344 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

Hysteresis-Based Mechanical StateProgramming of MEMS Mirrors

David Torres , Member, IEEE, Jun Zhang, Member, IEEE, Sarah Dooley,

Xiaobo Tan, Fellow, IEEE, and Nelson Sepúlveda , Senior Member, IEEE

Abstract— The programming of different mechanical states ofa MEMS mirror is presented here. The mirror is based on a smartmaterial called vanadium dioxide (VO2) and actuated throughJoule heating. The system exhibits dynamical and hysteresisnonlinearity. While hysteresis is largely deemed undesirable,in this paper, the hysteretic behavior of the VO2 is exploited toprogram tilt angles of the device. The programming is realizedby applying a constant current of 7.02 mA (with an electricalpower of 5.1 mW) to pre-heat the VO2 film and adding electricalpulses to move between states. Furthermore, the capability ofprogramming any particular mechanical state across the phasetransition of the VO2 is demonstrated. A Preisach model isadopted to characterize the hysteretic relationship between theinput current and the pitch angle at the steady state, andto compute the input pulse for generating the desired pitchoutput via an iterative algorithm. Experiments are conducted todemonstrate the effectiveness of the proposed approach, wheretwo sequences of pitch angle values, with a triangular profile anda random profile, respectively, are successfully programmed withan average error of 0.03°. [2017-0284]

Index Terms— MEMS-mirrors, vanadium dioxide, phase-change materials, programmables MEMS, hysteresis.

I. INTRODUCTION

M ICROELETROMECHANICAL mirrors are devicesthat use an electric signal to move a platform to redirectan incident light beam to a desired direction. These deviceshave been used for different applications including opticalphase arrays [1], [2] spectroscopy [3], track-positioning [4],scanners [5], optical displays [6], [7], and medical imag-ing [8]–[10]. The actuation mechanisms implemented on these

Manuscript received November 16, 2017; revised January 6, 2018; acceptedFebruary 10, 2018. Date of publication March 1, 2018; date of currentversion April 2, 2018. This work was supported by the National ScienceFoundation under Grant ECCS 1306311 and Grant CMMI 1301243. Thework of D. Torres was supported in part by the SMART Scholarship by theDepartment of Defense of USA and in part by the Cooperative Research andDevelopment Agreement between AFRL’s Sensors Directorate (AFRL/RY)and Michigan State University under Grant 15-075-RY-01. SubjectEditor O. Solgaard. (David Torres and Jun Zhang contributed equally to thiswork.) (Corresponding author: David Torres.)

D. Torres and S. Dooley are with the Sensors Directorate, Air ForceResearch Laboratory, Wright-Patterson AFB, Dayton, OH 45433 USA(e-mail: david.torres-reyes.1@us.af.mil; sarah.dooley@us.af.mil).

J. Zhang is with the Department of Electrical and Computer Engineering,University of California at San Diego, San Diego, CA 92093 USA (e-mail:j5zhang@ucsd.edu).

X. Tan and N. Sepúlveda are with the Department of Electricaland Computer Engineering, Michigan State University, East Lansing,MI 48840 USA (e-mail: xbtan@egr.msu.edu; nelsons@egr.msu.edu).

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

Digital Object Identifier 10.1109/JMEMS.2018.2807482

devices include: electrostatic (ES), piezoelectric (PE), elec-tromagnetic (EM), and electrothermal (ET). The actuationmethods determine the speed, input signal that is needed foractuation, and the power consumption of the device, while thetotal displacement relies on the combination of the design ofthe device and the actuation method. ES and PE use electrosta-tic fields to generate mechanical forces that produce displace-ment of the mirror platform. Although the electrostatic field isgenerated using high voltage (with reported values of 115 Vand 40 V for ES and PE in [11] and [12], respectively),the power consumed by these two actuation mechanisms isoften relatively low since a low current flow is requiredto charge the plates. While ES devices can be designedwith conventional materials, the PE designs use piezoelectricmaterials, where the most common material for micromirrorsis lead zirconate titanate (PZT) [12]–[15]. The addition of thismaterial increases the complexity of fabrication of the device,due to the cross contamination of the equipment. The speedof both actuation mechanisms is limited by the mechanicalresponse of each device, where the devices are commonlyoperated in resonance to increase the displacement range. TheEM mechanism depends on the interaction between two mag-netic fields, where Lorentz force governs [7]. A combinationof a permanent magnet and coils are used to generate the twomagnetic fields, where the magnitude of the electromagneticfield created by the coils is dependent on the input current.This current can reach values as high as 515.17 mA [16].Another approach is to integrate magnetic sensitive materialswith the device and use an external magnetic field to generatethe movement [17]. Many of these materials are incompatibleswith the conventional micro fabrication technique, resulting inthe use of unconventional fabrication techniques. Similar tothe ES actuation mechanism, the speed of the actuation willbe limited by the mechanical dynamics of the device.

The ET mechanisms use temperature as the stimulus foractuation, where the conventional configuration consists ofa layer structure of two materials (bimorph) with differentthermal expansion coefficients (TECs), such as a metalwith a large TEC and glass material (SiO2) with a lowerTEC. This method of actuation has the lowest speed ofall since the speed depends on the thermodynamics of thedevice, which is usually below the mechanical dynamics ofthe structure. The change in temperature is created by Jouleheating with reported values for full actuation reaching upto about 350 ◦C with a voltage of 4.5 V consuming a powerof 41 mW [2].

1057-7157 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

https://orcid.org/0000-0002-6496-8243https://orcid.org/0000-0002-9676-8529

TORRES et al.: HYSTERESIS-BASED MECHANICAL STATE PROGRAMMING OF MEMS MIRRORS 345

In [18] we presented the first smart material-embeddedMEMS mirror based on the ET mechanism, where the stresswas generated due to the phase transition of vanadium diox-ide (VO2). The fabrication process of the device is describedin detail in [18], where conventional fabrication techniqueswere used without a problem. Using VO2 reduces the rel-atively large temperature change requirements in traditionalTEC-based ET mechanisms from 300 ◦C to less than 100 ◦C,resulting in a reduction in power consumption. The devicereported in [18] consisted of a mirror platform surroundedby 4 actuator legs, where the average power consumed bythe actuators during individual actuation was 6.5 mW andthat for the generation of the piston-like vertical displacementwas 26.1 mW.

VO2 is a smart material that has been used to design differ-ent devices, such as Variable Optical Attenuators (VOA) [19],actuators (both electro- and photo-thermal) [20], [21], andmicromechanical resonators [22]. These applications exploitthe solid-to-solid phase transition of the material, where theoptical [23], electrical [24], and mechanical [25] properties ofthe material change abruptly. The transition can be inducedwith an increase of temperature with a transition temperatureof 68 ◦C. The required range of temperature for full actuationis less than 20 ◦C. During the transition the material showsa higher strain energy density compared to the commonactuation mechanisms, such as ES, EM, PE and conventionalET [26].

The material exhibits a nonlinear behavior, hystere-sis, between the mechanical property and the tempera-ture [27], [28]. There has been extensive effort to dealwith the hysteresis nonlinearity in VO2-based devices, fromusing self-sensing, where the relationship between the resis-tance of the VO2 and the displacement is used to facili-tate the feedback control of the system [29], to modelingcomplex non-monotonic hysteresis behavior using Preisachoperator [30] or Prandtl-Ishlinskii operator [31] for the purposeof inverse compensation. Furthermore, in [32] a compre-hensive modeling scheme for the VO2-based MEMS mir-ror was developed, where a Preisach operator was used tomodel the hysteretic behavior. Two noticeable differencesexist between [32] and this work: Firstly, in [32], a modelwas presented to capture the MEMS mirror dynamics andhysteresis. The goal was to model the mirror system andestimate the mirror mechanical states. The goal of this work isto control the mirror to generate desired mechanical state andachieve accurate mechanical programming function. Secondly,the inherent hysteresis of the system allowed for the use ofinputs in the form of pulses (not steps as was done in [32]).The use of pulses to control and program the device’s posi-tion through the system’s hysteresis presents new challenges.A generalizable and iterative algorithm is proposed to controlthe mirror positions for mechanical state programming.

Programmable devices are those that are capable of storinga measurable variable for a period of time. Basic examplesof programmable devices are capacitors, where electricalcharge is the stored variable [33], [34], and a minimumof two states, charged and discharged, are realized. Similarto this, MEMS devices are capable of storing electrical or

mechanical variables, such as voltage [35]–[37], resonantfrequencies [22] or displacements [38]–[40]. In [37], MEMSand CMOS technologies are combined to create devicescapable of storing voltage values in two different states,by charging/discharging a floating gate of a transistor witha mechanical switch. Programmed displacement values can bestored in an external circuitry, which works as the system’smemory by saving the input signal for each desired state.An example of this is the digital mirror presented by TexasInstrument, Digital Micromirror Device (DMD) [6], where thedevice includes a CMOS memory circuit. Another approachis creating bi-stable MEMS devices, where two states areprogrammed and accessed using an electrical or mechanicalstimulus [41], [42]. The disadvantage of these approachesis that systems are limited to two states. To overcome thislimitation, a different approach for programming mechanicaloutputs has been implemented by exploiting the hysteresisnonlinearity observed in certain MEMS devices.

In [38] a cantilever beam covered with a thin film ofVO2 was used to demonstrated the programming of differentmechanical states. This was done by maintaining the tem-perature of the sample near the transition temperature (pre-heating temperature) and applying laser pulses (photothermalactuation) to move to different states, where the magnitudeof the laser pulse will determine the new mechanical state.This technique was also implemented in VO2-based resonatorsin [22], where the programmable variable is the resonantfrequency of the system. This was further explored in [39],where Joule heating with a monolithically integrated resistiveheater was used for actuation. The hysteresis-enabled approachnot only allows for programming of multiple states, but alsoenables achieving and maintaining the desired states in anenergy-efficient manner. This is because only electrical currentpulses (on top of a pre-heating current level), instead of con-tinuous high-magnitude inputs, are required to move aroundthe states. While hysteresis is largely deemed undesirable forVO2 and other smart material-based actuators, in this work,it is exploited to achieve programming of the mechanical statesof MEMS mirrors.

The work in [22], [38] and [39] was focused on demon-strating that the mechanical states of a VO2 MEMS devicecan be maneuvered with laser or current pulses; however, howto compute the magnitude of the pulse to achieve a desiredmechanical state has not been addressed in these papers. Solv-ing the latter problem with a principled approach is preciselythe contribution of the current paper, where programmingthe pitch angle of a VO2 MEMS mirror, actuated via Jouleheating, is used as a case study. In our approach, the Preisachoperator is used to capture the hysteresis of the VO2 mirrorbetween the pitch angle output and the current level at thesteady state. Then using the hysteresis model, an iterativealgorithm is developed to compute the pulse magnitude giventhe desired output value. Finally, a constant current will beapplied to the device, which will drive it to the preheated stage;and the calculated pulses magnitude will be added to achievethe desired output. Experiments are conducted to verify theeffectiveness of the model and the algorithm for programmingmechanical states.

346 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

Fig. 1. SEM images of the VO2-based MEMS mirror, where the rota-tional axes are labels for the actuated leg [32]. The mirror platform is600 µm × 600 µm, and the device size is 1.048 mm2, resulting in a fillfactor of 32.8%. The insert shows the patterned metal traces in the bimorphregions.

II. EXPERIMENTAL PROCEDURES

A. Design and Operation of VO2-Based MEMS Mirrors

The design of the VO2-based MEMS mirror (shownin Fig. 1) is presented in [18]. The device uses four individualactuators (legs) to move a square mirror platform. The legsare connected to each side of the mirror platform, whichresults in a tilting movement upon the actuation of a singleleg, or the vertical platform displacement upon the simulta-neous actuation of all the legs (piston movement). Each leghas monolithically integrated metal traces that are used asresistive heaters to induce the transition of the VO2 on thelegs by Joule heating. The legs are mechanically connected tothe mirror platform with only a 1.4 µm thick SiO2 layer. Thisis highlighted as mechanical connection in Fig. 1 and Fig. 2.The SiO2 layer was used to thermally isolate the actuatorswith the platform and between actuators. Thermal isolation isneeded in order to prevent the heat produced for the actuationof one leg from inducing actuation of another leg.

B. Fabrication of VO2-Based MEMS Mirrors

The device presented here follows the same fabricationprocess flow as in [18], with a change in the metal layersto increase yield. In this work, the metal traces and resistiveheaters for the devices are chromium (20 nm) and gold(110 nm), where the Cr layer is used only for adhesionpurposes. The devices used in [18] were titanium (Ti) and plat-inum (Pt). The new metal layers (Cr/Au) mitigate significantlythe delamination issues observed when using Ti/Pt, increasingthe yield per wafer from 12.5% to 75%. We believe that thelower yield with Ti/Pt was caused by the high intrinsic stressof the evaporated Ti/Pt on SiO2, which has been reported tobe as high as 340 MPa (compressive) [43], [44], while Cr/Auhas an intrinsic stress of 250 MPa (tensile) [45], [46].

Fig. 2. Fabrication process flow of the VO2-based MEMS mirror observedin a cross-section view of the device. a) PECVD of SiO2 on both side ofthe wafer at 300 ◦C, b) deposition of VO2 by PLD, c) PECVD of SiO2at 250 ◦C, d) metallization lift-off of the metal layer (Cr/Au), e) PECVDof SiO2 at 250 ◦C, f) RIE on top SiO2 layer to access the metal layer,g) RIE on top SiO2 to define the structure and expose the Si, h) RIE ofbackside SiO2, i) DRIE of the backside Si layer, j) DRIE of topside Si layerto release the structure, and k) isotropic etch of the Si layer (XeF2 gas) tocreate the bimorph section of the legs and the mechanical connection betweenthe legs and the platform is shown.

The fabrication process flow of the device is shown in Fig. 2and is conducted on a double-side polished 2-inch silicon (Si)wafer with thickness of 300 µm. This wafer is needed toensure a uniform backside Si etch. SiO2 (1 µm thick) is grownon both sides of the wafer by Plasma Enhanced ChemicalVapor Deposition (PECVD) at a temperature of 300 ◦C. Oneof the SiO2 layers (back side) is used as a hard mask forthe anisotropic etch of Si, while the other (top side) is usedto promote the crystalline orientation of polycrystalline VO2growth [22], which increases the maximum deflection causedby the phase transition of the VO2 [39]. The VO2 is grown byPulse Laser Deposition (PLD) and patterned with ReactiveIon Etch (RIE) following the recipe from [18]. After thedeposition of the VO2, all thermal process are conductedwith temperatures lower than 250 ◦C to avoid degradation ofthe material due to film oxidation. Before the metallization,another 200 nm SiO2 layer is grown to insulate the VO2from the metal layer. The metal is deposited and patternedby lift-off technique – the Cr is used as an adhesion layerbetween the Au layer and the SiO2 layer. A final SiO2 layer(200 nm) is deposited to reduce convection thermal losses fromthe resistive heater to the ambient. Two SiO2 etch steps areperformed by RIE to expose the metal contact and pattern themirror device exposing the Si layer. Another 1 µm SiO2 etchby RIE is performed on the backside of the wafer to exposethe Si layer. A 250 µm Si anisotropic etch by Deep ReactiveIon Etch (DRIE) is performed on the back side of the wafercreating the thickness of the mirror platform and the frames ofthe legs. The structure is released with another Si anisotropicetch by DRIE for the remaining 50 µm, but from the top sideof the wafer. The final step is a Si isotropic etch to remove the

TORRES et al.: HYSTERESIS-BASED MECHANICAL STATE PROGRAMMING OF MEMS MIRRORS 347

Fig. 3. Experimental setup used to measure the movement of the mirrorplatform, using a laser scattering technique with a low power infrared laserand a 2D PSD, and a digital camera to calibrate the output voltage of the PSD.

Si layer from beneath the bimorph by XeF2 gas. This processis timed to etch the Si beneath the bimorph and reduce anyover etch on the frame.

C. Experimental Setup

Fig. 3 shows the experimental setup used to measure themirror platform movement. The device chip is wire-bondedto a rectangular IC package and soldered to a PCB board,creating the electrical connections between the metal contactsand the data acquisition system (DAQ). All the electricalsignals are controlled through LabVIEW using a NationalInstrument-DAQ. The NI-DAQ is composed of a NI cRIOcontroller (NI-cRIO-9076), 2 input modules (NI-9201: ±10 V,12-Bit, 8-Channel, and NI-9222: ±10 V, 16-Bit, 4-Channel)and 1 output module (NI-9269: ±10 V, Isolated, 4-Channel).The DAQ system is capable of achieving speed of 100ksamples per second per channel. The input current is generatedusing a BJT NPN transistor (2n3904) in an emitter followerconfiguration. A low power infrared laser (λ = 985 nm) witha beam diameter of 240 µm focused on the mirror platformin combination with two-dimensional (2D) position sensingdetectors (PSD THORLABS PDP90A) is used to track themovement of the mirror platform. An optical setup is used tocalibrate the output of the PSD (voltage) to displacement. Theoptical setup is composed of a digital camera (Nikon 1 J1) andan external objective lens (10X Mitutoyo Plan Apo InfinityCorrected Long WD Objective lens). The resolution of theoptical setup is 0.5 µm/pixel with a speed of 30 Hz. Videoscaptured by the camera are analyzed using Tracker videoanalysis and modeling software tool (Version 4.94, DouglasBrown, physlet.org/tracker).

All the experiments are performed in air and at room tem-perature. Although the sample temperature was not controlledduring testing, fluctuations were very small. A variation in theambient (or background) temperature would affect the actua-tion range of the device. One way to compensate for a small

variation in the ambient temperature would be implementinga close-loop control system. This was demonstrated in [39],where the change in resistance of the metal traces caused bythe Joule heating were used as the feedback variable.

In this study, we consider the tilt movement of the device.This is achieved by actuating only one leg, which creates a 2Dmovement. The 2D motion is generated by the asymmetricalconnection between each actuator and the mirror with respectto the center of the platform. This movement is captured bythe 2D-PSD, and separated into two components (see Fig. 1):(i) pitch - rotation of the platform along the “Pitch Axis,” and(ii) roll - rotation of the platform along the “Roll Axis.” Theaxes are selected to describe the movement of the platformindependent of which actuator is being used. In addition, therewill be a cross-talk between legs at the moment of excitationdue to the mechanical coupling between them. This will bemore prominent when more than one leg is actuated at thesame time. In this work, we minimize the cross talk betweenthe actuators by only actuating one leg at a time, which makesthe other legs behave like passive springs.

III. THE ALGORITHM OF PROGRAMMINGMECHANICAL STATES

Exploiting the hysteresis of the system, an algorithm forprogramming any selected tilt angle of the VO2-based MEMSmirror is proposed. Without loss of generality, this studyfocuses on programing the pitch angle.

A. Problem Statement

Similar to [39], the input to the mirror system is a combi-nation of a constant current and a sequence of current pulses.The constant current is a pre-heating input to the mirror,and the current pulses are applied to change the angle ofthe system. The pre-heating input corresponds to a steady-state temperature where the possible pitch angle values, dueto hysteresis, encompass the significant chunk of the outputrange. The durations of the current pulses are chosen to bethe same, and are much longer than the time constants of theheating and mechanical dynamics. In the presented system,the heating dynamic is the dominant mechanism with reportedvalues of less than 15 ms [32] and the current pulse is heldfor 5 s. The latter ensures that the output reach its steady-statevalue by the end of each pulse.

Fig. 4(a) illustrates the definition of the input pulses, wherethe pre-heating current is denoted as i0, and different cur-rent pulses are denoted i1, i2, · · · , in , etc. After each pulse,the current returns back to i0. Under the pulse ik , 1 ≤ k ≤ n,the current input sequence consists of three steps in the orderof i0, ik, i0. This input sequence of the pulse ik is denoted asi0 → ik → i0.

The corresponding angle output is illustrated in Fig. 4(b).It is noted that the angle outputs, θ0, θ1, · · · , θn , are steady-state values. They are obtained when the angle reachesthe steady-state value while the input returns to i0 at timet0, t1, · · · , tn , respectively, as shown in Fig. 4(a). Due tohysteresis, the newly attained angle output under the inputsequence i0 → ik → i0 will be different from the value prior

348 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

Fig. 4. (a) An input sequence consists of the pre-heating current and differentcurrent pulses; (b) The corresponding angle outputs under the input sequence.

to the application of the pulse. The angle first reaches θ0 underthe pre-heating current i0, then reaches θ1 under the inputsequence of the pulse i1, i0 → i1 → i0, and further reachesθ2 under the input sequence of the pulse i2, i0 → i2 → i0.Under the input sequence of the pulse ik , i0 → ik → i0,the angle reaches θk .

Given the current hysteresis memory and a desired pitchangle value, θd , the goal of programming the pitch angle is tocalculate the value of pulse id such that the steady-state anglewill be equal or close to θd .

B. System Modeling

The MEMS mirror is actuated by four legs, each being aVO2-coated microactuator. Different mirror poses with com-binations of pitch, roll, and piston motions can be achieved byvarying the actuation schemes of the legs. For the clarity ofpresentation, we focus on the single leg-actuated pitch motionin this paper. The roll motion can be similarly captured withthe same model formulation for the pitch motion but withdifferent model parameter values [32].

The following equation can be used to model the dynamicsof the pitch motion:

J θ̈ + Gθ̇ + kθ = a�[

ATτths + 1 i

2(·); ζ0](t), (1)

where J is the moment of inertia, G is the rotational dampingcoefficient, k is the rotational spring constant, θ is the pitchangle of the mirror platform generated during actuation, a isthe distance between the actuation force and the axis ofrotation, and � is the actuation force, which can be capturedby a hysteresis with the actuation power as input. As shown inthe brackets of (1), the Joule heating dynamics is modeled as

a first-order linear system, where AT is the gain of the thermaldynamics, τth is the time constant of the thermal dynamics, sis the Laplace variable, i is the current input, i(·) denotes theinput history, i(τ ), 0 ≤ τ ≤ t, and ζ0 is the initial condition ofthe hysteresis memory. More details of the model formulationand analysis can be found in [30] and [32].

VO2-coated microactuators exhibit non-monotonic hystere-sis, which results from two competing mechanisms: one is thestress generated due to the thermal expansion difference ofthe materials forming the bimorph structures, and the other isthe stress induced by the phase transition [30]. The actuationforce � in (1) is expressed as

�[·] = �P [·] + �T (·), (2)where �P [·] is the phase transition-induced force, and �T (·)is the differential thermal expansion-induced force. Thephase transition-induced force is monotonically hysteretic andcaptured by a Preisach operator. The differential thermalexpansion-induced force is modeled as the following linearterm:

�T (i2(t)) = −c1i2(t), (3)

where c1 is a constant term related to thickness, modulus ofelasticity, and thermal expansion coefficients of VO2 layerand SiO2 layer, and the negative term is introduced since thethermal expansion-induced force has an opposite direction asthe phase transition-induced force [30], [32].

Under quasi-static conditions, the mechanical and thermaldynamics of the mirror system are not considered. The modelof the mirror can be simplified as

θ(t) = a · ATk

· �[i2(·); ζ0](t). (4)

C. Modeling the Phase Transition-Induced Forceby a Preisach Operator

The Preisach operator is one of the most effective hysteresismodels and has been widely adopted [30], [32], [47], [48].In this study, we adopt the Preisach operator to model thephase transition-induced force of the mirror system consider-ing its proven modeling capability:

�P [i2(·); ζ0](t)=∫P0μ(β, α)γβ,α[i2(·); ζ0(β, α)](t)dβ dα+c0,

(5)

where P0 is the Preisach plane P0 �= {(β, α) : i2min ≤ β ≤α ≤ i2max}, [i2min, i2max] defines the phase transition range,μ is the Preisach density function, γβ,α is the basic unit ofthe model called hysteron, ζ0(β, α) ∈ {1,−1} is the initialcondition of the hysteron, and c0 is a constant bias. The outputof the hysteron is determined by the input history and iseither 1 or −1:

γβ,α[i2(·); ζ0](t) =⎧⎨⎩

+1 if i2(t) > α−1 if i2(t) < βζ0 if β ≤ i2(t) ≤ α.

(6)

TORRES et al.: HYSTERESIS-BASED MECHANICAL STATE PROGRAMMING OF MEMS MIRRORS 349

Fig. 5. An illustration of the Preisach operator. The Preisach density functionis uniformly discretized with a discretization level of N .

Based on the outputs of the hysterons, P0 is divided intotwo regions. The boundary of the two regions is called thememory curve and denoted as ψ . The memory curve ψ fullyrepresents the input history and determines the output of thePreisach operator. Thus the initial memory curve, ψ0, can beadopted to rewrite the formulation of the Preisach operator as�P [i2(·);ψ0](t), and the system model is rewritten as

θ(t) = a · ATk

· �[i2(·);ψ0

](t). (7)

Practical model implementation often involves discretizationof the weight function μ to obtain a finite number of parame-ters [30]. At time n, the output of the discretized Preisachmodel is expressed as

�P [i2(·);ψ0](n) =N∑

i=1

N+1−i∑j=1

μi j si j (n)+ c0, (8)

where ui j is the weight for cell (i, j) on the Preisach plane,i = 1, 2, · · · , N , j = 1, 2, · · · , N − i + 1, N is the dis-cretization level, si j (n) is the signed area for cell (i, j), whichis fully determined by the input up to time n. {u}i j are themodel parameters and determine the phase transition-inducedforce �P . The weights {u}i j are identified by the experimentalhysteresis data. Fig. 5 shows an example of discretization ofPreisach density function. The parameters are usually eitherconstrained to be non-negative or non-positive such that thePreisach model can describe monotonic hysteresis.

D. The Algorithm for Programming the Pitch Angles

Let the initial memory curve be ψ0 and the pre-heatinginput be i0. The goal of programming the pitch angle state isto calculate a pulse id , such that if the input sequence i0 →id → i0 is applied, the angle is equal or close to θd :

θd = a · ATk

· �[i20 → i2d → i20 ;ψ0

]. (9)

The following algorithm is proposed to obtain the pulseinput id . The main idea of the algorithm is to obtain therequired pulse by iteratively calculating and updating the

estimated pulse input and the corresponding angle value.If θd = θ0, there is no need to apply an pulse input andid = i0; If θd > θ0 or θd < θ0, the algorithm can be describedas follows:

• Step 1:1) j := 0;2) Let i ( j ) = i0;3) Under the input sequence of pulse i ( j ),

i0 → i ( j ) → i0, compute the angle outputθ( j ) as

θ( j ) = a · ATk

· �[i20 → (i ( j ))2 → i20 ;ψ0

]. (10)

• Step 2:1) j := j + 1;2) Update the estimate of the desired pulse input as

i ( j ) = i ( j−1) + δ(θd − θ( j−1)), (11)where δ is a small positive constant;

3) Under the updated input sequence i0 → i ( j ) →i0, compute the angle output θ( j ) based on theformulation in Eq. (10);

4) If |θ( j ) − θd | ≤ �, � > 0 is an arbitrary smallconstant, then go to Step 3; otherwise go back toStep 2.

• Step 3: id := i ( j ), the current pulse is id and the inputsequence is i0 → id → i0, stop.

It is verified that this algorithm can converge to the correctinput, and the input sequence with a larger pulse results in alarger steady-state output. The proposed algorithm works forboth the case of θd > θ0 and the case of θd < θ0. If θd > θ0,then a positive pulse is required, id > i0; If θd < θ0, then thepulse is negative, id < i0.

It is noted that the proposed mechanical programmingalgorithm is not limited to the presented VO2 MEMS mirrorsystem. The proposed algorithm can be adapted for mechanicalprogramming of systems with hysteresis behaviors, such asMEMS mirrors and microactuators with different mechanismsand actuated by other smart materials.

IV. MODELING AND PROGRAMMING RESULTS

A. Model Identification

The experimental measurement of the quasi-static hysteresisbetween the pitch angle and the square of the current isobtained, as shown in Fig. 6. A triangular step sequenceof decreasing magnitude is applied as the input, as shownin Fig. 6(a). To ensure a steady state measurement, each stepis held for 5 seconds. The full range of the current is [0, 11]mA, and the full range of the pitch angle is [−0.35, 1.87] deg.The hysteresis curve exhibits non-monotonic behavior. Forexample, the pitch angle firstly decreases, then increases, andfinally decreases when the current is monotonically increasedfrom 0 to 11 mA.

The hysteresis model is identified based on the experimentaldata. More details of the identification algorithm can be foundin [32]. Fig. 6(b) shows the modeling performance for thehysteresis between the pitch angle and the current input.

350 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

Fig. 6. The (a) input sequence, (b) modeling performances, and(c) the modeling error of the hysteresis between the pitch angle and thesquared term of current input.

Fig. 6(c) shows the corresponding modeling error. The averageerror is 0.0146 deg. Considering that the overall pitch anglerange is 2.18 deg, the proposed model is considered to beable to accurately capture the non-monotonic hysteresis of themirror system.

B. Model Verification Under Pulsed Inputs

To verify that the model can effectively predict the mir-ror angle under current pulses, additional experiments areconducted. A randomly-chosen sequence of current pulses,as shown in Fig. 7(a), is applied to the MEMS mirror.The amplitude of the DC current for pre-heating the systemis 7.03 mA. Each pulse is held for 1 s while the intervalbetween pulses is 5 s. The pulse width is selected to belarger than the thermal time response of the actuator, wherein [32] it was demonstrated to be less than 15 ms determined

Fig. 7. (a) A current pulse input for model verification; (b) The pitch outputunder the current input and corresponding steady-state values; (c) The pitchangle verification performance; (d) The verification error of the pulse inputbased on the proposed model.

by the property of VO2. The corresponding measured pitchangles and steady-state values are provided in Fig. 7(b).Moreover, the figure also shows the creep behavior of thesystem that was reported in [32]. It is shown that the adopted

TORRES et al.: HYSTERESIS-BASED MECHANICAL STATE PROGRAMMING OF MEMS MIRRORS 351

Fig. 8. (a) The calculated current pulse input for programming pitch angles tofollow a triangular-shaped sequence; (b) The corresponding pitch output andsteady-state values under the applied input; (c) The pitch angle programmingperformance; (d) The programming error of the proposed scheme.

mirror is based on thermal mechanism and has relativelyslow responses comparing to micromirrors using electrostaticactuation. As discussed in [32], the slow responses are likelydue to the pseudo-creep effect caused by the added stressfrom the legs that are not actuated. With creep, the mirrorangles reach steady-state values in 3-5 seconds. The experi-mental measurements are compared with the model-predictedvalues. Fig. 7(c) shows the model verification performance

Fig. 9. (a) The calculated current pulse input for programming pitch angles tofollow a random sequence; (b) The corresponding pitch output and steady-statevalues under the applied input; (c) The pitch angle programming performance;(d) The programming error of the proposed scheme.

and Fig. 7(d) shows the corresponding verification error. Theaverage error is 0.0256 deg. The effectiveness of the proposedmodel is confirmed, and the model can accurately estimate thesystem output under random current pulses.

C. Programming Results

The proposed algorithm for programming arbitrary mirrorangles is experimentally validated. A desired angle sequence

352 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

with different values is prescribed. The values of the requiredcurrent input pulses are computed based on the algorithmprovided in Section III-C. The computed pulse sequence isapplied to the mirror system, and the experimental results ofthe steady-state angles are compared with the prescribed ones.

The programming of a triangular-shaped angle sequence isfirst conducted as a case study. The DC current for pre-heatingthe system is 7.03 mA. With this pre-heating current, the rangeof angle can be programmed is [0.03, 0.67] deg. The desiredangle sequence is shown in Fig. 8(c), which covers theoverall range of angles that can be programmed. Based on theproposed algorithm, the computed current pulse is providedin Fig. 8(a). The corresponding pitch angle trajectory andsteady-state values are shown in Fig. 8(b). Fig. 8(c) showsthe programming performance and Fig. 8(d) shows the corre-sponding programming error. The average programming erroris 0.0313 deg. The effectiveness of the proposed algorithmfor programming mirror states is confirmed since the error iscomparable to the modeling and estimation accuracy, over thepitch angle range of over 2 deg (Fig. 6(b)). It is found thatwhen the desired pitch sequence is increasing, the obtainedpitch values are larger than the desired values; when thedesired pitch sequence is decreasing, the obtained pitch valuesare smaller than the desired values. This is likely caused by thecreep effect. No clear trend is found between the magnitudesof the error and the pulse amplitudes.

Finally, the programming of a random angle sequence(shown in Fig. 9(c)), with 20 different states, is conducted.The DC current for pre-heating the system is also chosen to be7.03 mA. According to the proposed programming algorithm,the current pulse is computed and presented in Fig. 9(a). Thecorresponding pitch angle trajectory and steady-state valuesare shown in Fig. 9(b). Fig. 9(c) shows the programmingperformance and Fig. 9(d) shows the corresponding error. Theaverage programming error is 0.0287 deg. The effectivenessof the proposed algorithm for programming mirror statesis further strengthened since programming of random pitchangles is demonstrated with similar accuracy as in modelidentification. While the values of the programming error aresmall, the same trend as the programming of triangular-shapedangle sequence is noticed: when the desired pitch sequenceis increasing, the obtained pitch values are larger than thedesired values; when the desired pitch sequence is decreasing,the obtained pitch values are smaller than the desired values.

V. CONCLUSION AND FUTURE WORK

In this work a systematic approach to programming themechanical states of VO2-based MEMS mirrors was proposedby exploiting the hysteresis inherent in VO2. The programingof the output values was performed by applying input pulses.The non-monotonic hysteretic behavior was modeled using aPreisach operator in combination with a linear term, with anaverage modeling error of 0.0146 deg. The model was able topredict the pitch angle under pulsed inputs with an averageerror of 0.0256 deg. An iterative algorithm was developedto find the pulse values for achieving the desired outputsequence. The proposed approach was successfully verified

with experimental results on programming two pitch anglesequences that had different profiles. Although programmingthe pitch angle of a VO2 MEMS mirror was used as a casestudy, the approach is applicable to other types of MEMSdevices that exhibit hysteresis, for example, those actuated byother smart materials.

In future work, the study of fast mechanical programmingconsidering the creep effect will be considered. A possiblesolution is to model the creep effect of the system and incor-porate the analysis into the proposed programming algorithmto compute the pulse input. Another future direction is to pro-gram the mirror to follow trajectories rather than quasi-staticangles. To obtain dynamical programming of mirror angles,the proposed pulse algorithm cannot be directly adopted.Feedback control with external position sensors is needed.

ACKNOWLEDGMENT

The fabrication of the MEMS Mirror was partially done atthe Lurie Nanofabrication Facility at University of Michigan.The SEM image was taken at the Center for AdvancedMicroscopy at Michigan State University.

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David Torres (S’15–M’18) received the B.S. degreein electrical and computer engineering from theUniversity of Puerto Rico, Mayagüez, Puerto Rico,in 2012, and the Ph.D. degree in electrical and com-puter engineering from Michigan State University,East Lansing, MI, USA, in 2017. He is an Elec-tronics Engineer with the Sensors Directorate, AirForce Research Laboratories, Wright-Patterson AirForce Base, Dayton, OH, USA. His current researchinterests include design, fabrication, and characteri-zation of micro-mirror arrays actuators and control

of hysteretic systems. He received the Science, Mathematics and Researchfor Transformation Scholarship for Service Program from the Department ofDefense.

Jun Zhang (S’12–M’16) received the B.S. degreein automation from the University of Science andTechnology of China, Hefei, China, in 2011, and thePh.D. degree in electrical and computer engineeringfrom Michigan State University, East Lansing, MI,USA, in 2015. He is currently a Post-DoctoralScholar in electrical and computer engineering withthe University of California at San Diego, SanDiego, CA, USA. He received the Student BestPaper Competition Award at the ASME 2012 Con-ference on Smart Materials, Adaptive Structures and

Intelligent Systems, and the Best Conference Paper in Application Award atthe ASME 2013 Dynamic Systems and Control Conference, and was namedthe Electrical Engineering Outstanding Graduate Student at Michigan StateUniversity from 2014 to 2015.

His current research interests include modeling and control of smart mate-rials, nonlinear system modeling, control, and design of artificial muscles forrobotics.

354 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 27, NO. 2, APRIL 2018

Sarah Dooley received the B.S. degree in physicsfrom Xavier University in 2002, and the M.S. degreein electro-optics engineering from the University ofDayton in 2004. She is an Electronics Engineerwith the Sensors Directorate, Air Force ResearchLaboratory, Wright-Patterson Air Force Base. She iscurrently involved in the field of non-mechanicalbeam-steering, including the design, fabrication,and characterization of MEMS micro-mirror arrays.She also collaborates with several institutions on thedevelopment of novel plasmonic and liquid crystal

beam-steering techniques, MEMS optical switches, and MEMS frequencygeneration technology.

Xiaobo Tan (S’97–M’02–SM’11–F’17) receivedthe B.Eng. and M.Eng. degrees in automatic con-trol from Tsinghua University, Beijing, China,in 1995 and 1998, respectively, and the Ph.D. degreein electrical and computer engineering from theUniversity of Maryland at College Park, CollegePark, MD, USA, in 2002.

From 2002 to 2004, he was a Research Associatewith the Institute for Systems Research, Universityof Maryland at College Park. He joined the facultyof the Department of Electrical and Computer Engi-

neering, Michigan State University in 2004, where he is currently an MSUFoundation Professor. His research interests include modeling and controlof systems with hysteresis, electroactive polymer sensors and actuators,bio-inspired underwater robots and their application to environmental sensing,and soft robotics.

He has co-authored one book, Biomimetic Robotic Artificial Muscles, andover 90 journal papers and holds two U.S. patents. He was a recipient of theNSF CAREER Award in 2006, the MSU Teacher-Scholar Award in 2010,the MSU College of Engineering Withrow Distinguished Scholar Awardin 2018, and several best paper awards. He has served as the Program Chair ofthe 2011 International Conference on Advanced Robotics, the Finance Chairof the 2015 American Control Conference, and the General Chair of the2018 ASME Dynamic Systems and Control Conference. He has served asan Associate Editor/Technical Editor for the Automatica, the IEEE/ASMETRANSACTIONS ON MECHATRONICS, and the International Journal ofAdvanced Robotic Systems.

Nelson Sepúlveda (S’05–M’06–SM’11) receivedthe B.S. degree in electrical and computer engineer-ing from the University of Puerto Rico, Mayagüez,Puerto Rico, in 2001, and the M.S. and Ph.D.degrees in electrical and computer engineering fromMichigan State University (MSU), East Lansing, MI,USA, in 2002 and 2005, respectively. During hislast year of graduate school, he attended SandiaNational Laboratories as part of the fellowship fromthe Microsystems and Engineering Sciences Appli-cations Program.

In 2006, he joined the faculty of the Department of Electrical and Com-puter Engineering, University of Puerto Rico. He was a Visiting FacultyResearcher with Air Force Research Laboratories in 2006, 2007, 2013, and2014, the National Nanotechnology Infrastructure Network in 2008, and theCornell Center for Materials Research 2009, the last two being the NationalScience Foundation (NSF) funded centers at Cornell University, Ithaca, NY,USA. In 2011, he joined the faculty of the Department of Electrical andComputer Engineering, MSU, where he is currently an Assistant Professor.His current research interests include smart materials and the integration ofsuch microelectromechanical systems, with particular emphasis on vanadiumdioxide thin films and the use of the structural phase transition for thedevelopment of smart microactuators.

Dr. Sepúlveda received the NSF CAREER Award in 2010 and the MSUTeacher Scholar Award in 2015.

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