IIC--7 Precise Positioning Control of High-order Pitching Mode for High-Precision Stage Yushi Seki, Hiroshi Fujimoto (The University of Tokyo), Hideaki Nishino, Kazuaki Saiki (Nikon) Abstract Precision stages are demanded to grow in size for increased production efficiency. But stages are getting larger, its rigidity are decreasing. Then, the pitching of stage causes a problem on the control. Conventional method is using the feedback control of the pitching angle θ 2 between carriage part and table part. But, the pitching angle θ is caused between guide and carriage part. Then, The feedback control using the pitching angle θ θ 2 of the entire stage is proposed. Finally, simulations and experiments are performed to show the advantages of the proposed method. (High-precision stage, MIMO, Feedback Control, Pitching, Tracking Error )., Hz () (2) V oice Coil Motor(VCM) / (AF/LV) (3) 4 2 ( 2) (4) θ 2 (4) θ θ θ θ 2 θ θ 2 2 4 Fig.. Two-mass Four-degree Model. 2. (x ) L f M M 2 O x /6
Table. Stage model parameters. Mass of pole part m.4 kg Mass of table part M 22 kg Thrust viscous constant c x 2. 2 N s/m Twist dumping constant of joint µ θ. kn m s Spring constant k 4.5 2 MN/µm Viscous constant c 3 kn s/µm Twist spring constant of joint k θ 9. kn m/rad Inertia of pole part I m 89 kg m 2 Inertia of table part I M 2.4 3 kg m 2 Distance from C M to joint L.3 m Distance from C m to joint L 2.5 m Distance from C M to X 2 L x2.6 m Distance from C M to F L f. m Distance between air pads L p.5 m Gravity acceleration g 9.8 m/s 2 y z F VCM y T x θ y C z θ C 2 θ θ 2 X 2 2 cos θ i, sin θ i θ i, θi 2 = (i =, 2) x () θ θ 2 (2) (3) F = (M m)ẍ c xẋ m(l L 2) θ ml 2 θ2 () F (L L 2 L f ) T = {I M I m m(l L 2 ) 2 } θ (I m ml 2 2 ml L 2 ) θ 2 m(l L 2 )ẍ {2kl 2 mg(l L 2)}θ mgl 2θ 2 2cl 2 θ(2) F (L 2 L f ) T = (I m ml 2 2 ml L 2) θ (I m ml 2 2) θ 2 µ θ θ2 mgl 2 θ (k θ mgl 2 )θ 2 (3) 2 2 (7) (4) (6) F (s) = (a s 2 b s)x(s) a 2s 2 θ (s) a 3s 2 θ 2(s)(4) F (s)(l L 2 L f ) T = a 2 s 2 X(s) (a 4 s 2 b 3 s c ) θ (s) (a 5 s 2 c 2 )θ 2 (s) (5) F (s)(l 2 L f ) T = a 3 s 2 X(s) (a 5 s 2 c 2 )θ (s) (a 6s 2 b 2s c 3)θ 2(s) (6) a = M m a 2 = m(l L 2) a 3 = ml 2 a 4 = I M I m m(l 2 L 2 2 2L L 2) a 5 = I m ml 2 2 ml L 2 a 6 = I m ml 2 2 b = c x c = 2kl 2 mg(l L 2 ) b 2 = µ θ b 3 = 2cl 2, c 2 = mgl 2 (7) c 3 = k θ mgl 2 (4) (6) (8) (2) (4) D e(s) = {(a a 4 a 2 2)a 6 (a 2a 5 a 3a 4)a 3 a a 2 5 a 2 a 3 a 5 }s 6 {(a a 6 a 2 3)b 3 (a a 4 a 2 2)b 2 (a 4 a 6 a 2 5) b }s 5 {(a a 4 a 2 2)c 3 (a 2 a 3 2a a 5 a 2 a 3 )c 2 (a a 6 a 2 3)c (a b 2 a 6b )b 3 a 4b b 2}s 4 {(a b 3 a 4b )c 3 2a 5b c 2 (a b 2 a 6b )c b b 2b 3}s 3 {(a c b b 3) c 3 a c 2 2 b b 2 c }s 2 (b c c 3 b c 2 2)s (8) X(s) F (s) = [{(a2a6 a3a5)l f a 4a 6 a 2 5}s 4 (a 2b 2L f a 6b 3 a 4b 2)s 3 {(a 2c 3 a 3c 2)L f a 4c 3 2a 5c 2 a 6c b 2b 3}s 2 (b 3c 3 b 2c )s (c 2 2 c c 3)]/D e(s) = F X (9) θ (s) F (s) = [{(aa6 a2 3)L f a 2a 6 a 3a 5}s 3 {(a b 2 a 6 b )L f a 2 b 2 }s 2 {(a c 3 b b 2 )L f a 2c 3 a 3c 2}s b c 3L f ]/D e(s) = F θ () θ 2 (s) F (s) = [{(a 2a 3 a a 5 )L f a 3 a 4 a 2 a 5 }s 3 (a 5 b L f a 3 b 3 )s 2 (a c 2 L f a 2 c 2 a 3 c )s b c 2 L f ]/D e (s) = F θ2 () X(s) T (s) = [{(a2a6 a3a5)l f a 4a 6 a 2 5}s 4 (a 2b 2L f a 6 b 3 a 4 b 2 )s 3 {(a 2 c 3 a 3 c 2 )L f a 4 c 3 2a 5 c 2 a 6 c b 2b 3}s 2 (b 3c 3 b 2c )s (c 2 2 c c 3)]/D e(s) = T X(2) θ (s) T (s) = [{(a a 6 a 2 3)L f a 2 a 6 a 3 a 5 }s 3 {(a b 2 a 6 b )L f a 2 b 2 }s 2 {(a c 3 b b 2 )L f a 2 c 3 a 3 c 2 }s b c 3 L f ]/D e (s) = T θ (3) θ 2 (s) T (s) = [{(a 2a 3 a a 5 )L f a 3 a 4 a 2 a 5 }s 3 (a 5 b L f a 3b 3)s 2 (a c 2L f a 2c 2 a 3c )s b c 2L f ]/D e(s) = T θ2 (4) (5) 2/6
Magnitude [db] 8 2 4 2 Frequency response of plant measured model 2 3 2 X(s) θ (s) θ 2(s) 2 = Fig. 2. F X F θ F θ2 Model fitting. T X T θ T θ2 [ F (s) T (s) ] (5) (5) C 2 L x2 L, L 2 (6) 2 P X P X = {F X T X F θ (L L 2 L x2 ) 3. 3 F θ2 (L 2 L x2)} (6) F VCM T 2 x θ 2 2 2 x θ θ 2 2 2 F F PTC F B 3(a), 3(b) 3 2 4 (7) (8) (7) F X (8) T θ 4 PID C x (s), PD C y(s) (9), (2) C x (s) ω p Hz, C y (s) ω p 8 Hz.3 ms P nx(s) = a xs 2 a x2s (7) Ref Ref PTC - FB Cx PTC FF Force FB Force F FB Torque T - FB Cx Fig. 3. P ny(s) = FB Force F 3 Plant FB C y (a) Conventional FF Force FB Torque T Plant FB C y (b) Proposed Precise positioning control system. X X θ 2 θ θ 2 a ys 2 a y2s (8) C x (s) = b xs 2 b x2 s b x3 (9) s(s c ) C y (s) = a x = 2.9 a x2 = 2. 2 b x = 6ω 2 pa x c b x2 = 4ω 3 pa x c b x3 = ω 4 pa xc c = a x 4ω p a x a x2 bys s c 2 b y2 (2) a y = 2.5 2 a y2 =. a y3 = 4. 2 b y = ω 3 pa y(4c ω p) b y2 = 6ω 2 pa y c b y c 2 = 3 3 (PTC) a y 4ω pa y a y2 5 PTC (5) 2 n p x[k n p] 4. V CM θ 2 θ θ θ 2 θ θ 2 θ θ 2 3/6
Magnitude [db] 5 2 Frequency response of nominal plant 2 Plant model Position [m].2.5..5 Trajectories.5.5 2 2.5.2 2 Velocity [m/s].5..5 3 2.5.5 2 2.5 Magnitude [db] r(t) 5 2 25 2 3 4 (a) P nx Frequency response of nominal plant 2 2 (b) P ny Fig. 4. r[i]=x pd [i] S (T Multilate FFC r ) Fig. 5. 5 Nominal plant fitting. u [i] H M (T u ) P n [z] C[z] y [k] e[k] - Plant model u [k] u[k] u(t) H Plant (T u ) Perfect Tracking Control system. S (T y ) 6 X (θ θ 2) 7(a) X ±.57 nm 7(b) (θ θ 2) ±.55 µrad θ 2 θ (θ θ 2 ) 8(a) X ±.62 nm 8(b) (θ θ 2 ) ±.5 µrad θ 2 θ (θ θ 2 ) 2 x y(t) Vel * [m/s] θ θ 2 (θ θ 2 ) x [nm] 2 Table 2. Fig. 6. 6 Target trajectories. () Tracking error (Simulations). Conv. Prop. Tracking error X ±.57 nm ±.62 nm Tracking error (θ θ 2 ) ±.55 µrad ±.5µrad.2.5..5 Simulation result.5.5 2 2.5 2 Conventional 2.5.5 2 2.5 (a) x Positioning error Conventional.5.5 2 2.5.5.5 2 2.5.5.5 2 2.5 Fig. 7. (b) θ y Positioning error 7 Conventional method..5 nm, θ θ 2 θ θ 2 4/6
Vel * [m/s].2.5..5 Simulation result Proposed.5.5 2 2.5 θ θ 2 (θ θ 2 ) x [nm] 2 2.5.5 2 2.5 time[sec] (a) x Positioning error Proposed.5.5 2 2.5.5.5 2 2.5.5.5 2 2.5 5. 5 (b) θ y Positioning error Fig. 8. 8 Proposed method. 9(a) 2 2 x nm 9(b) VCM 4 z VCM θ 2 AF/LV θ θ 2 (4) VCM AF/LV θ 5 2 VCM AF/LV T θ (2), (22) C x (s) 4 (23), (24) C x (s) ω p Hz, C yc(s) C yp(s) ω ye 8 Hz P nyc (s) = a yc s 2 a yc2 s (2) (a) Overview of Nano-stage 2 (b) Image of the table part 9 2 Fig. 9. Experimental nano stage 2. P nyp (s) = a yp s 2 a yp2 s (22) C yc (s) = b ycs s c 3 b yc2 (23) C yp (s) = b yps b yp2 (24) s c 4 a yc = 2.5 2 a yp = 2.5 2 a yc2 =. a yc3 = 4. 2 b yc = ω 3 yea yc (4c 3 ω ye ) b yc2 = 6ω 2 yea yc c 3 b yc c 3 = a yc 4ω yea yc a yc2 5 3 a yp2 =. a yp3 =.6 2 b yp = ω 3 yea yp (4c 4 ω ye ) b yp2 = 6ω 2 yea yp c 4 b yp c 4 = a yp 4ω yea yp a yp2 6 (b) θ 2 θ θ θ 2 θ θ 2 3 X ± 422 µm (θ θ 2 ) ±39 µrad X ±3 µm (θ θ 2) ±2 µrad x 76% θ y 48% F B θ θ 2 5/6
Magnitude [db] 5 5 2 Frequency response of plant 2 Measurement Vel * [m/s] x [µm].2 Conventional Proposed.2 2 3 4 5 6 7 5 Experimental results Magnitude [db] 5 5 2 3 Fig.. Table 3. 2 (a) P nyc (VCM sensor) Frequency response of plant 2 2 (b) P nyp (AF/LV sensor) Measurement ( ) Nominal plant fitting (Experiments). 3 ( ) Tracking error (Experiments). Conv. Prop. Tracking error X ± 422 µm ± 3 µm Tracking error (θ θ 2 ) ± 39 µrad ± 2 µrad 6. θ 2 FB FB 7. ( 268628) θ θ 2 θ θ 2 5 2 3 4 5 6 7 5 (a) x Positioning error 5 2 3 4 5 6 7 5 5 2 3 4 5 6 7 2 4 2 3 4 5 6 7 Fig.. (b) θ y Positioning error Experimental results. Y. Seki, H. Fujimoto, A. Hara, T. Yamanaka, and K. Saiki, Basic examination of simultaneous optimization of mechanical and control design for gantry-type precision stage modeled as two-mass 4-DOF system, in Proc. The 36rd Annual Conference of the IEEE Industrial Electronics Society, pp. 872-877 (2). 2 S. Wakui, Roles of an Active Anti-Vibration Apparatus in Precision Positioning J. JSPE, vol. 73, no. 4, pp. 45-49, (27). 3 K. Sakata, H. Fujimoto, T. Ohtomo, and K. Saiki, Experimental verification on auto focus and leveling control of scan-stage using driving force and surface shape of the stage in Proc. IEE of Japan Technical Meeting Record, IIC-8-44, (28). 4 K. Sakata, H. Fujimoto, A. Hara, T. Ohtomo, and K. Saiki, Design Fabrication and Control of 4-DOF High- Precision Stage, The th IEEE International Workshop on Advanced Motion Control, pp. 2-24, (2). 5 H. Fujimoto, Y. Hori, and A. Kawamura, Perfect tracking control based on multirate feedforward control with generalized sampling periods, IEEE Trans. Industrial Electronics, vol. 48, no. 3, pp. 636-644 (2). 6/6