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Topology Optimal Compliant Microgripper∗

Shyh-Chour HUANG∗∗, Chi-Ming LEE∗∗, Chien-Ching CHIU∗∗ and Wei-Liang CHEN∗∗

This study proposes a MEMS microgripper design based on a compliant mechanism,utilizing the multi-input method to obtain greater output force and displacement. The com-pliant mechanism is very effective because the mechanism has flexible pieces that transmitforce or deliver motion. The design domain is formed by the ground structure parameteri-zation of the optimal topology. The goal is to obtain the optimal topology layout throughcomputer simulation. This study combines insights from the topology optimization of thecompliant mechanism and the piezoelectric microactuator to design a microgripper and toanalyze outputs and displacement with different parameters under topology optimization.

Key Words: Microgripper, MEMS, Topology Optimization, Compliant Mechanism

1. Introduction

In recent years, Micro-Electro-Mechanical System(MEMS) is an important engineering field involving elec-trical and mechanical functions in hybrid microscaledevices in the range of several millimeters to severalnanometers.

The potential advantage of miniaturization usingMEMS technology lies in its ability to scale down macrocomponents into mass-produced, compact, and unitizedmicro system devices. MEMS manufacturing are supe-rior to conventional manufacturing because MEMS prod-ucts have superior sensing and actuating capabilities. Atthe moment, MEMS devices on the market include pres-sure sensors, accelerometers, inkjet print heads and opti-cal switching devices with microdevices(1), microrelays(2),microgrippers(3) and micromotors(4) currently under de-velopment. The novel MEMS products will have a widevariety of applications in engineering, physics, biology,chemistry and other fields.

As the trend towards miniaturization continues, mi-crogrippers will become indispensable tools for handling,manipulating, and assembling micro components fromvarious application fields. Microgripping devices are nec-essary in many applications, particularly in the assemblingprocess of microscale parts, which may be fashioned froma various material using a number of different microma-chining processes.

∗ Received 13th February, 2006 (No. 06-5022)∗∗ Department of Mechanical Engineering, National Kaoh-

siung University of Applied Sciences, 415 Chien-KungRoad, Kaohsiung, Taiwan 807, R.O.C.E-mail: shuang@cc.kuas.edu.tw

In this paper, we deal with a design problem of a mi-crogripper actuated by a piezoelectric microactuator. Theproblem is formulated as a topology optimization problemand is solved by a compliant mechanism. In topology op-timization, we define the design domain firstly, and thendivide it into a large number of finite elements. These el-ements are then treated as variables giving the truss ele-ment called ground structure parameterization. Next, op-timization theory is used to determine which elements areretained in the design domain.

This microgripper is driven by a piezoelectric mi-croactuator that causes movement in the compliant mech-anism. The compliant mechanism gives optimum shapeand size when the objective function is minimized. Thedesign procedures are depicted in Fig. 1.

2. Compliant Mechanisms

Mechanisms that obtain entire or partial motion fromthe deflections of their compliant parts are termed “com-pliant mechanisms”(5). Depending on the configura-tion, G.K. Ananthasuresh and Mary I. Frecker classify(6)

compliant mechanisms into two kinds: lumped com-plaint mechanisms; and distributed compliant mechanisms(Fig. 2). In lumped compliant mechanisms, some linksor rigid linkage joints may be replaced by elastic mem-bers. By contrast, in a distributed compliant mechanism,all links and joints are elastic. Compliant mechanismspossess advantages such as reduced weight, wear, fric-tion, noise, and backlash. Morereover, they do not requirerigid-body joints(7) – (9). Their assembly is also simplified,and they can even be fabricated in one piece. This studyconsiders a type of distributed compliant mechanism formicrogripper design.

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Fig. 1 Design procedure

(a) Lumped (b) Distributed

Fig. 2 Classification of compliant mechanisms(10)

2. 1 Topology optimizationFrame elements are deformable in both the longitudi-

nal and transverse directions and therefore can incorporatebending modes. Ground structure parameterization is rec-ommended for the topology synthesis using truss, beamor frame elements. The ground structure parameterizationis a set of elements in a grid of points where each pointis connected to every other point. The ground structureparameterization is shown in Fig. 3.

2. 2 Optimization problemThe study considers an arbitrary design domain with

the generic loading and boundary conditions shown inFig. 4. F1 is the input force; P2 is the point of output de-formation along the prescribed direction. Ananthasuresh(1994) has proposed a spring model to approximate thisforce using a spring constant Ks, which represents the re-action forces at the output port. For linear finite elementmodels, the output deformation or the mutual strain energycan be used.

The work by the applied external force is often used

Fig. 3 Design domain on the ground structure

Fig. 4 Design domain and problem specifications

as a measure of the stiffness of a structure. The optimiza-tion can be expressed as a function called “Strain Energy”(SE)(11).

SE=∫

V

12σεdV (1)

The matrix form of the SE is expressed as:

SE=12{U}T [K]{U} (2)

Where σ and ε are the stress and strain, respectively,in Eq. (1) and [K] is global stiffness matrix and {U} isglobal displacement vector in Eq. (2).

The problem pays attentions to the output displace-ment at a specific port. The optimization can be expressedas a function called “mutual strain energy” (Mutual StrainEnergy; MSE). A unit virtual force is applied at an outputport in the direction of the desired deformation. MSE isused as measure of flexibility(11).

MSE=1 ·δout =

∫

V

σdεdV (3)

The matrix form is expressed as:

MSE= {V}T {K}{U} (4)

Where the σd is stress field in Eq. (3), [K] is globalstiffness matrix and {V} is global displacement vectorwhen the unit virtual force is applied. Obtaining the gra-dient of SE and MSE (the object function) is necessary toresolve the optimization problem. Starting from a station-ary condition, the function gradients must vanish at an op-

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timal. The objectives in Eqs. (2) and (4) are differentiatedas follows:

∂SE∂xi=−1

2{Ui}T [Ki]

xi{Ui}=−SEi

xi(5)

∂MSE∂xi

=−{Vi}T [Ki]xi{Ui}=−MSEi

xi(6)

Where the xi is width variable of i-th truss element,the SEi and MSEi are the strain energy and mutual strainenergy of i-th element in Eqs. (5) and (6), respectively.

When the maximum stiffness of the structure to theapplied load in desired, the optimization problem is justreduced to the problem to minimize a single object func-tion SE. When the structure must exhibit maximum stiff-ness and must generate displacement at a specific port, theoptimization problem must be a multi-criteria object func-tion of minimum SE and maximum MSE. This study con-siders the output displacement at two specific ports. WhenMSE is negative and SE positive, it implies that when φ(xi)is minimized, MSE is maximized and SE minimized. Al-ternatively, MSE and SE can be combined in a ratio-typeformulation as:

minimize :ϕ(xi)=MS ESE

(7)

Subject to:

g1(xi)=−xi≤0 (8)

g2(xi)= xi−1≤0 (9)

h(x)=n∑

i=1Lixi−AC ≤0 (10)

Where Li is the length variable of the truss element,n is the truss number from the design domain and Ac isthe area limit of the truss. SE is the strain energy when anexternal force is applied, while MSE represents the mutualstrain energy when a unit virtual force is applied at theoutput ports. In this context, the thickness of the device isfixed, so the only variables are the width and the length.The terms xi and Ac are design variables: xi represents thewidth of every truss element, Ac represents the limitationof the area of the cross section of truss element.

The optimization result for truss width is xi (0< xi <

1). When the width, or the cross-sectional dimension ofthe element goes to zero, then that element should be re-moved. The optimal topology configuration thus appearsonce the truss width is known. This optimum structure, or“optimal mechanism,” will transform the motion and forceat specific input and output ports. It represents the optimaldesign compromise between flexibility and stiffness. Inother words, the compliant mechanism is able to generatelarger displacement at specific ports while retaining suffi-cient stiffness to support the load.

2. 3 Optimization methodThe optimization problem model of Eqs. (7)–(10) has

a nonlinear form. This study uses Sequential Linear Pro-gramming (SLP) to solve our problem(12). The concept ofthe SLP is to use first-order linear terms in a Taylor’s seriesexpansion to approximate the original nonlinear problem.The problem may be represented as an expansion:

minimize : ϕ(x)=n∑

i=1

∂ϕ(xi)∂xi∆xi (11)

Subject to:

g1i(xi)=g1i(xi0)+∂g1i(xi)∂xi

∆xi ≤0 (12)

g2i(xi)=g2i(xi0)+∂g2i(xi)∂xi

∆xi ≤0 (13)

h(x)=n∑

i=1h(xi0)+

∂h(xi)∂xi∆xi<0 (14)

When the nonlinear problem is transformed into a lin-ear problem, the new variables are ∆xi. Solving this linearproblem should give an initial value xi0, giving values forthe new variables ∆xi. If we define xi0 as updated by ∆xi

this gives xi0+∆xi. Substituting the new xi0 into the prob-lem, we solve for ∆xi again. Optimization is achieved by

Fig. 5 The solution procedures of optimum problem

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iterating the above process until ∆xi is smaller than a lim-ited value, whereupon the series converges. The solutionprocedure of the optimization problem is shown in Fig. 5.

3. PZT Actuator

The piezoelectric ceramic element is a componentthat transfers energy between electrical and mechanicalstates, making it useful across a range of applications suchas sensors or actuators. Two basic phenomena, namely,characteristic of piezoelectric materials, permit them tobe used as sensors and actuators for intelligent structures.First, piezoelectric materials are able to generate an elec-trical response when mechanically stressed (sensors). Sec-ond, they yield an inversely high precision motion whenan electrical field is applied to them (actuators).

In this study, we uses a PZT-5H driver with the sizeof 3 000 µm×2 000 µm area and a thickness of 500 µm togenerate force. The conceptual drawing of PZT actuatoris shown in Fig. 6. Figure 7 illustrates the force and dis-placement relationship with applied voltage. The mate-rial properties are listed in Ref. (13), where Eqs. (15), (16),and (17) are the flexibility matrix, the piezoelectric matrix,and the dielectric matrix, respectively.

Our microactuator exerts about 6 mN of force when20 volts are applied.

Fig. 6 A conceptual drawing of PZT microactuator

Fig. 7 The relation between output and voltage

[S Emn]=

16.5 −4.78 −8.45 0 0 00 16.5 −8.45 0 0 00 0 20.7 0 0 00 0 0 43.5 0 00 0 0 0 43.5 00 0 0 0 0 42.6

×10−12 m2

N(15)

[dmn]=

0 0 0 0 741 00 0 0 741 0 0−274 −274 593 0 0 0

×10−12 C

N(16)

[εTi j]=

2.7 0 00 2.7 00 0 2.9

×10−8 Fm

(17)

4. Design the Microgripper

The design definition of microgripper is analyzed inthis section. A topology optimization design domain de-fines the configuration of the mechanism. The parame-ters defined by the design domain include boundary con-ditions, domain scale, size of finite elements, force ap-plied at input port, number and direction of output ports,and spring constants Ks shown in Fig. 8, we can define thesymmetricity. We consider the system having two inputsand two outputs. Other details of domain definitions areshown in Table 1.

4. 1 Topology optimization of the microgripperThe whole design domain and the input and output

relation under topology optimization in the design of thecompliant microgripper are illustrated in Fig. 9 (a), whileFig. 9 (b) shows one half of the design domain with iden-tical input forces in order to obtain the optimal form.

Fig. 8 The design domain scheme of the microgripper

Table 1 Design domain specifications

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(a) (b)

Fig. 9 Compliant gripper optimization with: (a) Designdomain and boundary conditions of the microgripper.(b) Equivalent model

Fig. 10 Topology optimization result

Fig. 11 The object function convergence

After defining the design domain and determining themicroactuator specifications, the next step is to find a solu-tion for the optimization problems Eqs. (7)–(10) to obtainthe optimal topology. The ANSYS software package isused to obtain SE and MSE; while MATLAB is used tosolve the optimization problem.

MEMS fabrication methods include depositing andetching. In order to simplify fabrication, and to providesufficient flexibility and thickness for the compliant mech-anism, the SU-8 photoresist was chosen. The SU-8 hasgood mechanical and chemical properties, with a Young’smodulus of E = 4.4 GPa and a Poisson’s ratio of ν =0.22(14).

Based on the aforementioned specifications of the de-sign domain, the results of roughly 500 iterations of topol-ogy optimization are illustrated in Fig. 10. The variationin the element sizes depicted in Fig. 10 represents the var-ious truss widths, where each truss element has a differentwidth ranging from 0 to 1. The topology shape of Fig. 10is as the basis design of microgripper and build analysis

Fig. 12 The variation in SE

Fig. 13 The variation in MSE

Fig. 14 3D model of the microgripper

model.The variation of optimization object function is

shown in Fig. 11. After 500 iterations, the object func-tion converges. The SE and MSE of the object function atoutput spring constant 100 N/m are shown in Figs. 12 and13.

In Fig. 14, the 3D model of the microgripper is il-lustrated. After obtaining the topology optimization pro-file, we used ANSYS software to construct the calculationmodel and run simulations on it. In the results of the topol-ogy optimization simulation shown in Fig. 15, the outputdisplacement would move approximately 25 µm in the y-direction at input force 6 mN.

4. 2 Optimization resultsThis paper describes a new way to use the topology

optimization of compliant mechanisms in the design of

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Fig. 15 Microgripper simulation

Fig. 16 Results with different numbers of inputs

Fig. 17 Results with different numbers of inputs

a microgripper. This microgripper offers easily satisfieddrive actuator requirements, optimal structure, and suit-ability for MEMS fabrication. Our topology optimizationproblem contains a large number of parameters affectingthe topology optimization result, including the input force,and the output port spring constant.

The variation of outputs under various input forces isdepicted in Figs. 16 and 17. In this study, the maximumnumber of input was four. Microgrippers with larger num-bers of input would have increased output results, whilethe material from which the microgripper was made placesan upper limit on output values. Figure 18 shows the

Fig. 18 Optimization results by design domain with differentnumbers of input forces

Fig. 19 The effects of different spring constants

Fig. 20 The effects of different spring constants

topology optimization results under various inputs. If mi-crogrippers with a larger number of inputs have greaterstiffness, there will be small changes in the optimizationresults.

The effects of topology optimization on systems withdifferent output port spring constant Ks are shown inFigs. 19 and 20. The MSE curve of Fig. 19 decreases withlarge spring constant values, but the decrease in MSE issmaller than increase in spring constant value, so the out-put force of the microgripper will still increase with large

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spring constant values. Consequently, the spring constantprimarily determines the amount of material the clips cangrip; in other words, it determines the stiffness of the twoclips of the gripper, in turn affecting the output force.However, the output force would remain around 580 µNif Ks is greater than 100 N/m, because the stiffness ofclips determines the upper limit. The SE curve of Fig. 20increases with large spring constant values, because thelarger Ks values increase the stiffness of the clips, but adecrease in the stiffness of the left portion of the gripperthat supports the input force would result. Greater stiff-ness causes smaller SE, Thus, therefore the microgripper

Fig. 21 Optimum results with different spring constant K

(a) (b)

Fig. 22 The SEM photo of the microgripper: (a) Top view and (b) Front view

would have more output displacement (Fig. 19). The out-put requirements for the microgripper can be controlledby varying the spring constant values under topology op-timization (Fig. 21).

5. Microgripper Fabrication and Measurement

Microgrippers have been successfully fabricated byPhotolithography(15). The main task of the process is totransfer the mask pattern onto thin films deposited onwafer surfaces, performed by finishing the common se-quencing steps in a photolithography room. These stepsinvolve: wafer cleaning; spin coating of the photoresist;soft baking; mask alignment; exposure; and development.Figure 22 is an SEM photo of a microgripper. The thick-ness of the structure is 500 µm; the size of the microgipperis 1 100 µm×1 000 µm.

After fabrication, we measured output displacementof a clip part of the microgripper. Figure 23 shows theschematic diagram of the experimental setup. The instru-ment is set up on a vibration isolation table. The outputdisplacement of the microgripper clip under different ap-

Fig. 23 Experimental setup for the microgripper

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Fig. 24 Microgripper displacement at various applied voltages

plied voltages is shown in Fig. 24. Figure 24 shows thatdisplacement increases as the voltage increases.

6. Conclusion

Our distributed compliant microgripper was designedby topology optimization. The compliant mechanism usesan SU-8 photoresist element. This micro compliant micro-gripper could produce about 16 µm in output displacementwhen a 20 volt is applied, and has an output spring con-stant of Ks= 100 N/m. Further work to be done includescorrecting errors in the simulation, assembly and charac-terization of the designed gripper, and the expansion oftopology optimization algorithms so that they are able toincorporate PZT and structure behaviors.

Acknowledgment

The authors would like to thank the National ScienceCouncil of the Republic of China, Taiwan for financiallysupporting this research under Contract No.NSC 93-2212-E-151-007.

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