Wataru Ohnishi a) Student Member, Hiroshi Fujimoto Senior Member Koichi Sakata Member, Kazuhiro Suzuki Non-member Kazuaki Saiki Member.

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1 IEEJ International Workshop on Sensing, Actuation, and Motion Control Proposal of 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 High-precision stages are widely used in the semiconductor and flat panel industry. Because these stages have si degrees of freedom to control, coupling forces can deteriorate its control performance and stability. This paper proposes a novel decoupling control method from motion to θ y motion using multiple actuators considering the misalignment of the actuation point, Center of Gravity (CoG), and Center of Rotation (CoR). The proposal of this paper consists of three steps: 1) the detailed modeling of the and θ y motions, 2) proposal of a changeable actuation point stage, 3) the derivation of the optimal actuation position. The validity of the proposal method is demonstrated by eperiments. 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 si degrees of freedom (, y, z, θ, θ y, θ z ) to control (3). For this purpose, a dual stage which 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 shown in Fig. 1. In such multi-input multi-output (MIMO) systems, coupling force degrades the control performance. This paper focuses on the and θ y motion.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, and the measurement point of the translational position are not at the same points, coupling between and θ y motion 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. Our research group designed a novel structure fine stage which has two Voice Coil Motors (VCMs) in the direction shown in Fig. 2. By changing thrust distribution ratio, the height of the virtual actuation point can be placed arbitrarily. Net this paper proposes a decoupling control method using multiple actuators. This method focuses on the relationship between the numerator of the transfer functions (zeros) and kinematic parameters such as positions of CoG, CoR, the actuation point, and the measurement point. Finally, a method to improve the performance considering the height of the aca) Correspondence to: ohnishi@hflab.k.u-tokyo.ac.jp The University of Tokyo 5-1-5, Kashiwanoha, Kashiwa, Chiba, Japan Nikon Corporation 47-1, Nagaodaityou, Sakae, Yokohama, Japan Linear motor for the coarse stage Relative position sensor Linear motor for the coarse stage Fig. 1. Fine stage Relative position sensor Coarse stage Picture 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 placed arbitrarily. tuation point and the bandwidth of the feedback (FB) controller is proposed. The effectiveness of the proposed method is demonstrated by eperiments. Although the method in this paper models the motion and the θ y motion as an eample, it can also be applied to other coupling such as the y and θ motions. Furthermore, this method could be used for other applications such as aircrafts, automobiles, and satellites, all of which have a misalignment of its CoG, its CoR, its actuation point, and its measurement point. c 215 The Institute of Electrical Engineers of Japan. 1

2 z y VCM(f y1 ) VCM(f 2 ) VCM(f z4 ) VCM(f 1 ) (measured) (unmeasured) (measured) VCM(f z1 ) VCM(f z3 ) Gravity Canceller(f gc) Fine stage Center of rotation VCM(f y2 ) VCM(f z2 ) (a) Actuator arrangement of the fine stage. (unmeasured) z y Gravity canceller Encoder(z 4 ) Encoder(z 3 ) Fig. 5. Fine stage model of the and θ y motion. Table 1. Model parameters. Encoder( 1 ) Encoder( 2 ) Encoder (z 1 ) Encoder(z 2 ) Encoder(y) (b) Sensor arrangement of the fine stage. Fig. 3. Structure of the fine stage. y y z z Air Air Fig Modeling Fine stage Center of rotation Air gyro(ò ;ò y ;ò z ) Planar air bearing(;y) (a) Side view. Air bearing actuator(f gc;z) Coarse stage Center of rotation Air gyro(ò ;ò y;ò z) Planar air bearing(;y) Air bearing actuator(f gc;z) (b) Schematic. Structure of the gravity canceller. 2.1 Eperimental setup Our research group designed the dual stage shown in Fig. 1. The actuator and sensor arrangement of the fine stage is illustrated in Fig. 3. As shown in Fig. 2 and Fig. 3(a), the fine stage has two Voice Coil Motors (VCMs) in the direction. Symbol Meaning Value m Measured position of the fine stage g1 Position of the CoG of the planar air bearing and the air gyro g2 Position of the CoG of the fine stage θ y Measured attitude angle of the fine stage f Input force of the fine stage in the direction τ y Input torque of the fine stage in the θ y direction M 1 Mass of the planar air bearing and the air gyro.77 kg C 1 Viscosity coefficient in the g1 motion 43 N/(m/s) K 1 Spring coefficient in the g1 motion 11 N/m M 2 Mass of the fine stage 5.3 kg J θy Moment of inertia of the fine stage.1 kgm 2 C θy Viscosity coefficient of the fine stage in the θ y motion 1.6 Nm/(rad/s) K θy Spring coefficient of the fine stage in the θ y motion 12 Nm/rad L m Distance between the measurement point of m and the CoR.28 m L g2 Distance between the CoR and the CoG of the fine stage.51 m L f Distance between the CoR of the fine stage and the actuation point changeable The fine stage is supported by a 6-DOF air bearing gravity canceller (6). The picture and schematic of the gravity canceller is shown in Fig. 4. The gravity canceller compensates for the gravitational force eperienced 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 which support the (θ, θ y, θ z ), (, y) and (z)-directional motion, respectively. As shown in Fig. 4, the shape of the air gyro is like a hemisphere. The fine stage slides on the hemispheric surface of the air gyro with an air gap of a few µm. In this paper, the center of the hemisphere is called the center of rotation (CoR). In other words, the radius of the curvature of the air gyro decides the height of the CoR. 2.2 Lagrange s equations In this section, the Lagrange s equations are formulated based on the model shown in Fig. 5 and Table. 1. First, the relationship of g1, g2, and θ y is epressed by g2 = g1 + L g2 sin(θ y ), (1) 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 1ẋ 2 g M 2ẋ 2 g J θy θ 2 y, (3) U = 1 2 K 1 2 g K θyθ 2 y + L g2 M 2 g cos(θ y ), (4) 2

3 f (s) = [J θy + L f L m M 1 (L f L g2 )(L g2 L m )M 2 ]s 2 + (C θy + L f L m C 1 )s + K θy + L f L m K 1 L g2 M 2 g (15) θ y (s) f (s) = [L f M 1 + (L f L g2 )M 2 ]s 2 + L f C 1 s + L f K 1 τ y (s) = [L mm 1 + (L m L g2 )M 2 ]s 2 + L m C 1 s + L m K 1 (16) (17) θ y (s) τ y (s) = (M 1 + M 2 )s 2 + C 1 s + K 1 (18) =[(M 1 + M 2 )J θy + M 1 M 2 L 2 g2 ]s4 + [(M 1 + M 2 )C θy + (J θy + M 2 L 2 g2 )C 1]s 3 + [(J θy + M 2 L 2 g2 )K 1 + (M 1 + M 2 )(K θy M 2 L g2 g) + C θy C 1 ]s 2 + [C θy K 1 + C 1 (K θy L g2 M 2 g)]s + K 1 (K θy L g2 M 2 g) (19) B = 1 2 C 1ẋ 2 g C θy θ 2 y, (5) W = f [ g1 + L f 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 1ẋ 2 g M 2 [ẋg1 + L g2 cos(θ y ) θ y ] J θy θ 2 y 1 2 K 1 2 g1 1 2 K θyθ 2 y L g2 M 2 g cos(θ y ). (7) The 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 i denotes generalized coordinates, q 1 = g1 and q 2 = θ y. Finally, according to (1) (8), nonlinear equations (9) and (1) are obtained: ẍ g1 (M 1 + M 2 ) + C 1 ẋ g1 + K 1 g1 +M 2 L g2 [ cos(θy ) θ y sin(θ y ) θ 2 y] = f, (9) M 2 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 L f cos(θ y ). (1) 2.3 Linearization Assuming cos(θ y ) 1, sin(θ y ) θ y, θ 2 y, (9) and (1) are linearized as follows: (M 1 + M 2 )ẍ g1 + C 1 ẋ g1 + K 1 g1 + M 2 L g2 θ y = f, (11) (M 2 L 2 g2 + J θy) θ y + C θy θ y + K θy θ y + M 2 L g2 (ẍ g1 gθ y ) = τ y + f L f. (12) 2.4 Transformation to measurable coordinate The generalized coordinate g1 cannot be measured. Thus, g1 is converted to m by f (s) = g1(s) f (s) + L θ y (s) m f (s), (13) τ y (s) = g1(s) τ y (s) + L θ y (s) m τ y (s). (14) 2.5 Transfer functions According to (11) (14), 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)). transfer functions (15) (19) are obtained. In this paper, (15) (19) are also epressed as m (s) g 11 (s) g 12 (s) G(s) = g 21 (s) g 22 (s) = f (s) τ y (s) (2) θ y (s) θ y (s) f (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 Due to the spatial limitation of the fine stage, the VCM for the direction cannot be placed at the desired actuation point. In this section, a new fine stage structure which can shift the virtual actuation point by means of thrust distribution of multiple actuators is proposed. This is based on a simple idea shown in Fig. 2. The two VCMs generate force f 1 and f 2 in the direction by thrust distribution law, f 1 = a f, f 2 = (1 a) f, (21) where a denotes the thrust distribution ratio. Here, the height of the virtual actuation point L f is defined as L f = al f 1 + (1 a)l f 2, (22) where L f 1 and L f 2 denote the heights of the actuation point of f 1 and f 2 from the CoR. Although this structure doubles the amount of wiring, and the mass of the fine stage becomes slightly heavier, this structure has advantages as described below. By changing the 3

4 (a) g 11 (s). (b) g 12 (s). (c) g 21 (s). (d) g 22 (s). without and path through path through path through and (e) Legend Fig. 7. Coupling path of G(s). The characteristics of G(s) is defined by the height of the actuation point L f, the measurement point L m, and the CoG of the fine stage L g2. Control object Control object Precompensator Precompensator Fig. 8. Multi-input multi-output feedback system with the precompensator K(s). thrust distribution ratio a, the height of the virtual actuation point L f can be placed at desired point such as the position of the CoG and the CoR. Moreover, during operation, the CoG shifts caused by 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 by means of placing the virtual actuation point at the optimal position. 4. Integrated design of mechanism and control The coupling path of G(s) can be illustrated by Fig. 7. Fig. 7 clearly shows the relationship between the position of the zeros of G(s) and the position of the actuation point L f. 4.1 Equivalence between the precompensator k 21 and the actuation point L f Fig. 8 shows the MIMO system with a precompensator. 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 by Fig. 9. Substitution (25) for Fig. 8 Control object Fig. 1. Control object Q(s) with the precompensator k 21. Because the precompensator k 21 and the actuation point L f are connected in parallel, varying k 21 results in a variation of L f. q 11 (s) Q(s) = q 21 (s) g 11 (s) G(s) = g 21 (s) q 12 (s) q 22 (s), K(s) = k 11 (s) k 12 (s) k 21 (s) k 22 (s), g 12 (s) g 22 (s). (24) In this section, the precompensator shown in (25) is considered. 4

5 CoRdriven (CoR driven).33.51(cog driven).78 Fig. 11. CoR-driven case (L f = ). The coupling term is L g2 M 2 s 2 / which has a low gain in the low frequency. CoGdriven Fig. 12. CoG-driven case (L f = L g2 ). The coupling term is L f (M 1 s 2 + C 1 s + K 1 )/ which has a low gain in the high frequency considering M 1 M 2 (see Table. 1). 1 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. 1. Here, the precompensator k 21 and the position of actuation point L f are connected in parallel. Hence, there is a linear relationship between the precompensator k 21 and the actuation point L f ; varying k 21 results in a variation of L f. 4.2 Decoupling g 21 (s) using the changeable actuation point L f By changing the actuation point L f, Fig. 7 shows that coupling transfer function g 21 (s) can be suppressed without any impact of g 12 (s) and g 22 (s). In this section, CoR-driven case and CoG-driven case are proposed. CoR-driven case (Fig. 11) If the L f is set as, this means the actuation point is placed at the CoR, L f (M 1 s 2 + M 2 s 2 + C 1 s + K 1 ) become. Now, the coupling term is only L g2 M 2 s 2 /. Then, the coupling gain of g 21 in low frequency is suppressed. CoG-driven case (Fig. 12) If the L f is set as L g2, this means the actuation point is placed at the CoG, (L g2 L f )M 2 s 2 become. Now, the coupling term is only L f (M 1 s 2 + C 1 s + K 1 )/. Then, the coupling gain of g 21 in high frequency is suppressed considering M 1 M 2 (see Table. 1). The comparison between the CoR- and CoG-driven case is shown in Fig. 13. Fig. 13 shows that there is trade-off between CoR- and CoG-driven case in low and high frequency. If the virtual actuation point L f is selected to minimize the gain of g 21 (s) at the frequency of the sensitivity function peak, the coupling effect can be reduced. 5. Eperiment Measurements are performed by using the eperimental stage shown in Fig. 1. The virtual actuation point L f is set Fig. 13. Frequency characteristics of g 21 (s) (model parameters are shown in Table. 1). The height of the actuation point L f is set as.21,,.33,.51, and.78. The position of the CoG L g2 is fied as.51. There is a trade-off between CoR-driven case (L f = ) and CoG-driven case (L f =.51) in high and low frequency. L f =,.33, and.51 by using the thrust distribution law described in (21) and (22). The CoG of the eperimental setup is approimately by L f =.51. Then, L f =.51 is considered as CoG-driven case. Here, precompensator is not used (K(s) = I, I is the identity matri). 5.1 Frequency responses Frequency responses are shown in Fig. 14. The model formulated in (15) (19) is validated by the eperiments in Fig. 14(c). Frequency responses shown in Fig. 14(c) demonstrate that there is a trade-off between CoR-driven and CoG-driven case. Fig. 14(a), (b), and (d) prove changing L f does not affect g 11 (s), g 12 (s) and g 22 (s). 5.2 Step responses The block diagram of the step response eperiment is illustrated in Fig. 15. The nominal plants of the feedback controllers c 1 and c 2 are defined by 1 g 11n (s) =, (26) (M 1 + M 2 )s 2 + C 1 s + K 1 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 using (26) and (27). The bandwidths of the position loops are 1 Hz for m and θ y, respectively. As shown in Fig. 15, a 1 µm step reference is given for m and a zero constant reference is given for θ y, respectively. The eperimental result of θ y is shown in Fig. 16. The settling time of L f = (CoR-driven) is better than that of other conditions. This is because at sensitivity function peak around 1 Hz, the g 21 (s) gain of L f = (CoR-driven) is lower than that of L f =.51 (CoG-driven). 6. Conclusion This paper proposes a novel decoupling method using mul- 5

6 (a) g 11 (s) (b) g 12 (s) (c) g 21 (s) (d) g 22 (s). Fig. 14. Frequency responses of G(s) (eperiment, L f =,.33,.51). The variation of L f does not affect g 11 (s), g 12 (s), and g 22 (s). According to Fig. 13 and Fig. 14(c), the validity of the proposed model is demonstrated. 6, Control object Fig. 15. Block diagram of the step response eperiment. The coupling characteristics can be shaped by changing the virtual actuation point L f using multiple actuators. Angle [µrad] Time [s] Fig. 16. Time responses of θ y (eperiment, L f =,.33,.51). The settling time of L f = is better than other conditions because the coupling gain at 1 Hz is small (see Fig. 14(c)). tiple actuators in the direction based on precise model of and θ y motions. 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 matri and the misalignment. The validity of the proposed model and decoupling method are demonstrated by eperiments. References ( 1 ) S. Ozawa, The Current Lithography Technologies of the LCD Eposure System, IEICE Trans. C, vol. J84-C, no. 12, pp , 21. (in Japanese) ( 2 ) T. Oomen, R. van Herpen, S. Quist, M. van de Wal, O. Bosgra, and M. Steinbuch, Connecting System Identification and Robust Control for Net- Generation Motion Control of a Wafer Stage, IEEE Trans. Control Systems Technology, vol. 22, no. 1, pp , 214. ( 3 ) P. Yang, B. Alamo, and G. Andeen, Control design for a 6 DOF e-beam lithography stage, American Control Conference, vol. 3, pp , 21. ( 4 ) H. Butler, Position Control in Lithographic Equipment, IEEE Control Systems Magazine, vol. 31, no. 5, pp , 211. ( 5 ) Y. Choi, and D. Gweon, A High-Precision Dual-Servo Stage Using Halbach Linear Active Magnetic Bearings, IEEE/ASME Trans. Mechatronics, vol. 16, no. 5, pp , 211. ( 6 ) W. Ohnishi, H. Fujimoto, K. Sakata, K. Suzuki and K. Saiki, Design and Control of 6-DOF High-Precision Scan Stage with Gravity Canceller, 214 American Control Conference, pp , 214.