dahl model-based hysteresis compensation and precise

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Qingsong Xue-mail: [email protected]

Yangmin Li1

e-mail: [email protected]

Department of Electromechanical Engineering,Faculty of Science and Technology,

University of Macau,Avenue Padre Tomás Pereira S. J.,

Taipa Macao SAR 3001, China

Dahl Model-Based HysteresisCompensation and PrecisePositioning Control of an XYParallel Micromanipulator WithPiezoelectric ActuationThis paper presents a new control scheme for the hysteresis compensation and precisepositioning of a piezoelectrically actuated micromanipulator. The scheme employs aninverse Dahl model-based feedforward in combination with a repetitive proportional-integral-derivative feedback control algorithm along with an antiwindup strategy. Thedynamic model of the system with Dahl hysteresis is established and identified throughparticle swarm optimization approach. The necessity of using global optimization andhow to choose the model parameters to be optimized are addressed as well. The effec-tiveness of the proposed controller is demonstrated by several experimental studies on anXY parallel micromanipulator. Experimental results reveal that both antiwindup andrepetitive control strategies can improve the positioning accuracy of the system, and awell performance of the proposed scheme for both one-dimensional tracking and two-dimensional contouring tasks of the micromanipulator is achieved. Moreover, due to asimple structure, the proposed methodology can be easily generalized to other micro- ornanomanipulators with piezoelectric actuation as well. �DOI: 10.1115/1.4001712�

IntroductionRecently, micro- and nanomanipulators have become a hot re-

earch topic because they are fundamental components for a va-iety of devices applied in micro- and nanotechnologies, such ascanning probe microscope �SPM�, biological cell operator, andntegrated circuit chip assembly line. Most micro- and nanoma-ipulators employ a flexure-based mechanism and piezoelectricctuators �PZTs� to deliver a submicron or subnanometer reso-ution positioning. In addition to vacuum compatibility, flexure

echanism eliminates clearance, backlash, friction, and lubrica-ion requirements for the device. As a type of linear actuator, PZTs capable of positioning with �sub�nanometer resolution, largelocking force, high stiffness, and rapid response characteristics.evertheless, PZT introduces nonlinearity into the actuationainly due to its hysteresis property occurring at voltage-driven

trategy, which will attenuate the accuracy of the manipulator ifot carefully treated.

One regular way to overcome the nonlinearity is to model theysteresis first and then construct a feedforward compensationcheme. A number of hysteresis models are available in the litera-ure, such as the Preisach model �1,2�, Maxwell model �3�, Duhem

odel �4�, Prandtl–Ishlinskii model �5�, and Bouc–Wen model6�, etc. Based on the inverse hysteresis model, successful com-ensation of hysteresis effects with open-loop feedforward controlas been reported in Refs. �2,7,8�. Moreover, feedforward com-ined with feedback control has also been employed to suppressoth hysteresis and remaining nonlinearities in terms of creep orrift effects.2 The creep or drift means a time-variant change be-avior in displacement while applying a constant voltage to the

1Corresponding author.2http://www.piezo.ws/piezoelectric_actuator_tutorial/part_two.phpContributed by the Dynamic Systems Division of ASME for publication in the

OURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedarch 16, 2009; final manuscript received March 30, 2010; published online June

8, 2010. Assoc. Editor Qian Wang.

ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

PZT actuator. For example, the combination of a cascadedproportional-derivative �PD�/lead-lag feedback controller with in-verse Preisach model is presented in Ref. �9�, a modified Prandtl–Ishlinskii model combined with a sliding mode controller is usedin Refs. �10,11�, and a proportional-integral �PI� feedback controlaugmented with a feedforward hysteresis observer based on theBouc–Wen model is proposed in Ref. �12� to compensate for thenonlinearities aiming at a precise positioning control. The majorpoint of this approach lies in establishing and identifying a suit-able hysteresis model to describe the system behavior. Althoughthe hysteresis effect can be tremendously reduced by resorting tothe current-driven strategy instead �13�, extra complicated hard-ware is required to implement a current-based control.

In recent advances, an alternative method to deal with the non-linearity in a micro- or nanopositioner is to consider the hysteresisas an uncertainty or a disturbance and then design a robust closed-loop controller to tolerate it �14�. For instance, by identifying theplant transfer function at the home position of a nanopositioner, anH� robust controller is designed to operate over the entire work-space of the positioner in Ref. �15�. However, the dynamic re-sponse of the manipulator varies with respect to pose changes inthe workspace. Thus, based on the plant models identified in dif-ferent poses of a three degrees of freedom �3DOF� positioningplatform, a gain-scheduled H� controller is designed in Ref. �16�according to the magnitude variation. Besides, it has been shownthat sliding mode-based controller can also be employed to sup-press the unmodeled hysteresis effect for a piezo-driven flexuremechanism �17�. With a robust control approach, the modeling ofthe nonlinear hysteresis is not necessarily required. However, spe-cial efforts have to be paid to implement a success controller tocompensate for the complex nonlinearities. For example, suitableweighting functions need to be determined to implement an H�

control, and the bounds on uncertainties are usually necessitatedto design a sliding mode controller without chattering phenom-enon, which are all challenging works and deserve independent
investigations. Moreover, intelligent controllers based on neural
JULY 2010, Vol. 132 / 041011-110 by ASME

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etworks �NNs� have also been introduced in the hysteresis com-ensation of PZT �18�. However, the performance of a NN-basedontrol usually depends on the initial value selection for theeighting parameters, and different types of reference input re-uire different sets of parameters to achieve a satisfactory controlesult. Besides, some other strategies have been presented for theysteresis suppression as well �13,19,20�.

In the opinion of the authors, the smaller the system modelncertainty, the better the control performance for a piezo-drivenicro- or nanomanipulator. Thus, the hysteresis is modeled by a

impler model in this research, and a feedforward �FF� plus feed-ack �FB� compensation strategy is adopted for a precise position-ng of an XY parallel micromanipulator. Specifically, a new con-rol scheme with inverse Dahl model augmented with repetitiveroportional-integral-derivative �PID� control is proposed. Al-hough the Dahl model has been widely applied in friction mod-ling �21�, its application in piezoelectric hysteresis modeling istill limited. In the literature, a hysteresis observer based on aimplified Dahl model is presented in Ref. �22� for the hysteresisompensation in a piezo-driven system. Higher-order Dahl modelsased on classic Dahl model are constructed in Ref. �21� to predicthe hysteresis arising from bearing friction in disk drive actuators.n addition, based on the second-order Dahl model proposed inef. �21�, a hysteresis observer is constructed in Refs. �23,24� for

he feedforward compensation of hysteresis in a piezoelectricallyriven micromanipulator. In the current research, the second-orderahl model is employed to establish the nonlinear hysteretic sys-

em model due to its less number of parameters, and an inverseahl model is used to construct a feedforward compensation. Itill be shown that the nonsymmetric hysteresis loop can be betterescribed by the Dahl model than the Bouc–Wen model with theame number of model parameters. Concerning the feedback con-rol, the PID controller is still widely used nowadays in variousractical industrial fields owning to its simple control structurend ease of implementation and maintenance. In this paper, theID algorithm is used to realize the feedback control. Moreover,ntiwindup and repetitive control �RC� strategies are undertakeno improve its control performance. The effectiveness of the pro-osed control scheme for a precise position tracking control isemonstrated by experimental investigations on a two-axis micro-anipulator system.In the rest of the paper, the system architecture of an XY mi-

romanipulator with experimental setup is described in Sec. 2,hich serves as a test bed for this research. Then, the dynamicodel of the system with Dahl hysteresis is implemented and

dentified through the particle swarm optimization �PSO� ap-roach in Sec. 3. Based on the inverse Dahl system model, aeedforward combined with PID feedback control scheme is con-tructed in Sec. 4, where the antiwindup of the integrator andepetitive control for a periodic reference input are realized asell. Afterwards, extensive experiments are conducted in Sec. 5 toiscover the performance of the designed controllers. Finally,ome concluding remarks are summarized in Sec. 6.

Experimental Test Bed Description

2.1 Mechanical Architecture of the XY Stage. Theomputer-aided design �CAD� model of the XY parallel microma-ipulator is illustrated in Fig. 1�a�, where the monolithic XY stages constructed with four identical prismatic-prismatic �PP� limbsnd actuated by two PZTs through integrated displacement ampli-ers as detailed in Fig. 1�b�. Since the employment of only two PP

imbs is sufficient to generate an XY translation, four limbs aresed to construct a symmetric structure to reduce the temperatureradient and enhance the accuracy performance accordingly. It isnown that PZT cannot bear transverse loads due to the risk ofamage. The integrated compound bridge-type displacement am-lifier acts as an ideal prismatic joint and possesses larger ratio oftiffness in transverse direction than that in a working direction.
ence, the amplifier also acts as a decoupler with the roles of
41011-2 / Vol. 132, JULY 2010

transmitting the axial force of the actuator and preventing theactuator from suffering undesirable transverse motions and loadsas well. By this way, the two actuators are well isolated and pro-tected. Moreover, the ideal translations provided by compoundparallelogram flexures allow the generation of decoupled outputmotion of the stage. Different from a common decoupled XYparallel stage with output motion decoupling only, the presentedone has both input and output decouplings in virtue of actuationisolation and decoupled output motion. This totally decouplingproperty is necessary for the situations where the manipulator isunderactuated, and sensory feedback of the end-effector positionsis not permitted. In addition, the physical relation between thedisplacements of the PZT actuators and XY stage, i.e., the ampli-fication ratio of the amplifier, is analytically derived in previousworks �25� of the authors. More details about the working prin-ciple of the micromanipulator can also be found in Ref. �25�.

2.2 Experimental Setup. The experimental setup for the mi-cromanipulator system is graphically shown in Fig. 2. The mono-lithic XY stage is fabricated from a piece of light material Al7075-T651. Two 20 �m-stroke PZT �model PAS020 produced byThorlabs, Inc., UK� are adopted to drive the stage, and the PZTsare actuated through a two-axis piezo amplifier and driver�BPC002 from Thorlabs, Inc.� with a voltage ranging between 0 Vand 75 V. By mounting two blocks with smooth surface finish onthe output platform, the positions of the end-effector are measuredby two laser displacement sensors �Microtrak II, head model:LTC-025–02, from MTI Instruments, Inc., USA�. The analog volt-age outputs �within the range of �5 V� of the two sensors areconnected to a peripheral component interconnect �PCI�-baseddata acquisition �DAQ� board �PCI-6143 with 16-bit a/d convec-tors, from NI Corp., USA� through a shielded I/O connector block

Fig. 1 „a… CAD model of the assembled XY parallel microman-ipulator; „b… details of one limb with integrated compoundbridge-type mechanical amplifier

Fig. 2 Connection scheme of the experimental apparatus

Transactions of the ASME

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SCB-68 from NI Corp.� with noise rejection. The digital outputsf the DAQ board are then read by a personal computer �PC�hrough the PCI local bus. Based on the experimental apparatus, aC-based control will be undertaken in this paper.Since the sensitivity of the laser sensor is 2.5 mm/10 V and theaximum value of 16-bit digital signal corresponds to 10 V, the

esolution of the displacement detecting system can be calculateds

2.5 mm

10 V�

10 V

216 = 0.038 �m �1�

he step size in the resolution level of the sensor output can belearly identified from the acquired home position data �with a-kHz sampling rate�, as plotted in Fig. 3�a�. However, due to aonsiderable level of the noise, the resolution of the sensor islaimed as 0.1 �m by the manufacturer. In addition, for the at-est error signal of the laser sensor output shown in Fig. 3�a�, theoot mean square �rms� error can be calculated as 0.097 �m.

oreover, the frequency spectrum obtained by fast Fourier trans-orm �FFT� is described in Fig. 3�b�, which indicates that it is not

true white signal. Actually, the true white noise can only bebtained in the limit as the sampling rate goes to infinity and asime goes to infinity. That is, one can never work with true whiteoise, but rather a finite time-segment from a white noise, whichs band-limited to less than half of the sampling rate.3

In addition, the open-loop tests conducted in Ref. �25� showhat the micromanipulator has a workspace around an area of17�117 �m2 with a maximal cross-talk of 1.5% between thewo axes, which verifies the well-decoupled property of the XYtage.

3

ig. 3 „a… At-rest laser sensor output of the micromanipulatorystem acquired with a 5-kHz sampling rate; „b… frequencypectrum of the error signal at rest

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ournal of Dynamic Systems, Measurement, and Control

3 System Modeling and IdentificationDue to the two axial motions of the manipulator being well-

decoupled, they can be treated independently. Thus, two single-input-single-output �SISO� controllers can be employed for the x-and y-axes of the micromanipulator, respectively. For conciseness,only the treatment of the x-axis motion is presented in the follow-ing discussions.

3.1 Dynamic Modeling. In the current research, instead ofinvestigating the individual PZT actuator, the entire micromanipu-lator system including the components of mechanical amplifiers,compound parallelogram flexures, and PZT actuators is studied.Considering the mechanical part as linear and of second order, andthe nonlinearity arising from the PZT actuator, the dynamic modelof whole system with nonlinear hysteresis can be established asfollows:

Mx + Dx + Kx = Tu − Fh �2�

where the parameters M, D, K, and x represent the equivalentmass, damping coefficient, stiffness, and x-axis displacement ofthe XY micromanipulator, respectively. In addition, T is the piezo-electric coefficient, u denotes the input voltage, and Fh indicatesthe hysteretic effect of the system in terms of force.

Traditionally, the system model of Eq. �2� can be identified byfinding the model parameters through the experiments. Specifi-cally, dividing Eq. �2� by M, the left hand side equation can beobtained as a linear second-order system in the form

x + 2��nx + �n2x =

T

Mu −

1

MFh �3�

where the natural frequency �n and damping ratio � can be deter-mined from the open-loop step response of the system.

For illustration, the time-domain open-loop step response of thex-axis motion with an input voltage of u=75 V is plotted in Fig.4�a�. Based on the step response data, the frequency response isobtained by FFT, as shown in Fig. 4�b�. The damping ratio andnatural frequency of the mechanical system including the XYstage and PZT actuators can be respectively calculated as

� =1.5759

2 � 12.0806= 0.065 �4�

�n = 2� � 160 �Hz� = 1.005 � 103 �rad/s� �5�

Consequently, the time constant of the second-order underdampedsystem can be derived as

� =1

��n= 0.0153 �s� �6�

As far as the selection of the sampling frequency is concerned,several criteria should be taken into account simultaneously. First,in order to capture the system behavior precisely and to preventintroducing too many noises, a sampling frequency larger than tentimes of the natural oscillation frequency �160 Hz� of the me-chanical part is preferred. Second, in view of the time constant ofthe system �26�, the sampling frequency should be not lower than3 /� �196 Hz�. Considering these constraints, the sampling fre-quency is chosen as 5 kHz in the above experimental tests.

Besides, the equivalent stiffness K and piezoelectric coefficientT can be derived via static analysis. Then, the equivalent mass Mand damping coefficient D can be obtained in sequence.

3.2 Hysteresis Modeling. It has been shown that the hyster-etic force term Fh can be predicted more accurately by the second-order than the first-order Dahl model for the bearing friction indisk drive actuators �21�. In the current research, the second-order
Dahl model with state space representation, as reported in Ref.
JULY 2010, Vol. 132 / 041011-3

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21�, is adopted to describe the piezoelectric hysteresis force ofhe entire micromanipulator system including the effect of me-hanical amplifier

V = AhVx + Bhupx �7a�

Fh = ChV �7b�

here the variable x denotes the x-axis displacement, V�q1 q2�T is an intermediate state vector for the second-orderodel, up is recommended as a constant in Ref. �21� �up=30 is set

n this research�, and the matrices are

Ah = � 0 1

− a2 − sgn�x�a1�, Bh = �0

1�, Ch = �b1 sgn�x�b0�

�8�here the hysteresis model parameters can be derived based on

he relations

a1 =2

Tdln�Pn−1

Pn� �9�

a2 = �2�

Td�2

+ �a1

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�10�

b1 = Gdca2 �11�

b0 = S0 �12�

here Td denotes the period of the damped oscillation of timeesponse of the system, Pn represents the overshoot value of theth peak with respect to the steady state value of the response.

ig. 4 Open-loop step response of the x-axis motion with annput voltage of 75 V: „a… time-domain plot; „b… frequency re-ponse obtained by FFT

dditionally, Gdc and S0 are the dc gain and initial slope of the

41011-4 / Vol. 132, JULY 2010

response, respectively.For the time-domain step response, as shown in Fig. 4�a�, the

dc gain Gdc=117.7622 /75 and initial slope S0=0 can be observed.Then, the parameter a1 can be calculated as

a1 =2

0.0058� ln�192.5302 − 117.7622

170.2524 − 117.7622� = 121.9874 �13�

Afterwards, other model parameters a2, b1, and b0 are computedin sequence, as shown in Table 1. It can be observed from Eq. �7�that the Dahl hysteretic force Fh depends on the output velocity ofthe system only.

3.3 Model Identification

3.3.1 Model Implementation. Generally, with four dynamicparameters �M, D, K, and T� and four hysteresis parameters �a1,a2, b1, and b0� obtained based on the above conventional ap-proach, the whole system �Eq. �2�� can be determined finally. Ablock diagram of the dynamic system is illustrated in Fig. 5�a�,where the details of the Dahl hysteresis model are shown in Fig.5�b�. Based on the block diagrams, the system model can be easilyimplemented with MATLAB/SIMULINK software. Alternatively, withan integrated consideration of Eqs. �2� and �7� along with theselection of a state vector x= �x , x ,q1 ,q2�T, the entire dynamicsystem including the hysteresis effect can be expressed in nonlin-ear state space as follows:

Table 1 Identified Dahl model parameters for the XY microma-nipulator system

Parameter Value Unit

M 0.1828 kgD 2.5973�103 N s/mK 2.6065�104 N/mT 0.0468 C/ma1 121.9874 -a2 1.1773�106 -b1 1.8485�106 -b0 0 -

Fig. 5 „a… Block diagram of the entire dynamic model for cal-culating the output displacement x� from the input voltage u;„b… details of the Dahl model for calculating the hysteresis term

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A = 0 1 0 0

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0 up − a2x − a1x sgn�x�, B =

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M

0

0 ,

C = �1 0 0 0 � �15�espectively.

Given the input voltage u, the output displacement x of theystem can be solved by Eq. �14� based on numerical approachesuch as the Runge–Kutta method �using the function “ode45” inATLAB environment�.However, a preliminary simulation reveals that the output of theodel identified by the aforementioned method does not match

xperimental data closely. The reason lies in that the conventionalodel identification with different amplitudes of the input voltage

esults in different sets of model parameters mainly attributed tohe hysteresis effects introduced by the PZT. Consequently, it isecessary to identify the system model by resorting to an alterna-ive way based on numerical optimization.

3.3.2 Fitness Function Selection. Preliminary study showshat once a set of dynamic parameters is fixed, the simulationesults almost remain unchanged even with very different valuesor the hysteresis parameters. On the other hand, with any set ofysteresis parameters, the simulated hysteresis loop varies clearlyith respect to different sets of dynamic parameters. This implies

hat the shape of the system hysteresis loop is more sensitive tohe dynamic parameters rather than the hysteresis parameters.herefore, once the four hysteresis parameters are identifiedbove, it is reasonable to optimize the four dynamic parameters ofhe model to match to experimental results. Although the eightarameters can also be optimized simultaneously, it is at the ex-ense of a longer calculation time to obtain an expected result.

Thus, the goal of optimization is to determine a set of modelarameters, which makes the Dahl model output match the experi-ental results with the minimum deviation. In particular, the fit-

ess function for minimization is chosen as

f�M,D,K,T� =�1

n�i=1

n

�xi − xi��2 �16�

.e., rms of the Dahl model output �xi�� deviations with respect to

he experimental results �xi�, where n denotes the total number ofamples.

3.3.3 Model Identification With PSO. In view of the currentptimization problem, it is extremely difficult to express the fit-ess function into an analytical form, i.e., there is no analyticalxpression for the objective function. It follows that the fitnessunction is not differentiable and there are no explicit expressionsor Hessian matrix and gradient descent. Thus, it can be regardeds a nonlinear discontinuous optimization problem, which cannote solved by a standard successive quadratic programming algo-ithm utilizing a local search procedure based on the gradient ofhe fitness function. Instead, it can be worked out by a directearch method such as the well-known Nelder–Mead simplex al-orithm. However, this technique heavily depends on good start-ng points, and may fall into local optima. In contrast, as a global

ethod for solving both constrained and unconstrained optimiza-ion problems, the PSO can be employed to solve a variety of
ptimization problems, which are not well suited for standard op-

timization algorithms. These include the problems where the ob-jective function has discontinuous, nondifferentiable, stochastic,or highly nonlinear features.

As a form of swarm intelligence, PSO is a relatively new algo-rithm proposed in Ref. �27�. It is a population-based stochasticoptimization technique inspired by the social behavior of birdflocking or fish schooling. Since PSO was originally introducedfor the optimization of continuous nonlinear functions, it has beensuccessfully applied to many other problems such as discrete op-timization, artificial neural network training, fuzzy system control,and mobile robot navigation. A PSO system is initialized with apopulation of random solutions and it searches for optima by up-dating generations, which makes it similar to evolutionary com-putation techniques such as genetic algorithm �GA�. However,compared with the GA, PSO has no evolutionary operators interms of crossover and mutation. Hence, from the viewpoint ofprogramming, the advantages of PSO are easy to implement andfewer adjustable parameters. Moreover, it is shown that PSO issuperior over other methods such as the direct search approachand GA in terms of optimization performance �28,29�. Therefore,the PSO is adopted for the model identification in the currentresearch.

In PSO, the population is called a swarm and the individuals�i.e., the search points� are called particles. Each particle moveswith an adaptable velocity within the search space, and retains amemory of the best position it has ever found. Finally, it knowswhere the best solution encountered by any other particles in thesearch space, and will then modify its direction toward its ownbest position and the global best position, which will providesome forms of convergence in searching. Regarding aC-dimensional search space and a swarm consisting of N par-ticles, the ith particle is represented by a C-dimensional vectorXi= �xi1 ,xi2 , ¯ ,xiC�, the velocity of this particle is aC-dimensional vector Vi= �vi1 ,vi2 , ¯ ,viC�, and the best previousposition encountered by this particle is described by Pi= �pi1 , pi2 , ¯ , piC�. Let g represent the index of the particle whichattains the best previous position among all the particles in theswarm, and k denote the iteration counter. Then, the swarm ismanipulated in accordance with the following equations �28�:

Vi�k + 1� = wVi�k� + c1r1�Pi�k� − Xi�k�� + c2r2�Pg�k� − Xi�k��17�

Xi�k + 1� = Xi�k� + Vi�k + 1� �18�

where w is the inertial weight, c1 and c2 are the accelerationconstants called cognitive and social parameters, respectively, r1and r2 are random numbers uniformly distributed between 0 and1, and the particle index i=1,2 , ¯ ,N. The selection of the aboveparameters has been widely studied in the relevant literature �28�.

The current model identification procedure is carried out off-line as follows. First, with a voltage signal covering the full inputrange �0–75 V� applied to the PZT, the experimental output dataof the stage displacement are collected and stored. Then, the sys-tem model with Dahl hysteresis is implemented according to theblock diagrams in Fig. 5, and the model output is generated by thesimulation. Afterwards, the dynamic parameters are optimized bya PSO algorithm, which minimizes the fitness value to match thesimulation results to the experimental data. The optimizationproblem is stated below:

Minimize: f�M,D,K,T� represented by Eq. �16�

Variables to be optimized: M, D, K, T

Subject to: M � �0.1,0.2�, D � �1000,5000�, K

� �10000,50000�, and T � �0.01,0.1�

In order to apply PSO for the model identification, several funda-
mental parameters are required to be assigned at first. In the cur-
JULY 2010, Vol. 132 / 041011-5

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ent four-dimensional optimization problem, a particle can be de-cribed by Xi= �xi1 ,xi2 ,xi3 ,xi4� with the particle velocity Vi

�vi1 ,vi2 ,vi3 ,vi4�, which corresponds to a set of the system dy-amic parameters �M ,D ,K ,T�. In the nth generation, there are Narticles popn= �X1 ,X2 , ¯ ,XN�, where the population size N iset to be 10 for the current problem. The inertia weight w deter-ines the impact of previous velocities on the current velocity,

nd its initial and final values are selected as 0.9 and 0.4, respec-ively, where 500 epoches are allowed to take from the initialalue to the final one linearly. In addition, the local and globalcceleration constants are assigned as c1=2.0 and c2=2.0, respec-ively. As far as the termination criterion is concerned, three itemsre set. One criterion is the maximum number of iterations �2000�or the optimization procedure, another one is the minimum globalrror gradient �10−6�, which is the error between two neighboringBest, and the third one is the maximum number of iterationsithout error change, which is chosen as 500.The optimization is undertaken in MATLAB environment. The

elation between the fitness value and generation is described inig. 6, and the identified model parameters are tabulated in Table. In addition, the input voltage applied to the PZT for the dataollection is shown in Fig. 7�a�. To demonstrate the generalizationroperty of the identified system model, only the first one-third0–4 s� experimental data are used in the optimization procedure.

ith the identified model parameters, the model output and errorsre shown in Figs. 7�c� and 7�b�, respectively.

3.3.4 Comparison of the Model Output. It has been shown thathe Bouc–Wen hysteresis model can also be employed to describehe plant dynamics in a simple manner �30,31�

mx + bx + kx = k�du − h� �19�

h = du − u h h n−1 − �u h n �20�

here the parameters m, b, k, and x represent the mass, dampingoefficient, stiffness, and x-axis displacement of the XY microma-ipulator, respectively, d is the piezoelectric coefficient, u denoteshe input voltage, and h indicates the hysteretic loop in terms ofisplacement, whose magnitude and shape are determined by thearameters , , �, and the order n, where n governs the smooth-ess of the transition from elastic to plastic response. For thelastic structure and material, n=1 is usually assigned.

As simpler hysteresis models, both Bouc–Wen and Dahl modelsan describe the system with the same number of parameterseight�. For the purpose of comparison, a Bouc–Wen hysteresisodel of the system as identified by the PSO approach in the

revious works �31� of the authors is adopted in this research. The

ig. 6 Convergence procedure of the PSO for dynamic modeldentification

dentified model parameters are as follows �31�: n=1, m

41011-6 / Vol. 132, JULY 2010

=0.1283 kg, b=1.5800�103 N s /m, k=1.9567�103 N /m, d=1.7339�10−6 m /V, =0.3575, =0.0364, and �=0.0272, andthe model errors are also described in Fig. 7�b�.

It can be observed that the Dahl model represents the systemmore accurately with a better generalization property. From thispoint of view, the Dahl model is more superior to the Bouc–Wenmodel in dealing with nonsymmetric hysteresis. Even so, thereexists the maximum model error about 5 �m for the Dahl model,as shown in Fig. 7�b�, which accounts for 4.2% of the total travelrange of the manipulator. To compensate for the model errors andother uncertainties, a controller design for the micromanipulatorsystem is carried out in Sec. 4.

4 Controller DesignThe objective of motion control is to force the output platform

of the micromanipulator to track a given position trajectory. Oncethe trajectory is assigned, the determination of the input voltageapplied to the PZT is the target of the controller design.

4.1 Feedforward Compensation. Instead of designing a

Fig. 7 „a… A 0.25-Hz input voltage signal applied to the PZT; „b…errors of the identified Bouc–Wen and Dahl models with re-spect to the experimental results; „c… hysteresis loop obtainedby the experiment and the Dahl model

Dahl model-based hysteresis observer as in Refs. �23,24�, a feed-


ftm

cm

wpawFi

eecpataeb

wdBp

s

Fc

Fs

J

orward compensation is proposed in this paper. This simplifieshe controller implementation process by eliminating the require-

ent of the velocity observer design.For a given position trajectory xd, the expected voltage input

an be calculated by the inverse Dahl model from the systemodel in Eq. �2�, i.e.

uFF =1

T�Mxd + Dxd + Kxd + Fh� �21�

here the hysteretic force term Fh is expressed by Eq. �7� witharameters described by Eqs. �9�–�12�. The inverse Dahl model islso implemented with a block diagram, as depicted in Fig. 8,here the details of the Dahl model are represented by Fig. 5�b�.urthermore, the block diagram of the feedforward compensation

s shown in Fig. 9.

4.2 Feedforward Plus Feedback Compensation. Due to thexistence of errors for the identified model output with respect toxperimental results as discussed in Sec. 3, the hysteresis effectannot be totally compensated by the feedforward voltage com-uted by Eq. �21�. Therefore, an additional feedback control isdopted to compensate for the model imperfection and other dis-urbances of the system. The concept is illustrated in Fig. 9, where

PID feedback controller is employed due to its robustness andase of implementation properties. The feedback control input cane written as

uFB�t� = Kp�e�t� +1

Ti�

0

t

e��d� + Tdde�t�

dt � �22�

here t represents the time variable, and the tracking error isefined as e�t�=xd�t�−x�t� with x denoting the measured position.esides, the three control parameters Kp, Ti, and Td denote theroportional gain, integral time, and derivative time, respectively.

By adopting an incremental PID algorithm, the overall controlignal is given in a discretized form

ig. 8 Block diagram of the inverse Dahl system model for thealculation of the input voltage uFF from the given position xd

ig. 9 Block diagram of the FF plus feedback FB control
cheme

u�tk� = uFB�tk� + uFF�tk� = u�tk−1� + Kp�e�tk� − e�tk−1�� +KpT

Tie�tk�

+KpTd

T�e�tk� − 2e�tk−1� + e�tk−2�� + uFF�tk� �23�

where k is the index of time series, T represents the sampling timeinterval, u�tk−1� is the control command in the previous time step,and the feedforward term uFF�tk� is generated from a lookup table,which is established off-line based on the inverse Dahl systemmodel.

4.3 Antiwindup Strategy. Due to the limits of voltage �0–75V� applied to the PZT, a saturation function is added to restrict thesignal of controller output between the two input extremes. How-ever, the interaction of integration and saturation may cause thephenomenon of windup for the PID controller �32�. When thecontrol signal reaches to the actuator limits, the feedback closedloop is broken, and the system runs like an open loop since theactuator will remain at its limit independently of the controlleroutput. Under such situation, the error will continue to be inte-grated if a feedback controller contains an integral action. It fol-lows that the integral term may become very large or wind up.Although the integral action is alleviated by the incremental algo-rithm implemented above, the windup still deserves a carefultreatment especially for a precise motion control.

In the current research, the back calculation antiwindup schemeis employed, as shown in the block diagram in Fig. 10. It isobserved that an additional feedback loop is added in the system,which is formed by generating a voltage error between the actualinput signal �v� to the PZT and the controller output �u�, i.e.

eu = v − u �24�Once actuator saturation occurs, the integral term in the controlleris reset based on eu dynamically with a tracking time constant Tt.On the other hand, the error signal eu remains zero when there isno saturation. Therefore, it will not affect the normal operationwhen the actuator does not saturate. When eu is different fromzero, the speed for the controller output reset depends on thefeedback gain 1 /Tt. Generally, the smaller the tracking time con-stant, the quicker the reset integrator. Even so, it has been recom-mended that the tracking time constant Tt should lie between theintegral time constant Ti and derivative time constant Td. Hence,Tt is suggested to be chosen as Tt=�TiTd. Moreover, the controlleroutput signal with the antiwindup scheme can be calculated by thediscretized equation

u�tk� = u�tk−1� + Kp�e�tk� − e�tk−1�� +KpT

Tie�tk� +

T

Tteu�tk�

+KpTd

T�e�tk� − 2e�tk−1� + e�tk−2�� + uFF�tk� �25�

4.4 Repetitive Control Strategy. For many applications such

Fig. 10 PID controller with back calculation antiwindupscheme

as the SPM, the position tracking of a periodic scanning path is

JULY 2010, Vol. 132 / 041011-7

upvscoaaRps

0

sually required. As a simple learning control, the RC scheme is aromising approach to achieve a precise tracking. The major ad-antage of RC lies in that it gives the control input using the errorignal information in the previous period. Thus, the control signalan be adjusted repetitively by the RC for the tracking of a peri-dic reference motion. Although it is has been presented as earlys in Ref. �33� based on the internal model principle, the rigorousnalysis and synthesis of RC systems were not proposed until inefs. �34,35�. Since then, the RC has been applied widely for theerformance enhancement and disturbance rejection of controlystems �36,37�. It is observed that the RC scheme is very similar

Fig. 11 Block diagram of the FF plus FB with RC strategy

Fig. 12 Sinusoidal motion tracking results of „a…–„c… feedfo
feedforward plus feedback compensation with a 0.2-Hz input ra
41011-8 / Vol. 132, JULY 2010

to the iterative learning control �ILC� �38�. Actually, comparing tothe fact that the control input in ILC is updated only once in eachcycle and the plant is reset at the beginning of each iteration, thecontrol input in RC is updated continuously without the need ofreset.

It has been shown that it is possible to simply plug the repeti-tive control into the system while keeping the existing controller�35�. In the current research, the designed repetitive controller isplugged in parallel with the PID controller, as illustrated in Fig.11, where Q�s� denotes the Laplace transform for the transferfunction of a periodic function generator e−Ls cascaded with afirst-order low-pass filter

Q�s� =e−Ls

1 + Tqs�26�

where L is the period of the periodic reference input and Tq de-notes the time constant of the filter.

In view of the effect of zero-order hold �ZOH�, the sampled-data transfer function of Q�s� can be obtained as follows

Q�z� = �1 − z−1�Z� e−Ls

s�1 + Tqs�� =1 − e−�T/Tq�

�z − e−�T/Tq��zN �27�

where N=L /T and the sign Z represents the z-transform operation.Therefore, the sampled-data transfer function for the RC output

uRC�t� with respect to error signal e�t� can be derived as

ard compensation, „d…–„f… PID feedback control, and „g…–„i…
rw te

wogstrbf

sDiK

w

J

URC�z�E�z�

=KqQ�z�1 − Q�z�

�28�

ith Kq denoting a positive gain of the repetitive controller. Inrder to keep the stability of the system �35�, the repetitive controlain Kq should be maintained within 0�Kq�2. Experimentaltudies show that the larger the control gain is, the faster theracking error converges, which are coincident with the resultseported in Ref. �37�. However, too large gain leads to the insta-ility of the system. Thus, a trade-off between the tracking per-ormance and system stability is required in the selection of Kq.

For illustration, the period of the reference input and time con-tant of the filter are selected as L=5 and Tq=0.2 s, respectively.iscretizing the transfer function of Eq. �28� with a sampling time

nterval T=0.02 s allows the generation of the control input �withq=1.0 turned by trial and error in experiments�

URC�z−1�E�z−1�

= z−250

�0.0952z−1 − 0.0861z−2

1 − 1.81z−1 + 0.8187z−2 − 0.0952z−251 + 0.0861z−252

�29�

Fig. 13 Motion tracking results of feedforward plus feedbwith shrunk saturation limits whereas without antiwindupstrategy „ut…

hich implies that the discretized controller output signal uRC has


a period of N=L /Ts=250.According to Eq. �29�, the control signal can be derived in a

discretized format

uRC�tk� = 1.81uRC�tk−1� − 0.8187uRC�tk−2� + 0.0952uRC�tk−N−1�

− 0.0861uRC�tk−N−2� + 0.0952e�tk−N−1� − 0.0861e�tk−N−2��30�

which indicates that the control commands in the former twosteps, and the previous period and tracking errors in previousperiod are utilized to determine the signal of the current controlinput.

The performances of the controller designed above are verifiedby experimental studies conducted in Sec. 5.

5 Experimental Results and DiscussionsBased on the hardware available, a PC-based real-time control

is implemented in this research. According to the discrete-timecontrol theory for sampled-data systems, more than 20 times ofthe closed-loop system bandwidth is preferred for the samplingfrequency of discrete control �39�. Because a maximal closed-loopsampling rate of 50 Hz �corresponding to a sampling interval of0.02 s� can be achieved by the hardware, a maximal bandwidth of

controller: „a…–„c… with normal actuator saturation; „d…–„f…tegy; „g…–„i… with shrunk saturation limits and antiwindup

ackstra

2.5 Hz is expected from the control system. It is noticeable that

JULY 2010, Vol. 132 / 041011-9

mi

ostctoetAst06fihtd

tvtttatawttftmtcss

0

uch higher sampling rate and system bandwidth can be obtainedf a digital signal processor �DSP�-based control is implemented.

First, the effectiveness of combined FF and FB compensationver either of the single approaches is tested with a 0.2-Hz sinu-oidal reference motion input covering almost the whole range ofhe workspace �0–110 �m�, and the experimental results areompared in Fig. 12. It can be observed from Fig. 12�c� that withhe FF open-loop compensation based on the inverse Dahl modelnly, the rising and falling curves repeat each other well. How-ver, the actual-desired position relationship is still nonlinear dueo a maximal tracking error of 5.69 �m, as shown in Fig. 12�b�.dditionally, from the closed-loop PID FB control results, as

hown in Figs. 12�d�–12�f�, it is observed that the maximum con-rol error is as large as 4.44 �m, which leads to a phase lag of.063 rad and a maximal hysteresis loop width accounting for.24% of the whole travel range. In contrast, with the combinedeedforward plus feedback �FF+FB� approach, the phase shiftings significantly reduced to 0.006 rad, which is negligible and theysteresis effect is suppressed to a low level of 1.76%. Moreover,he peak-to-peak tracking error is maintained within �1.5 �m, asepicted in Figs. 12�g�–12�i�.

Normally, to demonstrate the effect of the antiwindup strategy,he input voltage should go beyond the range of 0–75 V to acti-ate the saturation function. However, such a test has a potentialo damage the PZT with input signal switching frequently betweenhe two voltage extremes. To avoid such a harmful situation, thewo limits of the actuator saturation are temporarily shrunk to 6 Vnd 28 V, respectively. With the normal actuator saturation, theracking results are shown in Figs. 13�a�–13�c�. Due to the newlyssigned saturation limits, the FF+FB tracking performance iseakened, as shown in Figs. 13�d� and 13�e�. It is observed that

he maximal error trends to a large value of 6.35 �m, which is siximes worse than that in Fig. 13�b�. The large tracking error arisesrom the integrator of the PID controller, which begins to integratehe position error at the beginning of the saturation process at the

oment t1, as indicated in Fig. 13�d�. Thus, the integral term startso integrate the error with a speed relying on the integral timeonstant Ti, which results in an increasing PID control effort, ashown in Fig. 13�f�. Once the desired position enters inside the

Fig. 14 High frequency motion tracking results of „a…

aturation range at the moment t2, the PID control effort reaches

41011-10 / Vol. 132, JULY 2010

the maximum value, as shown in Fig. 13�f�. At the same time, theunnecessary large control effort begins to cause a large trackingerror, as described in Fig. 13�e�. Only after the stored controleffort is completely released at the moment t3, the motion trackingcan be returned to the normal state, as reflected in Figs. 13�d� and13�f�. On the contrary, when the antiwindup strategy is used, theintegral term in the PID controller is reset smoothly by a calcu-lated voltage error based on the tracking time constant Tt. In thissense, the integrated position error is counteracted by the voltagetracking integrator, which results in a smaller control effort of thePID controller due to a reverse control effort of the tracking inte-grator in the saturation process, as plotted in Fig. 13�i�. Conse-quently, a better tracking result is obtained, as reflected in Fig.13�h�.

Third, to discover the contribution of repetitive control, a0.5-Hz sinusoidal motion, as shown in Fig. 14�a�, is tracked bythe FF+FB controller with and without the repetitive control item.Due to a relatively high input frequency, the maximum trackingerror as large as 2.29 �m is resulted by the controller withoutrepetitive control, as shown in Fig. 14�b�. On the other hand, thetracking results with the repetitive item are illustrated in Figs.14�d�–14�f�. As expected, due to the reason that the repetitivecontrol employs the error signal in the previous period to derivethe current control signal uRC, the tracking error has been gradu-ally reduced. The control efforts, as plotted in Fig. 14�f�, indicatethat the control item uRC increases period by period while theoverall controller output remains a constant, and the correspond-ing tracking error is reduced gradually until the fourth period, asdepicted in Fig. 14�e�. It can be seen that the tracking error hasbeen maintained within the range of �1.5 �m after three periodsby the effect of repetitive learning control.

Moreover, in order to demonstrate the cooperative tracking ofx- and y-axes for the micromanipulator, a circular contouring testis carried out with different input rates and circle sizes within theworkspace. For instance, with an input rate of 0.1 Hz, the con-touring results for three circles with radii of 10 �m, 20 �m, and40 �m are illustrated in Fig. 15�a�. It can be found that as thecircle size increases, the tracking results get worse due to a con-stant input rate. Concerning a smaller circle with a 10 �m radius,

without and „d…–„f… with repetitive control term „uRC….
–„c…
the tracking results are detailed in Figs. 15�b�–15�d�. It is seen that


t�pmdawlrcqaiicim

eP=v

Tt

J

he tracking errors in both the x- and y- axes are maintained within0.44 �m, which is almost the noise level of the adopted dis-

lacement sensing system. It is additionally obtained that theaximum peak-to-peak �p-p� contouring error is 0.96 �m in two-

imensional �2D� workspace, as implied in Fig. 15�d�. Moreover,s the input frequency increases, the contouring is performed asell, and the statical comparisons of the control results are tabu-

ated in Table 2. In the table, the terms Max, Mean, and SDepresent the maximum, mean, and standard derivation of the p-pontouring errors, respectively. Because of a fixed sampling fre-uency, the position control step size in a time interval increasess the circle size rises. Thus, it can be observed that for a specificnput rate, the tracking performances get worse as the circle sizencreases. On the other hand, for a specified circular motion, theontouring errors get larger as well when the input frequency isncreased. This is caused by the bandwidth limit of the imple-

ented digital controller.It is remarkable that the above tracking experiments for differ-

nt amplitudes of input signal are conducted with the same set ofID control parameters �Kp=0.120, Ti=0.009 s, and Td0.044 s�. Extra contouring results reveal that, with a specificalue of input frequency and a larger circle size, better control

Fig. 15 „a… Circular contouring results wi„b…–„d… contouring results for a circle of a

able 2 Two-dimensional peak-to-peak circular contouring erhe proposed controller

Radius��m�

0.1Max Mean SD Max

10 0.957 0.021 0.184 1.40620 1.499 0.063 0.300 2.25040 2.347 0.052 0.472 4.152


performance can be obtained by tuning the PID parameters bigger.From this point of view, ideal PID parameters should be designedin accordance with the amplitude of the reference input, which isleft as a future work for the research. It is also noticeable that nodigital filter is used in the above experiments due to a relativelylow level of the sensor noise. Better results may be obtained byfiltering high-frequency noises out of the sensor output data. Fur-thermore, a better tracking result of the micromanipulator can beachieved with a faster sampling rate even with the same controlalgorithm proposed in this paper.

6 ConclusionThis research is focused on the hysteresis modeling and com-

pensation for a piezo-driven XY parallel micromanipulator aimingat a precise positioning. It is found that PSO is an efficient methodto identify the system model parameters, and the second-orderDahl model is capable of describing the nonsymmetric piezoelec-tric hysteresis more accurately than the Bouc–Wen model does.Experimental studies show that the combined inverse Dahl model-based feedforward and PID feedback control can well compensatefor the nonlinearity of the system than either of the stand-alone

n input rate of 0.1 Hz and different radii;�m radius.

„�m… of the XY parallel micromanipulator characterized with

Input rate�Hz�

0.2 0.5Mean SD Max Mean SD

0.095 0.311 2.606 0.570 0.485 0.210 0.480 4.118 1.030 0.831 0.272 0.853 7.048 1.649 1.518

th a10-

rors

JULY 2010, Vol. 132 / 041011-11

ceatrtp

mvffMot

A

mEm

R

0

ontrol schemes. Moreover, the back calculation antiwindup strat-gy improves the tracking performance of the controller in case ofctuator saturation, and the plugged repetitive controller enhanceshe positioning accuracy of the micromanipulator when periodiceference signal is specified. The experimental results demonstratehe effectiveness of the proposed controller for both 1D and 2Dositioning of the two-axis micromanipulator.

Due to a simple model and control structure, the proposedethodology is attractive from practical implementation point of

iew. The tracking performance of the micromanipulator can beurther enhanced by implementing higher closed-loop samplingrequency with the real-time controller proposed in this paper.

oreover, the proposed methodology can be easily extended tother types of micro- or nanomanipulators with piezoelectric ac-uation as well.

cknowledgmentThe authors appreciate the fund support from the research com-ittee of the University of Macau under Grant No. UL016/08-Y2/ME/LYM01/FST and Macao Science and Technology Develop-ent Fund under Grant No. 016/2008/A1.

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dahl model-based hysteresis compensation and precise

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