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/Proceedingsfthe1996IEEE

InternationalonferencenRoboticsndAutomation .. / ._Minneapolis, Minnesota - April 1996Behavioral Implications of ezoelectri Stack Actuatorsfor Control of Mlcromanipuladon - - _...=_NASA-CR-205000 Michael Goldfarb (C._ .,_ ./,'/Y ?:_ /,.2

Nikola Celanovic

Department of Mechanical EngineeringVanderbilt UniversityNashville, TN 37235

Abstract

A lumped-parameter model of a piezoelectric stackactuator has been developed to describe actuatorbehavior for purposes of control system analysis anddesign, and in particular for microrobotic applicationsrequiring accurate position andor force control. Inaddition to describing the input-output dynamic behavior,the proposed model explains aspects of non-intuitivebehavioral phenomena evinced by piezoelectric actuators,such as the input-output rate-independent hysteresis andthe change in mechanical stiffness that results fromaltering electrical load. The authors incorporate ageneralized Maxwell resistive capacitor as a lumped-parameter causal representation of rate-independenthysteresis. Model formulation is validated by comparingresults of numerical simulations to experimental data.1 Introduction

Piezoelectric ceramics transduce energy betweenthe electrical and mechanical domains. Application of anelectric field across the ceramic creates a mechanicalstrain, and in a similar manner, application of amechanical stress on the ceramic induces an electricalcharge. Since these devices are monolithic and have nosliding or rolling parts, they exhibit no significantmechanical stiction. A typical lead-zirconate-titanate(PZT) piezoelectric actuator, for example, can performstep movements with a resolution on the order of ananometer and a bandwidth on the order of a kilohertz.Consequently, piezoelectric ceramics are well suited foruse as precision microactuators for micropositioningdevices or micromaniputators. Such applications requirecontrol design which can provide both accurate positiontracking performance and suitable stability robustness.The purpose of the model presented herein is to map therelationship between voltage and charge at the electricalport of the PZT to force and displacement at themechanical port in a lumped parameter form that can berepresented by a finite number of ordinary differential

equations. This type of formulation provides both generalinsight into PZT behavior as well as a specific causalmathematical representation for purposes of model-basedcontrol system analysis and design.2 The Standard on Piezoelectricity

The most widely recognized description ofpiezoelectric ceramic behavior was published by astandards committee of the Institute of Electrical andElectronics Engineers [4]. The linearized constitutiverelations formulated by this commitee essentially state thatmaterial strain and electrical displacement (charge per unitarea) exhibited by a piezoelectric ceramic are bothlinearly affected by the mechanical stress and electricalfield to which the ceramic is subjected. Severalresearchers have utilized the IEEE constitutive relations toderive a piezoelectric actuator model. Basing modelderivation on the IEEE constitutive relations, however,requires several simplifying assumptions and fails toexplicitly describe the extreme nonlinearities that arepresent in all piezoelectric ceramics, rendering theresulting description too approximate for use indeveloping suitable position or force control. Others havemodelled the behavioral nonlinearities of piezoelectricceramic with algorithmic methods that provide littlephysical insight into actuator behavior and have limiteduse with respect to model-based control system design.3 Model Formulation

The fundamental component of a piezoelectricstack actuator is a wafer of piezoelectric materialsandwiched between two electrodes. Prior to fabrication,the wafer is polarized uniaxiaily along its thickness, andthus exhibits significant piezoelectric effect in thisdirection only. A typical piezoelectric stack actuator isformed by assembling several of the wafer elements inseries mechanically and connecting the electrodes so that

0-7803-2988-4/96 $4.00 1996 IEEE 226

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the wafers are in parallel electrically, as illustrated inFigure 1. Since piczoccr,'unic is a known dielectric, onewould expect such a configuration to exhibit capacitivebehavior. The electrical behavior of a IrZT stackactuator, however, is significantly more complex. Forpurposes of controller design, one of the mostinconvenient aspects of the actuator behavior is the rate-independent hysteresis exhibited between voltage anddisplacement as well as between force and displacement,as shown in Figure 2. If not specifically addressed, thistype of behavior can cause closed-loop limit cycling andpossibly instability.

Experimental observation indicates that the rate-independent hysteresis exhibited in Figure2 is notpresent between the endpoint displacement of the PZTstack actuator and the net electrical charge delivered tothe actuator. Additionally, dynamic observationindicates that endpoint displacement as a function ofelectrical charge is well-approximated by second-orderlinear dynamics, as shown in the measured data ofFigure 3.

The quasi-static force-displacement relationshipof a PZT stack actuator is shown in Figure 4. Asillustrated in the figure, rate-independent hysteresis isobserved only when the electrode leads are shorted+When the leads are open and current cannot flow throughthe ceramic, the actuator exhibits no static hysteresis.This evidence, along with the absence of static hysteresisbetween displacement and charge illustrated byFigure 3(a), suggests that the rate-independent hysteresislies solely in the electrical domain between the appliedactuator voltage and resulting charge. The hysteresis cantherefore be characterized by an electrical resistor withconstitutive relations that, unlike a typical resistor, relatevoltage to charge rather than current. This type of rate-independent dissipation is commonly experiencedmechanically as Coulomb friction. This analogy is thebasis for describing the static hysteresis exhibited by thepiezoelectric actuator. Figure 5 illustrates a single elasto-slide element which consists of a masslcss linear springand a masslcss block that is subjected to Coulombfriction. For a displacement input of sufficientamplitude, the relationship between the applied force andthe endpoint displacement will exhibit rudimentaryhystcrctic behavior. If several of these elasto-slideelements are put in parallel, each subjected to anincrementally larger normal force, the simplerelationship of Figure 5 becomes a piccewise linearapproxim:ltion of the continuous hysteresis exhibited bythe piezoelectric actuator, as illustrated in Figure 6. Thisconstruction was initially h)rmulatcd by the

-- Elm:_

w _====_ *

Figure I. Illustration ofa piezoelectric stack actuator.

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Figure2. Measured quasi-static relationships betweenapplied voltage and endpoint displacement andbetween applied force and endpoint displacement.

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.41a ...... -_ .............. _ _-_.+ ...........+ i+i++.......(_) (_)

Figure3. Column (a): Charge input versusendpointdisplacementutputforPZT stackactuatoror100and I000Hz sinusoidalnputs.Column (b): Frequencyresponseof endpointdisplacementutputtoactuatorhargeinput.Themarks representeasured datapointsand thecontinuouslinesrepresent linearsecond-ordersystem.

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PZT FotceiDisplacomen! Behawor1000

900 ... .... ..... .... ..... ..

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dis plac em ent (micr ons)

Figure 4. Quasi-static force displacement relationship for aPZT stack actuator with open leads (dashed line)and with closed leads (solid line).

mathematician and physicist James C. Maxwell in themid-1800's, and in the limit as the number of elasto-slipelements becomes infinite, the model is referred to asGeneralized Maxwell Slip [2]. Though originally amechanical formulation, the energy-based constitutiverelations of the Maxwell slip model are not domainspecific, and can therefore represent any rate-independent hysteretic relationship between ageneralized effort and generalized displacement in alumped parameter causal form. Consequently, inaddition to force and displacement, the generalizedMaxwell model can represent rate-independent hysteresisbetween voltage and charge, temperature and entropy,and magnetomotive force and magnetic flux. Note alsothat the number of elasto-slip elements has no bearing onthe order of the model, since the blocks are all masslessand the springs are in parallel.

The PZT stack actuator model resulting from theaforementioned observations is shown in schematic formin Figure 7. The generalized Maxwell resistivecapacitance, which is represented by the MRC element,resides in the electrical domain and therefore relates theelement's electrical voltage to charge. The PZT modelhas two ports of interaction, a voltage-current port on theelectrical side and a force-velocity port on the mechanicalside. With respect to the mechanical side of thetransformer, since the actuator model is concerned onlywith endpoint displacement in a band within the firstmechanical mode of vibration, the piezoelectric stack isa.,;sumcd to have a lumped m:L';Sand ,'l linear materialstiffness :rod d:unping.

Figure 5.

Figure 6.

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Force-displacement behavior of several elasto-slide elements in parallcl, each subjected toincreasing normal forces.

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"lilt bchaviorf the actuator is thercfore described by:vi, = v,_ + v, (1)q = ttr + Cv, (2)troi + b.i"+ "k__ Ft + F,._t (3)F, = nv, (4)

where q is the total charge in the ceramic, vi, is theactuator input voltage, v,_ is the voltage across theMaxwell capacitor (which is a function of q), v, is theback-emf from the mechanical side, C is the linearcapacitance in parallel with the transformer, n is theelectromechanicai transformer ratio, m, b, and k are themass, damping, and stiffness of the ceramic, x is thestack endpoint displacement, F, is the transduced forcefrom the electrical domain, and F,= is the force imposedfrom the external mechanical load.

4 Simulation and ExperimentThe model was parameterized for a

commercially available piezoelectric stack actuator (NECmodel #AE0505DI6). This actuator operates at inputvoltages between zero and 150 volts (in the direction ofwafer polarization), which corresponds to an endpointdisplacement range of approximately 25 microns. Uponmeasuring the mass m of the PZT, the mechanicalstiffness k and damping b can be determined byobservation of the charge-displacement dynamics that areshown in Figure 3. Measurement of the open-leadstiffness and knowledge of the DC gain between chargeand displacement are sufficient to determine the linearelectrical capacitance C and the transformer ratio n. Themodel parameters utilized for the simulations that followare given in Table 1. After determining the parametersof Table 1, the Maxwell capacitor parameters can bedetermined by propagating experimental results throughthe model. The parameters defining the Maxwellcapacitor for the simulations presented herein are givenin Table 2. As indicated by the table, these simulationswere run with ten elasto-slip elements in the generalizedMaxwell resistive capacitor. A real-time application maybe better served by fewer elements; since the number ofelements does not affect the order of the model, however,the increased computational overhead from addedelements is minimal.

Figure 8 shows the measured and simulatedendpoint displacement response of the modeled actuatorto a 90 volt 100 IIz triangle-wave voltage input in theabsence of any external mechanical load. As indicatedby the pious, the model faithfully represents the voltageinput to endpoint displacement output behavior of the

Vin

Figure 7.

.j_ MRC+l..iL1

v I *rc, * / IiL. J

Fl= IIV

Schematic representation of the PZT stackactuator model. The capacitor and resistorcontained within the MRC element are bothnonlinear elements.

Mdel I Symbl ]arameterrR

s_iffness kdampin_ blinear Ccapacitanceu'aasformer nratio

Numerical Value

0.00375 k_6xl0 _ N/m150 N-slm1.2x10 "6 F

10 C/m

Table 1. Model parameters utilized in simulation of theNEC model #AE0505D16 piezoelectric actuator.

, ,. ale.) ;.=_. (v.)1 2.0 0.22 0.6 0.33 0.3 0.34 0.26 2.65 0.06 0.96 0.l 2.07 0.05 1.58 0.03 1.29 0.1 7.010 0.5 80.0

Table 2. Generalized Maxwell capacitor parametersutilized in the model. The stiffnesses andbreakaway forces correspond with the mechanicalschematic of Figure 6.

PZT actuator. Note in the plots that the displacementoutput evinces distortion on both the rising and fallingslopes and maintains an amplitude-dependent offset.Further observation indicates that this is not typicaldynamic distortion. There is no discernible phase lagbetween the input and output, as indicated by the relativepositions of the waveform peaks, and there ate nosignificant filtering effects, evidcnced by the fact that thepeaks arc neither rounded nor otherwise distorted. These

4

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observations indicate the existence of a rate-independenthysteresis. This hysteretic behavior is clearly displayedin Figure 9, which shows both the measured andsimulated data of Figure 8 plotted as voltage input versusdisplacement output. Any dynamic system will exhibitan ellipsoid-shaped input-output hysteresis, provided theinput is of sufficient frequency to create discernible phaselag in the output. This type of dynamic hysteresis ischaracterized by a smooth curve relating the output toinput, such as that shown in Figure 3. The rate-independent hysteresis of Figure 8, however, isdistinguished from a dynamic hysteresis by the distinctdiscontinuities exhibited at both extremes. Modelaccuracy is additionally demonstrated by the data ofFigure 10, which shows the measured and simulatedresponse of the actuator to a linearly decaying 100 Hertzsinusoidal voltage input.

90 Volt PZT Input Response for gOVolt Input1580 ........... i ........

E 6 ........ i ...........q>_ ..... _....

20 !

O0 2O 40t (rr_e)

10

_s00 20 40

t (reset)

Figure 8. Measured (solid line) and simulated (dashed line)endpoint displacement response of the P'ZT to a90 volt I00 Hz triangle-wave input. Thedifference between the measured and simulateddata is difficult to discern.

Figure 9.

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20

Hysteresisor90 Volt Input

00 5 10 15displacementmicrons)Measured (solid line) and simulated (dashed line)voltage versus displacement for a 90 volt 100 IIztriangle-wave voltage input. The differencebetween the measured and simulated data isdifficult to discern.

Figure 11 shows the simulated displacementresponse to a 10 llz sinusoidal external force for caseswith the electrical leads open and shorted. The simulatedbehavior accurately reflects the quasistatic measuredbehavior illustrated by Figure 4. In addition to providinga structure for numerical simulation, the lumped-parameter model also provides insight into how theelectrical properties of the PZT reflect into themechanical domain. As shown by the measured andsimulated data of Figures 4 and 11 respectively, the PZTactuator exhibits significantly greater stiffness when theleads are open than when the leads are shorted. Thischange in stiffness can be demonstrated by observing thatboth the linear and the Maxwell capacitors in theelectrical domain reflect as stiffnesses in the mechanicaldomain. This is similar to the electrical resistance of aDC motor appearing as mechanical damping when themotor leads are shorted. Linearizing the Maxwellcapacitor and deriving expressions for mechanicalstiffness in both the open and shorted-lead configurationsyields

'1n"k o =k+-- (5)Cn 2kS = k +-- (6)C._+C

where ko is the open lead stiffness, k, is the shorted leadstiffness, k is the mechanical domain stiffness, C is thelinear electrical capacitance, n is the transformer ratio,and C,, is the linearized Maxwell capacitor. Theinclusion of C= in the second term of the latterexpression clearly indicates the decreased shorted leadstiffness.

5 Implications for Actuator ControlComstock [1] and Newcomb and Flinn [3]

observed that utilizing charge input to commandendpoint displacement of a PZT stack actuator resulted inbetter closed-loop performance than when using voltageinput. The model presented herein elucidates the reasonfor performance improvement. Controlling actuatorcharge prevents the nonlinear Maxwell capacitor fromindependently storing energy, and therefore effectivelyremoves it from the input-output behavior of the plant.Controlling charge thus renders the PZT stack a linearsecond-order system. With respect to force control, themodel indicates that endpoint force can be commandeddirectly by constraining the stack to have zerodisplacement and controlling the input charge.

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Thoughchargecontrolof a F-Z'Tactuatorcircumventshe nonlinearbehaviorof piezoelectricceramicndenablesheuseof linearcontrolechniques,thesimplicityf linearcontrolsboughtttheexpenseftheincreasedlectronicomplexityequiredoreffectiveactuators to utilizetheformulationor purposesfmodel-basedonlinearontrol.Nonlinearontrolchargecontrol. Oneof theprimaryreasonsor derivinga

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. .. .. .. .. . i . . .. .. .. .. . i . .. .. .. .. .. .. .. .. .. .. :

....... ji .......... i.......... ....... .+,................ , ...... !..........

I,5 10 15 20 2S

displacement (microns)

Figure I I. Simulated PZT behavior showing force versusendpoint displacement for a 10 Hz sinusoidal ex-ternal force input for cases with the electrical leadsopen (dashed line) and shorted (solid line).

lumped-parameter real-time description of the PZTtechniques provide specific methods of addressingnonlinear behavior, and can therefore offer a means ofachieving stable, high performance voltage control. Suchan approach would obviate the need for complexelectronic hardware, but would require a moresophisticated controller than a charge control approachand therefore would likely entail a greater computationalcost. Ascertaining the specific cost versus benefit of thetwo approaches will be a topic of future investigations.6 Summary and Conclusions

The model presented accurately represents thebehavior of a piezoelectric stack actuator in a lumped-parameter real-time representation, and can therefore beutilized for purposes of model-based control analysis anddesign. The static hysteresis evinced by the PZT actuatorwas identified as a rate-independent dissipation and wasfaithfully represented by a generalized elasto-slip modelwhich was originally formulated by J.C. Maxwell in thenineteenth century. Despite the presence of thisnonlinearity, the relationship between charge delivered tothe PZT and endpoint displacement of the stack wasobserved to have simple second-order linear characteris-tics. Effective closed-loop control of actuatordisplacement can therefore be achieved either byincorporating a nonlinear controller that commandsactuator voltage or by utilizing a linear controller thatcommands actuator charge.Acknowledgments

Support for this work was provided by NASAGrant No. NAGW-4723. The authors gratefully acknow-ledge this support.References

[1] Cornstock, R., "Charge Control of Piezoelectric Actuatorsto Reduce Hysteresis Effects," United States Patent#4.263,527. Assignee: The Charles Stark Draper Labora-tory. Cambridge, MA. 1981.

[21 Lazan, Benjamin J., Damping of Materials and Members inStruct, ralMechanics. Pergamon Press, London, 1968.[31 Newcomb, C. and Flinn, I., "Improving the Linearity ofPiezoelectric Ceramic Actuators," Electronics Letters,1982, Vol. 18, No, 1I, May, pp. 442-444.

[4] Standards Committee of the IEEE Ultrasonics,Ferroelectrics, and Frequency Control Society, An Ameri-can National Sumdard." IEEE Standard on Piezoelectric.ity, The Institute of Electrical and Electronics Engineers,1987, ANSI/IEEE Std 176-1987, New York.