IOP PUBLISHING NANOTECHNOLOGY

Nanotechnology 19 (2008) 315501 (10pp)

A flexure-based electromagnetic linearactuatorTat Joo Teo, I-Ming Chen, Guilin Yang and Wei Lin

School of Mechanical and Aerospace Engineering, Nanyang Technological University, SingaporeandSingapore Institute of Manufacturing Technology, Agency for Science Technology and Research, Singapore

E-mail: tjteo@pmail.ntu.edu.sg/tjteo@SIMTech.a-star.edu.sg

AbstractThis paper introduces a novel nano-positioning actuator with large displacement and drivingforce, termed a flexure-based electromagnetic linear actuator (FELA). It mainly comprises anelectromagnetic driving scheme and flexure-supporting bearings that provide infinitepositioning resolution and highly repeatable motion. In this work, analytical modeling of theproposed electromagnetic scheme and flexure mechanism is presented. Solutions obtained fromeach model are evaluated by the experimental studies conducted on a FELA prototype. Thisprototype achieves a stroke of 4 mm with a positioning accuracy of ±10 nm. With direct forcecontrol, it generates various force profiles with a force–current ratio of 60 N A−1 and anaccuracy of ±0.3 N. Such capabilities make FELA a promising solution for realizing ultra-highprecision layer-over-layer fabrication in the nano-imprinting process.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Nano-imprint lithography (NIL) was first introduced in 1995as a low-cost mass-production process that delivers nano-sized features [1] based on the principle of mechanicalprinting. Using a hot embossing technique, NIL transfersnano-sized patterns onto a single-layer substrate to form animpression with sub-100 nm features. In fact, fabrication of10 nm (in width) features on a single-layer impression hasalready been well demonstrated [2]. Consequently, NIL hasbeen reported as a promising approach for semiconductor-component fabrication as conventional optical lithographyruns into technical limits when producing transistors andinterconnections of sizes below 45 nm due to the wavelengthlimitation and diffraction effects of the photons. Unfortunately,the requirements for large imprinting forces and elevatedtemperature have made NIL unsuitable for fabricating layer-over-layer impressions. Hence, NIL is only used to fabricatesub-100 nm features on single-layer substrates [3]. Thediscovery of step-and-flash imprint lithography (SFIL) in1999 successfully demonstrated multi-layer-interconnectionfabrication through such a nano-imprinting approach [4]. By

eliminating the heat needed for embossing and loweringthe imprinting force requirements [5], it was reported thatSFIL has successfully fabricated a well-aligned two-layerinterconnected impression with features of sub-100 nm inwidth and sub-micron height [6].

A SFIL system consists of two major components: amicro-resolution Z -stage that carries the template and a nano-alignment flexure stage that holds the substrate for imprintingprocess [6]. As illustrated in figure 1, the Z -stage stamps thetemplate onto the substrate, while the flexure stage, with twodegrees of freedom (DOF) in θx and θy orientation, passivelyaligns the substrate to the template. In the stamping process,bringing the template in contact with the substrate may requirea traveling range of a few millimeters in layer-over-layerfabrication. Consequently, a stepper motor and linear ball-screw with pitch size of a few microns are employed to drivethe Z -stage. However, such micro-resolution ball-screws limitthe overlay alignment resolution to ∼0.5 μm [4]. One majorreason for this limitation is the lack of an appropriate actuatorsuitable for driving a SFIL system. Such an actuator musthave positioning accuracy of nanometers for performing ultra-high alignment and several millimeters of travel for layer-over-

Preprint version to IOP Publishing Ltd 1

http://dx.doi.org/10.1088/0957-4484/19/31/315501mailto:tjteo@pmail.ntu.edu.sghttp://stacks.iop.org/Nano/19/315501

Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 1. (a) Conventional nano-imprinting lithography stage with(b) and (c) passive angular alignment strategies.

layer fabrication. It must also provide a continuous imprintingforce of ∼100 N, which is required in the SFIL process.However, initial studies recognized that existing actuatorshave limitations in meeting these stringent requirements. Inparticular:

(i) Piezoelectric (PZT) actuators, which are commonly usedfor nano-scale positioning, have limited strokes of up toseveral hundreds microns [7].

(ii) PZT-driven actuators that use a high-pitch screw actuatingshaft to achieve millimeters of displacement have poorrepeatability due to backlash and Coulomb friction.Others that use the magnetostrictive clamping method [8],the inchworm [9] or the impact driving technique [10]have low payload capacities, i.e. 2 mm, aforce–current ratio of ∼60 N A−1 and an actuating speed of�100 mm s−1. In this paper, a current–force analytical modelof a EDM and a static force–displacement analytical model ofthe FBM are presented. Experimental studies are conductedon the prototype of EDM and FBM to validate the accuracyof the established analytical models. These prototypes areassembled to form a complete FELA assembly, which will beimplemented with position and force control schemes to verifyits claimed capabilities.

2. Current–force modeling of EDM

2.1. Lorentz-force actuation

The magnitude of the Lorentz force is determine by themagnetic flux density, B , input current, i , and orientation of thefield and current vector. With a total length of coil, l, placed inthe magnetic field, it can be estimated by

F = i∫

dl × B. (1)

Based on equation (1), the current and force relationship isaffected by the total length of the coil and the magnetic fluxdensity within the effective air gap. In this analysis, the totallength of the coil is fixed and the magnetic flux density varieswithin the effective air gap. Assuming the current rate ofchange is very small, i.e. di/dt → 0, equation (1) suggeststhat an accurate current–force model requires a good predictionof the magnetic flux density along the X–Y plane (refer tofigure 3).

2.2. Magnetic field solution

For a current-free environment, the magnetic field within theair gap of such a configuration can be reduced to a scalar

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 3. A symmetric diagram of a DM configuration.

potential, Φ, via the Laplace equation [14]

∇2Φ = 0. (2)Assuming that the PMs are uniformly magnetized and themagnitude of the magnetic field in the z-axis is similar to that inthe x-axis. A 3D problem can be reduced to a 2D problem. Asshown in figure 3, a DM configuration is decomposed into fiveregions: Region I, half of the air gap of a DM configuration;Region II, air gap between PM-1 and the stator; Region III,half of the PM-1 of a DM configuration; Region IV, air gapbetween PM-2 and the stator; Region V, half of the PM-2 ofa DM configuration. Equation (2) can be reduced to a 2DDirichlet boundary-value problem. With appropriate boundaryconditions, the solution of the scalar potential that representseach region is obtained. Thus the 2D magnetic flux densitysolution of the air gap is derived as,

BTotalI (x, y) = −4μ0M

π

∑ 1(2n − 1) sin

[(2n − 1)π

2lx

]

×{

1

UIcosh

[(2n − 1)π

2l(y − c)

]

+ 1UII

cosh

[(2n − 1)π

2ly

]}(3)

where

UI = sinh[(2n − 1)π

2l(a − c)

]coth

[(2n − 1)π

2la

]

− cosh[(2n − 1)π

2l(a − c)

](4)

and

UII = sinh[(2n − 1)π

2lb

]coth

[(2n − 1)π

2l(b − c)

]

− cosh[(2n − 1)π

2lb

](5)

M is the magnetization and μ0 is the permeability in space.Based on equations (1) and (3), a complete current–forceanalytical model for the proposed EDM of a FELA can beestablished.

2.3. Experimental investigations

In this work, an EDM with a symmetrical DM configurationwas fabricated to validate the accuracy of the analytical results

obtained from equation (3) and the current–force model.In the prototype EDM, AWG24 wire (diameter 0.45 mm)is used based on the estimated operating temperature of120 ◦C and 2 A of input current to the FELA. Each DMconfiguration is formed by two rare-earth PMs (NdFeB typeN45M) with residue magnetic flux density of 1.33 T and amaximum operating temperature of 120 ◦C. The effectiveair gap within the DM configurations is designed to be11 mm to optimize the output force obtained from theEDM prototype. The experiment setup mainly comprisesa Hall-sensor probe (Lakeshore, model MFT-2903-VH) anda gaussmeter (Lakeshore, model 460) for measuring themagnetic flux density within the air gap of the DMconfiguration. On the other hand, the parameters used inequation (3) to obtain the field solution include M = 692.33 ×103 A m−1, μ0 = 4π×10−7 Wb A−1 m−1, a = 7.5×10−3 m,b = 18.5 × 10−3 m, c = 26 × 10−3 m, g = 3 × 10−3 mand l = 30 × 10−3 m. The numerical field solution obtainedfrom the finite-element simulation (ANSYS 10) is also usedfor comparison.

The magnetic flux density measured experimentally alongthe X–Y plane (refer to figure 3) is plotted against theresults obtained from equation (3) and the numerical analysis,respectively (figure 4). Comparisons between the experimentalresults (figure 4(a)) and the analytical results (figure 4(b)) showthat the magnetic flux leakage between 0 and 5 mm of theeffective air gap is well predicted by the analytical model.From 5 to 27 mm of the effective air gap, the differences ofmagnetic flux density between the experimental and analyticalresults are

Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 4. A comparison between the magnitudes of the magnetic flux density obtained from (a) experimental results, (b) the analytical modeland (c) numerical simulations.

an input current of 0.1 A. Based on such linear relationships,the prototype EDM is capable of generating an output forceof >100 N at 2 A. The analytical results obtained from theestablished current–force model were also plotted against theexperimental results (figure 5). Based on parametric analysisof the EDM, it shows that the analytical model predicts acurrent–force ratio of 7 N per 0.1 A to be generated from theEDM. The slight differences of 0.4 N can be explained by theforce-torque sensor used to conduct this experiment. To obtainthe actual force at the actuating direction without sensing theforces in the other directions, it is necessary to have a perfectmounting between the moving air-core coil and this six-axistransducer. However, it is difficult to ensure a perfect mountingin practice. Another reason is the difference between the lengthof coil assumed during the analytical modeling and the actuallength of the coil used in practice. Imperfect coiling of theair-core coil can lead to a smaller amount of coil being usedin practice, causing the EDM to generate a lower current–force ratio. Nevertheless, the proposed analytical model givesa well-predicted current–force relationship that is useful forrapid parametric analysis and design, and proof-of-conceptassessment of a EDM with DM configurations.

3. Force–displacement model

3.1. Pseudo-rigid-body modeling

A pseudo-rigid-body (PRB) modeling technique [15] is usedto describe the behavior of the force–deflection relationship ofthe flexure joints by modeling them as kinematic rotary jointswith torsional springs. The flexure joints of a FBM, which arethe unclamped portions of a shim, are described as a small-length flexure pivot. Consequently, the torque, T , generated at

Figure 5. Predicted current–force relationship from the proposedanalytical model versus the actual current–force experimental data.

the middle of the flexure length can be expressed as

T =(

E I

L

)� (6)

where � is the PRB deflection angle of the pivot, E I is theflexure rigidity and L is the length of the flexure pivot [16].

3.2. Force–displacement relationship

A FBM, which adopts a bi-stable mechanism design, can bedescribed as a pair of symmetrical linear spring configurationswith flexure joints, as shown in figure 6. Each of these linearspring configurations can be represented by a planar four-bar

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 6. (a) A bi-stable FBM is represented as (b) a symmetricalfour-bar linkage. (c) A schematic diagram of a four-bar linkage.

mechanism. Based on figure 6(c), forward kinematics analysisderives the velocity coefficients as (scalar form) and can beexpressed as

{θ̇3/θ̇2θ̇4/θ̇2

}=

{r2 sin(θ4 − θ2)[r3 sin(θ3 − θ4)]−1r2 sin(θ3 − θ2)[r4 sin(θ3 − θ4)]−1

}(7)

where θ2 is the primary variable and θ3 and θ4 are the secondaryvariables. The length of each rigid link is represented as r1, r2,r3 and r4.

In this analysis, the total virtual work, δWSYS, of a linearspring configuration based on the FBM is summarized as

δWSYS =1∑

i=1�Fi · δ �Zi +

4∑i=1

�Ti · δ �ψi (8)

where the first term represents the virtual work due to a drivingforce, �F , acting on the rigid link with a virtual displacement,δ �Z . The second term represents the virtual work due to thetorsional springs with �Ti = −k(ψ). The virtual deflection isexpressed as δψi = θi − θi(o), where θi(o) is the initial angle ofthe un-deflected torsional spring. Based on equations (6), (7)and the virtual work principle, δWSYS = 0, the static force–displacement relationship of the FBM is derived as

Fin = −16E I (sin−1[δx( L2 + r2)−1] − θ2o)

Pr2 sin{sin−1[δx( L2 + r2)−1]}(9)

where Fin is the input force required to achieve the desireddisplacement, δx , in the actuating direction.

3.3. Experimental investigations

In this work, a 100 mm × 70 mm × 70 mm (length × width×height) FBM is developed. Its mainly consists of foursymmetrical linear springs, which are used to hold thetranslating air-core coil of the EDM. Each linear spring has

Figure 7. A FELA prototype.

Figure 8. A comparison between the predicted stiffness of FBMfrom proposed analytical model and the actual stiffness obtainedfrom experiments.

flexure joints made from 0.08 mm thick stainless steel shims.The middle section of each shim is clamped with a pair of21 mm long clamping blocks, leaving a unclamped portionof 3 mm at both ends of the shims as short-length flexurejoints. Such an assembly forms a partially compliant bi-stable mechanism that becomes the main supporting structureof a FELA (figure 7) with an EDM embedded inside. ThisFBM is also used to validate the accuracy of the establishedforce–displacement analytical model. The experimental setupconsists of a THRUST DC linear amplifier (model TA-115,max. 48VDC, 8 A), an ATI F/T sensor (model Mini-40,max. 240 N, resolution 0.01 N) and a MicroE-Systems opticallinear encoder (model M3500Si, resolution 5 nm/count). Theforce relationship of the FBM is experimentally obtainedby moving the air-core coil to a designated position whilerecording the force and position from the F/T sensor andencoder, respectively. Both experimental and analytical resultsare plotted in figure 8. The analytical results obtainedfrom the established model indicate an actuating stiffness of1.02 N mm−1 for the FBM. These results almost tally withexperimental results as 1 N mm−1 of actuating stiffness isregistered from the actual FBM. Consequently, the established

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Nanotechnology 19 (2008) 315501 T J Teo et al

analytical model has been shown to be suitable for predictingthe static force–displacement relationship of the FBM.

4. Nanometric positioning control

A position control scheme is implemented on the FELA tovalidate its performances in terms of positioning accuracy,displacement range and actuating speed. This closed-loopcontrol is mainly governed by a proportional, integral anddifferential (PID) controller, and a position feedback as shownin figure 9(a).

4.1. Dynamic modeling

A FELA is treated as a linear mass-spring-damper system asillustrated in figure 9(b). In this system, the mass, m, is theweight of the moving air-core coil and the stiffness of theflexure joints is represented by spring stiffness, k. Unlikeconventional flexure mechanisms, damping of this system iscaused by a force generated through the induced eddy currentfrom the relative motion of the moving core and the PMs.Hence, the transfer function of FELA, G F (s), is

G F (s) = Xa(s)Fc(s)

= 1ms2 + bs + k (10)

where Fc(s) is the command force and Xa(s) is the actualposition. On the other hand, the transfer function of aconventional PID controller, Gc(s), is

Gc(s) = Fc(s)E(s)

= Kp(

1 + 1Tis

+ Tds)

(11)

where E(s) is the error between the desired and actual values,Kp is the proportional gain, Ti is the integral time and Td isthe derivative time. Consequently, the transfer function of theentire closed-loop control system is

Xa(s)

Xd(s)= Kc Kp(Tds

2 + s + T −1i )ms3+ (b+ KcKpTd)s2 + (k + Kc Kp)s +KcKpT −1i

(12)where Xd(s) is the desired position and Kc is the compensationgain, which accounts for the quantization and amplifier gain ofthe open-loop system.

4.2. Controller design

To determine the control parameters, equation (12) is comparedagainst the transfer function of a standard third-order system soas to formulate the relevant equations that represent Kp, Ti andTd. In this work, the transfer function of a standard third-ordersystem is expressed as

G(s) = ω2n

(s + γ )(s2 + 2ζωns + ω2n)(13)

where ωn represents the undamped natural frequency of thesystem and ζ represents the damping ratio of the system. Thetransfer function of a standard third-order system is derived byadding an additional real pole, γ , to the transfer function of a

Figure 9. (a) Block diagram of a proposed position control system.(b) Schematic diagram of a mass-spring-damper system.

standard second-order system. The value of the additional polemust be at least five times the value of the undamped naturalfrequency to ensure that it has a minimum effect on the systemcharacteristic. Hence, a comparison between the coefficientsof equations (12) and (13) yields

Kp = m(ω2n + 2ζωnγ )− k

Kc(14)

Ti = Kc Kpmγω2n

(15)

Td = m(γ + 2ζωn)− bKc Kp

. (16)

In this work, the rise time, Tr, and the damping ratio areused to determine the system performance. Using these twoparameters, the undamped natural frequency is obtained using

ωn = tan−1[ζ−1(√1 − ζ 2)]

Tr√

1 − ζ 2 . (17)

Prediction of the PID control parameters throughequations (14)–(16) requires the values of the mass, springand damper. In this work, the MATLAB System IdentificationToolbox is used to identify these values based on the outputstep response of the FELA. Initially, a series of step inputswas first given to the FELA to obtain the output step responseof the open-loop system. These step inputs are the commandforce, which include 1.986 23×10−7 N, 1.986 23×10−6 N and9.931 16×10−6 N. Subsequently, the output response obtainedfrom each step input is recorded at a sampling time of 1 ms.Both input and output data are fed into the MATLAB toolboxto estimate a transfer function of the open-loop system. As aresult, the estimated transfer function, Gest(s), is given as

Gest(s) = 279.9(1 × 10−6)s2 + (2.254 × 10−3)s + 1 (18)

where 279.9, 1 × 10−6, 2.254 × 10−3 and 1 represent theratio between compensation gain, mass, damping and springstiffness, respectively.

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 10. Positioning accuracy of the end-effector at a neutralposition.

Figure 11. Repetitive 20 nm steps generated from FELA with aclosed-loop positioning control.

4.3. Results

For position control, a rise time of 0.4 ms and damping ratioof 0.6 is used to determine the value of natural frequency,which is in turn used to estimate the PID control parametersusing equations (14)–(17). As a result, Kp = 0.2064, Ti =0.474 91 and Td = 0.271 93 are obtained. At this stage it isnoticeable that the Kp of 0.2064 is lower than the Kp valuesused on other conventional electromechanical modules. Inaddition, it is observed that higher Kp values cause the FELAto oscillate. This is because a moving air-core coil, whichhas a low moment of inertia, requires low Kp values to avoidlarge overshoot in the transient response or uncontrollableoscillations. Subsequently, these PID control parameter valueswere input to the PID controller, which is written in the FPGAenvironment via a National Instrument (NI) FPGA controllercard (model PCI-7833R, max. 3 Mgates, processing speed25 ns/command). This FPGA controller allows the PID servo-loop to run at 10 kHz, while the trajectory generator runs inthe NI LabVIEW environment with a control frequency of1 kHz. The rest of the hardware includes an industrial PCwith a P4 processor, a Trust 48 V DC linear amplifier and

Figure 12. FELA performs ±2 mm of a large displacement strokewith high accuracy demonstrated at both ends of the stroke.

Figure 13. Positional accuracy of the end-effector at 2 mm.

a MicroE-Systems optical linear encoder with a resolution of5 nm/count.

A Renishaw laser interferometer (model RLE10) isemployed to verify the smallest and largest achievable steps,accuracy and repeatability of the FELA. With the PID servo-control, a positioning accuracy of ±10 nm was obtained atthe end-effector of the FELA (figure 10). A 20 nm repetitivestep is performed by the FELA and is plotted in figure 11.It shows a positioning accuracy of ±10 nm at every step andvalidates that the smallest achievable step from FELA is 20 nm.The maximum stroke of the FELA is performed and plottedin figure 12. It shows that the FELA achieves a ±2 mmstep and settles quickly after 0.7 s. With a constant error of50 nm, the positioning accuracy at 2 mm is ±10 nm peak-to-peak (figure 13). To verify the repeatability of FELA, fiverepeat runs to each targeted position, i.e. 5 μm and 2 mm, areconducted. For the targeted position at 5 μm, all five runsperformed by the FELA fall within ±1.5σ (figure 14). With aconstant error of 24 nm, this shows a positioning repeatabilityof ±10 nm. As for the targeted position at 2 mm, all fiveruns performed by the FELA fall within ±1.5σ and indicate apositioning repeatability of ±20 nm (figure 15) with a constanterror of 72 nm. All constant errors obtained at the end-effector

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 14. Positional repeatabilities at a targeted position of 5 μm.

Figure 15. Positional repeatabilities at a targeted position of 2 mm.

can be compensated through error-mapping techniques. Mostimportantly, the positioning accuracy of FELA is very muchlimited by the encoder resolution of 5 nm/count. Based onfigure 10, the accuracy of ±2 counts suggests that an encoderwith a higher resolution will further improve the positioningaccuracy of the FELA.

5. Direct force control

One of the most important features of this nano-positioningactuator is its ability to achieve direct force control for nano-imprinting tasks. Using a PID control scheme, the ATIforce/torque (F/T ) sensor becomes the main feedback sourcefor the PID controller when the FELA operates under the directforce control mode (figure 16(a)). In this mode, the analogsignal from the F/T sensor is directly fed into the FPGAcontroller to achieve a signal-processing frequency of 100 kHz.

5.1. Dynamic modeling

In a direct force control mode, the FELA and the F/Tsensor become an integrated system as shown in figure 16(b).In this work, the F/T sensor is treated as a linear spring-damper where the spring stiffness, ks, represents the stiffness

Figure 16. (a) Block diagram of the proposed direct force controlsystem. (b) Schematic diagram of a mass-spring-damper system withan additional pair of spring-dampers representing the force sensor.

of the strain gauge within the F/T sensor and the damper,bs, represents the friction between the F/T sensor, and theworkpiece. Hence, the transfer function of the F/T sensor is

Xa = Fa(s)ks

(19)

where Fa(s) is the actual force detected from the end-pointof the F/T sensor. Consequently, the transfer function theintegrated system is

Fa(s)

Fc(s)= ks

ms2 + (b + bs)s + k + ks . (20)

In the case of direct force control, the transfer function of theentire closed-loop control system becomes

Fa(s)

Fd(s)= Kc Kpks(Tds

2 + s + T −1i )ms3+(bt +KcKpTd)s2+(kt + Kc Kp)s+KcKpT −1i

(21)where Fd(s) is the desired force, bt = b + bs and kt = k + ks.

5.2. Controller design

A comparison between the coefficients of equations (13)and (21) yields

Kp = m(ω2n + 2ζωnγ )− kt

Kc(22)

Ti = Kc Kpmγω2n

(23)

Td = m(γ + 2ζωn)− btKc Kp

. (24)

A set of command forces, which includes 9.931 16×10−6,1.986 23×10−5 and 1.390 36×10−4, is given to the integratedsystem and a set of output forces is recorded. These sets ofdata are fed into the MATLAB System Identification Toolboxand the estimated transfer function is

Gest(s) = 4.145 × 105

(2.9591 × 10−5)s2 + (89.426 × 10−3)s + 1 (25)

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 17. A 10 N force profile generated by FELA with ±0.15 N of accuracy.

Figure 18. A 60 N force profile generated by FELA with ±0.3 N of accuracy.

where 4.145 × 105, 2.9591 × 10−5, 89.426 × 10−3 and 1represent the ratio between compensation gain, m, bs and ks,respectively.

5.3. Results

For direct force control, the settling time of the transientresponse is more crucial than the rise time. This is because thetime to reach steady-state is more important in a nano-imprintprocess. Hence, a settling time of 0.5 ms and a damping ratioof 0.6 are used to estimate the PID control parameters usingequations (22)–(24). As a result, Kp = 0.2686, Ti = 0.060 39

and Td = 0.037 40 are obtained from the analyses. These PIDparameters are used by the PID controller when performinga direct force control. A 10 N force profile generated by theFELA is plotted in figure 17. It shows that FELA holds at 10 Nfor about 40 s with an accuracy of ±0.15 N. A 60 N forceprofile generated by the FELA is plotted in figure 18 with anaccuracy of ±0.3 N. In both cases, a simple PID scheme can beemployed due to the linearity between the input current and theoutput force. With a DM configuration, a current of less than1 A is used to generate the 60 N profile. Low current ensureslow heat generation and low thermal expansion.

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Nanotechnology 19 (2008) 315501 T J Teo et al

Figure 19. Changes in temperature during 60 N force profiling.

A thermal sensor from Pico Technology is used torecord the heat generated from the air-core coil for everysecond during the force control operation. Figure 19 plotsthe measured temperature when the FELA is generating acontinuous thrust force of 60 N for 15 min. It shows that thetemperature rises from 22 to 32 ◦C at a rate of 0.0173 ◦C s−1and saturates at 32 ◦C after 11 min. Hence, a 60 N profileforce control has a maximum temperature increase of 10 ◦C.With the air-coil holder length of 44 mm and the coefficientof thermal expansion of 24 × 10−6 ◦C−1, any 60 N profilingoperation that takes less than 11 min requires compensationof 18.2688 nm s−1 in the actuating (x-axis) direction. For60 N profiling operations that require more than 11 min,a maximum length extension of 10.56 μm in the actuating(x-axis) direction must be compensated. Such temperaturecompensations are additional means for ensuring the accuracyof a feature’s depth during a direct force control imprintingoperation.

6. Conclusions

This paper introduced a novel nano-positioning actuator,termed a FELA, which mainly comprises an electromagneticmoving air-core coil which is supported by the flexurebearings. Analytical modeling of the magnetic field andstatic current–force relationship of the electromagnetic drivingcomponent was discussed. Experimental investigations wereconducted and validated the accuracy of both proposedanalytical models. In addition, a static force–displacementanalytical model of the flexure bearings was derived and itsaccuracy also verified experimentally. Dynamic analyses of theFELA were conducted analytically and experimentally for thepurpose of designing a PID controller to realize both positionand direct force servo-controls. Detailed control modeling wasdiscussed to estimate the appropriate PID control parametersfor both control schemes. Based on the position servo-control, FELA achieves a positioning accuracy of ±10 nm(limited by the encoder resolutions) over a displacement of

4 mm. It achieves a smallest output step of 20 nm and apositioning repeatability of ±1.5σ . With a direct force control,this compact-sized FELA is capable of generating any forceprofile with a continuous thrust force of 60 N A−1. Hence,FELA is useful for realizing the stamping of templates ina nano-imprinting process. Consequently, a FELA, whichoffers nanometric accuracy, millimeters of strokes and largecontinuous force offers a promising solution for assistingcurrent SFIL systems in achieving high alignment accuracy inlayer-over-layer fabrication.

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http://dx.doi.org/10.1063/1.114851http://dx.doi.org/10.1016/S0167-9317(96)00097-4http://dx.doi.org/10.1063/1.118625http://dx.doi.org/10.1002/(SICI)1521-3757(19980302)110:53.0.CO;2-Xhttp://dx.doi.org/10.1016/S0141-6359(01)00068-Xhttp://dx.doi.org/10.1016/S0141-6359(02)00147-2http://dx.doi.org/10.1115/1.2919359http://dx.doi.org/10.1115/1.2826101

1. Introduction2. Current--force modeling of EDM2.1. Lorentz-force actuation2.2. Magnetic field solution2.3. Experimental investigations

3. Force--displacement model3.1. Pseudo-rigid-body modeling3.2. Force--displacement relationship3.3. Experimental investigations

4. Nanometric positioning control4.1. Dynamic modeling4.2. Controller design4.3. Results

5. Direct force control5.1. Dynamic modeling5.2. Controller design5.3. Results

6. ConclusionsReferences