ELECTRI C MACHINES AND CONTROL Vol. 16 No. 9 Sep y x 20 m /s μm TP 271 A X Control of planar

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1 ELECTRI C MACHINES AND CONTROL Vol. 16 No. 9 Sep y x 2 m /s μm TP 271 A X Control of planar motor on the basis of the adaptive feedforward LI Xin YANG Kai-ming ZHU Yu YU Dong-dong Department of Precision Instruments and Mechanology Tsinghua University Beijing 184 China Abstract In order to solve the problem that under the motion with a large acceleration a constant of feedforward coefficient cannot improve the servo performance during acceleration and deceleration an a- daptive feedforward coefficient identification method on the basis of proportional differential PD control and the acceleration data was proposed. Using the least square method and according to the output of PD controller and the desired acceleration during motion the feedforward coefficient was corrected. Introducing the forgetting factor the feedforward coefficient was more available with the current position depend behavior. y-direction and x-direction trajectory tracking experiments with different acceleration were carried out and in the case of the maximal acceleration being 2 m /s 2 the maximal tracking error during acceleration and deceleration is. 56 μm. This method is completely based on the online experiment the calculation of the feedforward is realized without the knowledge of the parameters of the motor. Key words planar motor acceleration feedforward coefficient adaptive feedforward least square method trajectory tracking forgetting factor CB

2 96 16 multi - input PD multi - output MIMO k 1-1 k y x 1 1 Halbach A B C D 4 x A B y x y z y 1 兹 2 Y Z 3 C D 直线电机 A 直线电机 C 空气轴承激光尺 Halbach 永磁阵列平面镜 1 iterative learning Fig. 1 Structure of planar motor 11 - control ILC 17 ILC Mq τ d = u 1 M = diag m m J τ d q = x y θ z T u = F X F Y T θ T u 2 proportional differential PD F AX PD 2 1 F BX = F X F F PD CY 4 Y F T θ 2 1 DY F AX F BX F CY F DY 2 4 Z X 线圈骨架直线电机 D 直线电机 B

3 9 97 Fig. 2 F CY 2 F AX F BX X Y Z E DY Force direction of each linear motor 2 PD 6 β = m2 C PD 3 VBK t β m C D /A V B K t N = 2n n ILC PD out PD out mq d β F β = min N ε 2 k 5 β ε k = PD out k - βq d k q d k k 5 β = q T dq d q T dpd out = N q PD out k 3 K t 3 β PD 6 3 PD k k PD out = PD out 1 PD out N = PD a 1 PD a n β k = β k - 1 Δβ k 7 PD b 1 PD b n 4 Δβ k k PD a i PD b N 6 自适应前馈 q d q d e PD 控制器茁 (k)= 茁 (k) 驻茁 (k) 驻茁 (k)=f(pd out(k)) u 2=q d(k) 茁 (k) u 2 u 1 F x F Y T 兹 - - F X1 F X2 F Y1 F Y2 驱动器电流分解 i A i B i C i D q=[x Y 兹 ] X=X 1 Y=(Y 1Y 2)/2 兹 =(Y 1-Y 2)/l 激光尺信号 X 1 激光尺信号 Y 1 激光尺信号 Y 2 Fig. 3 3 Adaptive feedforward control schematic diagram PD out k Δβ k Δβ k

4 98 16 F Δβ k = min η 2 i Δβ k k η i = PD out i - Δβ k q d i i = 1 8 Φ k = q d 1 q d 2 q d k T Y k = k P k P k = k k P k [ k q d i PD out i ] P k = [ ] k d i = k d i P k d i = Δβ k = P k [ k q d i PD out i ] = 13 P k [ k q d i PD out i q PD out k ] [ ] Δβ k - 1 = P k - 1 k q d i PD out i 14 k q d i PD out i = P k - 1 Δβ k Δβ k = P k P - 1 k - 1 Δβ k - 1 P k q PD out k 12 Δβ k = P k P - 1 k - Δβ k - 1 P k q PD out k Δβ k = Δβ k - 1 P k q PD out k - P k Δβ k - 1 P k = Φ T k Φ k - 1 = Φ T k - 1 Φ k = P - 1 k = 2 k P k P2 k P k Δβ k β k = β k - 1 Δβ k Δβ k = Φ T k Φ k - 1 Φ T k Y k 9 Δβ = Δβ k - 1 K k η k } 2 PD out 1 PD out k T K k = P k q 2 P k - 1 q P k = P k Δβ k η k = PD out k - Δβ k q d k Δβ k = [ d i ] [ q d i PD out i ] = 2 k k 2 8 F Δβ k = min Δβ k ρ k -i η 2 i 22 ρ β k = β k - 1 Δβ k Δβ = Δβ k - 1 P k q η k } 23 K k = P k - 1 q ρ P k - 1 } P k = 1 ρ 1 - K k q P k y x 4 5 a(m/s 2 ) v(m/s) s/m 1 加速度速度位移 Fig. 4 4 y Tracking trajectory in y-direction

5 9 99 a(m/s 2 ) v(m/s) s/m 2 加速度速度位移 x Fig. 5 Tracking trajectory in x-direction 7 x PD m = 5. kg Fig. 7 Output of PD controller in x-direction. 8 μm K t = 48. N /A motion experiment. 2 ms ρ = ρ = 1 ρ =. 9 3 β β = m2 y 8 C = VBK t = x 9 6 PD y 6 x 7 PD 控制路输出 5% 2%5-2%5 加速段 PD 控制器输出 -5% % 2%5 减速段 PD 控制器输出 -2%5-5% Fig. 6 y PD Output of PD controller in y-direction motion experiment 6 y q PD out k β = = N x β = 4 q PD out k = N PD 控制路输出 y 方向运动加速段前馈系数变化 x 方向运动减速段前馈系数变化 1% 5% 加速段 PD 控制器输出 -5% % -5% 减速段 PD 控制器输出 % 自适应前馈籽 =1. 16 自适应前馈籽 =.9 14 局部放大 (a) 加速段前馈系数变化自适应前馈籽 = (b) 减速段前馈系数变化自适应前馈籽 = Fig. 8 y Feedforward coefficient curve in y-direction motion experiment

6 1 16 x 方向运动减速段前馈系数变化 x 方向运动加速段前馈系数变化局部放大自适应前馈籽 =1. 自适应前馈籽 = (a) 加速段前馈系数变化自适应前馈籽 = 自适应前馈籽 = Fig. 9 (b) 减速段前馈系数变化 x Feedforward coefficient curve in x-direction motion experiment 3 6 (b) 减速段轨迹跟踪误差 1 y 3 Fig. 1 Tracking errors during acceleration and deceleration in y-direction motion experiment K t y Table 1 The maxima tracking errors during x acceleration and deceleration /μm 加速段轨迹跟踪误差 e/ 滋 m 减速段轨迹跟踪误差 e/ 滋 m 无加速度前馈标准计算方法基于 PD 控制计算方法自适应计算方法籽 =1. 自适应计算方法籽 = (a) 加速段轨迹跟踪误差无加速度前馈标准计算方法基于 PD 控制计算方法自适应计算方法籽 =1. 自适应计算方法籽 = y x /μm PD ρ = ρ =

7 μm PD ρ = 1. ρ =. 9 ρ = μm. 56 μm 加速段轨迹跟踪误差 e/ 滋 m 减速段轨迹跟踪误差 e/ 滋 m 无加速度前馈标准计算方法基于 PD 控制计算方法自适应计算方法籽 =1. 自适应计算方法籽 = (a) 加速段轨迹跟踪误差无加速度前馈标准计算方法基于 PD 控制计算方法自适应计算方法籽 =1. 自适应计算方法籽 = (b) 减速段轨迹跟踪误差 4 JUNG K S BAEK Y S. Precision stage using a non-contact planar actuator based on magnetic suspension technology J. Mechatron- 11 ics x 5 CHO H S. Analysis and design of synchronous permanent-magnet Fig. 11 Tracking error during acceleration and deceleration in x-direction motion experiment planar motors J. IEEE Transactions on Energy Conversion ILC magnets J 6 ILC 8 CORNELIS M M. Magnetically levitated planar actuator with mov- ing magnets dynamics m /s 2. 4 m /s μm 2 2 m /s 2. 8 m /s μm ρ = μm 1 KIM W J TRUMPER D L. High-precision magnetic levitation stage for photolithography J. Precision Engineering GAO W DEJIMA S YANAI H et al. A surface motor-driven planar motion stage integrated with an XYθz surface encode for precision positioning J. Precision Engineering JANSEN J W. Magnetically levitated planar actuator with moving magnets electromechanical analysis and design D. Einhoven Technische Universiteit Einhoven BOEIJ J. Modeling ironless permanent-magnet planar actuator structures J. IEEE Transactions on Magnetics JANSEN J W. Magnetically levitated planar actuator with moving IEEE Transactions on Industry Applications 28 commutation and control design D. Einhoven Technische Universiteit Einhoven HU Tiejun. Extended range six DOF high-precision positioner for wafer processing J. IEEE Transactions on Mechatronics 26

8 J. J LI Bingqiang LIN Hui. PMSM periodical torque ripple minimization using iterative learning control J. Electric Machines ZHOU Gan HUANG Xueliang. Control stragies of 3-degree-of- and freedom synchronous permanent-magnet planar motors J. Electric Machines and Control OUYANG P R ZHANG W J GUPTA M M. An adaptive switching learning control method for trajectory tracking of robot manipulators J. Mechatronics AHN Hyosung CHEN Yangquan MOORE K L. Iterative learning control brief survey and categorization J. Applications and Reviews DIJKSTRA B G. Iterative learning control with applications to a wafer stage D. Zuid-Holland Delft University of Technology PMSM Control HEERTJES M TSO T. Nonlinear iterative learning control with applications to lithographic machinery J. Control Engineering Practice MOORE K L CHEN Yangquan AHN Hyosung. Iterative learning control a tutorial and big picture view J. IEEE Transactions on Mechatronics TAYEBI A. Adaptive iterative learning control for robot manipulators J. Automatica

34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1

34 22 f t = f 0 w t + f r t f w θ t = F cos p - ω 0 t - φ 1 2 f r θ t = F cos p - ω 0 t - φ 2 3 p ω 0 F F φ 1 φ 2 t A B s Fig. 1 22 2 2018 2 Electri c Machines and Control Vol. 22 No. 2 Feb. 2018 1 2 3 3 1. 214082 2. 214082 3. 150001 DOI 10. 15938 /j. emc. 2018. 02. 005 TM 301. 4 A 1007-449X 2018 02-0033- 08 Research of permanent