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Optimal design of a flexure hinge based XYφ wafer stage
ELSEVIER
Optimal design of a flexure hinge based XY0 wafer stage Jae W. Ryu,* Dae-Gab Gweon,* and Kee S. Moont *Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yusong-Gu, Taejon, Korea and tDepartment of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, Ml USA
Optimal design of a XY8 micromotion stage is presented. The stage consists of a monolithic flexure hinge mechanism with three piezoelectric actuators. This paper describes the procedures of selecting parameters for the optimal design. In particular, it presents a mathematical formulation of the optimization problem. Based on the solution of the optimization problem, the final design of the stage is also presented. Experimental results indicate that the design procedure is effective, and the designed stage has the total range of 41.5 pm and 47.8 pm along the X- and Y-axes, respectively, and the 0 1997 maximum yaw motion range of 322.8 arcsec (1.565 mrad). Elsevier Science Inc. Keywords: XY0 micromotion tric actuator; optimal design
stage; flexure hinge mechanism;
Introduction Because of the growing levels of integration of semiconductor products, requirements for the positioning precision of wafer stages are continuously increasing. For the fabrication of VLSI circuits, for example, the positioning accuracy of 0.01 pm is necessary. 1 In addition, recent use of large sized wafers demands longer travel range of the wafer stages. The emphasis has, therefore, necessitated the development of a wafer stage with high accuracy and long travel for wafer positioning. XY tables with DC servo motor and lead screw (ball screw or sliding screw) have been popular for wafer positioning. Although they have a relatively long travel range, it has been known that they have the nonlinear spring effect of balls and stickslip. 2,3 Such problems are difficult to eliminate; therefore, submicrometer positioning accuracy is rarely achieved by the XY tables with DC servomotor and lead screw. In practice, for this reason, the Address reprint requests to Dr. Dae-Gab Gweon, Korea Advanced Institute of Science and Technology, Department of Mechanical Engineering, 373-Kusong-Dong, Yusong-Gu; Taejon 305-701, South Korea.
Precision Engineering 21:18-28, 1997 0 1997 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010
piezoelec-
wafer positioning usually uses a two-stage positioning scheme. 4-6 In the first stage, coarse positioning is achieved by the XY table with DC servomotor and lead screw. In the second stage, fine positioning is achieved by a high-precision micromotion stage. The micromotion stage usually has short travel range but high positioning resolution. In fact, the micromotion stage is required to have sufficient travel range to compensate for the position errors that occurred during the coarse positioning. It is also important that the micromotion stage has the capability to correct yaw motion error from the coarse XY positioning. The yaw error can also be caused by wafer fixturing devices. In particular, the need for long travel range for coarse positioning also requires the large yaw motion range for the fine positioning stage. For example, friction drives in electron beam lithography system can cause more than +360 arcsec (50.1”) of the yaw motion error.’ To correct the yaw motion error, parallel linkage-type micromotion stages have been used because of their simple structure. These, however, have limited yaw range of less than 36 arcsec. In this paper, the design of a flexure hinge-
0141-6359/97/$17.00 PII SO141-6359(97)00064-O
Ryu et al.: Flexure hinge based on XY0 wafer stage ing bodies are denoted by the numbers 1 through 13 (noncircled). Note that the number 1 body is the output body and will be activated by the three piezoelectric actuators acting on bodies 5, 9, and 13. The complete design of the XYO design is shown in I;igure 3. Modeling
L Figure 1 Double chanical lever
compound
flexure
pivoted
me-
based micromotion stage with a large yaw motion of 322.8 arcsec is presented. It utilizes three piezoelectric actuators and a monolithic flexure hinge mechanism. The flexure hinge mechanism uses three double compound mechanical levers to amplify each actuator motion. The performance of the flexure hinge mechanism is heavily influenced by selection of such parameters as lever length, hinge thickness, and hinge radius.8 This paper describes a procedure for selecting parameters to optimize the design of the XY8 stage. In particular, the optimal design procedure to maximize the yaw motion utilizing a mathematical model of the mechanical system is presented. Structure
of the XY0 micromotion
stage
The XY0 stage consists of three piezoelectric actuators and a monolithic flexure hinge mechanism. The flexure hinge mechanism can be represented as a joint mechanism between an output body (wafer holder) and the actuators. Such mechanisms provide almost no backlash or stickslip friction and provide smooth, continuous displacement positioning. Typically, the flexure hinge mechanism contains a number of hinges and mechanical levers. Figure I shows the double compound mechanical lever employed for the XYO stage design. The displacement generated by a piezoelectric actuator is 15 pm. Therefore, to amplify the actuator displacement to obtain a large yaw motion, use of the compound mechanical lever to provide mechanical displacement gain without significant motion loss is required. Figure 2 also shows a schematic of the complete XYB stage design. It is clear from the figure that the design utilizes three mechanical input levers radially disposed with 120” pitch. The symmetric design reduces the effect of temperature gradient on the structure. In this figure, the movPRECISION
ENGINEERING
of the stage motion
Let us assume that the levers and the hinges in the flexure hinge mechanism be modeled as rigid bodies connected through translational/rotational springs. Let us also regard a piezoelectric actuator be a translational spring. Then, the XY0 stage mechanism can be assumed to be a springmass system. In this study, the XY0 stage employs a right circular single-axis flexure hinge, and it is shown in Figure 4. The spring rates of the hinge have been calculated by using Paros and Weisbord equations for a single-axis flexure hinge9 and torsional equation of a rectangular bar.iO From Lagrange’s equation, the motion of the XY0 micromotion stage can be represented by M%+Kx=F
(1)
where M, K, and F are the system mass matrix, stiffness matrix, and force vector, respectively. l1 The system displacement vector x is defined by x =
[q’q . . . , q’:
. . . , cpy-
(2)
where Nb is the number of moving bodies in the system and qi is the displacement vector of the origin of the body i or qi = [xi, yi, zi, Oi, t$, Of]*. The XY0 stage is a planar motion mechanism; therefore, each rigid body has only three degrees of freedom, (x, y, 0,). However, in this study, the model considers three additional degrees of freedom (x, 0, fly) for the analysis of tilt stiffness. Note that the size of the displacement vector x is (6 X 13) X 1, because there are 13 rigid bodies in the system, and each body has six degrees of freedom. Similarly, the force vector F also has the same size (6 X 13) X 1. The mass matrix M and the stiffness matrix K have the same size of (6 X 13) X (6 X 13). The natural frequency of the system can be calculated from the given information of mass and stiffness of the system. Consider the following eigenvalue equation. IK- w%[ = 0
(3)
Note that the positive square root of the solutions of Equation (3) are the natural frequencies of the system. The stiffness of the XY0 stage can be obtained from the displacement caused by external load at the center of the output body. For example, con19
Ryu et al.: Flexure hinge based on XXI wafer stage
moving
??
output
??
??
??
Figure 2
Schematic
diagram
P’, . . . , PqT
hinges
: 0,
... ,@$
(6, - &K&,,)x,
(7)
= F, - K,,K&
or wx’ = F’
(8)
where
f’ = [I N, 0, 0, 0, 0, 01 i-2,3,
. . . , N,,3
K’ = K,, - K,,K;;K,,
(5)
(9)
F’ = F, - K,,K;;F,
and fi is the resolved force vector of the body i. The stiffness of the system for the other directions can be obtained using the same procedure. To calculate the displacement of the output body attributable to a displacement by the actuators, let q1 be the output displacement vector, and let the input displacement vectors be $, qg, $3, respectively. Also, let x1 = [qlT, q5T, q T, q13 IT and x, represent the displacement vectors of the other bodies excluding bodies 1, 5, 9, and 13. Then, Equation (4) becomes
Equation
(10)
(8) can be rewritten as
(11) where XL = q1 and xi = [SST, qgT, q13’jT. In the equation, the subscripts o and i represent the output and the input bodies, respectively. Given that the load applied to the output body FL is known, the displacement vector of the output body x: can be given by x’0 = &,-‘(F,
(6) 20
: 5, 9, 13
From which
(4)
where f’=O,
body : 1 (XYe stage)
input bodies flexure
: 1, ... , 13
of the XY0 stage mechanism
sider a 1 N load applied to the output body. The direction of the load is, for example, X. Then, the stiffness of the system can be calculated from the following linear simultaneous equations. Kx = [P,
bodies
Also,
- Kb;x:)
(12)
the required force at the input bodies is JULY
1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
Figure 3
Complete
design
F,, = -F; = -(K;,x’,
of the XY8 stage mechanism
+ K:;x;)
(13)
Equations (12) and (13)) provide a basic and direct relationship between input and output of the system. Given the input displacement by actuators and the load applied to the output body, the ,” Fz
and coordinate
resulting output displacement and the required input force can be calculated. However, the inverse relationship between the input and the output also must be obtained. In particular, the required input displacement of the actuators must be calculated to get a desired displacement of the output body. Consider Equation (12)) which can be represented by K;;x; =
C-&
systems
(6x31~1
F’, (6x1)
-
v’,x;
(14)
(6x1)
From Equation (14), it is clear that there are 18 unknown variables and only six equations; therefore, there is no unique solution. In this study, a different coordinate system is considered to adjust the number of unknown variables and the equations. fipre 5 shows the transformed coordinate system. It may be noted that the direction of local coordinate x2has been aligned to the direction of piezoelectric actuation. Hence,
Figure 4 PRECISION
Right circular
single-axis
ENGINEERING
flexure
hinge
-i X -i Y -i il z 21
Ryu et al.: Flexure hinge based on XYO wafer stage for understanding the effect of such design parameters as hinge radius, width, thickness, lever ratio, and link length. In this section, the optimal selection of the design parameters is presented. As shown in Fig-LLreI, the stage uses a double compound flexure pivoted mechanical lever to amplify the length change of the piezoelectric actuator. Ideally, the flexure hinge is a revolute joint and the mechanical gain G is
(23)
Figure 5
Coordinate
transform
= Re,zXi
From Equation
-__ from xyz to xyz (15)
(S),
KR = F
(16)
where, i?= RTK’R
(17)
F=
(18)
R’F
R = diag(l,
1, R;,,,
R_;,Z,3)
(19)
Because the stage has only plane movement (z” = Bx = 0; = 0: = 0)) and the input body moves only along with the direction of piezoelectric actuation, Equation (16) becomes
(20) where the output displacement vector, x, = [x1, y’, and the input displacement vector, xi = [35, 07 2%;“9 -g3]T . From the Equation (20) and for a given F,, the input and output relationship can be found. For example, let F, = 0, then, xi= -KZli’KLX& = T;,x,
(21)
Note that the matrix Ti, is a transformation matrix, and it gives the input and output relationship between xi and x,. Ti, = - Ki: Ka,
Optimal
(22)
design of the stage
In the previous sections, the model of the stage has been described. This model provides the basis 22
It can be noted that the gain relies only on the lever length ratio. However, in actual case, the flexure hinge provides not only rotations but also has translational compliances. Therefore, the effect of these unwanted hinge compliances on the output displacement must be assessed. Figure 6 shows simulation results for the purpose. Fig-ure 6(a) describes the effect of the gain G and the hinge thickness ton the output displacement and required input force Fin when the input displacement u is 1 pm. In the case, there is no load applied to the output body. It is very clear from (a) that the design with higher gain provides larger output displacement for a given hinge thickness. However, when there is a static load applied to the output body, higher gain does not always provide larger output displacement. For example, from (b), the gain of about 50 provides the maximum output displacement when 1 N of static load Fwas assumed at the output body. Fig-ure 7 shows simulation results of the effect of the gain and the hinge radius R on the output displacement and required input force Fi, when the input displacement u is 1 pm. Similarly, higher gain provides larger output displacement for a given hinge radius in the case of no static load. In the case of a given static load (IN), the gain of about 50 provides the maximum performance for most cases. From Figures 6 and 7, the output-to-input displacement ratio is smaller than the ideal gain G. It may be explained that the translational compliances, especially the longitudinal compliance (x-axis in Figure 41, as well as the rotational compliance about z-axis have an effect on the output-to-input displacement ratio. From the figures, furthermore, it is clear that the output displacement and the required input force are a more sensitive to the load when the hinge has a smaller hinge thickness for a specific hinge radius or a larger hinge radius for a specific hinge thickness, because these provide a smaller y-directional stiffness. The previous simulation approach provides a rough tool to achieve the optimal flexure hinge JULY 1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
lool-----l 80
z f=O.5
80 3 A
L&I 40
0
1=2.5
0
t=2.5
$ 150 5 P .100 * s! '5 50 K
H.5
20
200
i3
50
f=1.5
f=O.5
0 kz 0
100
50
1
G
(4 100 80. 80. 3h
ti2.5
40. M.5
G (b) F = 1 N
Figure 6 Simulated y/u and 6, versus G with tvariations L = L, + f,, + L, + 2b, R = 4 km, b = 20 mm)
design. In this paper, a rigorous approach is taken to optimize the design parameters by using mathematical programming. Figure 8 shows the design variables for the hinge mechanism used for the XY8 stage. Note that the hinge radius is a design variable for all the hinges considered. In practice, a lack of flexibility to change the drilling diameter causes manufacturing problems for the hinge design with various hinge radii. The objective of the optimization problem is to maximize the yaw motion of the output body (minimize l/Q:). The problem includes a number of constraints. For example, the maximum stress at the hinge point should be less than the yield stress. There are also constraints for the system size, minimum tilt stiffness, and the natural frequency of the system. First, consider the maximum stress constraint. Because the maximum stress occurs at the hinge point with minimum thickness
(u = 1 km, L, = L, = 10 mm, L, = 10 - 80 mm,
(26)
M, = k&3
Because the maximum rotational movement occurs when each piezoelectric actuator reaches its maximum elongation, the maximum allowable rotation at the hinge is
s,e < -
t2b
(27)
6k,K, uy
In Equation (27) Sfis a safety factor, and o,is the yield stress of the hinge material. Table 1 shows the given constant values. As denoted in Figure 8, let 13: be the rotational movement of body i with respect to the Zaxis, let k: be the rotational stiffness of hinge i, let gi be a stress concentration factor of hinge i, and let ti be the minimum thickness of hinge i, respectively. Now, for the six hinges, the six constraints for the maximum stress are
6 MA umax
=
__
t2b
(28)
where MZ is the moment at the hinge, and K, is a stress concentration factor given by Smith et ali2 K, =
2.7t + 5.4R 8R+
t
+ 0.325
(25)
Let 12,be the rotational stiffness of the hinge, then PRECISION
ENGINEERING
(29)
g, =
‘b
s,le;l - -ciy
(30)
23
Ryu et al.: Flexure hinge based on XY0 wafer stage 100,
250,
I
(a)F=O
‘OOl-----l
250r------
100 G
lb) F=lN Figure 7 Simulated y/u and 6, versus G with R variations L=L1+L,+L,+20,t=1.5mm,b=20mm)
Second, consider the constraints for the system size. Based on the description of variables in Figure 8, there can be four additional constraints for the problem. They are g,=(L+c)-c,
C,-
g,=
[,,+;+;j
g,,=
(31)
[c-
(2R+;)]
(32)
- L,
i,,+$+;j - L,
(34)
where C, and C, are constants, and the values are given in Table 1. Third, consider the constraints for the minimum tilt stiffness value. Based on Equation (4) and moment equation, two additional constraints for X and Y moments can be obtained by
c,+o 1
g,,=
(35)
x
g,,=
&&a Y
where C, and C, are also given in Table 1. Finally, consider the constraint for the natu24
(u = 1 pm, L, = L, = 10 mm, L, = 10 - 80 mm,
ral frequency of the system. The first natural frequency w1 can be obtained from Equation (3) and must satisfy the following constraint. g,, = c, - 01 < 0
(37)
It is assumed that only body 1 (output body) has its mass and the others are massless. Note that this study did not consider the constraints for the force required to move the stage nor the load applied to the actuators. In fact, it is known that the piezoelectric actuators provide similar displacements, regardless of the load at a given applied voltage. In summary, the described optimization problem can be written as minimize 1 /e,
subject to LJi< 0
i= 1, . . .
I
13
and 1.0 < R< 7.0 lO
3.0
4.5 < L,< 50
j=l
, ..., 6
k=1,...,3 JULY 1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
Design
variables
All hinges Figure
8
Design
: R, b, tl,
have same
t2r
15
For the solution method formulated in the previous section, a sequential quadratic programming (SQP) method and MATLAB were used. The SQP method uses an iterative procedure, and it generates a quadratic programming (QP) subproblem at each iteration and updates the estimate, the Hessian of the Lagrangian.13 The method does not always guarantee the global minimum. Therefore, in this study, four different starting points have been used to find the minimum value. It, in fact, has provided the same minimum values, regardless of the starting points. The obtained optimum design parameters are given in Table 2. It may be noted that for manufacturing reasons, the parameter value used in actual design is not exactly the same as the theoretical optimum values. Table
1
Constants
t,
f5,
ii,
LI,
L
L3r
d,
c
variables
15
Design
t,
R and b.
Table 3 shows the calculated characteristics of the stage based on the optimum values of the design parameters. From the table, it is very clear that the maximum yaw motion is 439.5 arcsec (2.131 mrad) when all the piezoelectric actuators produce the displacement of 15 km. This large yaw motion may allow successful yaw error compensation for wafer positioning applications. The table also indicates that the maximum travel along the X and Y-axes are ‘7’7.2 and 88.9 pm, respectively. Figure 9 shows the simulation results of straight line motions along the X-axis and Y-axis. From the figures, it can be seen that the stage can also amplify the actuator motion successfully. A prototype XY0 stage has been constructed based on the optimal design procedure described (Figure IO). The performance of the prototype stage has been tested at KAIST. Experimental and simulation results for the maximum motion range of the XY0 stage are shown in Figure 11. From the exper-
used for the stage design
Constants
Sr
uy (MM
Ci (mm)
CZ (mm)
C, (Nm/arcsec)
C, (Nm/arcsec)
Cj (Hz)
Values
10
270
150
30
0.1
0.1
200
PRECISION
ENGINEERING
25
Ryu et al.: Flexure hinge based on XW wafer stage Table 2
Design variable
sets Design variable sets Optimum and design values
Start points Design variables (mm)
s Sl
Ss2
s s3
Ss4
Sopt
‘design
b R
10.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 4.5 4.5 4.5 15.0 15.0
20.0
15.0 4.0 1.5 1.5 1.5 1.5 1.5 1.5 50.0 46.0 50.0 20.0 30.0
10.0
20.000 3.2770 0.5000 2.4065 1.0069 3.0000 0.5000 3.0000 18.790 27.611 29.180 42.676 36.804
20.0 3.3 0.5 2.4 1.0 3.0 0.5 3.0 18.8 27.5 29.0 41.7 36.9
t t3 t4 5 t6
;
d3 c
7.0 3.0 3.0 3.0 3.0 3.0 3.0 50.0 50.0 50.0 50.0 50.0
imental results, the stage has the total motion range of 41.5 pm, 47.8 pm, and 322.8 arcsec along X, Y, and 8, directions, respectively. These are quite different from the simulation results listed in Table 3. It can be noted that the simulation results were calculated using a assumption that a piezoelectric actuator has a maximum elongation of 15 Frn. However, in the experiments, the maximum Table 3
Characteristics
7.0 3.0 0.5 3.0 0.5 3.0 0.5 50.0 4.5 50.0 15.0 15.0
-
length change of the piezoelectric actuator is approximately 12 pm. Such a difference can be explained that the effective expansion of a piezoelectric actuator under load with an external spring (hinge mechanism) is smaller than that of the actuator under static load14 and can be the reason the XY8 stage has smaller motion range than was expected. The difference between the experimen-
of the stage
Design variable sets Optimum and design values
Start points
Sdesign
(arcsec) cnax
KY (N/Fm) K, (N/pm) 4, (Nm/arcsec) her (Nm/arcsec) I& (Nm/arcsec) 01 (Hz) Active design variables Active constraints Violated constraints
26
1336.8 227.57 90.084 49.459 103.92 57.024 1.2069 6.9927 1.2069 6.9927 2.9252 4.2322 0.0081 0.0075 0.0081 0.0075 0.0024 0.1487 52.901 306.79 All 99,
9io
All None
AH but 9s gs>sil, glz
330.37 40.166 46.312 6.8293 6.8293 5.8827 0.0308 0.0308 0.0487 196.38
Ll,
s
440.70 77.244 88.891 15.773 16.013 18.182 0.1000 0.1006 0.2035 229.98
All None
None g7, &
50.211 3.6431 4.0866 2.2828 2.5250 5.0366 0.0031 0.0031 0.0052 127.71
65129129g13
g-e
511>
439.46
77.172 88.806 15.627 15.864 18.228 0.1000 0.1007 0.2025 229.30 b, tl> t4, None
gm
g13
None
ts>t6
None
JULY 1997 VOL 21 NO 1
_
Ryu et al.: Flexure hinge based on XYB wafer stage
(a) when the stage moves along X axis Figure 9
Behavior
(b) when the stage moves along Y axis
of the stage mechanism
tal and simulation results may also be introduced from the machining error of the XYCI stage and measurement error as well as the modeling error of T,, in Equation (22). The first natural frequency of the prototype stage is also measured. From the experiment, the stage has the first natural frequency of 156 Hz. It can be noted that the pre-
Figure 10
Photograph
PRECISION
ENGINEERING
of a prototype
dieted first natural frequency of 229.3 Hz, shown in Table ?, was calculated using a assumption that all bodies excluding body 1 (output body) are massless. This assumption, as well as the modeling error of stiffness matrix K, can cause the difference between the predicted and observed first natural frequency.
XYB stage 27
Ryu et al.: Flexure hinge based on XY0 wafer stage
,im
77.2
0
5
10 w
15
(a) X directional motion Figure 11
Achieved
0
5
(w4
motion
10
u3 WI
(b) Y directional motion
(c) 6, directional motion
range
Conclusions An optimal design procedure for a XY8 stage has been presented. The design has focused on the development of a precision stage with large yaw motion. It has been shown that the performance of the flexure hinge mechanism is heavily influenced by the design parameter selection. This paper has provided modeling and analysis of such important design parameters as hinge thickness, hinge radius, and hinge width. In particular, a mathematical formulation of the optimization problem has been described and solved using an SQP method. Simulation results have demonstrated that the optimally designed stage can provide about 10 times larger yaw motion than previous conventional parallel linkage type micropositioning stages. The optimal design procedure can be applied to the general design problems of micromotion stages with monolithic flexure hinge mechanism.
2
3
4
5
6
7
8
9 IO 11
Acknowledgment This work was funded by the Korea Science and Engineering Foundation (KOSEF). The authors thank KOSEF for their financial support (940200-09-01-3). References 1
28
15
~2 (w)
Taniguchi, N. “Current status in, and future trends of, ultraprecision machining and ultrafine materials processing,” Ann CARP, 1983,32, 573-582
12
13 14
Futami, S., Furutani, A., and Yoshida, S. “Nanometer positioning and its microdynamics,” Nanotechnology, 1990, 1, 31-37 Otsuka, J., Fukada, S., Kawase, Y., lida, N., and Aoki, Y. “Ultraprecision positioning using lead screw drive (2nd report),” Int J Japan Sot Prec Eng, 1993,27, 142-147 Moriyama, S., Harada, T., and Takanashi, A. “Precision X-Y stage with a piezo-driven fine-table,” Bull Japan Sot Prec Eng, 1988, 22, 13-17 Taniguchi, M., Ikeda, M.,‘Inagaki, A., and Funatsu, R. “Ultra precision wafer positioning by six-axis micromotion mechanism,” Int J Japan Sot Prec Eng, 1992,26,35-40 Sakuta, S., Ogawa, K., and Ueda, K. “Experimental studies on ultra-precision positioning,” Int J Japan Sot Prec Eng, 1993,27, 235-240 Kendall, R., Doran, S., and Weissmann, E. “A servo guided X-Y-theta stage for electron beam lithography,” J Vat Sci Techno/ B, 1992,9,301 S-3023 Scire, F. E. and Teague, E. C. “Piezodriven 50.km range stage with subnanometer resolution,” Rev Sci Instrum, 1978, 49, 1735-1740 Paros, J. M. and Weisbord, L. “How to design flexure hinge,” Machine Design, 1965, 37, 151-157 Timoshenko, S. and Goodier, J. N. Theory of Elasticity, 3rd ed. New York: McGraw-Hill, 1970, pp. 309-313 Ryu, J. W. “6-Axis ultraprecision positioning mechanism design and positioning control,” Ph.D. diss., KAIST, Taejon, South Korea, 1997 Smith, S. T., Chetwynd, D. G., and Bowen, D. K. “Design and assessment of monolithic high-precision translation J Ph ys E: Sci Instrum, 1988,20,977-983 mechanisms,” Grace, A. Optimization Toolbox User’s Guide. The MathWorks Inc., 1990 Tojo, T. and Sugihara, K. “Piezoelectric-driven turntable with high positioning accuracy (1st report)-Operation principle and basic performance,” Bull Japan Sot Prec Eng, 1989,23,65-71
JULY 1997 VOL 21 NO 1
Optimal design of a flexure hinge based XY0 wafer stage Jae W. Ryu,* Dae-Gab Gweon,* and Kee S. Moont *Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yusong-Gu, Taejon, Korea and tDepartment of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, Ml USA
Optimal design of a XY8 micromotion stage is presented. The stage consists of a monolithic flexure hinge mechanism with three piezoelectric actuators. This paper describes the procedures of selecting parameters for the optimal design. In particular, it presents a mathematical formulation of the optimization problem. Based on the solution of the optimization problem, the final design of the stage is also presented. Experimental results indicate that the design procedure is effective, and the designed stage has the total range of 41.5 pm and 47.8 pm along the X- and Y-axes, respectively, and the 0 1997 maximum yaw motion range of 322.8 arcsec (1.565 mrad). Elsevier Science Inc. Keywords: XY0 micromotion tric actuator; optimal design
stage; flexure hinge mechanism;
Introduction Because of the growing levels of integration of semiconductor products, requirements for the positioning precision of wafer stages are continuously increasing. For the fabrication of VLSI circuits, for example, the positioning accuracy of 0.01 pm is necessary. 1 In addition, recent use of large sized wafers demands longer travel range of the wafer stages. The emphasis has, therefore, necessitated the development of a wafer stage with high accuracy and long travel for wafer positioning. XY tables with DC servo motor and lead screw (ball screw or sliding screw) have been popular for wafer positioning. Although they have a relatively long travel range, it has been known that they have the nonlinear spring effect of balls and stickslip. 2,3 Such problems are difficult to eliminate; therefore, submicrometer positioning accuracy is rarely achieved by the XY tables with DC servomotor and lead screw. In practice, for this reason, the Address reprint requests to Dr. Dae-Gab Gweon, Korea Advanced Institute of Science and Technology, Department of Mechanical Engineering, 373-Kusong-Dong, Yusong-Gu; Taejon 305-701, South Korea.
Precision Engineering 21:18-28, 1997 0 1997 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010
piezoelec-
wafer positioning usually uses a two-stage positioning scheme. 4-6 In the first stage, coarse positioning is achieved by the XY table with DC servomotor and lead screw. In the second stage, fine positioning is achieved by a high-precision micromotion stage. The micromotion stage usually has short travel range but high positioning resolution. In fact, the micromotion stage is required to have sufficient travel range to compensate for the position errors that occurred during the coarse positioning. It is also important that the micromotion stage has the capability to correct yaw motion error from the coarse XY positioning. The yaw error can also be caused by wafer fixturing devices. In particular, the need for long travel range for coarse positioning also requires the large yaw motion range for the fine positioning stage. For example, friction drives in electron beam lithography system can cause more than +360 arcsec (50.1”) of the yaw motion error.’ To correct the yaw motion error, parallel linkage-type micromotion stages have been used because of their simple structure. These, however, have limited yaw range of less than 36 arcsec. In this paper, the design of a flexure hinge-
0141-6359/97/$17.00 PII SO141-6359(97)00064-O
Ryu et al.: Flexure hinge based on XY0 wafer stage ing bodies are denoted by the numbers 1 through 13 (noncircled). Note that the number 1 body is the output body and will be activated by the three piezoelectric actuators acting on bodies 5, 9, and 13. The complete design of the XYO design is shown in I;igure 3. Modeling
L Figure 1 Double chanical lever
compound
flexure
pivoted
me-
based micromotion stage with a large yaw motion of 322.8 arcsec is presented. It utilizes three piezoelectric actuators and a monolithic flexure hinge mechanism. The flexure hinge mechanism uses three double compound mechanical levers to amplify each actuator motion. The performance of the flexure hinge mechanism is heavily influenced by selection of such parameters as lever length, hinge thickness, and hinge radius.8 This paper describes a procedure for selecting parameters to optimize the design of the XY8 stage. In particular, the optimal design procedure to maximize the yaw motion utilizing a mathematical model of the mechanical system is presented. Structure
of the XY0 micromotion
stage
The XY0 stage consists of three piezoelectric actuators and a monolithic flexure hinge mechanism. The flexure hinge mechanism can be represented as a joint mechanism between an output body (wafer holder) and the actuators. Such mechanisms provide almost no backlash or stickslip friction and provide smooth, continuous displacement positioning. Typically, the flexure hinge mechanism contains a number of hinges and mechanical levers. Figure I shows the double compound mechanical lever employed for the XYO stage design. The displacement generated by a piezoelectric actuator is 15 pm. Therefore, to amplify the actuator displacement to obtain a large yaw motion, use of the compound mechanical lever to provide mechanical displacement gain without significant motion loss is required. Figure 2 also shows a schematic of the complete XYB stage design. It is clear from the figure that the design utilizes three mechanical input levers radially disposed with 120” pitch. The symmetric design reduces the effect of temperature gradient on the structure. In this figure, the movPRECISION
ENGINEERING
of the stage motion
Let us assume that the levers and the hinges in the flexure hinge mechanism be modeled as rigid bodies connected through translational/rotational springs. Let us also regard a piezoelectric actuator be a translational spring. Then, the XY0 stage mechanism can be assumed to be a springmass system. In this study, the XY0 stage employs a right circular single-axis flexure hinge, and it is shown in Figure 4. The spring rates of the hinge have been calculated by using Paros and Weisbord equations for a single-axis flexure hinge9 and torsional equation of a rectangular bar.iO From Lagrange’s equation, the motion of the XY0 micromotion stage can be represented by M%+Kx=F
(1)
where M, K, and F are the system mass matrix, stiffness matrix, and force vector, respectively. l1 The system displacement vector x is defined by x =
[q’q . . . , q’:
. . . , cpy-
(2)
where Nb is the number of moving bodies in the system and qi is the displacement vector of the origin of the body i or qi = [xi, yi, zi, Oi, t$, Of]*. The XY0 stage is a planar motion mechanism; therefore, each rigid body has only three degrees of freedom, (x, y, 0,). However, in this study, the model considers three additional degrees of freedom (x, 0, fly) for the analysis of tilt stiffness. Note that the size of the displacement vector x is (6 X 13) X 1, because there are 13 rigid bodies in the system, and each body has six degrees of freedom. Similarly, the force vector F also has the same size (6 X 13) X 1. The mass matrix M and the stiffness matrix K have the same size of (6 X 13) X (6 X 13). The natural frequency of the system can be calculated from the given information of mass and stiffness of the system. Consider the following eigenvalue equation. IK- w%[ = 0
(3)
Note that the positive square root of the solutions of Equation (3) are the natural frequencies of the system. The stiffness of the XY0 stage can be obtained from the displacement caused by external load at the center of the output body. For example, con19
Ryu et al.: Flexure hinge based on XXI wafer stage
moving
??
output
??
??
??
Figure 2
Schematic
diagram
P’, . . . , PqT
hinges
: 0,
... ,@$
(6, - &K&,,)x,
(7)
= F, - K,,K&
or wx’ = F’
(8)
where
f’ = [I N, 0, 0, 0, 0, 01 i-2,3,
. . . , N,,3
K’ = K,, - K,,K;;K,,
(5)
(9)
F’ = F, - K,,K;;F,
and fi is the resolved force vector of the body i. The stiffness of the system for the other directions can be obtained using the same procedure. To calculate the displacement of the output body attributable to a displacement by the actuators, let q1 be the output displacement vector, and let the input displacement vectors be $, qg, $3, respectively. Also, let x1 = [qlT, q5T, q T, q13 IT and x, represent the displacement vectors of the other bodies excluding bodies 1, 5, 9, and 13. Then, Equation (4) becomes
Equation
(10)
(8) can be rewritten as
(11) where XL = q1 and xi = [SST, qgT, q13’jT. In the equation, the subscripts o and i represent the output and the input bodies, respectively. Given that the load applied to the output body FL is known, the displacement vector of the output body x: can be given by x’0 = &,-‘(F,
(6) 20
: 5, 9, 13
From which
(4)
where f’=O,
body : 1 (XYe stage)
input bodies flexure
: 1, ... , 13
of the XY0 stage mechanism
sider a 1 N load applied to the output body. The direction of the load is, for example, X. Then, the stiffness of the system can be calculated from the following linear simultaneous equations. Kx = [P,
bodies
Also,
- Kb;x:)
(12)
the required force at the input bodies is JULY
1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
Figure 3
Complete
design
F,, = -F; = -(K;,x’,
of the XY8 stage mechanism
+ K:;x;)
(13)
Equations (12) and (13)) provide a basic and direct relationship between input and output of the system. Given the input displacement by actuators and the load applied to the output body, the ,” Fz
and coordinate
resulting output displacement and the required input force can be calculated. However, the inverse relationship between the input and the output also must be obtained. In particular, the required input displacement of the actuators must be calculated to get a desired displacement of the output body. Consider Equation (12)) which can be represented by K;;x; =
C-&
systems
(6x31~1
F’, (6x1)
-
v’,x;
(14)
(6x1)
From Equation (14), it is clear that there are 18 unknown variables and only six equations; therefore, there is no unique solution. In this study, a different coordinate system is considered to adjust the number of unknown variables and the equations. fipre 5 shows the transformed coordinate system. It may be noted that the direction of local coordinate x2has been aligned to the direction of piezoelectric actuation. Hence,
Figure 4 PRECISION
Right circular
single-axis
ENGINEERING
flexure
hinge
-i X -i Y -i il z 21
Ryu et al.: Flexure hinge based on XYO wafer stage for understanding the effect of such design parameters as hinge radius, width, thickness, lever ratio, and link length. In this section, the optimal selection of the design parameters is presented. As shown in Fig-LLreI, the stage uses a double compound flexure pivoted mechanical lever to amplify the length change of the piezoelectric actuator. Ideally, the flexure hinge is a revolute joint and the mechanical gain G is
(23)
Figure 5
Coordinate
transform
= Re,zXi
From Equation
-__ from xyz to xyz (15)
(S),
KR = F
(16)
where, i?= RTK’R
(17)
F=
(18)
R’F
R = diag(l,
1, R;,,,
R_;,Z,3)
(19)
Because the stage has only plane movement (z” = Bx = 0; = 0: = 0)) and the input body moves only along with the direction of piezoelectric actuation, Equation (16) becomes
(20) where the output displacement vector, x, = [x1, y’, and the input displacement vector, xi = [35, 07 2%;“9 -g3]T . From the Equation (20) and for a given F,, the input and output relationship can be found. For example, let F, = 0, then, xi= -KZli’KLX& = T;,x,
(21)
Note that the matrix Ti, is a transformation matrix, and it gives the input and output relationship between xi and x,. Ti, = - Ki: Ka,
Optimal
(22)
design of the stage
In the previous sections, the model of the stage has been described. This model provides the basis 22
It can be noted that the gain relies only on the lever length ratio. However, in actual case, the flexure hinge provides not only rotations but also has translational compliances. Therefore, the effect of these unwanted hinge compliances on the output displacement must be assessed. Figure 6 shows simulation results for the purpose. Fig-ure 6(a) describes the effect of the gain G and the hinge thickness ton the output displacement and required input force Fin when the input displacement u is 1 pm. In the case, there is no load applied to the output body. It is very clear from (a) that the design with higher gain provides larger output displacement for a given hinge thickness. However, when there is a static load applied to the output body, higher gain does not always provide larger output displacement. For example, from (b), the gain of about 50 provides the maximum output displacement when 1 N of static load Fwas assumed at the output body. Fig-ure 7 shows simulation results of the effect of the gain and the hinge radius R on the output displacement and required input force Fi, when the input displacement u is 1 pm. Similarly, higher gain provides larger output displacement for a given hinge radius in the case of no static load. In the case of a given static load (IN), the gain of about 50 provides the maximum performance for most cases. From Figures 6 and 7, the output-to-input displacement ratio is smaller than the ideal gain G. It may be explained that the translational compliances, especially the longitudinal compliance (x-axis in Figure 41, as well as the rotational compliance about z-axis have an effect on the output-to-input displacement ratio. From the figures, furthermore, it is clear that the output displacement and the required input force are a more sensitive to the load when the hinge has a smaller hinge thickness for a specific hinge radius or a larger hinge radius for a specific hinge thickness, because these provide a smaller y-directional stiffness. The previous simulation approach provides a rough tool to achieve the optimal flexure hinge JULY 1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
lool-----l 80
z f=O.5
80 3 A
L&I 40
0
1=2.5
0
t=2.5
$ 150 5 P .100 * s! '5 50 K
H.5
20
200
i3
50
f=1.5
f=O.5
0 kz 0
100
50
1
G
(4 100 80. 80. 3h
ti2.5
40. M.5
G (b) F = 1 N
Figure 6 Simulated y/u and 6, versus G with tvariations L = L, + f,, + L, + 2b, R = 4 km, b = 20 mm)
design. In this paper, a rigorous approach is taken to optimize the design parameters by using mathematical programming. Figure 8 shows the design variables for the hinge mechanism used for the XY8 stage. Note that the hinge radius is a design variable for all the hinges considered. In practice, a lack of flexibility to change the drilling diameter causes manufacturing problems for the hinge design with various hinge radii. The objective of the optimization problem is to maximize the yaw motion of the output body (minimize l/Q:). The problem includes a number of constraints. For example, the maximum stress at the hinge point should be less than the yield stress. There are also constraints for the system size, minimum tilt stiffness, and the natural frequency of the system. First, consider the maximum stress constraint. Because the maximum stress occurs at the hinge point with minimum thickness
(u = 1 km, L, = L, = 10 mm, L, = 10 - 80 mm,
(26)
M, = k&3
Because the maximum rotational movement occurs when each piezoelectric actuator reaches its maximum elongation, the maximum allowable rotation at the hinge is
s,e < -
t2b
(27)
6k,K, uy
In Equation (27) Sfis a safety factor, and o,is the yield stress of the hinge material. Table 1 shows the given constant values. As denoted in Figure 8, let 13: be the rotational movement of body i with respect to the Zaxis, let k: be the rotational stiffness of hinge i, let gi be a stress concentration factor of hinge i, and let ti be the minimum thickness of hinge i, respectively. Now, for the six hinges, the six constraints for the maximum stress are
6 MA umax
=
__
t2b
(28)
where MZ is the moment at the hinge, and K, is a stress concentration factor given by Smith et ali2 K, =
2.7t + 5.4R 8R+
t
+ 0.325
(25)
Let 12,be the rotational stiffness of the hinge, then PRECISION
ENGINEERING
(29)
g, =
‘b
s,le;l - -ciy
(30)
23
Ryu et al.: Flexure hinge based on XY0 wafer stage 100,
250,
I
(a)F=O
‘OOl-----l
250r------
100 G
lb) F=lN Figure 7 Simulated y/u and 6, versus G with R variations L=L1+L,+L,+20,t=1.5mm,b=20mm)
Second, consider the constraints for the system size. Based on the description of variables in Figure 8, there can be four additional constraints for the problem. They are g,=(L+c)-c,
C,-
g,=
[,,+;+;j
g,,=
(31)
[c-
(2R+;)]
(32)
- L,
i,,+$+;j - L,
(34)
where C, and C, are constants, and the values are given in Table 1. Third, consider the constraints for the minimum tilt stiffness value. Based on Equation (4) and moment equation, two additional constraints for X and Y moments can be obtained by
c,+o 1
g,,=
(35)
x
g,,=
&&a Y
where C, and C, are also given in Table 1. Finally, consider the constraint for the natu24
(u = 1 pm, L, = L, = 10 mm, L, = 10 - 80 mm,
ral frequency of the system. The first natural frequency w1 can be obtained from Equation (3) and must satisfy the following constraint. g,, = c, - 01 < 0
(37)
It is assumed that only body 1 (output body) has its mass and the others are massless. Note that this study did not consider the constraints for the force required to move the stage nor the load applied to the actuators. In fact, it is known that the piezoelectric actuators provide similar displacements, regardless of the load at a given applied voltage. In summary, the described optimization problem can be written as minimize 1 /e,
subject to LJi< 0
i= 1, . . .
I
13
and 1.0 < R< 7.0 lO
3.0
4.5 < L,< 50
j=l
, ..., 6
k=1,...,3 JULY 1997 VOL 21 NO 1
Ryu et al.: Flexure hinge based on XY0 wafer stage
Design
variables
All hinges Figure
8
Design
: R, b, tl,
have same
t2r
15
For the solution method formulated in the previous section, a sequential quadratic programming (SQP) method and MATLAB were used. The SQP method uses an iterative procedure, and it generates a quadratic programming (QP) subproblem at each iteration and updates the estimate, the Hessian of the Lagrangian.13 The method does not always guarantee the global minimum. Therefore, in this study, four different starting points have been used to find the minimum value. It, in fact, has provided the same minimum values, regardless of the starting points. The obtained optimum design parameters are given in Table 2. It may be noted that for manufacturing reasons, the parameter value used in actual design is not exactly the same as the theoretical optimum values. Table
1
Constants
t,
f5,
ii,
LI,
L
L3r
d,
c
variables
15
Design
t,
R and b.
Table 3 shows the calculated characteristics of the stage based on the optimum values of the design parameters. From the table, it is very clear that the maximum yaw motion is 439.5 arcsec (2.131 mrad) when all the piezoelectric actuators produce the displacement of 15 km. This large yaw motion may allow successful yaw error compensation for wafer positioning applications. The table also indicates that the maximum travel along the X and Y-axes are ‘7’7.2 and 88.9 pm, respectively. Figure 9 shows the simulation results of straight line motions along the X-axis and Y-axis. From the figures, it can be seen that the stage can also amplify the actuator motion successfully. A prototype XY0 stage has been constructed based on the optimal design procedure described (Figure IO). The performance of the prototype stage has been tested at KAIST. Experimental and simulation results for the maximum motion range of the XY0 stage are shown in Figure 11. From the exper-
used for the stage design
Constants
Sr
uy (MM
Ci (mm)
CZ (mm)
C, (Nm/arcsec)
C, (Nm/arcsec)
Cj (Hz)
Values
10
270
150
30
0.1
0.1
200
PRECISION
ENGINEERING
25
Ryu et al.: Flexure hinge based on XW wafer stage Table 2
Design variable
sets Design variable sets Optimum and design values
Start points Design variables (mm)
s Sl
Ss2
s s3
Ss4
Sopt
‘design
b R
10.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 4.5 4.5 4.5 15.0 15.0
20.0
15.0 4.0 1.5 1.5 1.5 1.5 1.5 1.5 50.0 46.0 50.0 20.0 30.0
10.0
20.000 3.2770 0.5000 2.4065 1.0069 3.0000 0.5000 3.0000 18.790 27.611 29.180 42.676 36.804
20.0 3.3 0.5 2.4 1.0 3.0 0.5 3.0 18.8 27.5 29.0 41.7 36.9
t t3 t4 5 t6
;
d3 c
7.0 3.0 3.0 3.0 3.0 3.0 3.0 50.0 50.0 50.0 50.0 50.0
imental results, the stage has the total motion range of 41.5 pm, 47.8 pm, and 322.8 arcsec along X, Y, and 8, directions, respectively. These are quite different from the simulation results listed in Table 3. It can be noted that the simulation results were calculated using a assumption that a piezoelectric actuator has a maximum elongation of 15 Frn. However, in the experiments, the maximum Table 3
Characteristics
7.0 3.0 0.5 3.0 0.5 3.0 0.5 50.0 4.5 50.0 15.0 15.0
-
length change of the piezoelectric actuator is approximately 12 pm. Such a difference can be explained that the effective expansion of a piezoelectric actuator under load with an external spring (hinge mechanism) is smaller than that of the actuator under static load14 and can be the reason the XY8 stage has smaller motion range than was expected. The difference between the experimen-
of the stage
Design variable sets Optimum and design values
Start points
Sdesign
(arcsec) cnax
KY (N/Fm) K, (N/pm) 4, (Nm/arcsec) her (Nm/arcsec) I& (Nm/arcsec) 01 (Hz) Active design variables Active constraints Violated constraints
26
1336.8 227.57 90.084 49.459 103.92 57.024 1.2069 6.9927 1.2069 6.9927 2.9252 4.2322 0.0081 0.0075 0.0081 0.0075 0.0024 0.1487 52.901 306.79 All 99,
9io
All None
AH but 9s gs>sil, glz
330.37 40.166 46.312 6.8293 6.8293 5.8827 0.0308 0.0308 0.0487 196.38
Ll,
s
440.70 77.244 88.891 15.773 16.013 18.182 0.1000 0.1006 0.2035 229.98
All None
None g7, &
50.211 3.6431 4.0866 2.2828 2.5250 5.0366 0.0031 0.0031 0.0052 127.71
65129129g13
g-e
511>
439.46
77.172 88.806 15.627 15.864 18.228 0.1000 0.1007 0.2025 229.30 b, tl> t4, None
gm
g13
None
ts>t6
None
JULY 1997 VOL 21 NO 1
_
Ryu et al.: Flexure hinge based on XYB wafer stage
(a) when the stage moves along X axis Figure 9
Behavior
(b) when the stage moves along Y axis
of the stage mechanism
tal and simulation results may also be introduced from the machining error of the XYCI stage and measurement error as well as the modeling error of T,, in Equation (22). The first natural frequency of the prototype stage is also measured. From the experiment, the stage has the first natural frequency of 156 Hz. It can be noted that the pre-
Figure 10
Photograph
PRECISION
ENGINEERING
of a prototype
dieted first natural frequency of 229.3 Hz, shown in Table ?, was calculated using a assumption that all bodies excluding body 1 (output body) are massless. This assumption, as well as the modeling error of stiffness matrix K, can cause the difference between the predicted and observed first natural frequency.
XYB stage 27
Ryu et al.: Flexure hinge based on XY0 wafer stage
,im
77.2
0
5
10 w
15
(a) X directional motion Figure 11
Achieved
0
5
(w4
motion
10
u3 WI
(b) Y directional motion
(c) 6, directional motion
range
Conclusions An optimal design procedure for a XY8 stage has been presented. The design has focused on the development of a precision stage with large yaw motion. It has been shown that the performance of the flexure hinge mechanism is heavily influenced by the design parameter selection. This paper has provided modeling and analysis of such important design parameters as hinge thickness, hinge radius, and hinge width. In particular, a mathematical formulation of the optimization problem has been described and solved using an SQP method. Simulation results have demonstrated that the optimally designed stage can provide about 10 times larger yaw motion than previous conventional parallel linkage type micropositioning stages. The optimal design procedure can be applied to the general design problems of micromotion stages with monolithic flexure hinge mechanism.
2
3
4
5
6
7
8
9 IO 11
Acknowledgment This work was funded by the Korea Science and Engineering Foundation (KOSEF). The authors thank KOSEF for their financial support (940200-09-01-3). References 1
28
15
~2 (w)
Taniguchi, N. “Current status in, and future trends of, ultraprecision machining and ultrafine materials processing,” Ann CARP, 1983,32, 573-582
12
13 14
Futami, S., Furutani, A., and Yoshida, S. “Nanometer positioning and its microdynamics,” Nanotechnology, 1990, 1, 31-37 Otsuka, J., Fukada, S., Kawase, Y., lida, N., and Aoki, Y. “Ultraprecision positioning using lead screw drive (2nd report),” Int J Japan Sot Prec Eng, 1993,27, 142-147 Moriyama, S., Harada, T., and Takanashi, A. “Precision X-Y stage with a piezo-driven fine-table,” Bull Japan Sot Prec Eng, 1988, 22, 13-17 Taniguchi, M., Ikeda, M.,‘Inagaki, A., and Funatsu, R. “Ultra precision wafer positioning by six-axis micromotion mechanism,” Int J Japan Sot Prec Eng, 1992,26,35-40 Sakuta, S., Ogawa, K., and Ueda, K. “Experimental studies on ultra-precision positioning,” Int J Japan Sot Prec Eng, 1993,27, 235-240 Kendall, R., Doran, S., and Weissmann, E. “A servo guided X-Y-theta stage for electron beam lithography,” J Vat Sci Techno/ B, 1992,9,301 S-3023 Scire, F. E. and Teague, E. C. “Piezodriven 50.km range stage with subnanometer resolution,” Rev Sci Instrum, 1978, 49, 1735-1740 Paros, J. M. and Weisbord, L. “How to design flexure hinge,” Machine Design, 1965, 37, 151-157 Timoshenko, S. and Goodier, J. N. Theory of Elasticity, 3rd ed. New York: McGraw-Hill, 1970, pp. 309-313 Ryu, J. W. “6-Axis ultraprecision positioning mechanism design and positioning control,” Ph.D. diss., KAIST, Taejon, South Korea, 1997 Smith, S. T., Chetwynd, D. G., and Bowen, D. K. “Design and assessment of monolithic high-precision translation J Ph ys E: Sci Instrum, 1988,20,977-983 mechanisms,” Grace, A. Optimization Toolbox User’s Guide. The MathWorks Inc., 1990 Tojo, T. and Sugihara, K. “Piezoelectric-driven turntable with high positioning accuracy (1st report)-Operation principle and basic performance,” Bull Japan Sot Prec Eng, 1989,23,65-71
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