(Manuscript received June 3, 2015, revised Nov. 17, 2015)

IEEJ Journal of Industry Applications Vol.5 No.2 pp.141 147 DOI: 10.1541/ieejjia.5.141 Decoupling Control Method for High-Precision Stages using Multiple Actuators considering the Misalignment among the Actuation Point, Center of Gravity, and Center of Rotation Wataru Ohnishi a) Student Member, Hiroshi Fujimoto Senior Member Koichi Sakata Member, Kazuhiro Suzuki Non-member Kazuaki Saiki Member (Manuscript received June 3, 2015, revised Nov. 17, 2015) Paper High-precision stages are widely used in the semiconductor and flat panel industries. Because six degrees of freedom have to be controlled in these stages, coupling forces can degrade their control performance and stability. This paper proposes a decoupling control method from the scanning motion x to the pithing motion θ y using multiple actuators considering the misalignment of the actuation point, the center of gravity (CoG), and the center of rotation (CoR). The method proposed in this paper consists of three steps: 1) a detailed modeling of the the scanning motion x and the pitching motion θ y, 2) a changeable actuation point stage, and 3) an integrated design of mechanism and control. The validity of the proposed method is demonstrated by experiments. Keywords: multiple actuator, decoupling control, integrated design of mechanism and control, high-precision stage 1. Introduction High-precision scan stages play an important role in the manufacturing processes for semiconductors and liquid crystal displays (1) (2). In these applications, high-precision stages have to control six degrees of freedom (DOFs: x,y,z,θ x,θ y, and θ z ) (3). For this purpose, a dual stage that consists of a short-stroke 6-DOF fine stage and a long-stroke coarse stage is widely used (4) (5). Our research group designed a dual stage, which is shown in Fig. 1. In such multi-input multi-output (MIMO) systems, the coupling force degrades the control performance. This paper focuses on the scanning motion x and the pitching motion θ y. If the center of gravity (CoG) of the fine stage, the center of rotation (CoR) of the fine stage, the actuation point of the translational force f x, and the measurement point of the translational position x are not at the samepoints, coupling betweenthe scanning motion x and the pitching motion θ y occurs. It is difficult to match these four points because of mechanical constraints such as spatial limitation and shifting of the CoG due to load mass variation. This paper proposes a detailed model of the scanning motion x and the pitching motion θ y considering the misalignment among the CoG, the CoR, the actuation point, and the measurement point. In general, a precise model is important for the design of high-performance feedforward a) Correspondence to: Wataru Ohnishi. E-mail: ohnishi@hflab.k. u-tokyo.ac.jp The University of Tokyo 5-1-5, Kashiwanoha, Kashiwa, Chiba 227-8561, Japan Nikon Corporation 47-1, Nagaodaityou, Sakae, Yokohama 244-8533, Japan Fig. 1. Photograph of the 6-DOF high-precision stage Fig. 2. Side view of the fine stage. By changing the thrust distribution ratio a (see (21) and (22)), the height of the virtual actuation point can be determined arbitrarily (FF)/feedback (FB) controllers (4) (6) (7). Next, on the basis of the detailed model, this paper proposes a block diagram that clearly shows the relationship between the misalignment and the zeros of the transfer functions. A 6-DOF high precision stage that has two VCMs (voice coil motors) in the x direction is then designed. With a thrust distribution, the height of the virtual actuation point can be placed arbitrarily. Additional actuator approach for higher bandwidth FB is provided in Refs. (8) and (9). In this paper, by utilizing the detailed model and multiple actuators, an integrated design of c 2016 The Institute of Electrical Engineers of Japan. 141

(a) Actuator arrangement of the fine stage. Fig. 5. Fine stage model of the scanning motion x and the pitching motion θ y Table 1. Model parameters (b) Sensor arrangement of the fine stage. Fig. 3. Structure of the fine stage Symbol Meaning Value x m Measured position of the fine stage x g1 Position of the CoG of the planar air bearing and the air gyro x g2 Position of the CoG of the fine stage θ y Measured attitude angle of the fine stage f x Input force of the fine stage in the x direction τ y Input torque of the fine stage in the θ y direction M x1 Mass of the planar air bearing and the air gyro 0.077 kg C x1 Viscosity coefficient in the x g1 motion 430 N/(m/s) K x1 Spring coefficient in the x g1 motion 11000 N/m M x2 Mass of the fine stage 5.3kg J θy Moment of inertia of the fine stage 0.10 kgm 2 C θy Viscosity coefficient of the fine stage in the θ y motion 1.6Nm/(rad/s) K θy Spring coefficient of the fine stage in the θ y motion 1200 Nm/rad L m Distance between the measurement point of x m and the CoR 0.028 m L g2 Distance between the CoR and the CoG of the fine stage 0.051 m L fx Distance between the CoR of the fine stage and the actuation point changeable Fig. 4. (a) Side view. (b) Schematic. Structure of the gravity canceller mechanism and control (10) (11) is proposed for decoupling control between the scanning motion x and the pitching motion θ y. This method focuses on the relationship between the numerator of the transfer functions (zeros) and kinematic parameters such as the positions of the CoG, the CoR, the actuation point, and the measurement point. Finally, a method to improve the performance considering the height of the actuation point and the bandwidth of the feedback (FB) controller is proposed. The effectiveness of the proposed method is demonstrated experimentally. 2. Modeling 2.1 Experimental Setup Our research group designed the dual stage shown in Fig. 1. The actuator and sensor arrangement of the fine stage is shown in Fig. 3. As shown in Fig. 2 and Fig. 3(a), the fine stage has two VCMs in the x direction. The fine stage is supported by a 6-DOF air bearing gravity canceller (12). The picture and schematic of the gravity canceller are shown in Fig. 4. The gravity canceller compensates for the gravitational force experienced by the fine stage and supports its 6-DOF without friction. The gravity canceller is composed of three parts: the air gyro, the planar air bearing and the air bearing actuator that supports the (θ x,θ y,θ z ), (x,y) and (z)-directional motion, respectively. As shown in Fig. 4, the air gyro is shaped like a hemisphere. The fine stage slides on the hemispheric surface of the air gyro with an air gap of a few micrometers. In this paper, the center of the hemisphere is called the CoR. In other words, the radius of the curvature of the air gyro determines the height of the CoR. 2.2 Lagrange s Equations In this section, Lagrange s equations are formulated on the basis of the model shown in Fig. 5 and Table 1. First, the relationship between x g1, x g2,andθ y is expressed by x g2 = x g1 + L g2 sin(θ y ), (1) 142 IEEJ Journal IA, Vol.5, No.2, 2016

f x (s) = [J θy + L fx L m M x1 (L fx L g2 )(L g2 L m )M x2 ]s 2 +(C θy +L fx L m C x1 )s+k θy +L fx L m K x1 L g2 M x2 g (15) θ y (s) f x (s) = [L fxm x1 + (L fx L g2 )M x2 ]s 2 + L fx C x1 s + L fx K x1 (16) τ y (s) = [L mm x1 + (L m L g2 )M x2 ]s 2 + L m C x1 s + L m K x1 (17) θ y (s) τ y (s) = (M x1 + M x2 )s 2 + C x1 s + K x1 (18) = [(M x1 + M x2 )J θy + M x1 M x2 L 2 g2 ]s4 + [(M x1 + M x2 )C θy + (J θy + M x2 L 2 g2 )C x1]s 3 +[(J θy + M x2 L 2 g2 )K x1 + (M x1 + M x2 )(K θy M x2 L g2 g) + C θy C x1 ]s 2 +[C θy K x1 + C x1 (K θy L g2 M x2 g)]s + K x1 (K θy L g2 M x2 g) (19) ẋ g2 = ẋ g1 + L g2 cos(θ y ) θ y. (2) The kinetic energy T, the potential energy U, the dissipation function B, and the work W are defined as follows: T = 1 2 M x1ẋ 2 g1 + 1 2 M x2ẋ 2 g2 + 1 2 J θy θ 2 y, (3) U = 1 2 K x1x 2 g1 + 1 2 K θyθ 2 y + L g2 M x2 g cos(θ y ), (4) B = 1 2 C x1ẋ 2 g1 + 1 2 C θy θ 2 y, (5) W = f x [ xg1 + L fx sin(θ y ) ] + τ y θ y. (6) According to (3) and (4), the Lagrangian L = T U is given by L = 1 2 M x1ẋ 2 g1 + 1 2 M x2 [ẋg1 + L g2 cos(θ y ) θ y ] 2 + 1 2 J θy θ 2 y 1 2 K x1x 2 g1 1 2 K θyθ 2 y L g2 M x2 g cos(θ y ). (7) Lagrange s equations are given by ( ) d L L + B = W (i = 1, 2), (8) dt q i q i q i q i where q 1 and q 2 denote the generalized coordinates, q 1 = x g1 and q 2 = θ y, respectively. Finally, according to (1) (8), the following equations (9) and (10) are obtained: ẍ g1 (M x1 + M x2 ) + C x1 ẋ g1 + K x1 x g1 +M x2 L g2 [ cos(θy ) θ y sin(θ y ) θ 2 y] = fx, (9) M x2 L g2 [ẍg1 cos(θ y ) g sin(θ y ) + L g2 θ y cos 2 (θ y ) L g2 θ 2 y sin(θ y )cos(θ y ) ] + J θy θ y + C θy θ y + K θy θ y = τ y + f x L fx cos(θ y ). (10) 2.3 Linearization Assuming cos(θ y ) 1, sin(θ y ) θ y, θ 2 y 0, (9) and (10) are linearized as follows: (M x1 + M x2 )ẍ g1 + C x1 ẋ g1 + K x1 x g1 + M x2 L g2 θ y = f x, (11) (M x2 L 2 g2 + J θy) θ y + C θy θ y + K θy θ y + M x2 L g2 (ẍ g1 gθ y ) = τ y + f x L fx. (12) Fig. 6. Block diagram of G(s) according to (15) (19). The height of the actuation point can be changed by the thrust distribution ratio a (see (21) and (22)) 2.4 Transformation to Measurable Coordinate The generalized coordinate x g1 cannot be measured. Thus, x g1 is converted to x m by f x (s) = x g1(s) f x (s) + L θ y (s) m f x (s), (13) τ y (s) = x g1(s) τ y (s) + L θ y (s) m τ y (s). (14) 2.5 Transfer Functions According to (11) (14), transfer functions (15) (19) are obtained. In this paper, (15) (19) are also expressed as x m (s) g 11 (s) g 12 (s) G(s) = g 21 (s) g 22 (s) = f x (s) τ y (s) (20) θ y (s) θ y (s) f x (s) τ y (s) According to (15) (19), G(s) can also be modeled as the block diagram shown in Fig. 6. This block diagram clearly shows the coupling structure. 3. Proposal of a Changeable Actuation Point Stage Owing to the spatial limitation of the fine stage, the VCM for the x direction cannot be placed at the desired actuation point. In this section, a fine stage structure that can shift the virtual actuation point by means of the thrust distribution of multiple actuators is proposed. This is based on a simple idea shown in Fig. 2. The two VCMs generate force f x1 and f x2 in the x direction by the thrust distribution law, f x1 = af x, f x2 = (1 a) f x, (21) 143 IEEJ Journal IA, Vol.5, No.2, 2016

(a) g 11 (s). (b) g 12 (s). (c) g 21 (s). (d) g 22 (s). (e) Legend. Fig. 7. Coupling path of G(s). The characteristics of G(s) are defined by the height of the actuation point L fx, the measurement point L m, and the CoG of the fine stage L g2 where a denotes the thrust distribution ratio. Here, the height of the virtual actuation point L fx is defined as L fx = al fx1 + (1 a)l fx2, (22) where L fx1 and L fx2 denote the heights of the actuation point of f x1 and f x2 from the CoR. Although this structure doubles the amount of wiring, and the mass of the fine stage increases slightly, this structure has the following advantages. By changing the thrust distribution ratio a, the height of the virtual actuation point L fx can be placed at a desired point such as the position of the CoG and the CoR. Moreover, during the operation, the CoG shifts owing to load mass variation. Because the transfer functions (15) (19) depend on the height of the CoG (L g2 ), the coupling characteristics change dynamically. Even in this case, this structure can reduce coupling forces if the virtual actuation point is placed at the optimal position. 4. Integrated Design of Mechanism and Control ThecouplingpathofG(s) is shown in Fig. 7, which clearly shows the relationship between the numerators of G(s) and the position of the actuation point L fx. 4.1 Equivalence between the Precompensator k 21 and the Actuation Point L fx Figure 8 shows the MIMO system with a precompensator K(s). Here, the control object Q(s) in Fig. 8 is obtained by Q(s) = G(s)K(s), (23) where Q(s), K(s), and G(s) are defined as q 11 (s) q 12 (s) Q(s) = q 21 (s) q 22 (s), K(s) = k 11 (s) k 12 (s) k 21 (s) k 22 (s), g 11 (s) g 12 (s) G(s) =. (24) g 21 (s) g 22 (s) Fig. 8. Multi-input multi-output (MIMO) feedback system with the precompensator K(s) Fig. 9. Substitution (25) for Fig. 8 In this section, the following precompensator is considered. 1 0 K(const) = k 21 1 (25) Substituting (25) for Fig. 8, Fig. 9 is obtained. In this case, the control object Q(s) can be represented by Fig. 10. Here, the precompensator k 21 and the position of the actuation point L fx are connected in parallel. Hence, there is a linear relationship between the precompensator k 21 and the actuation point L fx ; varying k 21 results in variation in L fx. 144 IEEJ Journal IA, Vol.5, No.2, 2016

Fig. 10. Control object Q(s) with the precompensator k 21. Because the precompensator k 21 and the actuation point L fx are connected in parallel, varying k 21 results in a variation in L fx Fig. 11. CoR-driven method (L fx = 0). The coupling term is L g2 M x2 s 2 /, which has a low gain in the low frequency range 4.2 Decoupling g 21 (s) Using the Changeable Actuation Point L fx Figure 7 shows that by changing the actuation point L fx, the coupling transfer function g 21 (s) can be suppressed without any impact on g 12 (s) andg 22 (s). In this section, CoR-driven method and CoG-driven method are proposed. CoR-driven method (Fig. 11) If L fx is set as 0, this means the actuation point is placed at the CoR, L fx (M x1 s 2 + M x2 s 2 +C x1 s + K x1 ) becomes 0. Now, the coupling term is only L g2 M x2 s 2 /. Then, the coupling gain of g 21 in low frequency range is suppressed. CoG-driven method (Fig. 12) If L fx is set as L g2, this means the actuation point is placed at the CoG, (L g2 L fx )M x2 s 2 becomes 0. Now, the coupling term is only L fx (M x1 s 2 + C x1 s + K x1 )/. The coupling gain of g 21 in high frequency range is thus suppressed considering M x1 M x2 (see Table 1). The comparison between the CoR- and the CoG-driven method is shown in Fig. 13. It is clear from the figure that there is a trade-off between the CoR- and the CoG-driven method in low and high frequency ranges. This observation suggests that it is better to select a virtual actuation point L fx that has a small coupling gain g 21 (s) at the frequency of the sensitivity function peak of the θ y closed loop. This is because the output of g 21 (s) can be considered a disturbance of the θ y closed loop as shown in Fig. 8. 5. Experiment Measurements are performed by using the experimental stage shown in Fig. 1. The virtual actuation point L fx is set at L fx = 0, 0.033, and 0.051 by using the thrust distribution law described in (21) and (22). The CoG of the experimental setupisdesignedasl fx = 0.051. L fx = 0.051 is then considered the CoG-driven method. Here, a precompensator is not used (K(s) = I, I is the identity matrix). 5.1 Frequency Responses The frequency responses are shown in Fig. 14. The model formulated in (15) (19) is validated by the experiments in Fig. 14(c). The frequency responses shown in Fig. 14(c) demonstrate that there is a trade-off between the CoR-driven and CoGdriven methods. Figures 14(a), (b), and (d) show that the variation in L fx does not have a significant effect on g 11,g 12 and g 22. 5.2 Step Responses The block diagram of the step response experiment is shown in Fig. 15. The nominal plants Fig. 12. CoG-driven method (L fx = L g2 ). The coupling term is L fx (M x1 s 2 + C x1 s + K x1 )/, which has a low gain in the high frequency range considering M x1 M x2 (see Table 1) Fig. 13. Frequency characteristics of g 21 (s) (model parameters are shown in Table 1). The height of the actuation point L fx is set as 0.021, 0, 0.033, 0.051, and 0.078. The position of the CoG (L g2 )isfixedas 0.051. There is a trade-off between the CoR-driven method (L fx = 0) and the CoG-driven method (L fx = 0.051) in high and low frequency ranges of the feedback controllers c 1 and c 2 are defined by 1 g 11n (s) =, (26) (M x1 + M x2 )s 2 + C x1 s + K x1 1 g 22n (s) =, (27) J θy s 2 + C θy s + K θy where g 11n is the nominal plant of c 1 and g 22n is the nominal plant of c 2. The feedback controllers c 1 and c 2 are PID compensators designed by pole assignment (13) using (26) and (27). The bandwidths of the position loops are 10 Hz for x m and θ y, respectively. As shown in Fig. 15, a 100 μm step reference is given for x m and a zero constant reference is given for θy, respectively. The experimental result of θy is shown in Fig. 16. The settling time of L fx = 0 (CoR-driven) is better than that of other 145 IEEJ Journal IA, Vol.5, No.2, 2016

(a) g 11. Fig. 15. Block diagram of the step response experiment. The coupling characteristics can be shaped by changing the virtual actuation point L fx using multiple actuators (b) g 12. Fig. 16. Time responses of θ y with 10 Hz bandwidth feedback controller (experiment, L fx = 0, 0.033, 0.051). The settling time for L fx = 0 is better than under other conditions because at sensitivity function peak of 10 Hz, the g 21 gain for L fx = 0 (CoR-driven) is lower than that for L fx = 0.051 (CoG-driven) (c) g 21. conditions. This is because at a sensitivity function peak of 10 Hz, the g 21 (s) gainofl fx = 0 (CoR-driven) is lower than that of L fx = 0.051 (CoG-driven). 6. Conclusion This paper proposes a decoupling method using multiple actuators in the x direction on the basis of a precise model of the scanning motion x and the pitching motion θ y. This precise model considers the misalignment of the CoG, CoR, actuation point, and measurement point. According to this model, a new block diagram is illustrated. This block diagram clearly shows the relationship between the numerator of the transfer function matrix and the misalignment. The validity of the proposed model and the decoupling method that takes the height of the actuation point into consideration are demonstrated experimentally. References (d) g 22. Fig. 14. Frequency responses of G (experiment, L fx = 0, 0.033, 0.051). The variation in L fx does not affect g 11,g 12, and g 22 significantly. The validity of the proposed model is demonstrated by Fig. 13 and Fig. 14(c) ( 1 ) T. Oomen, R. van Herpen, S. Quist, M. van de Wal, O. Bosgra, and M. Steinbuch: Connecting System Identification and Robust Control for Next- Generation Motion Control of a Wafer Stage, IEEE Trans. on Control Systems Technology, Vol.22, pp.102 118 (2014) ( 2 ) S. Ozawa: The Current Lithography Technologies of the LCD Exposure System, IEICE Trans. C, Vol.J84-C, No.12, pp.1227 1231 (2001) (in Japanese) ( 3 ) P. Yang, B. Alamo, and G. Andeen: Control design for a 6 DOF e-beam 146 IEEJ Journal IA, Vol.5, No.2, 2016

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Salgado, Control System Design (2000) Wataru Ohnishi (Student Member) received the B.S. and M.S. degrees from the University of Tokyo in 2013 and 2015, respectively. He is currently working towards the Ph.D. degree in the Department of Electrical Engineering and Information Systems, the University of Tokyo, Japan. He is also a research fellow (DC1) of Japan Society for the Promotion of Science from 2015. He received the IEEJ Excellent Presentation Award in 2014 and the Dean s Award for Outstanding Achievement from the Graduate School of Frontier Sciences, the University of Tokyo in 2015. His research interest is highprecision motion control. He is a student member of the Institute of Electrical and Electronics Engineers. Koichi Sakata (Member) received the B.S., M.S., and Ph.D. degrees in the Department of Physics, Electrical and Computer Engineering from Yokohama National University, Japan in 2006, 2008, and 2011, respectively. He was a research student in the University of Tokyo, Japan during April 2010 to March 2011. He was also a research fellow (DC2) of Japan Society for the Promotion of Science during April 2009 to March 2011. Since April 2011, he has been with Nikon Corporation. His interests are in motion control and nanoscale servo control. He works on the design of a motion control and highprecision servo control of large-scale stage. Dr. Sakata is a member of IEEE. Kazuhiro Suzuki (Non-member) was born in 1984 in Tokyo, Japan. He received the B.S., M.S., degree from Sophia University, Tokyo, Japan, in 2008, 2010. Since 2010, he has worked for Nikon Corporation. Currently he works on the high-precision servo control of largescale stage. Kazuaki Saiki (Member) received the B.S. degree in the Department of Electronic Engineering from the University of Electro-Communications in 1981. Since April 1981, he has been with Nikon Corporation. He received the Ph.D. degree in the Department of Physics, Electrical and Computer Engineering from Yokohama National University, Japan in 2011. He was also a research student in the University of Tokyo, Japan during April 2010 to March 2011. He works on the design of a motion control and high-precision servo control of largescale stage. Hiroshi Fujimoto (Senior Member) received the Ph.D. degree in the Department of Electrical Engineering from the University of Tokyo in 2001. In 2001, he joined the Department of Electrical Engineering, Nagaoka University of Technology, Niigata, Japan, as a research associate. From 2002 to 2003, he was a visiting scholar in the School of Mechanical Engineering, Purdue University, U.S.A. In 2004, he joined the Department of Electrical and Computer Engineering, Yokohama National University, Yokohama, Japan, as a lecturer and he became an associate professor in 2005. He is currently an associate professor of the University of Tokyo since 2010. He received the Best Paper Award from the IEEE Transactions on Industrial Electronics in 2001 and 2013, Isao Takahashi Power Electronics Award in 2010, and Best Author Prize of SICE in 2010. His interests are in control engineering, motion control, nano-scale servo systems, electric vehicle control, and motor drive. Dr. Fujimoto is a member of IEEE, the Society of Instrument and Control Engineers, the Robotics Society of Japan, and the Society of Automotive Engineers of Japan. 147 IEEJ Journal IA, Vol.5, No.2, 2016