Journal of the Korean Physical Society, Vol. 51, No. 4, October 2007, pp. 1419∼1426

Study on a Spatial 3-Degree-of-Freedom Micromanipulator EmployingDisplacement Magnifying Four-Revolute-Flexure Hinges

Sang-Joon Yoon, Jae-Yong Park and Won-Gu Lee

Department of Mechanical Engineering, Graduate School, Hanyang University, Seoul 133-791

Seog-Young Han∗

School of Mechanical Engineering, Hanyang University, Seoul 133-791

Byung-Joo Yi

School of Electrical Engineering Computer Science, Hanyang University, Ansan 426-791

(Received 10 May 2007)

The micromanipulators basically require sufficient workspace and high natural frequency. How-ever, previous designs have difficulty in satisfying the required workspace and natural frequencysimultaneously, because the previous micromanipulators are coupled designs. In this study, a previ-ous design was analyzed from the viewpoint of the axiomatic design. Then, a new design parameter,displacement magnifier, which transforms a coupled design into a decoupled design, was suggested.And a design procedure based on an axiomatic design was suggested. A spatial 3-degrees-of-freedomparallel-type micromanipulator was chosen as an exemplary device. According to the suggested de-sign procedure, the micromanipulator having the required natural frequency was designed in the firststep, and then the displacement magnifier satisfying the required workspace was designed sequen-tially. As a consequence, the micromanipulator has a natural frequency of 500 Hz and workspaceof −0.5◦ × 0.5◦. To verify the effectiveness of the manipulator and displacement magnifiers, simu-lations and experiments were performed. It is verified that the manipulator and the displacementmagnifiers implemented work very well for satisfying the required natural frequency and workspace.

PACS numbers: 87.80.FeKeywords: Micromanipulator, Displacement magnifier, Natural frequency, Workspace, Axiomatic design

I. INTRODUCTION

Ultra-precision positioning is an essential technologyin many fields, such as scanning electron microscopy,X-ray lithography, and micromachining. With the fur-ther development of these fields, there is an increasingdemand for ultra-precision positioning devices that canprovide high displacement resolution and wide motionrange [1–4].

Generally, a parallel-type manipulator has an inher-ently higher precision than a serial-type manipulator.Therefore, parallel structures are mainly adapted to de-sign micromanipulators. Recently, there have been manystudies on the analysis and design of micromanipulatorswith flexure hinges. The flexure hinge has numerous ad-vantages, such as smoothness of movement, zero back-lash, no stick-slip, and high precision. The use of flexurehinge mechanisms is generally the most appropriate ap-proach to micromanipulators [5, 6]. Micromanipulators

∗E-mail: syhan@hanyang.ac.kr; Fax: +82-2-2298-4634

are usually driven by a piezo translator (PZT) that hassmooth motion, infinite resolution, high stiffness, andhigh speed response. However, the maximum stroke ofa PZT commonly used is only about 0.1 % of its length;thus it is not adequate for the demanded workspace [2–6].

If vibrations are to be suppressed and task workspacerequirements satisfied for micromanipulators, the manip-ulators basically require a high natural frequency andsufficient workspace. However, previous micromanipu-lators have a coupled design because high natural fre-quency and sufficient workspace cannot be independentlychanged [3–6]. Therefore, this paper suggests a new de-sign parameter, the displacement magnifier, and a newdesign procedure based on a decoupled design in an ax-iomatic design [7]. A spatial 3-degree-of-freedom (3-DOF) parallel-type micromanipulator was chosen as anexemplary device. By using the proposed design pro-cedure, we designed a micromanipulator having the re-quired natural frequency as a first step; then, we de-signed a displacement magnifier satisfying the requiredworkspace. The required natural frequency and theworkspace were established as 500 Hz and −0.5◦ × 0.5◦

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Fig. 1. A spatial 3-DOF micromanipulator [5].

Fig. 2. Modeling of the spatial micromanipulator [5].

in the directions of α and β, respectively. To verify theeffectiveness of the micromanipulator and the displace-ment magnifier, we performed simulations by using afinite element method (FEM), and to corroborate theavailability of the displacement magnifier, we performedexperiments.

II. A SPATIAL 3-DOFMICROMANIPULATOR

The micromanipulator shown in Figure 1 was chosenas an exemplary device in this study. It is a parallel-type

Fig. 3. Model of the ith serial chain [5].

mechanism composed of an upper plate and three serialchains. Each chain consists of a spherical flexure hinge, arevolute flexure hinge, and a prismatic joint of PZT, andis fabricated as one-module. The three chains are 120◦apart from each other. The symmetric structure reducesthe effect of the temperature gradient and disturbance.This mechanism has three degrees of freedom to the di-rections of α, β and z in space. The spherical and therevolute flexure hinges behave like mechanical joints bytheir compliances [5].

The modeling of the joints and the coordinates of themanipulator are shown in Figures 2 and 3. R and rare the radii of the lower and upper plates, respectively.The revolute and the spherical flexure hinges are real-ized as 1-DOF and 3-DOF, respectively. Hi denotes thedisplacement of a prismatic joint. In Figure 2, Sp

¯u and

Sp¯l indicate the upper and the lower lengths of a spher-

ical flexure hinge, respectively. Re¯u and Re

¯l denote

the upper and lower lengths of a revolute flexure hinge,respectively. Pr

¯l indicates the length of PZT. Pbt is

a position vector, which locates the center of the up-per plate relative to the global coordinate located at thelower plate. The transformation matrix of the local co-ordinate frame attached to the upper plate can be de-scribed by Eq. (1) with respect to the global coordinateframe:

T bt∼=

n1 o1 a1 xn2 o2 a2 yn3 o3 a3 z0 0 0 1

. (1)

The output vector iϕ of the proposed mechanism is com-posed of the positions x, y, z and the x-y-z Euler anglesα, β and γ. Since the mobility number of this system isthree, the independent output vector uind, is determinedas α, β and z, and the remaining components are treatedas elements of the dependent output vector udep. The

Study on a Spatial 3-Degree-of-Freedom Micromanipulator· · · – Sang-Joon Yoon et al. -1421-

input vector iϕ of the ith serial chain is defined as

iϕ = [Hi,i θ2,

i θ3,i θ4,

i θ5]T

∼= [iφ1,i φ2,

i φ3,i φ4,

i φ5]T . (2)

Now, we analyze the inverse kinematics of the pro-posed mechanism. The inverse kinematics calculates theinput vector of each chain for a given uind = [α, β, γ]T .P i denotes the position vector of the point Pi in the ith

chain with respect to the global coordinate frame and isdefined as

P 1 = [R 0 H1 + Pr¯l + Re

¯l]T ,

P 2 = [−R/2√

3R/2 H2 + Pr¯l + Re

¯l]T ,

P 3 = [−R/2 −√

3R/2 H3 + Pr¯l + Re

¯l]T . (3)

Furthermore, the position vector bi of the point Bi withrespect to the local coordinate fixed on the upper plateis described as

b1 = [b1x 0 b1z]T ,

b2 = [−b1x/2√

3b1x/2 − b1z]T ,

b3 = [−b1x/2 −√

3b1x/2 − b1z]T . (4)

where blx = r + Sp¯u cos θtr, blz = r + Sp

¯u sin θtr, and

θtr=45◦. The position of Bi can be described with re-spect to the global reference coordinate frame as follows:[

Bi

1

]= [T b

t ][

bi

1

]i = 1, 2, 3. (5)

As the links B1P1, B2P2 and B3P3 are constrained bythe revolute joints to move in the planes Y = 0, Y =−√

3X, and Y =√

3X, respectively, the constraint equa-tions imposed by the revolute joints are

yc = −n2b1x + a2b1z,

n2 = o1,

xc = −12(o2b1x − n1b1x − 2a1b1z). (6)

Then, by using x-y-z Euler angles, the elements of Eq.(1) can be expressed as

n1 = cos β cos γ,

n2 = sinα sinβ cos γ + cos α sin γ,

n3 = − cos α sinβ cos γ + sinα sin γ,

o1 = − cos β sin γ,

o2 = − sinα sinβ sin γ + cos α cos γ,

o3 = cos α sinβ sin γ + sinα cos γ,

a1 = sinβ,

a2 = − sinα cos β,

a3 = cos α cos β. (7)

From Eqs. (6) and (7), the Euler angle γ about the zaxis of the global coordinate frame is obtained as

γ = tan−1

(− sinα sinβ

cos α + cos β

). (8)

Fig. 4. Geometric solution of iθ2 [5].

Substituting γ and the given Euler angle α, β into Eq.(7), all the elements of T b

t are obtained: then, the centerposition of the top platform xc and yc are calculatedfrom Eq. (6), and the position of Bi is obtained fromEq. (5). Thus, the length between points Bi and Pi canbe obtained as

BiPi ={(Bix−Pix)2+(Biy−Piy)2+(Biz−Piz)2}1/2,(9)

where Bix, Biy, Biz, Pix, Piy, and Piz denote the elementsof the position vectors of points Bi and Pi with respectto the global coordinate frame. In Eq. (9), BiPi is aconstant value, and Bix, Biy, and Biz are known values.Therefore, Hi is obtained by substituting Eq. (3) intoEq. (9):

Hi = Biz − Pr¯l −Re

¯l

−{BiPi2 − (Bix − Pix)2 − (Biy − Piy)2}1/2. (10)

The revolute joint variable iθ2 of the ith chain is ob-tained geometrically from Figure 4 as follows:

iθ2 = π/2−i θd −i θ. (11)

Also, the spherical joint variables of the ith chain, iθ3,i θ4,

and iθ5, are obtained from Eq. (12):

i[T 21 ]i[T 1

b ]i[T b1 ]i[T t

5 ] =i [T 23 ]i[T 3

4 ]i[T 45 ], (12)

where i[T kj ] is a matrix that transforms the kth coordi-

nate to the jth coordinate of the ith chain. The left-sideof Eq. (12) is known, and the right-side is a function ofiθ3,

i θ4, and iθ5. Let the left side be defined as

i[T 21 ]i[T 1

b ]i[T b1 ]i[T t

5 ] ∼=

e1 f1 g1 i1e2 f2 g2 i2e3 f3 g3 i30 0 0 1

. (13)

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Fig. 5. Schematic diagram of a displacement magnifier.

By equating Eq. (12) to Eq. (13), we obtain the jointvariables as

iθ3 = tan−1

(g3

g3 cos θtr + g2 sin θtr

),

iθ4 = tan−1

(g1 cos θtr + g2 sin θtr

(g2 cos θtr − g1 sin θtr)× g3 cosi θ3

),

iθ5 = tan−1

(e2 − e1

f2 − f1

). (14)

Consequently, all the elements of the input vector iϕhave been obtained for the given independent output vec-tor uind.

III. AXIOMATIC DESIGN

Two of the most important functions of the mi-cromanipulators are a high natural frequency and asufficient workspace. Therefore, the FRs (functionalrequirements) of the mechanism in the axiomatic design[7] can be written as follows:

FR1: sufficient workspace,FR2: high natural frequency.

A spatial 3-DOF micromanipulator also has the sameFRs. For FR1, the DP1 (design parameter) is relatedwith the link structure. For FR2, DP2 is related withthe mass of the system. The DPs of the mechanism canbe written as follows:

DP1: use of many or long links,DP2: mass reduction of system.

Previous micromanipulators have had a coupleddesign because the two FRs cannot be independentlychanged. If the independence axiom is to be satisfied,the design matrix must be either diagonal or triangular.Therefore, instead of the previous DP1, a new DP1 issuggested in this study as follows:

DP1: use of a displacement magnifier

A new DP1 doesn’t affect FR2 at all. It is possibleto change a full matrix into a triangular matrix with a

Table 1. Material properties of a spatial micromanipula-tor.

Material SM45C

Density (ρ) 7400 kg/m3

Modulus of Elasticity (E) 109 GPa

Poisson’s Ratio (v) 0.26

Yield Stress (σγ) 250 MPa

decoupled design that satisfies the independence axiom.The schematic diagram of a displacement magnifier isshown in Figure 5. It is composed of four revolute flexurehinges and five links. bs, Rs and ts are the width, radius,and thickness of the hinge, respectively. ls and θs are thelength between two hinges and an angle of inclination, re-spectively. The input and the output displacements aredefined as the displacement of the PZT and the mag-nified displacement through the displacement magnifier,respectively. For the given input displacement by thePZT, the output displacement comes out as follows:

Output displacement =√l2s−(ls cos θs− Input displacement)2− ls sin θs

2. (15)

A typical PZT has a large force capacity, but it has asmall stroke. Thus, the displacement magnifier can beemployed to increase the stroke of the PZT without af-fecting the force capability. This study suggests a newDP1 to satisfy the independence axiom. A designer sat-isfies FR1 and FR2 at the same time when the DPs canbe determined independently and sequentially. The wayto satisfy the required FRs is to change DP2 for changingthe value of FR2 first, and then to change DP1, whichdoes not affect the value of FR2, for changing the valueof FR1. Therefore, a micromanipulator having the re-quired natural frequency is designed in the first step;then a displacement magnifier that satisfies the requiredworkspace is designed.

IV. DESIGNS OF THE MANIPULATOR ANDTHE DISPLACEMENT MAGNIFIER

1. Optimum Design of the Micromanipulator

The objective function of the optimum design processis to minimize the mass of the micromanipulator. Designvariables were chosen as the radius of the upper plate, thedisplacement, the length, the radius, the thicknesses ofthe spherical and revolute flexure hinges, and the heightof the revolute flexure hinge, because the natural fre-quency depends on these variables. The material prop-erties of the manipulator are shown in Table 1. Since themechanism has a symmetric structure, optimum design

Study on a Spatial 3-Degree-of-Freedom Micromanipulator· · · – Sang-Joon Yoon et al. -1423-

Table 2. Constants used for design of the micromanipula-tor [8].

C1 C2 C3 C4 C5Constants Sf

(mm) (mm) (mm) (N/µm) (Hz)

Value 9 500 300 5 30 500

was performed for only one serial chain. If the optimiza-tion problem is to be solved, the constraints must beconsidered.

First, the yield stress of the mechanism must be con-sidered. Since the maximum stress occurs at the hinge,the hinge’s stress must be smaller than the yield stressσγ . The constraints on the allowable stresses of hingesare described as

g1∼4 = Sf |θi| −t2i b

6kiKtiσγ < 0, i = 2, 3, 4, 5, (16)

where Sf is the safety factor, k is the rotational stiffnessof each hinge, and Kt is the concentration factor. Sec-ond, the constraints to be considered are on the sizes ofthe mechanism, and they can be expressed as

g5 = [(Sp¯u + Sp

¯l)×sin θtr + Re

¯u + Re

¯l]− C1, (17)

g6 = [(Sp¯u + Sp

¯l)× cos θtr + r]− C2. (18)

Third is the constraint on the structures of the mecha-nism. A manipulator can obtain a larger workspace inspite of a small input when (Re

¯u + Re

¯l) is greater than

(Sp¯u + Sp

¯l), which can be expressed as

gγ = (Sp¯u + Sp

¯l)− (Re

¯u + Re

¯l). (19)

Fourth, the structure of a revolute flexure hinge satisfiesRe

¯t � Re

¯b and Re

¯t � Re

¯r, and that of a spherical

flexure hinge satisfies Sp¯t � Sp

¯r. Re

¯t, Re

¯b, and Re

¯r

indicate the thickness, height, and radius of a revoluteflexure hinge, respectively. Likewise, Sp

¯t and Sp

¯r de-

note the thickness and radius of a spherical flexure hinge.Those can be described as

g8 =C3−Re

¯r

Re¯t, g9 =C3−

Re¯b

Re¯t, g10 =C3−

Sp¯r

Sp¯t. (20)

Fifth, the constraints on the stiffness of the hinge mustbe considered and can be expressed as

g11 = ki − C4, i = 2, 3, 4, 5. (21)

Finally, the constraint on the natural frequency of themanipulator must be considered and is given as

g12 = C5 − frequency, (22)

where Ci(i = 1, 2, 3, 4, 5) are constants, given in Table 2.In summary, the optimization formulation of a ma-

nipulator satisfying the required natural frequency iswritten as follows:

Minimize mass

Table 3. Optimization results for the micromanipulator.

Design Variable Previous and optimal values

(mm) Previous Optimum Actual

r 15 5 5

Sp¯u 11.87 15 15

Sp¯l 16.27 15 15

Re¯u 11.88 15 15

Re¯l 30 30 30

Sp¯r 1.5 2.7060 2.71

Re¯r 1.65 2.0356 2.04

Sp¯t 1 0.5412 0.55

Re¯t 0.7 0.3128 0.32

Re¯b 4 5 5

Table 4. Constants used for design of the displacementmagnifier.

Design ts Rs bs ls

Value (mm) 0.32 2.34 5 11.68

Subject to gi < 0 i = 1, · · · , 12

1 < r < 4015 < Sp

¯u < 50

15 < Sp¯l < 50

15 < Re¯u < 50

15 < Re¯l < 50

2 < Sp¯r < 5

2 < Re¯r < 5

0.3 < Sp¯t < 2

0.3 < Re¯t < 2

5 < Re¯b < 10

Find x = [r Sp¯u Sp

¯l Re

¯u Re

¯l Sp

¯r

Re¯r Sp

¯t Re

¯t Re

¯b]

A sequential quadratic programming (SQP) algorithmwas employed to solve the optimization problem above.The result of the optimum solution is compared with thatof the previous design (Figure 1) summarized in Table3. Optimum values are obtained by using MATLAB [9].To satisfy the constraints, as well as the technical limitsof fabricating the micromanipulator, we controlled thedesign variables by raising the numbers to two decimalplaces.

2. Design of the Displacement Magnifier

The required task space of the spatial micromanipula-tor is established, and the rotational angles about x and

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Fig. 6. Magnification rate vs θs for the spatial manipula-tor.

Fig. 7. FEM model of the flexure hinges. (a) A revoluteflexure hinge (b) A spherical flexure hinge

y axes are −0.5◦ ≤ α and β ≤ 0.5◦ at the upper plate.The spatial manipulator has the same value at the sameradius from the upper plate. Therefore, the design of themanipulator is direction independent, so the β rotationalangle was considered in this study. The manipulator isto be designed by using the optimum values. If the re-

Fig. 8. Natural frequency analysis of the optimum design.

quired task space is to be satisfied, a displacement of135.4 µm at the first hinge is required. The magnifica-tion rates (out/in) are largely dependent on the angle ofinclination θs. Figure 6 shows the curve for the magnifi-cation rate vs θs by Eq. (15), where in and out denotethe input and output displacements, respectively. Theconstants used for the design of the displacement mag-nifier are shown in Table 4. A displacement magnifierhaving a magnification rate of 3.01 (θs = 8.7◦) can bedesigned by use of the curve in Figure 6 when a P-841.30PZT having the maximum stroke of 45 µm is used. Inthis study, however, a displacement magnifier having amagnification rate of 3.25 (θs = 8.2◦) was designed byconsidering errors in the manufacturing process.

V. SIMULATIONS & EXPERIMENTS

1. Simulations

Simulations for the natural frequency were performedfor the optimum and the actual designs shown in Table3. Since the links and the flexure hinges in the modelingof the finite element method were assumed to be rigidand to be a rotational spring, respectively, a coarse meshand a fine mesh were applied to the links and the flexurehinges, respectively, as shown in Figure 7.

Figure 8 shows the natural frequency analysis of theoptimum design. The natural frequencies of the optimumand the actual designs were obtained as 505.32 Hz and509.48 Hz, respectively, satisfying the natural frequencyrequirement. On the other hand, the maximum stresson each hinge occurs when the upper plate is rotatedto 0.5◦ in the α and β directions, respectively. For thetwo directions, the maximum stresses were obtained as227 MPa and 147 MPa. Since the maximum stressesare smaller than the allowable stress, the required task

Study on a Spatial 3-Degree-of-Freedom Micromanipulator· · · – Sang-Joon Yoon et al. -1425-

Fig. 9. Displacements for the theory and the simulation.

Fig. 10. Natural frequency analysis of a displacement mag-nifier.

space was verified to be free from yielding under the givenconstraints.

Simulations for the magnification rates, the naturalfrequency, and the maximum stress of the displacementmagnifier were performed for the suggested design inTable 4. The simulation results for the magnificationrates of the displacement magnifier are shown in Figure9, which shows the lines for both the theoretical dis-placements obtained from Eq. (15) and simulated re-sults from the finite element analysis. The stroke of thePZT was 45 µm. This allowed for an output range of0 ∼ 146.6 µm theoretically, but the simulated outputrange was 0 ∼ 142 µm. Thus, the simulation resultsshowed a 3.8 % disagreement. The natural frequencyanalysis is shown in Figure 10, and the value was ob-tained as 1570 Hz. The maximum stress was obtained as238 MPa, thus satisfying the required natural frequencyand the allowable stress requirement.

Fig. 11. Schematic diagram of the experiment setup.

Fig. 12. Newly manufactured spatial 3-DOF micromanip-ulator.

2. Experiments

Figure 11 shows a schematic diagram of the experi-ment setup. The DC power supply was amplified by apower amplifier to drive the PZT. Displacement magni-fiers driven by the PZT supplied the magnified displace-ment to the micromanipulator. The output displacementwas measured by using a linear variable differential trans-former (LVDT) with a 0.61-µm resolution. A PC wasused to display, store, and analyze the experimental re-sults.

The new spatial 3-DOF micromanipulator shown inFigure 12 was manufactured by using the actual designin Tables 3 and 4. It is a parallel-type mechanism com-posed of an upper plate, three serial chains, and threedisplacement magnifiers. Each chain consists of a spher-ical joint and a revolute joint, and a displacement mag-nifier is located at each base.

The experimental results obtained using the experi-ment setup are shown in Figure 13. The figure shows thedisplacements for both the experimental and the simu-

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Fig. 13. Experimental and simulated displacements.

lation results. The stroke of the PZT was 45 µm. Thisallowed for the output range of simulation results to be0 ∼ 142 µm, but the output range of experimental re-sults was 0 ∼ 138.6 µm, a 2.4 % difference. There areseveral possibilities for this: an error in the manufactur-ing process and the friction effect at a coupling part ofthe displacement magnifier and the serial chain. How-ever, the experimental results satisfy the task workspaceof 135.4 µm.

An experiment for the natural frequency of the newlymanufactured spatial 3-DOF micromanipulator was alsoperformed. The natural frequency was measured as 512Hz. Therefore, we verified that the manipulator satisfiedthe natural frequency requirement.

VI. CONCLUSIONS

In this study, a previous spatial 3-DOF parallel-typemicromanipulator design was analyzed from the view-point of the independence axiom in an axiomatic design.Then, a new design parameter, the displacement magni-fier which transforms a coupled design into a decoupled

design, was suggested, and a design procedure based onan axiomatic design was developed.

According to the suggested design procedure, we firstdesigned a micromanipulator having the required natu-ral frequency; then, we designed a displacement magni-fier satisfying the workspace requirement. The resultingspatial 3-DOF manipulator had a natural frequency of500 Hz and a workspace of −0.5◦ × 0.5◦.

To verify the effectiveness of the micromanipulatorand the displacement magnifier, we performed simula-tions and experiments were performed. We verified thatthe micromanipulator and the displacement magnifiersworked very well and satisfied the task workspace andthe natural frequency requirements.

ACKNOWLEDGMENTS

This work was supported by grant no. R01-2005-000-10238-0 from the Basic Research Program of the KoreaScience and Engineering Foundation.

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