A Novel Direct Inverse Modeling Approach for Hysteresis Compensation of Piezoelectric Actuator in Feedforward Applications

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 3, JUNE 2013 981

A Novel Direct Inverse Modeling Approach forHysteresis Compensation of Piezoelectric Actuator

in Feedforward ApplicationsYanding Qin, Yanling Tian, Dawei Zhang, Bijan Shirinzadeh, Member, IEEE,

and Sergej Fatikow, Member, IEEE

Abstract—The Prandtl–Ishlinskii (PI) model is widely utilized inhysteresis modeling and compensation of piezoelectric actuators.For systems with rate-independent hysteresis, the inverse PI modelis analytically feasible and it can be adopted as a feedforwardcompensator for the hysteretic nonlinearity of piezoelectric actu-ators. However, for the rate-dependent PI model, the applicablevalid inversion methodology is not yet available. Although simplyreplacing all the rate-independent terms in the conventional in-version law with the rate-dependent terms can achieve acceptableresults at very slow trajectories. However, a large theoretical mod-eling error is inevitable at fast trajectories, which is investigatedthrough simulations. This paper proposes a new direct approachto derive the inverse PI model directly from experimental data. Asno inversion calculation is involved, the proposed direct approachis efficient and the theoretical modeling error can be avoided. Inorder to validate the accuracy of the direct approach, a number ofexperiments have been implemented on a piezo-driven compliantmechanism by utilizing the inverse PI model as a feedforward con-troller. The tracking performance of the mechanism is significantlyimproved by the direct approach.

Index Terms—Compliant mechanism, hysteresis reduction,inverse modeling, Prandtl–Ishlinskii (PI), rate-dependent.

I. INTRODUCTION

P IEZOELECTRIC actuators have been widely employedin high-precision positioning applications as actuators be-

cause of their ultrahigh resolution, fast response, and high outputforce [1]–[12]. However, the inherent rate-dependent hysteresisnonlinearity greatly degrades the overall positioning accuracy.Great efforts have been directed toward compensating for the

Manuscript received July 26, 2011; revised December 5, 2011 and February9, 2012; accepted March 27, 2012. Date of publication May 7, 2012; date ofcurrent version January 18, 2013. Recommended by Technical Editor D. Sun.This work was supported in part by the National Natural Science Foundationof China under Grant 51175372, in part by the National Key Special Project ofScience and Technology of China under Grant 2011ZX04016-011, and in partby the Alexander von Humboldt Foundation under Grant 1140304.

Y. Qin, Y. Tian, and D. Zhang are with the School of Mechanical Engineer-ing, Tianjin University, Tianjin 300072, China (e-mail: [email protected];[email protected]; [email protected]).

B. Shirinzadeh is with the Robotics and Mechatronics Research Laboratory,Department of Mechanical and Aerospace Engineering, Monash University,Clayton, Vic. 3800, Australia (e-mail: [email protected]).

S. Fatikow is with the Division of Microrobotics and Control Engineer-ing, University of Oldenburg, 26111 Oldenburg, Germany (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2012.2194301

hysteresis. Generally, the approaches fall into the following cat-egories: 1) feedforward control, 2) feedback control, and 3)feedforward and feedback combined control. In the first cate-gory, feedforward control typically involves the hysteresis mod-eling and inversion calculation. In the second category, with themeasurement of the piezoelectric actuator’s output, the feed-back control can proceed without any hysteresis model. Betterperformance can be achieved if combining both the feedforwardand feedback controls, which belongs to the third category. Forfeedback control, all the available control methodologies can beused in hysteresis compensation [13]–[20]. However, the dis-advantages of feedback control are also distinct: the cost andcomplexity of the whole system significantly increase with theintroduction of sensors; and the sensor noise and disturbancesshould be properly processed so as not to destabilize the en-tire system; and the lack of embedded sensor with the requiredperformances for piezoelectric actuators further restrict the useof feedback control. In addition, if no feedback is available toadjust the output of the actuator, feedforward control is stillattractive as it provides a simple and effective way to achievesatisfactory results [21], [22]. This paper focuses mainly on theinverse hysteresis modeling in feedforward applications.

Although rate-dependent in nature, if the input changes suffi-ciently slowly, the rate-dependence of the piezoelectric actuatoris not obvious and it can be treated as rate-independent. Basedon this fact, a number of rate-independent hysteresis modelshave been proposed [23]–[29]. Some of these can also be ex-tended to describe the rate-dependent hysteresis [30], [31]. ThePrandtl–Ishlinskii (PI) model [32]–[38] is another widely usedhysteresis model. Its inversion can be analytically obtained andis also of PI type [33], [39], making it attractive in real-time ap-plications. The inverse PI model serves as a feedforward com-pensator for the hysteresis nonlinearity if it is cascaded withthe actual hysteretic system. The resulting relation from thedesired trajectory to the actual output is linear. The classical PImodel is a weighted superposition of several backlash operators,which is symmetric about its loop center. In order to account forthe asymmetry and saturation properties of different hystere-sis types, several methodologies have been established, suchas differentiating the thresholds of backlash operators on theascending and descending directions [40], choosing proper en-velope functions for backlash operators [41], and incorporatingnew components with the classical PI model [21], [42]. In thecase of rate-dependence modeling, it is common to dynamicallyupdate the weights or the thresholds of the backlash operators

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according to the input rate. Ang et al. and Tan et al. [21], [43]found in their tests that the slope of the measured hysteresisloop of a piezoelectric actuator varies almost linearly with theinput rate. Thus, they proposed modeling rate-dependence bylinearly varying the weights of the backlash operators accordingto the input rate. Janaideh et al. [44] modeled the relationshipbetween the thresholds of backlash operators and input ratethrough natural logarithm functions. Together with dynamicdensity function, their model can well predict the piezoelec-tric actuator’s output. In the meantime, the rate-dependence canalso be modeled into a static (rate-independent) hysteresis inconjunction with a dynamics part. Then, the hysteresis compen-sation is done by cascading the inverse dynamics model withthe inverse model of the static hysteresis [22], [45], [46].

The purpose of the hysteresis modeling is either hysteresisdescription or hysteresis compensation. Hysteresis descriptionmeans to find proper hysteresis models to predict the output ofthe piezoelectric actuator. While the hysteresis compensation isto find proper inverse hysteresis model that linearizes the outputof the piezoelectric actuator. The central task in the feedforwardcontrol-oriented applications is hysteresis compensation. Cur-rent research efforts are dedicated to finding proper inversionlaws to invert the corresponding hysteresis models. For all therate-independent PI models, the exact inversion law is availableby extending the conventional inversion law of the classical PImodel [40], [41]. However, great difficulties arise in the in-version calculation for the rate-dependent PI model, and theapplicable exact inversion law for the rate-dependent PI modelis not yet available. Ang et al. and Tan et al. [21], [43] proposeddirectly replacing all the rate-independent terms in the conven-tional inversion law with the rate-dependent terms. Althoughacceptable at low frequencies, their approximation operationwill lead to severe theoretical modeling error if the frequencyis high, which will be investigated in Section II through sim-ulations. Recently, Janaideh et al. [47] proposed an inversionformula for a rate-dependent PI model. However, their inversionformula only holds true when the threshold dilation conditionis satisfied, that is, the distances between the thresholds do notdecrease in time [47]. In addition, no applications of their inver-sion formula in hysteresis compensation were reported in theliterature.

Current research has the difficulty in the inversion calculationof the rate-dependent hysteresis. However, it is not required tobuild the hysteresis model and then invert it if directly obtain-ing the inverse hysteresis model in other ways is possible. Byseparation of the hysteresis description and the hysteresis com-pensation, this paper proposes a novel direct inverse modelingapproach that can derive the inverse rate-dependent hysteresismodel directly from experimental data. As a result, hysteresismodeling and complex inversion calculations are totally avoidedin the proposed approach. For simplicity, the proposed approachis referred to as direct approach, and all the other approachesare referred to as indirect approaches. It will be shown in Sec-tion II that the same methodologies in indirect approaches canbe readily migrated to build the inverse hysteresis model. Thus,the proposed direct approach is convenient for use and applica-tions for all the PI-based hysteresis compensations. Extensive

Fig. 1. Feedforward control based on inverse hysteresis model.

Fig. 2. Relationship between hysteresis model and its ideal inversion.

experimental testing has been conducted to verify the modelingefficiency and accuracy of the proposed direct approach.

II. DIRECT INVERSE HYSTERESIS MODELING APPROACH

In feedforward applications, the inverse hysteresis modelfunctions as a feedforward controller. If taking the piezoelec-tric actuator as an example, as depicted in Fig. 1, the hysteresiscompensation works as follows: for a given desired trajectory,denoted as yd , the inverse hysteresis model will generate aninput signal u which is physically applied on the piezoelectricactuator; the output of the actuator is denoted as y. If the inversehysteresis model is ideal, the output y should follow the desiredtrajectory yd , that is y = yd . We denote this ideal inverse hys-teresis model as ideal inversion. The input-output relationship ofthe ideal inversion is a key issue in the direct approach. Duringthe typical hysteresis modeling process, a certain input signalis generated and applied on the piezoelectric actuator. Thus,the input-output relationship of the ideal inversion is readilyavailable by plotting u against y, which is actually the reflectionof the piezoelectric actuator’s hysteresis loop about 45◦ line.Fig. 2 shows the measured hysteresis loop of a piezoelectricactuator, together with its ideal inversion. If we can model theideal inversion directly from the measured data, both the hys-teresis modeling and its complex inversion calculation can beavoided.

Like all other PI-based applications, the basic hysteresis loopis formed by the elementary backlash operator with the follow-ing equation:

y(t) = Hr [x](t) = max{x(t) − r,min{x(t) + r, y(t − T )}}(1)

and then,

y(0) = max{x(0) − r,min{x(0) + r, y0}} (2)

QIN et al.: NOVEL DIRECT INVERSE MODELING APPROACH FOR HYSTERESIS COMPENSATION 983

Fig. 3. Principle of direct inverse modeling approach. (a) Individual hysteresisloops. (b) Overall hysteresis loop.

where x(t) represents the input to the operator, y (t) denotesthe output of the operator, r is the threshold, y0 is the initialcondition, and the operator utilizes a sampling period of T. Ifthe piezoelectric actuator starts from its de-energized state, theinitial condition can be set to zero.

The elementary backlash operator is a rate-independent op-eration. The classical PI model is a weighted superpositionof several backlash operators. If we define the weight vec-tor as wh = [wh1 , wh2 , . . . , whn ]T , the threshold vector asr = [r1 , r2 , . . . , rn ]T and the elementary backlash operator vec-tor as Hr [x](t) = [Hr1 [x](t),Hr2 [x](t), . . . ,Hrn [x](t)]T , thenthe classical PI model can be expressed as follows:

y(t) = wTh · Hr [x](t) =

n∑

i=1

whiHri [x](t). (3)

A. Formulation of the Direct Approach

Typically, the PI model is used to build the hysteresis model.However, by introducing negative weights, the PI model can alsobe used to build the inverse hysteresis model directly. Fig. 3(a)shows hysteresis loops of five backlash operators with a thresh-old vector of r = [0, 0.1, 0.3, 0.6, 1]T and a weight vector ofwh = [10, –5, –5/3, –5/6, –0.5]T . Fig. 3(b) is the overall hys-teresis loop by combining these backlash operators according to(3). It forms a basic loop of the ideal inversion when comparedwith Fig. 2. This reveals the basic idea that the inverse hys-teresis model can be directly derived from the measured data.Because only the weights of backlash operators are modifiedand no other special treatments have been made to obtain theinverse hysteresis loop, all the available techniques in indirectapproaches can be directly transferred into the direct approach.

Inherent from the symmetry of the backlash operator, the in-verse hysteresis model in Fig. 3(b) is also symmetric about itsloop center. However, due to the asymmetry and saturation prop-erties of the measured hysteresis loop of piezoelectric actuators,additional treatment should be introduced such that the inversehysteresis model better fits the ideal inversion. We propose anovel saturation operator featuring a polynomial formulation. Itcan be cascaded with the inverse hysteresis model to reshapethe hysteresis loop. The polynomial has the following form of

Fig. 4. Polynomial versus one-sided dead zone operator.

m degree:

S(x) = cm xm + cm−1xm−1 + · · · + c1x (4)

where ci , (i = 1, 2, . . ., m) is the coefficient and m is the degreeof the polynomial.

Through literature review, saturation operators with multipleone-sided dead zone operators [21], [42] have also been pro-posed to model the asymmetry and saturation properties. Fig. 4plots the input-output relationship of a cubic polynomial, to-gether with a combination of six one-sided dead zone operatorsas comparison. Although piecewise linear when using one-sideddead zone operators, there are altogether five thresholds (thefirst threshold is zero) and six weights to be identified. How-ever, utilizing a polynomial as the saturation operator, there arefewer coefficients (three in this case) to identify and the resultis smooth. Compared with other methods that involve modify-ing the elementary backlash operator [40], [41], the polynomialprovides more freedom and more complex curve is possible byincreasing the degree of the polynomial.

A simple example is given in order to illustrate the effect ofthe polynomial on the hysteresis loop of the inverse model. Afifth degree polynomial with the following form is used as thesaturation operator:

S(x) = 0.0001x5 + 0.002x3 + 0.5x. (5)

Ten backlash operators with uniform thresholds of ri = 0.5(i-1), (i = 1, 2, . . ., 10), and weight vector of wh = [5, –1, –1, –1,–0.5, –0.5, –0.2, –0.1, –0.05, –0.01]T is superimposed to fab-ricate a basic inverse hysteresis model. Then, the saturationoperator defined in (5) is cascaded with the basic inverse hys-teresis model so as to introduce saturation characteristics. Thecontribution of the inverse hysteresis model is to generate a pre-dicted input for a given desired trajectory. As a result, it can bedirectly utilized as a feedforward compensator. The input-outputrelationship of the resulting inverse model is plotted in Fig. 5.As the inverse hysteresis model is a reflection of the hysteresismodel about the 45◦ line, its saturation property correspondsto almost vertical lines, which is well modeled by the polyno-mial as in Fig. 5. Furthermore, we can also change the envelopefunctions or the thresholds according to references [41], [44] tomodel more complex inverse hysteresis models.

From the aforementioned example, the structure of the inversehysteresis model in the direct approach can be illustrated inFig. 6, where the desired trajectory yd (t) is the input to theinverse hysteresis model, and the output (predicted input to the


Fig. 5. Effect of the polynomial on the loop of the inverse hysteresis model.

Fig. 6. Structure of inverse hysteresis model in the direct approach.

actual hysteretic system) is denoted as u(t). Then, the inverserate-independent hysteresis model can be expressed as follows:

u(t) = wTh · Hr [S(yd)] (t) =

n∑

i=1

whiHri [S(yd)] (t). (6)

B. Modeling Rate-Dependence of the Piezoelectric Actuator

The method in the work of Ang et al. and Tan et al. [21], [43]is adopted to model the rate-dependence of the piezoelectricactuator, which is linearly varying the weights of backlash oper-ators according to the input rate while keeping other componentsunchanged. Thus, the saturation operator in (4) is assumed to berate-independent and the linear relationship between the weightsof the backlash operators and input rate is defined as follows:

wh(yd) = kyd(t) + b (7)

where k = [k0 , k1 , . . . , kn ]T and b = [b0 , b1 , . . . , bn ]T , ki andbi (i = 0,1, . . . , n) are the slope and offset coefficients.

Replacing the rate-independent weights in (6) with the rate-dependent weights defined in (7), the rate-dependent inversehysteresis model can be expressed in the following:

u(t) = (kyd(t) + b)T Hr [S(yd)] (t)

=n∑

i=1

(kiyd(t) + bi)Hri [S(yd)] (t). (8)

C. Theoretical Modeling Error of the Indirect Approach

In applications, the threshold dilation condition [47] cannotbe guaranteed. As a result, the only practical inversion law in theindirect approach is to simply replace all the rate-independentterms in the conventional inversion law with rate-dependent

Fig. 7. Hysteresis loop at different frequencies.

terms [21], [43]. The theoretical modeling error of such an ap-proximation will be investigated here through simulations. Sup-pose there is a piezoelectric actuator system with rate-dependenthysteresis that can be modeled by the PI model. The hysteresismodel is built upon ten backlash operators. The threshold vectoris set to r = [0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5]T , the slopevector is set to k = [–2, 3, –1, 0.2, –0.4, 0.8, –1, 1, –0.7, –0.6]T

× 10−4 , and the offset vector is set to b = [77, 9.7, 0.66, 6.0, 10,6.2, 4.2, 8.2, 2.2, 9.9]T × 10−2 . The input-output relationshipsof the model under a sinusoidal signal with an amplitude often at different frequencies are shown in Fig. 7. Due to the rate-dependence, the hysteresis loop becomes larger as the frequencyincreases.

In the indirect approach, the weights of all the backlash op-erators in hysteresis model are first calculated by the use of thelinear relationship defined in (7). Through the conventional in-version law [40], [41], the thresholds and weights for the inversehysteresis model are derived. The inverse PI model is then fab-ricated and employed as a feedforward compensator as depictedin Fig. 1. As an example, the desired trajectory is set to be

yd(t) = 5.5 + 4.5 sin(2πft − 0.5π) (9)

where f is the frequency.Computational analyses are performed in the environment of

MATLAB/Simulink. Fig. 8 shows the overall input-output re-lationship of the indirect approach when the driving frequencyis 5, 10, 50, and 100 Hz. A 45◦ line (dotted) is also plotted inevery plot as comparison. If the indirect approach is ideal, theplot of the actual output against the desired trajectory shouldcoincide with the 45◦ line. For frequencies below 10 Hz, thetheoretical modeling error is negligible and the tracking per-formance is acceptable. However, for higher frequencies, thetheoretical modeling error becomes considerable. Thus, poortracking performance in the actual system is expected. The sim-ulation results reveal that the conventional inversion law canonly be extended to rate-dependent systems with an input ofsufficiently slow trajectories.

III. PARAMETER IDENTIFICATION EXPERIMENTS

A number of experiments have been conducted on a2-DOF decoupled piezo-driven compliant mechanism, whichis mounted on a Newport RS-4000 optical table to reduce theexternal vibration disturbance. As 1-DOF motion is adequate


Fig. 8. Hysteresis compensation performance of the indirect approach at dif-ferent frequencies. (Dashed is the 45◦ line as comparison). (a) 5 Hz. (b) 10 Hz.(c) 50 Hz. (d) 100 Hz.

Fig. 9. Experimental setup for the 2-DOF compliant mechanism. (a) Two-DOF-compliant mechanism. (b) Experimental setup.

for the research, only the piezoelectric actuator in X-axis is acti-vated. The experimental setup is shown in Fig. 9. Two PZS001piezoelectric actuators manufactured by THORLABS Companyare utilized to actuate the compliant mechanism. The actuatorhas a maximum displacement of 17.4 μm at the driving voltageof 150 V. The control signal is amplified by an E-505.00 piezocontroller from Physik Instrumente Company with an amplifi-cation factor of ten. The displacement of the moving platformis measured by two D-050.00 high-precision capacitive posi-

tion sensors from Physik Instrumente Company. The maximumdisplacement of the moving platform is approximately 8.1 μm.A DS1104 R&D board from dSPACE Company is utilized toprovide a real-time environment at a sampling rate of 10 kHz.

As the inverse rate-dependent hysteresis model defined in(8) contains too many unknown parameters, it will take sev-eral hours to identify all these parameters. Thus, we separatethe identification process into two parts: 1) identification of therate-independent saturation operator; and 2) identification ofthe rate-dependent weights. For piezoelectric actuators, the in-verse hysteresis loop can be approximated as rate-independentif the driving frequency is low. In this case, the rate-independentmodel defined in (6) can be utilized to build the inverse hys-teresis model. All the parameters in (6) can be identified fromthe experimental data by comparing the predicted input u(t)with the actual input u(t) and minimizing the following errorfunction:

E[u, u](wh , c, t) = u(t) − u(t) = wTh Hr [S(y)] (t) − u(t)

(10)where c = [cm , cm −1 , . . . , c1]T is the coefficient of the satura-tion operator defined in (4).

The identified result consists of the coefficients of the sat-uration operator as well as the rate-independent weights. Asthe weights are assumed to be rate-dependent in general, theidentified weights are discarded and only the coefficients of thesaturation operator are adopted. Substituting these coefficientsinto the rate-dependent model in (8), the slope and offset coeffi-cients can be identified similarly by minimizing the other errorfunction:

E[u, u](k,b, t) = u(t) − u(t)

= (kyd(t) + b)T Hr [Sd(y)] (t) − u(t).

(11)

It must be guaranteed that different sets of experimental dataare used in (10) and (11). In this paper, the measured data ofa single low-frequency sinusoidal signal is used in (10). Themeasured data of a superposition of multiple sinusoidal signalswith relatively higher frequencies are used in (11).

It is intuitively known that higher order of backlash operatorsand higher degree of the polynomial will achieve a smoother andmore accurate result. However, the computational time wouldbecome a severe problem in real-time applications, especiallyin applications with multiple piezoelectric actuators. A tradeoffhas to be made between the modeling accuracy and the compu-tation time. After a trial-and-error process, we manually makea choice of tenth-order backlash operators and fourth degreeof polynomial (n = 10 and m = 4). The threshold vector isassigned to r = [0 0.05 0.1 0.2 0.3 0.5 1.0 1.5 2.0 3.0]T . It isnonuniform with fine spacing at smaller values as the maximumtracking error typically occurs at the turning points and backlashoperators with smaller thresholds are more frequently used thanthose with larger thresholds. Because the piezoelectric actua-tors can only accept nonnegative voltages, there is no need todefine the backlash operator with thresholds larger than half ofthe maximum displacement of the moving platform.


TABLE IIDENTIFIED SLOPE AND OFFSET VECTORS

The parameter identification is conducted in MATLAB andthe function lsqcurvefit is chosen to minimize the error functionsdefined in (10) and (11) in the sense of least-squares. At least5k points are fitted in every identification process to guaranteethe accuracy. Different from the method used in Ang et al.’swork [21], all the parameters required for the inverse hysteresismodel can be identified through only two sets of experimentaldata. As no postprocessing, such as curve-fitting, is required,the proposed identification method is time-efficient. The actualtime consumption is within 20 min on a computer with a CPUof Intel Core2 Duo T5670 and a memory of 1 GB.

The saturation operator is identified through the experimentaldata of a 5 Hz, 10-V p-p amplitude sinusoidal signal actuationto ignore the rate-dependence. By minimizing the error functionof (10), the identified result is

S(x) = −7.53 × 10−4x4 + 1.45 × 10−2x3

−7.62 × 10−2x2 + 1.40x. (12)

Though the input rate of a single sinusoidal signal at higherfrequency (60 Hz as an example) covers a wide range, it is ob-served during our tests that the superposition of multiple sinu-soidal signals with different frequencies provides better resultsthan a single sinusoidal signal when identifying the slope andoffset vectors in (7). Thus, three sinusoidal signals with fre-quencies of 20, 50 and 80 Hz are superimposed in the followingequation to serve as control input for the identification of kand b:

u(t) = 5 + sin(40πt − 0.5π) + 2 sin(100πt − 0.5π)

+2 sin(160πt − 0.5π). (13)

The identified results are listed in Table I. Comparisons havebeen performed between the ideal inversion and the identifiedmodel. As shown in Fig. 10(a), the hysteresis loop of the iden-tified model fits the ideal inversion well except for the initialloading curve. From Fig. 10(b), we find that the maximum mod-eling error occurs within the initial loading curve. Beyond theinitial loading curve, the maximum modeling error lies within0.21 μm. As there is no feedback in the feedforward applica-tions, the tracking performance of the inverse hysteresis modelwill be limited by the modeling accuracy.

Fig. 10. Comparison between the ideal inversion and the identified model. (a)Hysteresis loops of the ideal inversion and the identified model. (b) Time plotof the measured input and the predicted input.

Fig. 11. Control system setup.

IV. MULTIPLE FREQUENCY TRAJECTORY TRACKING

The inverse rate-dependent hysteresis model obtained inSection III can be employed as a feedforward controller. Thecontrol system setup is illustrated in Fig. 11, where the physicalsystem is labeled as “plant.” Because the 2-DOF piezo-drivencompliant mechanism (shown in Fig. 9) is a less-damped sys-tem, any sudden change in the input will excite its first-ordermodal vibration. In order to obtain a smoother transient stateresponse at the beginning of the experiments, a digital filteringtechnique is applied to the reference trajectory through a fullpass filter defined by the following:

yd + 2ζwn yd + w2nyd = yr + 2ζwn yr + w2

nyr (14)

where yr is the reference trajectory, yd is the desired trajectoryto be tracked, the damping ratio is set to ζ = 1 to get an aperiodicresponse, and the natural frequency is set to wn = 200π rad/s.

The initial conditions for yd are set as yd = yd = 0. Whilethe initial conditions for yr are not constrained. The influence ofthe filter is significant at the beginning and will vanish rapidly,making sure that the desired trajectory always starts from zeroposition, regardless of the status of the reference trajectory.The desired trajectory is then applied to the feedforward con-troller, which generates the actual control input to the physicalsystem.


Fig. 12. Time plot of the trajectory tracking result. (a) Slow trajectory.(b) Fast trajectory. (c) Tracking errors.

Two reference trajectories with low and high-input rates areadopted to validate the performance of the proposed direct ap-proach. Their equations are given in the following:

yr = 4 + sin(2πt − π

2

)+ sin

(10πt − π

2

)+ sin

(18πt − π

2

)

(15)

yr = 4.5 + 1.5 sin(64πt − π

2

)+ sin

(80πt − π

2

)

+ sin(100πt − π

2

). (16)

Fig. 12 plots the trajectory tracking results for both the slowand fast trajectories. In both cases, the actual displacement fol-lows the desired trajectory well. However, a little negative driftis observed in the tracking of the slow trajectory. This mightcome from the modeling error of the inverse hysteresis modelas it is identified from the experimental data utilizing relativelyhigher frequencies than the slow trajectory. From Fig. 12(c), itis found that the maximum tracking error lies within the initialloading curve, which coincides with the modeling accuracy ofthe inverse hysteresis model plotted in Fig. 10. After the initialloading curve, the tracking error enters its steady state and thetracking error remains at a constant magnitude. The maximumtracking errors of the slow and fast trajectories are –0.22 and

Fig. 13. Hysteresis reduction. (a) Slow trajectory. (b) Fast trajectory.

–0.23 μm, respectively. In the case of hysteresis reduction, theinverse hysteresis model can significantly reduce the hysteresisof the compliant mechanism and improve the output linearity,as shown in Fig. 13.

As (13) is used to identify the rate-dependent weights, themodeling accuracy of the inverse hysteresis model beyond 80 Hzis not guaranteed. Furthermore, the mechanism’s dynamics andfirst-order modal vibration will become obvious for higher fre-quencies, which greatly affect the tracking performance. Thesetwo factors contribute to poor tracking performances for trajec-tories with frequency components beyond 80 Hz. This can beovercome by the use of vibration compensator or feedback con-trol methodologies. The mechanism’s tracking performance onfast trajectories with even higher frequency components will beinvestigated in the future work. Thus, it is not presented hereinto better concentrate on the contributions of this paper.

V. CONCLUSION

A novel direct inverse modeling approach for rate-dependenthysteresis compensation of piezoelectric actuators is establishedand its performance has been investigated. The established directapproach identifies all the parameters required for the inversehysteresis model directly from the experimental data withoutany inversion calculations. By carefully selecting the weightsof the backlash operators, all the available techniques in PI-based hysteresis modeling can be directly utilized in the directapproach. Thus, the established direct approach provides forease of use and wide application.

A polynomial saturation operator is proposed to model thesaturation property of the piezoelectric actuator. It is a smoothoperator and more complex shapes are possible by increasingthe degree of the polynomial. The rate-dependence of the piezo-electric actuator is modeled by linearly changing the weights ofthe backlash operators according to the input rate in this paper.However, as no inversion calculation is involved, a more accu-rate model for the rate-dependence is achievable by using morecomplex functions.

In feedforward control applications, the established inversehysteresis model can be employed as a feedforward controller.Trajectory tracking experiments have been performed on a2-DOF piezo-driven compliant mechanism to examine the per-formance of the direct approach. The experimental results


showed that the proposed direct approach can significantlyimprove the linearity of the output and reduce the trackingerrors.

ACKNOWLEDGMENT

The authors would like to thank Prof. X. Zhao of the Schoolof Mechanical Engineering, Tianjin University of Technology,for his support with the hardware.

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Yanding Qin was born in 1983. He received theB. Eng. and M.Sc. degrees in industrial designfrom Tianjin University, Tianjin, China, in 2005 and2007, respectively, where he is currently workingtoward the Ph.D. degree in mechanical manufac-ture and automation in the School of MechanicalEngineering.

From September 2009 to August 2010, he was aVisiting Scholar in the School of Mechanical En-gineering, Purdue University, West Lafayette, IN.His research interests include compliant mechanisms,

hysteresis modeling and compensation, dynamics modeling, and real-time con-trol methodologies.

Yanling Tian received the B.Eng. degree in mechan-ical engineering from Northwest Institute of LightIndustry, Xianyang, China, in 1997, and the M.Sc.and Ph.D. degrees in mechanical engineering fromTianjin University, Tianjin, China, in 2002 and 2005,respectively.

From 2005 to 2006, he was a Postdoctoral Re-search Fellow at Tianjin University, where he becamean Associate Professor. From 2007 to 2009, he wasa Postdoctoral Research Fellow in the Robotics andMechatronics Research Laboratory, Department of

Mechanical and Aerospace Engineering, Monash University, Australia. He wasa Visiting Scholar at Hongkong University of Science and Technology, China,and the University of Warwick, U.K., in 2001 and 2006, respectively. He wasalso a Visiting Professor at Tohoku University, Japan, in 2010. He has accom-plished a couple of government and industry-based projects, and published morethan 50 peer-reviewed papers. He obtained the prestigious Alexander von Hum-boldt Fellowship for experienced researchers in 2010. His research interestsinclude micro/nanomanipulation, mechanical dynamics, finite element method,surface metrology, and characterization.

Dawei Zhang received the B.Eng. degree in mechan-ical engineering from Shenyang University of Tech-nology, Shenyang, China, in 1984, and the M.Sc.and Ph.D. degrees in mechanical engineering fromTianjin University, Tianjin, China, in 1990 and 1995,respectively.

From 1984 to 1987, he was an Associate Engineerwith the Tianjin Institute of Power Source, China. Heis currently a Professor in the School of Mechani-cal Engineering, Tianjin University. His current re-search interests include micro/nanopositioning tech-

nique, structural dynamics, signal processing, high-speed machining method-ologies, and dynamic design of machine tools.

Bijan Shirinzadeh (M’06) received the B.E. andM.S.E. degrees in mechanical and aerospace engi-neering from the University of Michigan, Ann Arbor.He received the Ph.D. degree in mechanical engineer-ing from the University of Western Australia, Perth,Australia.

He has held various positions in academia and in-dustry. He is currently a Professor and the Directorof the Robotics and Mechatronics Research Labora-tory, which he established in 1994, in the Departmentof Mechanical and Aerospace Engineering, Monash

University, Melbourne, Australia. His current research interests include laser-based measurements and sensory-based control, micro/nanomanipulation sys-tems, systems kinematics and dynamics, haptics, medical robotics, and auto-mated manufacturing.

Sergej Fatikow (M’98) studied computer scienceand electrical engineering at Ufa Aviation Techni-cal University, Ufa, Russia, where he received theDoctoral degree in 1988.

From 1988 to 1990, he was a Lecturer at UfaAviation Technical University, where he became anAssistant Professor in 1996. In 2000, he became anAssociate Professor at the University of Kassel, Ger-many. Since 2001, he has been a Full Professor inthe Department of Computing Science, University ofOldenburg, Oldenburg, Germany, where he is also

the Head of Technology Cluster Automated Nanohandling at the Research In-stitute for Information Technology. He has published three books and morethan 60 journal papers and 180 conference papers on micro- and nanorobotics,nanohandling automation, and intelligent robot control. His research interestsinclude micro- and nanorobotics, automated robot-based nanohandling in scan-ning electron microscope, micro- and nanoassembly, atomic force microscopy-based nanohandling, sensor feedback on the nanoscale, and neurofuzzy robotcontrol.

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A Novel Direct Inverse Modeling Approach for Hysteresis Compensation of Piezoelectric Actuator in Feedforward Applications