RESEARCH PAPER

Hybrid linear microactuators and their control modelsfor mirror shape correction

Kirill Shcheglov & Xiaoning Jiang & Risaku Toda &

Zensheu Chang & Eui-Hyeok Yang

Received: 16 January 2009 /Revised: 11 May 2009 /Accepted: 20 May 2009# Springer-Verlag 2009

Abstract Future space-based imaging systems demandultra-lightweight mirrors, which would involve a largenumber of actuators to provide the needed surfacecorrection. These lightweight actuators are required to beintegrated with the mirrors to avoid a significant increase inoverall areal mass density. This paper presents thefabrication and testing of a linear microactuator and themodeling of an actuated mirror composed of such light-weight actuators. The linear microactuator is driven by acombination of a piezoelectric actuator block and electro-static comb drive units. A full nonlinear optimization modelof a mirror lattice was developed to simulate a lightweightprimary with embedded microactuators, which allows foran arbitrarily connected lattice with connector elementshaving an arbitrary stiffness and actuation response. Themodeling yielded a high precision estimation of the mirrorshape correction.

Keywords Linear actuator . Adaptive optics . Large stroke .

Bulk-micromachining . Active shape correction .

Segmented mirror

1 Introduction

The application of lightweight (<1 kg/m2) apertures tospace-based imaging will enable substantial performancegains for future space missions [1]. Previously, inflatablestructures using flexible polymeric membranes [2] andnanolaminate-based rigid-shell mirrors [3] were investi-gated. Practical aperture systems could involve segmentedmirrors whose large surface errors are actively correctedby embedded actuators [1]. A key element of the successof this technology relies heavily on the ability to developactuators that are lightweight and small to be integratedwith the mirrors, in order to reduce the overall areal massdensity of the telescope system. Several MEMS-basedlinear actuators with electrostatic [4–8] and thermo-elasticlinks [5] were reported. While these previously reportedMEMS actuators represent a significant improvement inthe area, electrostatic clutching requires continuouslysupplied voltage to hold actuator position, and ifambient heating or cooling occurs, thermal actuatorslinks may be randomly activated. Presented in this paperis an updated version of the authors’ recently developedself-latched microactuator [9], which contains furthermodeling and analysis of the actuator characteristics todetermine the feasibility for the active mirror application.Also presented is the modeling of the mirror shapecorrection using these embedded microactuators. A mirrormodel for general actuated lattice structures was formu-lated using a direct numerical optimization method.

J. Micro-Nano Mech.DOI 10.1007/s12213-009-0017-2

K. ShcheglovSensors in Motion Inc.,4858 Lincoln Ave #3,Los Angeles, CA 90042, USA

X. JiangTRS Technologies, Inc.,State College, PA, USA

R. Toda : Z. ChangJet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive,Pasadena, CA 91109, USA

E.-H. Yang (*)Stevens Institute of Technology, Castle Point on the Hudson,Hoboken, NJ 07030, USAe-mail: eyang@stevens.edu

2 Microactuator: modeling, fabricationand characterization

The actuator consists of two comb drive units, a slider, arail substrate and a miniaturized PMN-PT (lead magnesiumniobate-lead titanate, or Pb(Mg1/3Nb2/3)(1−x)TixO3) singlecrystal piezoelectric actuator-block. A schematic of thelinear actuator is shown in Fig. 1. The actuator is driven bya combination of electrostatic comb drive actuator units anda laterally placed PMN-PT actuator block (Fig. 2). Thedimensions of the PMN-PT actuator block are about2 mm×0.5 mm×4 mm, and the maximum stroke can beobtained along the 4 mm direction. The intertwining U-

letter shaped comb drive unit design is intended to enhancestability of the slider motion [9]. During the operationcycle, the slider is gripped by at least four clutches at atime; also, the slider is confined to linear motions only,mitigating undesired slider tilt and drag friction. The slideris gripped and its position is maintained when power isturned off, which is made possible by pre-stressing tetherbeams during the assembly. Figure 3 illustrates theoperation sequence of the actuator [9]. In this figure, instep (1), unit A is released by actuating the comb drivewhile the unit B remains clutched, whereas in step (2), theunit B and the slider is laterally pulled toward the right by aPMN-PT stack actuator. The unit A is clutched while theunit B is released in step (3), and subsequently, the unit B ispushed back by the PMN-PT in step (4). By repeating theactuation sequence for many times, large cumulative strokeis achieved. The detailed fabrication process is previouslyreported [9]. The comb drive structure is designed “initiallyunengaged” as shown in Fig. 4(a), and then engaged by theslider insertion, thereby narrowing the comb-tooth-gap toapproximately 1 µm (Fig. 4(b)) [9]. Once the slider isinserted, tethers are displaced by approximately 10 µm and,therefore, grip the slider without external power since theyare pre-stressed. The clamps are electrostatically pulledaway to release the slider by applying voltage to the combdrive.

A Finite Element Model (FEM) analysis was performedon the tethers in the clamp structure to estimate theclamping force. The commercially available ANSYSWorkbench finite element tool was used to create theFEM model. The tethers and clamp structure wereconstructed by using hexahedral (brick) elements. Figure 5shows a modeled geometry for the bending of the actuatortether. The estimated bending force on the tethers caused by10 μm displacement of the clamp during the slider insertionprocess is approximately 25 mN. On the other hand, therepulsion of the 10-µm wide tether perpetually pushes theslider with the force of 25 mN. Before the slider insertion,the comb gap is approximately 5 µm and the electrostaticforce is negligible. Once the slider is inserted, the comb gapnarrows to approximately 1 µm (1.2 µm was used in thecalculation) and the electrostatic force is significantlyincreased. For example, the estimated electrostatic force isapproximately 35 mN, by applying 200 V to the combdrive, which exceeds the estimated tether bending force of25 mN.

Individual parts were fabricated separately and manuallyassembled. The PMN-PT single crystal actuator block wasfabricated by stacking ~20 layers of of 6.5 mm×2 mm×0.15 mm PMN-PT plates and metal shims using epoxy.Figure 2(a) and (b) show images of a 6.5 mm×2 mm×4 mm PMN-PT stack. Figure 2(c) is a cross-section of aminiaturized stack right after dicing, where the PMN-PT

Clutch

Tether beam

Comb drive

Lid (glass)

Au wires

Slider

PMN-PT actuator-block

Driver Unit

Rail substratea

b

Fig. 1 Actuator schematic. The actuator is driven by a combination ofthe electrostatic comb drive and piezoelectric actuators. The combdrive unit is fabricated on an SOI wafer with a 100-µm-thick devicelayer to increase both the stiffness and the electrostatic force. Byapplying voltage to the comb drive, the clamps are electrostaticallypulled away to release the slider. a Overall structure b Driver unit

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layer and metal shim can be clearly observed withoutbonding defects between layers. Shim tabs were solderedfor lead wires attachment. The stack was then mounted ontoa dicing substrate using low temperature wax and subse-quently diced using a Computer Numerical Control (CNC)machine tool to form six 2 mm×0.5 mm×4 mm PMN-PTactuator blocks (Fig. 2(d)). Displacement of all actuator-blocks was measured using a Linear Variable DisplacementTransducer (LVDT) system under a driving voltage of 0–150 V at room temperature (Fig. 2(e)). The curved linerepresents the measured results with a fitted straight line,from which the average effective d33 (piezoelectric coeffi-cient) of the piezo material used in miniaturized actuators

was calculated approximately to be 1500 pC/N, which isslightly lower than that of bulk PMN-PT crystal resultingfrom the epoxy clamping effect. The hysteresis observedwas about 10%, which is comparible to that of normalpiezoelectric actuators (10%–20%). The average stroke ofthe actuator-blocks composed of 19–21 layers of PMN-PTthin plates (0.15 mm thick) is about 4 µm. The strokeresolution is depending on the minimum applied drivingvoltage. For example, a 26 nm stroke can be obtained byapplying 1 V.

The finished PMN-PT actuator-blocks were bonded tothe finished silicon components for final characteriza-tion. The fabrication process of the silicon components

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0 50 100 150 200

Driving Voltage (V)

Dis

plac

emen

t (um

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Fig. 2 PMN-PT single crystalstacks. a and b 6.5 mm×2 mm×4 mm stacks c Cross-sectionof a miniaturized actuator afterdicing d Miniaturized actuators.e Displacement vs. drivingvoltage

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has been reported previously [9]. A rail substrate wasmade by attaching side-rails to the base plate by epoxyadhesive. The slider was fabricated by slicing the siliconwafer using a dicing saw. After completing the fabricationof the actuator components, the driver units were mountedon the rail substrate. The slider was manually insertedbetween clamps using a probe needle. Finally, a PMN-PTactuator-block and a lid were attached using epoxyadhesive, and the assembled structure was wire-bonded.Figure 6 shows the finished device, assembled andmounted.

Actuator operation was tested using a LabVIEW-basedsetup consisting of interfaced power relays and powersupplies; the voltages applied to the comb drive and thepiezoelectric-stack actuators were 200 V and 20 V, respec-

(0) Unpowered latching mode

(1) Unit A release

(2) Unit B laterally moved

Unit A

Unit B

Fig. 3 Operation principle of the microactuator. By repeating theindividual actuation sequences, a large cumulative stroke is achieved.The step increment resolution can be adjusted by controlling thevoltage applied to the actuator block. This actuator is capable of zeropower latching; the slider is clutched and its position is maintainedwhen power is turned off

(3) Unit A clutched

(4) Unit B released; 1-cycle actuation completed

Fig. 3 (continued)

Tethers

Combdrivearrays

Clutch

Engaged comb Stopper

a

b

Fig. 4 Microscope images of clamp structures and comb drive arrays.a Before slider insertion b Engaged comb teeth after slider insertion

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tively. The cumulative stroke after 200-cycle actuation was178 µm, with an operation speed of approximately 1 cycle persecond. Table 1 summarizes the linear actuator performance.

3 Structure model of actuated mirror

The design of a segmented large aperture adaptive mirror is achallenging task. A number of system requirements must bemet, such as the actuation rate, the best achievable surfacefigure, performance over temperature and vibration, existenceof persistent structure vibration modes, controllability andinfluence functions, and other important system parameters.The ability to meet such requirements can be investigatedusing a lumped element model described below relating thestructure response to the actuation of individual actuatorelements. The model describes an arbitrarily connectedmechanical structure containing microactuators, mirrors, andflexure beams. Such a structure is quite general and candescribe the bulk of real world implementations of a largeaperture segmented adaptive mirror.

The model can be used to determine the mirrorconfiguration for an arbitrary input set of microactuatordisplacements, as well as determine the micro-actuatorcontrol parameters required for achieving the desired mirrorconfiguration. Dynamic behavior of the primary mirrorsuch as resonance frequencies and response functions canbe investigated as well. In the current effort, an exemplarylattice structure envisioned to support the mirror segmentswas built and analyzed to investigate the use of the above

Tethers

Clamp

Boundary Conditions: The ends of the four tethers were fixed

Fig. 5 FEM model of a clamp structures and four tethers attached.The contour plot shows the results of analysis for bending and rotationof tethers. Location of the maximum stress is clearly indicated. Thedeformed shape of the tethers was exaggerated for clarity

Slider

Au wires

Comb unit

Driver A Driver B

Clutch

Fig. 6 Image of a fabricated inner component of a linear actuator afterassembly and wire bonding

Table 1 Measured actuator performance

Target Demonstrated

Max. Freq. ~1 kHz 20-cycle/s a

Stroke >1 mm 178 µm @ 200-cycle b

Resolution <30 nm 50 nm c

Force >30 mN 48 mN d

Power 100 µW 0 W when latched

Size ~10 mm3 14×7×0.6

Mass 100 mg

a The higher-speed actuation (>20 Hz/cycle) could not be demonstrat-ed due to the frequency limit of the mechanical relay used forsupplying electrical AC signal to actuators. In principle, the actuatorstructure with PMN-PT and comb units can move at frequenciesexceeding 1 kHz.b The stroke of our actuator is limited only by the slier length andimposed force.c The measured resolution was limited by the image quality for imageprocessing. Actual resolution (minimum step size) is expected to bebetter.d The clamping force was modeled using ANSYS.

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described actuator in such an application. Of particularconcern was the limited maximum actuation force that theactuator was able to provide as well as the limited“holding” force (tens of mN). The modeling has shownthe with the stiff lattice structure we have described it ispossible to achieve large displacements while not exceedingthe maximum load on any particular actuator.

The model contains two types of elements: nodes andconnectors. Nodes are conceptual points in space and/orinfinitely stiff junction elements attached to connectors.Figure 7(a) shows a lattice structure consisting of theseelements, and a close-up of a small portion of it containingthree adjacent nodes. Such adjacent sets of three nodes inthe top layer are envisioned as being kinematic mountpoints for hexagonal mirror segments filling the mirror

aperture (Fig. 7(b)). Connectors are rod-like elements thatspan two nodes. Connectors have an arbitrary stiffness (a 6DOF spring constant) and an actuation capability which caneither be force-driven or displacement driven. Although themodel supports an arbitrary nonlinear actuator response(such as piezoelectric), in the present implementation eachconnector is assumed to be an actuator with the followingsimple energy function Eact ¼ k rj j þΔrð Þ2 where k repre-sents the spring constant of the connector in the axialdirection, |r| is the length distance between connectorendpoints, andΔr is the inchworm displacement. The responseof the structure to applied stimulus was calculated by fixing the“driven” parameters to their prescribed values and minimizingthe total structure energy with respect to the rest of theparameters. A simple elastic tensile stiffness was assumed for

(a)

(b)

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50

X50

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50

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ConnectorsSprings

Actuators

Fig. 7 Lattice and mirror struc-tures. a The modeled latticestructure and a schematic of asmall section showing the nodeand connector elements.b Supporting hexagonal mirrorsegments with the modeledstructure

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each connector. The bending and torsional stiffnesses were setto zero, corresponding to a frictionless hinged attachment ateach end. The three rotation angles for each node weretherefore not included into the calculation. The energyminimization method was used to calculate the structureactuation and response by numerically minimizing the totalstructure energy with respect to the sought parameters.

The structure model developed was a stiff latticesupported at three symmetric points. The dimensions (incentimeters) were chosen to roughly correspond to astructure for a segmented mirror with 1 foot segments.The stiffness of the effective connector spring constant waschosen to mimic a realistic light rod-like element made of atypical metal-like material (Elastic modulus in the 100 GParange; For reference, 7075 Aluminum has an elasticmodulus of 72 GPa, 6Al-4 V Titanium−115 GP, 304Stainless−200 GPa), approximately 20 cm in length and

10 mm2 in cross-section. The resulting stiffness wascalculated to be 107N/m or 1010 dyne/cm. For a nanometerstep, the maximum force on all actuators is around 1 mN.However if larger step sizes are desired, the stiffness of theconnector must be reduced proportionally, i.e., for a 1 μmmaximum step size the stiffness must be reduced by a factorof a thousand. This can be done by designing the connectorshape to have a reduced stiffness, such as by machiningflexures into a portion of it. For instance, with the stiffnessof 105N/m, the actuator with 10 mN force can have a100 nm maximum step size. The model lends itself both tocomputing the structure response to a set of control inputs(such as actuator displacements or voltages applied topiezoelectric elements), as well as to computing therequired command inputs to achieve a prescribed shape.

Figure 8 shows the response of the structure to acalculated set of actuator displacements achieving the same

Fig. 8 Response of the structureto a set of minimum energycommands calculated for thesame prescribed shape. aSegment displacement and bactuator force in dynes

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structure shape while forces on the actuators are nearlynegligible. Figure 9 shows structure control of the top layerof the structure to the Zernike modes. A Zernike mode wasconstructed on a lattice of top layer structure nodes. Theother nodes were required to remain in place. Actuatordisplacement commands were calculated and applied.Subsequently, the structure response to the commands wascomputed. The difference between the target and theachieved control result is dominated only by the round-offerror. The mirror modeling results described above showthat the microactuators reported in this paper can correct thecurvature and deformation of future segmented mirrors.

4 Conclusions

A self-latched linear actuator has been fabricated and tested,and a mirror lattice structure actuated using embedded

microactuators has been modeled. The measured cumula-tive stroke of the actuator after a 200-cycle actuation is178 µm. Further development is required to analyze theactuator push force, increase operation speed, improvelinearity and reliability, and improve packaging technique.A mirror model for general actuated lattice structures hasbeen built using a direct numerical optimization model. Themodel has yielded an arbitrarily connected lattice withconnector elements having an arbitrary stiffness andactuation response. The tested actuator performance andthe mirror modeling results show that the developedmicroactuators can correct the curvature and deformationof future segmented mirrors. The current form of the linearmicroactuator may be susceptible to lateral force and launchload. For practical applications in the future, the actuatortechnology described in this paper has to be furtherdeveloped to be more reliable for applications on the mirrorsystem, consisting of actuators and backing structures.

Fig. 9 Example of structure control: the top layer nodes required to move in the vertical direction to match Zernike modes composed over theappropriate grid

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Acknowledgement The research described in this paper waspartially carried out under Research and Technology Developmentprogram at the Jet Propulsion Laboratory, California Institute ofTechnology under a contract with the National Aeronautics and SpaceAdministration.

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