicable to realmechanical systems. Since the practicalimplementation of the method requires just theapplication of a digital filter, the application of themethod for residual vibration suppression is quitesimple and versatile. The corresponding filteringoperation can be performed either online or offline,quite independently from the type of the originalguidance. Thus, the method can be easily applied inpractice to any mechanical system, with any form oforiginal guidance, either derived throughmathematical approaches (e.g. path planning

methods), or input directly to the system (e.g.through direct operator commands).

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