IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 3, MAY/JUNE 2014 1809
Empirical Modeling of a SqueezeFilm Haptic Actuator
Christophe Winter, Miroslav Markovic, and Yves Perriard
Abstract—In the analysis of squeeze film haptic tactile feedbackactuators, attention is focused on establishing an empirical modelof the force generated on the user’s finger. The model is based onfinite-element simulations which take into account the real motionof the actuator. As analytical models exist for very few specialcases, a methodology is exposed to establish an empirical modelof the force created between a finger and an actuator aided byan optimal design-of-experiment plan. Then, an example of anactuator which generates a sinc profile amplitude of vibration istreated.
Index Terms—Haptic interfaces, nonlinear design for experi-ments, piezoelectric actuators, squeeze film modeling.
NOMENCLATURE
Design of ExperimentSymbol DescriptionD Design matrix.η Response function (empirical model).f(·) Function f of ·.k Number of factors in the model.N Total number of experiments.O Objective function.p Number of parameters in the model.θ Model parameter.θ̂ Estimate of the parameter θ.u uth experiment.ξ Model factor.y Measured value for a given setup condition.Squeeze FilmSymbol Unit DescriptionH [–] Normalized air gap between two surfaces.h [meters] Air gap between two surfaces.h0 [meters] Mean air gap thickness over one period.ha [meters] Vibration amplitude of the actuator.hv [meters] Maximal vibration amplitude of the actuator.Λ [–] Bearing number.ω [radians per second] Pulsation.
Manuscript received December 13, 2012; revised May 9, 2013 andSeptember 19, 2013; accepted October 11, 2013. Date of publicationNovember 4, 2013; date of current version May 15, 2014. Paper 2012-EMC-616.R2, presented at the 2012 International Conference on Electrical Machines,Marseille, France, September 2–5, and approved for publication in the IEEETRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric MachinesCommittee of the IEEE Industry Applications Society.
The authors are with the Integrated Actuators Laboratory (LAI), Schoolof Engineering (STI), Ecole Polytechnique Fédérale de Lausanne, 2002Neuchâtel, Switzerland (e-mail: [email protected]; [email protected]; [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/TIA.2013.2288415
P [–] Normalized pressure.p0 [pascals] Ambient pressure.pf [pascals] Overpressure in the air gap.φϕ [degrees] Surface position (angular coordinate).φr [meters] Surface position (radial coordinate).σ [–] Squeeze number.T [–] Normalized time.t [seconds] Time.X [–] Normalized position in the x-axis.x [meters] Position in the x-axis.Y [–] Normalized position in the y-axis.y [meters] Position in the y-axis.
I. INTRODUCTION
NOWADAYS, tactile feedback actuators are more and moreinteresting due to the huge success of touch screen inter-
faces. They are present in a large number of consumer elec-tronic equipment but still have a lack of feedback compared toreal buttons. Studies have shown the benefits of tactile feedback[1] with different technologies [2]. In that context, the researchis focused on vibrating devices. The device presented in Fig. 1,showing a tactile click-wheel interface as could be found infuture consumer MP3 players, with other similar examples [3],[4] or [5] has shown that a vibrating surface under a finger canmodify the perceived sensation of the surface. The vibrationcreates literally an air cushion under the user’s finger which canlead to losing completely the contact with the actuator surfaceand thus suppress all friction forces.
The phenomenon can be explained by the Reynolds squeezefilm theory. As presented in Fig. 2, if two surfaces close to eachother but still separated by a very thin air film are consideredand if one of the surfaces moves, the pressure between them
1810 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 3, MAY/JUNE 2014
Fig. 2. Schematic representation of the air film trapped between a fingerand a tactile feedback actuator (where p0 is the ambient pressure, pf is theoverpressure in the air film, h0 is the mean air film thickness, ha is the vibrationof the actuator surface, and l0 is the characteristic length of a finger).
changes. The motion can be either tangential or normal to thesurfaces. One common example is the computer hard disk drivewhich uses the very fast tangential speed of the reading headto create an air cushion that prevents any contact with thesurface of the disks. The same phenomenon was also observedin a surface acoustic wave motor for example [6]. For the caseof tactile feedback, the finger tangential speed is too slow togenerate any force and will be thus neglected, but an ultrasonicvibration of the touched surface is able to create the needed aircushion. The normal motion of the surface squeezes the thin airfilm under the finger which has effect to increase its pressureand generate the lifting force.
The air flow between the surfaces can be expressed as a com-bination of two flows known as Hagen–Poiseuille and Couetteflows [7]–[9]. From the air flow, the Navier–Stokes equation(or continuity equation), which guarantees conservation of thefluid particle mass, leads to the Reynolds equation of squeezefilm for a thin air gap [10]
∂
∂X
(H3P
∂P
∂X
)+
∂
∂Y
(H3P
∂P
∂Y
)=Λ
∂
∂X(HP )+σ
∂
∂T(HP )
(1)
where according to the definitions in Fig. 2, H = h/h0 is thenormalized air gap, P = p/p0 is the normalized pressure in theair gap, X = x/l0 and Y = y/l0 are the normalized positions,T = ωt is the normalized time, and Λ and σ are the bearing andthe squeeze number, respectively (which are two dimensionlessparameters describing the behavior of the fluid under setupconditions). This equation is explained in detail in [10], and nofurther developments will be done in this paper. Nevertheless,it is a nonlinear partial differential equation with a very limitednumber of known analytical solutions for very special cases. Itis however important to be able to evaluate the pressure insidethe air film as a function of the actuator motion to predict thereaction force capability of the feedback device.
One particular solution has been proposed in [11] with thehypothesis of a squeeze film created between two flat surfacesmoving normally one to the other (like a piston). This simplifiedmodel has also been verified numerically in [12] with a finite-element (FE) model which showed very accurate pressurevalues at the center of the surfaces. However, the simulationshighlighted a border effect nearby the surface boundaries, i.e.,the pressure distribution is not constant near the surface border,that is not described with the analytical model. Moreover, inorder to obtain a sufficient vibration amplitude, it is verycommon to excite the actuator at one of its eigenmodes and
Fig. 3. Measured amplitude of vibration over the surface of the click-wheelinterface.
take benefit of the mechanical resonance amplification. Func-tional demonstrators of that kind of devices can be found invarious studies as in [4] and [13] and as the one in Fig. 1.The displacement amplitude of the click-wheel actuator for itsworking eigenfrequency (around 37 kHz) is presented in Fig. 3,which is obviously far away from a pistonlike motion. In sucha case, the assumptions made in [11] are no longer satisfieddue to the nonpistonlike motion of the actuator surface. Thishas been highlighted in [14]. Analytic solutions do not existanymore; however, it is still possible to numerically solve (1)by FE simulation, taking into account the real motion of thesurface. Those simulations are unfortunately time consuming.
The aim of this paper is thus to explain the methodology usedto generate an empirical model (based on FE simulations) of thesqueezed film pressure force created by the real motion of an ac-tuator. The benefit of such an empirical model is, as it requires afew FE simulations, to be able to quickly predict all the workingpoints of the actuator within a given experimental space. A casestudy is presented, and explanations on the FE model are given.Then, principles of nonlinear design of experiment are appliedto significantly reduce the number of needed simulations bychoosing the best experimental points to perform, which willgive the best estimate of the model coefficient to describe theforce generated by the squeezed air film.
Finally, it is to be kept in mind that the design of the actuatorthat generates the vibration used to squeeze the trapped airfilm is not addressed here. The empirical model of the pressureforce that will be presented is therefore valid for the particulardeformation shape of the case study actuator presented in thefollowing section, whereas the methodology can be applied onany actuators.
II. VIBRATION AMPLITUDE CHARACTERISTICS
OF THE FEEDBACK ACTUATOR
This section presents the vibration shape considered for theempirical modeling and its numerical description. To provide arealistic example, the deformation is measured on an existingactuator. The actuator shown in Fig. 4 consists of a 12-layer
WINTER et al.: EMPIRICAL MODELING OF A SQUEEZE FILM HAPTIC ACTUATOR 1811
Fig. 4. Vibrating surface setup. (Left) Piezoelectric stack actuator composedof 12 layers of piezoceramic rings and a T-shaped aluminum holding piece.(Right) Top view of the setup with the sensing surface and the representation ofthe amplitude profile.
Fig. 5. Amplitude of vibrations over the top surface of the actuator measuredwith a Doppler laser vibrometer. The limit position of the finger is highlightedwith the red dashed circle.
piezoelectric stack that generates vibrations on the top surfaceof the setup and which will be touched by the finger. Thissurface, which has a diameter of 20 mm, is presented on theright side of the figure. A square surface of 1 cm2 is chosen torepresent the contact surface of a finger without fingerprints (ac-cording to [15] and [16]). This test surface can be moved overthe actuator surface. The vibration amplitude perpendicular tothe surface is also represented.
The shape of the measured amplitude of the actuator vibra-tion is presented in Fig. 5 with the allowed limit position ofthe finger marked by the red dashed circle. The amplitude ismeasured at a frequency of 43.27 kHz which corresponds tothe eigenmode of the stack actuator. The deformation amplitudeha is axisymmetric and can be empirically modeled as a sincfunction along the radius of the surface with a maximum valuehv at the center (hv is a function of the driving voltage of theactuator)
ha(r, ϕ, t) = hv sinc(109.8r) sin(ωt) (2)
with r, in meters, and ϕ, in radians, being the polar coordinatesand ω = 2π 43.27 kHz. This empirical model is compared tovibration measurement on the real actuator in Fig. 6. This model
Fig. 6. Empirical model of (meshed) the displacement of the actuator surfacewith (dots) the measured data points.
is then used in the FE model of the squeeze film for the variationof the air film thickness as a function of time and position.The FE model is used to compute the mean pressure at anypoint inside the air film by solving (1). The result depends onthe mean air film thickness h0 and on the vibration amplitudeha. This is the most time-consuming step that needs to bereplaced by an empirical model. The mean force applied on thetest surface, depending on its location, is then obtained by anumerical integration of the computed overpressure.
III. NONLINEAR DESIGN OF EXPERIMENTS AIDED BY GA
Design-of-experiment methods help to choose the experi-ments to perform in order to obtain an accurate evaluation ofthe parameters of some response η of a chosen known functiondescribing the experiment
η = f(ξ,θ) (3)
where the vector ξ represents one chosen set of the k factors(variables) of our model in the experimental space and thevector θ represents the p parameters to determine. Consider thedesign matrix D = {ξtu}, where the u th row is the set of factorschosen for the u th experiment (with a total of N experiments).If the response η is a linear function of the factors ξ andparameters θ, plenty of design matrices are already availableand well described in [17]–[20] for example and have showntheir efficiency [21]. In a more general case, the response η isa nonlinear function of either ξ or θ, and the problem becomesmore complex because the derivatives depend on the values ofthe parameters. Therefore, it is not possible to predict the effi-ciency of the design without knowing in advance the parametervalues. At this stage, there are two open questions: How is itpossible to compute the best estimate of the parameters andwhat are the best points to perform experiments (or is it possibleto determine them)?
To answer the first question and to obtain the best estimateθ (given a set of N chosen experiments), in the sense of leastmean square, the sum of the squares
N∑u=1
(yu − f(ξu,θ))2 (4)
1812 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 3, MAY/JUNE 2014
where yu is the u th measurement on the system, has tobe minimized. This computation is iterative for a nonlinearfunction and, due to the frequent presence of multiple localminima, needs a starting point close to the solution to avoidbeing trapped in a local minimum of the function. To avoidthis guessed starting point, a two-step minimum search isimplemented. The first step is to use a genetic algorithm (GA)to find a minimum of the objective function (4) [22]. The searchboundaries of the GA are set wide (in the following examples,the upper/lower limits are set to ±1× 106 and to ±100 withthe same results, whereas the parameters are on the order ofmagnitude of ±10), and therefore, it removes the need of aknown starting point near the optimum. The GA is then stoppedquite early (after a low number of iterations or a rough changebetween two generations). The result (which is a solution closeto the optimum but not the optimum) is then used in the secondpart as a starting point for a deterministic search of minimumto find the exact local minimum (near the starting point) ofthe nonlinear function. In the presented work, the GA toolboxof Matlab [23] was used for the first step, and the fminsearchfunction was used for the second one.
To answer the second question, and as presented in [24],it is possible to determine which points are to be chosen byminimizing the determinant |(FtF)
−1|, where the matrix F ={fru} is given by
fru =
[∂f(ξu,θ)
∂θi
]θ=θ∗
. (5)
The only thing needed is the first estimation θ∗ of the param-eters θ which is close to the true estimate θ0 (which is theestimate that needs to be computed but that is unknown inadvance) in order to obtain the best design matrix given thechosen number of experiments (at least the same number as theparameters). This process can be refined iteratively.
As a real measurement setup able to impose a mean air filmthickness and measure the pressure or the pressure force isnot available, all experimental points are simulated with a FEmodel, and the FE results are considered as correct [25]. Fromthis point, a measurement is to be understood as a FE simulationresult.
IV. APPLICATION ON THE HAPTIC FEEDBACK
DEVICE—CENTERED FINGER
A. First Experimental Plan—A Priori
As a first example, only the vibration amplitude and the meanair film thickness are considered, keeping the finger positionfixed at the center of the actuator surface. At this stage, oneneeds to have an idea of the phenomenon which needs to bedescribed. In the case of squeeze film effect, the followinganalytic solution of (1) is known from [11], describing the meanpressure (in pascals) in the air film for a particular case of apistonlike motion of the actuator surface:
pf = p0
⎛⎜⎜⎝
√1 + 3
2
(hv
h0
)2
√1−
(hv
h0
)2− 1
⎞⎟⎟⎠ (6)
Fig. 7. Space of the experiment used for the a priori design (60 simulations),where hv is the maximal value of the sinc function of the surface motion modeland h0 is the mean air gap.
where p0 is the ambient pressure, hv is the amplitude ofvibration, and h0 is the mean air film thickness. As the pressureforce is a product of the pressure and the surface area, theforce created by the squeeze film is proportional to the pressuregiven by (6). As the surface vibration shape is different, (6) isnot correct for the displacement of the current studied actuator.However, the following conjecture is formulated: Equation (6)can still describe the phenomenon with the studied actuator butwith modified coefficients. The following empirical model isthus chosen to describe the force (in newtons):
η =
√1 + θ0
(hv
h0
)2
√1− θ1
(hv
h0
)2+ θ2 (7)
where θi denotes the three parameters to evaluate and hv and h0
are the two factors of the experiment [or, as presented earlier,the two components of the factor vector ξ = (h0, hv)].
The chosen experimental space is presented in Fig. 7. Ascreening, i.e., a regular exploration of the complete domain ofthe experiment, counting a total of 60 simulations is performedand used to estimate the parameter vector θ∗ = (θ∗0, θ
∗1, θ
∗2)
t
using the nonlinear least squares (NLS) method in two stepspresented in Section III. Fig. 8 shows the comparison betweenthe FE results (dots) and the fitted model (meshed) based on the60 points. This yields the first estimate of the model parametervalues.
B. Optimal Experimental Plan
Following the methodology presented in [24], the optimalexperimental points to reduce the number of experiments toperform can now be selected. This will ensure to still have agood evaluation of the model parameters. As there are threeparameters, at least three experiments are needed. The partialderivatives of (7) can be written as
∂η
∂θ0=
h2v
2√h20 + θ0h2
v
√h20 − θ1h2
v
(8)
∂η
∂θ1=
h2v
√h20 + θ0h2
v
2
√(h2
0 − θ1h2v)
3(9)
∂η
∂θ2=1. (10)
WINTER et al.: EMPIRICAL MODELING OF A SQUEEZE FILM HAPTIC ACTUATOR 1813
Fig. 8. (Meshed) Fitted force model η using (dots) the 60 FE simulations ofthe force.
Fig. 9. Optimal design space of the experiment for 20 optimization runs.
The value of the determinant∣∣∣(F′F)−1∣∣∣ (11)
can then be evaluated, and the three optimal factors ξ1,2,3 thatminimize this determinant can be found. The GA toolbox ofMatlab is used to find the optimal three-point set. However,the trivial solutions consisting of twice or three times the sameexperimental point, which yields an infinite number of nulldeterminants, cannot be accepted. The objective function Oused to find the optimal experimental points is therefore slightlymodified to avoid such situations: A penalty is added to increasethe objective function as much as the chosen points are spatiallyclose to each other. For each evaluation of (11), the normsbetween the experimental points ξi are computed, and theirinverse values are added to the objective function
O =∣∣∣(F′F)
−1∣∣∣+ 1
‖ξ1ξ2‖+
1
‖ξ1ξ3‖+
1
‖ξ2ξ3‖. (12)
The optimization is repeated 20 times using model (7) and theparameter vector θ∗ found in the first a priori design. The givensearch boundaries are set according to the space of the ex-periment domain. The 20 optimal triplets found are presentedin Fig. 9.
The optimization highlighted five regions where optimalpoints are located. With a closer look and after identificationof each triplet, two optimal patterns can be found. Both are
Fig. 10. Optimal design space of the experiment—first candidate topology.
Fig. 11. Optimal design space of the experiment—second candidate topology.
Fig. 12. Space of the experiment. Optimal design: Five black dots. A prioridesign: Sixty white dots.
presented in Figs. 10 and 11. This allows assuming that bothpatterns are very close optimal values. Based on those results,the decision is made to use the two optimal position patterns asthe optimal design of experiment with therefore five measure-ment points instead of only three. This means that the neededsimulation number is reduced by 12 to compute the modelparameter values with this design, and based on [24], in keepingthe best candidates to minimize the error in the parameter value.
The space of the experiment with the optimal design becomesthe one shown in black dots in Fig. 12. The obtained parametervalues, as well as the ones of the first screening, are presentedin Table I. The fitted model is represented in Fig. 13 withthe five experimental points used to compute the parametervalues (black dots) and with all the measured points of thea priori design. These points are used in this particular case toperform the quality assessment of the fitted model because theyare already available. Nevertheless, if the quality assessmentprocess is performed, as many points as desired, which areevaluated inside the experimental space, can be used as control
1814 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 3, MAY/JUNE 2014
TABLE ITWO-FACTOR MODEL PARAMETER VALUES
Fig. 13. (Meshed) Fitted force model η using the five points of the optimalexperiment. (Black dots) Optimal design points. (White dots) Control points.
points and are not forced to be related to the previous screeningstep. The sum of the squares of the residue is slightly higherthan the result of the NLS based on the 60 points, but it is agood tradeoff as the FE simulation time is in the range of daysand therefore the reduction of the number of needed simulationsis really a predominant factor.
The relative error in percent between each simulation pointand the empirical model is computed. The error is shown inFig. 14. The white dots are an error within ±15% of thesimulation value. The error is very large for very small vibrationamplitude hv . This is mostly due to the absolute value of theforce which is very close to zero. Another criterion can alsobe used to compare the empirical model with the availablecontrol points: Considering all experiments as a vector space,the norm of the residual r = y − η represents the radius of ahypersphere containing the error between the empirical modeland the experiments. The smaller this hypersphere is, the betterthe model is. This can be expressed in percent relative to thenorm of the experiment vector y. This leads to an error of 3%for the model parameters evaluated with the 60 simulations and4.6% with the five optimal ones, which are considered to begood. This criterion allows evaluating the validity of the modelalmost everywhere but cannot guarantee avoiding locally hugerelative errors as have been presented in Fig. 14.
For the particular application of a tactile feedback, the modelis more than sufficient given the fact that the force applied bythe user’s finger during an exploratory task is between 0.5 and1 N [15]. The region of interest is therefore at low h0 andhigh hv. In the region of low hv where the model has locallyhuge error compared to simulations, the force represents around90 ppm of the targeted force of 1 N. That means that, evenwith a huge error, the force is evaluated to be four orders ofmagnitude smaller than the needed one and can be rounded tozero. This is indeed the expected behavior of the squeeze film as
limhv→0
(η) = 0. (13)
Fig. 14. Error between the simulated points and the empirical model (under±15%, the dots are in white; otherwise, they are in black).
The model is therefore considered as valid and guarantees agood estimation of the force, considering the application, inthe whole domain of the space of the experiment (i.e., hv =[0.2; 2] μm and h0 = [5; 10] μm).
Predictions can be obtained outside the space of the experi-ment by extrapolation; however, it is not possible to guaranteethe correctness of the result. The model results have to be con-trolled by FE simulation to ensure their validity. For example, avibration amplitude slightly higher is needed to achieve a forceof 1 N. By extrapolation, the empirical model determines theworking condition to obtain 1 N with the value hv = 2.2 μm(with h0 = 5 μm), whereas the FE simulation gives a force of0.99 N for that working point. For other applications, and if theerror is considered not acceptable, the chosen model describingthe phenomenon should be reworked (e.g., by increasing itsdegrees of freedom), or a smaller experimental space aroundthe particular point of interest must be chosen. It is possible toconclude that the empirical model is valid for the application todescribe the simulated phenomenon.
V. EMPIRICAL MODEL WITH FOUR
FACTORS—NONCENTERED FINGER
In practice, the position of the user’s finger on the surfaceis not forced to be at the surface center. The finger can bemoved within the allowed perimeters shown in Fig. 6. It ispossible to consider this effect in the empirical model by addingtwo more factors: the coordinates of the finger position. Thesame methodology is applied, but at this time, four factorsare taken into account: the mean air film thickness h0, thevibration amplitude hv, and the positions of the test surface onthe actuator (in polar coordinates) φr and φϕ. The followingempirical model η is considered:
η =
√1 + θ0
(hv
h0
)2
√1− θ1
(1 + θ3
φϕ
45 + θ4φr
)(hv
h0
)2+ θ2 (14)
with a space of the experiment presented in Table II.
WINTER et al.: EMPIRICAL MODELING OF A SQUEEZE FILM HAPTIC ACTUATOR 1815
TABLE IISPACE OF THE EXPERIMENT WITH FOUR FACTORS
Fig. 15. Selected design matrix in the plane of experiment φϕ − φr .
TABLE IIIPARAMETER VALUES FOR MODEL η WITH FOUR FACTORS
A first experimental plan is performed for the position of thetest surface given in Fig. 15. For each position, the five optimalexperiments presented in Section IV-B are used in the planeh0 − hv . The first estimate θ∗ presented in Table III is obtainedbased on 65 simulations.
The partial derivatives of the model (14) can then be evalu-ated as
∂η
∂θ0=
h2v
2h20
√1+θ0
h2v
h20
√1−θ1
(1+θ3
φϕ
45 +θ4φr
)h2v
h20
(15)
∂η
∂θ1=
h2v
(1+θ3
φϕ
45 +θ4φr
)√1+θ0
h2v
h20
2h20
√(1−θ1
(1+θ3
φϕ
45 +θ4φr
)h2v
h20
)3(16)
∂η
∂θ2=1 (17)
∂η
∂θ3=
θ1φϕh2v
√1+θ0
h2v
h20
90h20
√(1−θ1
(1+θ3
φϕ
45 +θ4φr
)h2v
h20
)3(18)
∂η
∂θ4=
θ1φrh2v
√1+θ0
h2v
h20
2h20
√(1−θ1
(1+θ3
φϕ
45 +θ4φr
)h2v
h20
)3(19)
Fig. 16. Five optimal experimental points in four dimensions.
and the minimum of the determinant |(FtF)−1| can be com-
puted. The five optimal experiments obtained are presented inFig. 16(a) and (b) in a 4-D experimental space. It is to be notedthat the locus of the optimal points in the hv − h0 plane followsthe same pattern as obtained in the previous section. However,these points are found by the optimization algorithm itself with-out external biasing based on the presented previous results.
The model parameters obtained with the optimal experimen-tal plan are presented in Table III. The global error of theempirical model with four factors evaluated using the five-pointoptimal experimental plan is 4.5% which is considered as verysatisfying.
VI. CONCLUSION
A methodology has been presented to establish an empiricalmodel of the force created by the squeeze film effect of a tactilefeedback actuator. An example application has been treated forwhich the number of needed experiments was reduced by afactor 12 due to the choice of an optimal design-of-experimentplan. The empirical model matches well with the availablesimulated points. Moreover, an empirical model taking intoaccount four factors was established to describe the force as afunction of the amplitude of vibration, the mean air gap betweenthe actuator and the user’s finger, and the location of the fingeron the surface. In this second application, the needed number ofsimulations is reduced by a factor of 13. Owing to the generatedmodel, it is possible to evaluate very fast if a working pointcondition of the actuator is able to create a sufficient force toproduce a tactile feedback.
1816 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 3, MAY/JUNE 2014
Within the field of friction feedback devices, the presentedoptimal design locus results can be used as a first experimentalplan to avoid a complete screening of the experimental spaceand then be refined iteratively, considering that the force re-sponse shape is similar to the one presented in this paper.Finally, even for other application fields, the presented three-step methodology is still applicable to compute the optimalexperimental plans and the model parameters.
ACKNOWLEDGMENT
The authors would like to thank Dr. J.-M. Fuerbringer fromthe Ecole Polytechnique Fédérale de Lausanne for the discus-sions that we had on design-of-experiment methods.
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Christophe Winter was born in Bern, Switzerland,in 1986. He received the M.Sc. degree in microengi-neering from the Ecole Polytechnique Fédérale deLausanne, Neuchâtel, Switzerland, in 2009, wherehe is currently working toward the Ph.D. degree inthe Integrated Actuators Laboratory (LAI), School ofEngineering (STI).
His research interests are in the fields of piezoelec-tric haptic actuators (mechanical and electronic drivedesign) and squeeze film modeling.
Miroslav Markovic was born in Arandjelovac,Serbia, in 1970. He received the B.S. degree fromthe School of Electrical Engineering, University ofBelgrade, Belgrade, Serbia, in 1996 and the Ph.D.degree from the Ecole Polytechnique Fédérale deLausanne (EPFL), Lausanne, Switzerland, in 2004.
He is currently a Project Leader with the Inte-grated Actuators Laboratory (LAI), School of En-gineering (STI), EPFL, Neuchâtel, Switzerland. Hisresearch interest is the optimization design of high-performance electric drives.
Yves Perriard was born in Lausanne, Switzerland,in 1965. He received the M.Sc. degree in microengi-neering and the Ph.D. degree from the Ecole Poly-technique Fédérale de Lausanne (EPFL), Neuchâtel,Switzerland, in 1989 and 1992, respectively.
He was a Cofounder and the Chief ExecutiveOfficer of Micro-Beam SA, which is involved inhigh-precision electric drive design. He is currentlythe Director of the Integrated Actuators Laboratory(LAI), School of Engineering (STI), EPFL, wherehe was a Senior Lecturer beginning in 1998, has
been a Professor since 2003, and was the Vice Director of the Institute ofMicroengineering from 2009 to 2012. His research interests are in the fieldsof new actuator design and associated electronic devices.
