Input Predictive Shaping for Vibration Control of Flexible Systems

Stanislao Grazioso1 Student Member, IEEE, Giuseppe Di Gironimo1,William Singhose3 and Bruno Siciliano2 Fellow, IEEE

Abstract— This paper presents the foundation of a new classof input shapers, designed using a predictive approach. Themethod is used to control the transient and residual vibrationsin flexible nonlinear systems with time−varying parameters.The motivation is the development of simple algorithms andarchitectures for controlling the motion in flexible nonlinearsystems with minimal modeling effort. The approach trainsan artificial neural network to obtain closed−form expressionsused for calculating, in real time, the amplitudes and the timelocations of the impulses required by a common input−shapingtechnique. In this work we use this idea to design a commandshaper for controlling the motion of the simplest flexible non-linear system, an overhead crane with a suspended payload. Wevalidate the approach using simulations and experiments. Thebenefits of such a control system will, in the end, enable usingthis method for controlling the motion of complex nonlinearsystems, resulting in almost zero vibrations.

Index Terms— Nonlinear dynamical systems, vibration con-trol, predictive control, command shaping.

I. INTRODUCTION

Accurate motion of flexible systems is a challenging problembecause it often results in high levels of vibrations. Classicalfeedback control methods may be used to reduce transientvibrations, but they can be expensive and difficult to imple-ment, as they require sensors information. Moreover, theyoften need a complete dynamical model of the controlledsystem and significative control power. If not properly de-signed, then feedback may lead to instability of the system.

Command−shaping methods [1] are proven techniques tocontrol the motion of flexible systems: shaping the referencecommand such that the vibratory modes of the system arecanceled reduces the vibrations. In these methods, the refer-ence signal is convolved with a sequence of impulses, namelyan input shaper. The rise time of the command is lengthenedby the duration of the shaper. Hence, to achieve high−speedmotion, it is important to minimize the shaper duration. Themain advantage of input shaping techniques is that the timingand the amplitude of the impulses are determined by solvinga set of constraint equations using only estimates of thesystem natural frequencies and damping [1].

1 Stanislao Grazioso and Giuseppe Di Gironimo are with Departmentof Industrial Engineering, University of Naples Federico II and CRE-ATE Consortium, 80125, Napoli, Italy, {stanislao.grazioso,giuseppe.digironimo}@unina.it

3 William Singhose is with Faculty of Mechanical Engineer-ing, Georgia Institute of Technology, Atlanta, GA 30332, USAsinghose@gatech.edu

2 Bruno Siciliano is with PRISMA Lab, Department of ElectricalEngineering and Information Technology, University of NaplesFederico II and CREATE Consortium, 80125, Napoli, Italy,bruno.siciliano@unina.it

shaped command system responsereference command

input predictive

shaper

flexiblenonlinearsystem

time

f(t)

f(t)

time

vibration

time

Fig. 1. Input predictive shaping process.

However, shapers can be sensitive to parameter estimation,i.e. if the frequencies and the damping are not well es-timated or they are time−varying, then the effectivenessof the method is reduced. For some shapers, residual vi-bration increases rapidly as the actual parameters deviatefrom the modeled or estimated parameters. One possiblestrategy to overcome this problem is the use of robust inputshapers [2], which allow a significative vibration suppressioneven in presence of uncertainity in the plant parameters.The disadvantage in this case is increasing the rise time,which becomes higher as the robustness of the shaper in-creases. Moreover, robust shapers still suffer in systems withtime−varying parameters, since they allow the suppressionof the vibrations in a pre−defined frequency interval. In sum,the use of robust shapers, even if it allows for vibrationsuppression in presence of uncertainity, it still presents twomain problems: (i) increasing of the rise time; (ii) requiringa−priori knowledge of the frequency range in which thesystem operates.

In this paper we propose a predictive approach to developa new class of input shapers, as illustrated in Fig. 1. Thereference command is convolved with two impulses whoselocation and amplitude is given by the input predictiveshaper. The shaped command acts on the nonlinear system,which results in almost zero residual vibrations. In this workwe obtain a closed−form predictive−adaptive input shaperfor vibration suppression in slewing flexible systems withtime−varying natural frequencies. Using a zero vibrationstructure we overcome the limitation in (i). The predictiveapproach allows the method overcoming the limitation in (ii).The input shaper results from an offline training procedureused to learn, in a data-based fashion, the response of thesystem when it is subjected to a wide range of differentinputs. Thus, it might be used to control the motion offlexible nonlinear systems in complex trajectory trackingtasks, without knowing a priori the range of frequencies inwhich the system will operate.

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Following the trend of the past years in using predictiveapproaches to develop model−based controllers (a goodexplaination of model predictive control is given in [3]),the current work aims at introducing a predictive approachalso in designing input shapers. The idea is to train aneural network to compute the expressions for the amplitudesand the time locations of the impulses in systems withtime−varying parameters. We refer to this class of shaper asinput predictive shapers (IPS). The current work develops azero−vibration input shaper using a predictive approach: thisshaper will be referred to as zero vibration − input predictiveshaper (ZV-IPS). Simulations and experiments to validatethe method have been performed on a single−pendulumoverhead crane.

The rest of the paper is organized as follows. In the nextSection we review related work in the field. In Section IIIwe present the theoretical backrgound beyond the IPS. InSection IV we report the simulations and the experimentsperformed to validate this methodology for controlling thevibrations in a trajectory tracking task. Section V introducesa robustness analysis. Section VI concludes the paper anddiscusses future developments.

II. LITERATURE REVIEW

The earliest work on systematic command shaping can befound in the posicast control introduced by OJM Smith [4].After many years, Singer and Seering [5] introduced the ideaof robust shapers by adding in the design process a new con-straint, which forces the derivative of the residual vibrationwith respect to the frequency to equal zero. They also set tozero higher−order derivatives. The result is the achievementof more robust shapers, which limited the residual vibrationsover a wider set of frequencies. But, this increases thelength of the command. After the idea of zero−derivative(ZD) shapers, Singhose et. al [6] introduced the conceptof extra−insensitive (EI) shapers, where the constraint ofzero vibration at the modeling frequency was replaced witha constraint which limits the vibration to a small value.Another robust technique, called specified−insensitive (SI)shaping, suppresses the vibration in a specified range offrequencies [7]. A comprehensive review on robust commandshaper can be found in the work of Vaughan et. al [2]. Adifferent approach is to follow an adaptive strategy in lieu ofrobust methods. The adaptation of the impulse amplitudesand time locations to the changing dynamic properties ofthe system has been studied by many researchers. Currently,there are mainly two ways to design adaptive input shapers.The first one is to use a frequency−domain identificationscheme to estimate the vibration frequencies of single−modeor multi−mode systems [8]. Another approach adapts theinput shaper directly from the system output [9]. In [10], anadaptive feedforward control, consisting of a finite impulseresponse filter and a command input, is used for motioncontrol with zero residual vibrations in robots with flexiblearms.More recent works are related to applications and the com-bined use of input shaping and other control techniques.

o

payload

cable

θ(t)l(t)

gg

Fig. 2. Overhead crane with suspended payload.

Several works use hybrid control schemes given by thecombination of input shaping with: model reference con-trol [11], linear quadratic regulators [12], PID [13], feedbacklinearization [14]. Input shaping validation as a controlscheme for vibration suppression has been proved for indus-trial robots [15]; robots with flexible arms [16], [12]; wear-able robotic arms [17]; overhead cranes [18], [19]; fuelingtransporting systems in nuclear plants [20]; nanopositioningtasks [21].This paper introduces a predictive approach in the context ofcommand−shaping methods. The importance of the themeis illustrated by the recent comparisons between modelpredictive control and input shaping [22], [23], [24]. Thecontribution of this work lies in providing the foundation forthe development of input predictive shaping. The results ofthe ZV-IPS make the use of IPS attractive in real industrialscenarios for controlling the vibrations in uncertain flexiblesystems with a minimal modeling effort.

III. INPUT PREDICTIVE SHAPING

This section provides the theoretical background whichleads to the development of the input predictive shap-ing method. We develop a closed−form ZV-IPS for asingle−pendulum overhead crane.

A. System dynamics

An overhead crane with a suspended payload is a nonlineardynamic system. The model used in this work is shown inFig. 2. The system is a planar gantry crane with a variablecable−length l(t) that simulates hoisting the payload. Thecable is modeled as an inflexible and massless rod pinnedto the trolley. The configuration of the system is specifiedby the horizontal position of the trolley, x, the length of thecable, l, and the swing angle of the cable with respect to thevertical axis z, θ . Using a Lagrangian approach, the equationof motion of the system takes the form of a second−ordernonlinear ordinary differential equation with time−varyingcoefficients:

lθ +2lθ +gsinθ + cosθ x = 0 (5)

where g is the acceleration due to gravity. In the following,we assume a linear variation of l, thus l is constant.

306

TABLE IANALYTICAL EXPRESSIONS FOR THE ZV-IPS OF AN OVERHEAD CRANE WITH SUSPENDED PAYLOAD.

A1 =2.065

e−0.6987l+0.9542l+1.300 +1− 5.306

e−0.3422l−0.7200l−3.123 +1− 9.254

e−0.4359l+0.8196l+2.543 +1+

1.355

e0.2482l+0.0055l+2.524 +1+5.709 (1)

A2 = 1−A1 (2)t1 = 0 (3)

t2 =2.438

e0.0979l−0.3130l−0.1394 +1+

20.62

e0.0992l−5.124l−4.502 +1− 5.566

e−0.1764l+0.0964l−1.143 +1+

32.78

e0.0290l−0.9456l−4.046 +1−49.55 (4)

0 π 2π 3π 4π

−0.5

0

0.5AAA111 AAA222

Time [s]

Posi

tion

[mm

]

A1 response A2 response Total response

Fig. 3. ZV input shaping process.

B. ZV−IPS Design

In this section we derive the mathematical expression forthe ZV-IPS. It has a ZV−like structure, which provides a lowrise time. A ZV input shaper is implemented by convolvingtwo impulses with the desired reference command. Whena first impulse is applied to a system, some vibrations areexcited. If a second impulse is applied at the correct time,when the system crosses the zero deflections, i.e. at the timecorresponding to the half of the system natural period, thenthe combined response of the two impulses results in zeroresidual vibrations; see Fig. 3 for its graphical interpretation.In the case of zero damping, the amplitudes and times of theshaper are given by [

Aiti

]=

[0.5 0.50 T

2

](6)

where T is the natural period of the first mode of the system.This means that amplitudes and times of the pulses are easilycalculated if the system has a fixed or time−invariant naturalfrequency.When the natural frequencies are time−varying, as in thecase of a single−pedulum overhead crane with varyingsuspension length, the amplitudes and the times of a ZVshaper may no longer be trivially calculated using (6). Infact, as we can see in Fig. 4, the zero crossing times forpayload deflection and velocity change as suspension lengthvaries during movement. In general, when the hoist cableis shortened, the deflection and the velocity of the payloadincrease, thus the natural frequency increases, reducing thezero crossing time. The opposite happens when the hoistcable is lengthened.

0 0.5 1 1.5 2

−5

0

5

·10−2

Payl

oad

Defl

ectio

n[m

m]

0 0.5 1 1.5 2

−2

0

2

·10−1

Time [s]

Payl

oad

Vel

ocity

[mm

s−1 ]

Cable shortening No changes Cable lengthening

Fig. 4. Zero cross for payload deflection and deflection rate inoverhead cranes with varying cable suspension length subjected toone impulse. Velocity of the cable is 0.5 ms−1.

In order to calculate the appropriate impulse amplitudesand times, the input−output relationship which relates thestarting cable length and velocity with the zero crossing timefor the payload deflection and velocity should be estabilished.During an offline phase, we compute in simulation thezero crossing time for payload deflection and velocity forstarting conditions ranging from 0.1 m to 25 m for the initialsuspension length, and from −2 ms−1 to 2 ms−1 for the initialhoist velocity. These conditions make this method valid formost industrial cranes. We observed the motion of the systemfor 10 s for each simulation, with a sampling rate of 0.01 s.We use these data to train an artificial neural network byusing the Levenberg-Marquardt algorithm [25]. Thus, we fitthe appropriate curve which provides the espressions for Aiand ti, with i = 1,2, as reported in Table I, where l and lindicate the measured cable length and velocity. This Tableprovides the amplitudes and the times of the impulses usedto convolve the reference command using the ZV-IPS in anoverhead crane with a suspended payload.

307

payload

payload

payload

Fig. 5. Experimental setup, the Georgia Tech bridge crane with the payload in its initial and final configurations.

IV. SIMULATIONS AND EXPERIMENTS

In this section we describe the simulations and the exper-iments performed.

A. Experimental setup

The zero vibration input predictive shaper was imple-mented on a three−axis system available in the AdvancedCrane Control Laboratory at the Georgia Institute of Tech-nology. Figure 5 shows the experimental setup with a sus-pended payload in the initial and final configurations. Anoverhead vision sensor, the SIEMENS SIMATIC VS 723-21, consisting of a 1024 by 768 pixel grayscale CCD anda high speed processor, is used to track the motion ofthe suspended payload. The system is programmed usingSIMOTION motion control system1.

B. ZV−IPS performances

Results of the ZV-IPS are shown in terms of transient andresidual vibrations of a payload when the overhead craneis subjected, in x and z−directions, to a certain movement,which makes the system change its frequencies. The idealreference velocity command in x− direction is a step functionof duration 3.315 s and value 2×10−1 ms−1, which makesthe trolley moving of 1.3 m. As shown in Fig. 6, the realreference velocity command acting on the experimental craneresults having a trapezoidal profile, of total duration 3.723 s,since we deal with actual commands. The real commandshaped using the developed ZV-IPS is shown in the samefigure: the result is a total duration of 4.59 s. All these datarefer to the performed experiments. In the ZV-IPS command,we can note that the durations of the two commands actingat the velocity 1×10−1 ms−1 are different: the first one of0.36 s, while the second one of 1.02 s. This happens becausethe z−coordinate corresponding to the initial crane hoistis smaller than the z−coordinate corresponding to the finalcrane hoist. In fact, as the frequency decreases, the rise timeof the command increases. The movement in z−direction isof 0.8 m, with constant velocity. Figure 7 displays, in theexperimental setup, the payload deflection corresponding tothe shaped velocity command. The amplitude of the residual

1https://www.siemens.com/

0 2 4 6

0

100

200

Time [s]

Trol

ley

Vel

ocity

[mm

s−1 ] Ref. (ideal) Ref. (real) ZV-IPS shaped

Fig. 6. Velocity commands.

0 2 4 6 8 10

50

0

-50

Time [s]

Payl

oad

Defl

ectio

n[m

m]

0 2 4 6 8 10

200

100

0

-100

-200

Trol

ley

Vel

ocity

[mm

s−1 ]

Fig. 7. Zero Vibrations Input Predictive Shaper (ZV-IPS). Experi-mental data for payload deflections and trolley velocity.

vibrations, as they are detected by the camera, is below2.5×10−3 m.In order to validate the effectiveness of the method, beforeshowing the comparison with existing shapers, Figure 8compares the payload deflections of the ZV-IPS commandwith the unshaped response. Moreover, the same figuredisplays the expected data from simulation: both simulatedand experimental data are in agreement, hence the developedsimulation environment is able to capture the response of theactual crane, but with a small phase shift, due to the factthat the crane operates in the digital domain which producesmeasurement and timing issues. The ZV-IPS significantlyimproves performance over the unshaped case, exhibiting a50% of vibration suppression in the transient case and 98%in the residual case.

308

0 2 4 6 8 10−100

0

100

Time [s]

Payl

oad

Defl

ectio

n[m

m]

Experiment Simulation Unshaped

Fig. 8. ZV-IPS: comparison of experimental and simulated data.

0 0.5 1 1.5 2 2.50

20

40

60

80 unshaped

ZV

ZV-IPS two-hump EI

Shaper Duration [s]

Res

idua

lV

ibra

tion

Am

plitu

de[m

m]

Fig. 9. Comparison of input shapers. Experimental data.

C. Comparison with existing shapers

We compared the performance of the ZV-IPS in terms ofboth shaper duration and residual vibration amplitude withrespect to the classical ZV shaper and the robust two−humpextra insensitive (EI) shaper [26]. Experiments were recordedfor the same trajectory. The comparison is illustrated inFig. 9. ZV-IPS shows the same residual vibrations of thetwo−hump EI shaper, but with a shaper duration similarto the ZV, the fastest non−negative shaper. Input PredictiveShaper, using only two informations from the cranes states(hoist cable length and velocity), measured on the fly, is agood compromise in obtaining almost zero residual vibra-tions having at the same time a fast response.

V. ROBUSTNESS ANALYSIS

The robustness analysis of the ZV-IPS shows the sensitiv-ity of the method with respect to the measurement errors ofhoist length and velocity, motion distance and speed.

A. Robustness w.r.t. measurement errors

In order to examine the robustness of the system interms of the measurement errors, Fig. 10 shows the residualvibrations for normalized hoist length and velocity (modeledover actual), for simulations and experiments. Fig. 10(a)resembles the shape of the sensitivity curve of a classicalZV shaper. The experimental data matches the data fromsimulation, with a slight shift that results in an overestimationof the length. A slight overestimation of the length will stillprovide good performance of the shaper (an overestimation

0.8 1 1.20

5

10

15

20

lm/la

resi

dual

vibr

atio

n[%

]

0.8 1 1.20

2

4

6

8

10

˙lm/la

Fig. 10. Payload sensitivity to measurements errors. Solid line(blue), simulation data; scatter (black), experimental data.

0 0.5 10

5

10

hoist move distance [m]

resi

dual

vibr

atio

n[%

]

-100 0 1000

5

10

hoist velocity [%]

Fig. 11. Payload sensitivity to motion profile. Experimental data.

of 20% still results in less than 10% of residual vibrations).This shift is either due to timing errors and measurementoffset. The same figure shows that the payload is moresensitive to underestimates of the length as compared tooverestimates, in the experimental data. Fig. 10(b) shows thatlarge deviations of velocity measurements do not influencesignificantly on the residual vibrations, at least for simulateddata. In case of experimental data, we still observe thisbehavior, even if the amplitude of the residual vibrations ishigher (due to timing and measurement issues). From bothfigures it is evident that the payload residual vibration ismuch more affected by length errors rather than velocityerrors.

B. Robustness w.r.t. motion profile

We determine the robustness of the payload response fordifferent moving profiles, by first varying the hoist cablelength between 2×10−1 m and 8×10−1 m and setting thevelocity at the maximum value (100 % = 2×10−1 ms−1),after varying the velocity and setting the hoist cable lengthat 5×10−1 m. Results are shown in Fig. 11. Figure 11(a)shows that the hoist distance does not have a significativeeffect on the residual vibration of the payload. The effectsof the hoist velocity is remarkable, as we can see in Fig.11(b): lowering the hoist velocity results in less residualvibrations. This is caused by natural dampening when thehoist is lowered and the addition of energy into the systemwhen the hoist is raised. In general, for all the movementstested in these experiments, the residual vibration remainedbelow 10%.

309

VI. CONCLUSIONS

A novel input shaper was designed to suppress the tran-sient and residual vibrations in a flexible nonlinear systemwith time−varying parameters. This shaper was developedusing a predictive−adaptive approach. It is predictive inthe sense that the time and amplitude of impulses of theshaper have been computed using a data−driven approach,by training an artificial neural network. It is adaptive in thesense that it can adapt in real−time the parameters on whichit depends, based on the actual system response. In this paperwe obtained the zero vibration input predictive shaper (ZV-IPS) for an overhead crane with a suspended payload. Themethod allows for a residual vibration suppression of 98%under certain testing conditions, and as much as 90% formost of the experiments performed. The ZV-IPS has beenvalidated both in simulations and experiments. It turns outto be robust in terms of measurement errors in hoist lengthand velocity, motion distance and velocity. A comparison ofthe ZV-IPS with classical and robust input shapers shows thatIPS allows (i) the same residual vibration suppression of therobust shapers; (ii) a shaper duration comparable to the fastershapers. This work might leverage the development of a newclass of shapers, the input predictive shapers (IPS). Futureworks of the authors will extend the predictive approach tomulti−mode shapers, for controlling the vibrations in flexibledistributed nonlinear systems, for which physical simulatorsexist [27] to generate realistic dataset for the offline trainingprocedure.

ACKNOWLEDGMENT

The first author would like to acknowledge EUROfusionfor its financial support under the grant EEG-2015/21 ”De-sign of Control Systems for Remote Handling of Large Com-ponents”, which has received funding from Horizon2020, theEU Framework Programme for Research and Innovation.

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