64 4 ρ θ R X x X Y ε' XY X Y 曲率圆 XY 期望轨迹 Y (X o,y o) XY 切线 XY 着 R 2 Korn 兹 Crosscoupld O CNC 2 Fig. 2 Contour rror of arbitrar contour trajctor

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1 ELECTRI C MACHINES AND CONTROL Vol. 4 No. 9 Sp. 2 XY 78 XY H ZPETC H ZPETC XY H TP 273 A 7 449X Contour rror analsis and normal crosscoupld control for dirct driv XY tabl WANG LimiWU ZhitaoSUN YibiaoJIN Fuing School of Elctrical EnginringShnang Univrsit of TchnologShnang 78China Abstract Load disturbancmchanical tim dla and paramtr mismatch btwn two axs will affct th contour accurac of XY tabl dirctl drivn b two prmannt magnt linar snchronous motors. Th contour coordination control stratg with H vlocit fdback controlzro phas rror tracking control ZPETC and normal crosscoupld control is proposd to dcras tracking rror and contour rror for improving contour accurac. H vlocit fdback control can supprss load disturbanc and obtain strong robustnsszpetc can implmnt prcis tracking basd on zropol and phas cancllations and normal crosscoupld control can improv th contour accurac and simplif th controllr dsign b raltim compnsation btwn two axs with th contour rror as th dirct controlld variabl. Simulation rsults show that th proposd control sstm posssss good tracking prformancstrong robustnss and high contour accurac. K words XY tabl H control zro phas rror tracking control contour rror normal crosscoupld control

2 64 4 ρ θ R X x X Y ε' XY X Y 曲率圆 XY 期望轨迹 Y (X o,y o) XY 切线 XY 着 R 2 Korn 兹 Crosscoupld O CNC 2 Fig. 2 Contour rror of arbitrar contour trajctor ε ε' = 槡 X a X o 2 Y a X o 2 ρ PMLSM XY X a = ρsin θ X o x Y a = ρcos θ Y o H ZPETC ε ε' = 槡 ρsin θ x 2 ρcos θ 2 ρ 2 H ZPETC ε ε' = cos θ ( 2ρ) sin θ x ( 2ρ) x 3 XY A A 2 A XY XY PMLSM XY R PMLSM dq P ε d i d = Y A 着 Fig. O R P 2 着 A 2 着 2 期望轨迹 Tracking rror and contour rror X 2 P X a Y a X o Y o 籽 n 着着 ^ P x ε ε 2 2 XY M v Bv = F F L = K f i q F L X 4 i q q B F F L K f M v i q v K f G s = v s i q s = Ms B 5

3 9 XY H 6 x = X X2 ] T X ZPETC X 2 ξ u z 7 5 G s q > q 2 > K ξ f G n s = M n s B H K H KG s x = A ΔA x B ξ B 2 ΔB u G 7 z = Cx Du } cl s = s G s G n s G s K 2 K s AB B 2 CD ΔA ΔB ΔA ΔB ΔA ΔB= EQF a F b 8 E F a F b Q z d d Q ΩΩ = Q t Q T t Q t I t I ZPETC c z c z = zd A c z B u c z 7 D T D C= B a c z B u a 2 3 I λ > Riccati A T X XA X B B Τ λ 2 EE Τ X C T C λ 2FΤ a F a ( XB 2 λ 2FT a Fb ) ( ) R 2 B Τ 2 X λ 2FT b Fa < 9 z k * k F b z = jωt = cos ωt jsin ωt X > R 2 = I λ 2 F T b K = R 2 B T 2 X λ 2 F T b F a u = KxK =K K 2 H Q Ω B u c z = b u c b u c z b u cm z m ξ z T zξ s T zξ s < δ t σ δ 2 t σ 2 σ > σ 2 > PMLSM δ t H σ σ B σ 2 K f Q = E = M M 6 δ 2 t δ t δ 2 t B /M /M σ2 PMLSM F a = ] F b = ] T B x = M δ t x M δ 2 t K f ξ M δ Q Ω ΔA ΔB 8 MAT 2 t u LAB 9 K 2. 2 ZPETC q XY z = q 2 x u Tomizuka G cl z = z d B a c z B u c z 2 A c z B a c z B u c z * k k k * k = Bu c z B u c z 4 B u c 2 z = jωt ω B u c jωt = R ω jim ω B u c 5 6 R ω = bu c b u c cos ωt b u cm cos mωt b u c b u c b u cm

4 66 4 Im ω = bu c b u c sin ωt b u cm sin mωt b u c b u c b u cm B u c z /B u c B u c z /B u c = R ω j Im ω R ω j Im ω= R 2 ω Im 2 ω 7 R ω j Im ωr ω j Im ω 3 X d Y d X a Y a ξ x ξ X Y 4 K x K X Y Ⅰ Ⅱ X Y Ⅲ Ⅳ X Y 2. 3 H Ⅴ K f 8 Ⅲ M nsb C x Ⅰ 4 C ε = C x C x 8 n n ZPETC K Yd u Ⅱ K 2 ^ε K Ⅳ ^ε 2 3 ^ε ε Fig. 3 T x ] sin θ 3 ε = cos θ x sin θ = cos ] θ 9 2 XY M n = n. kgb = 8 N s /mk f = 29 N /A = x ] sin θ n = cos ] θ ^ε n Y d t = sin π t. 5 mm ^ε = n 2 2 ^ε. s ε B /M /M n ^ε = ε n 2 ^ε n t t = t x t ] n = n x 槡 t n ] 2 x t 2 = t x 槡 t 2 x t 2 C x = n x C = n X d t n 4 s Y 6 N 23 H ZPETC XY ZPETC x x n x K x G ccc u x n x Ⅴ K f K 2x K x K f K f 孜 x 孜 MsB MsB M nb XY Block diagram for XY tabl control sstm M =. M n M n B K f ` M X Y X d t = sin πt mm σ =. 25σ 2 =. 25 q = 5q 2 = 25λ = 3. 9 MAT ^ε LAB H K x K = K 2x K 2 = ^ε = ε n = n n s X 5 N X a Y a

5 9 XY 67 2 K x = 6. 5K = 7 XY 2 3 MATLAB c2d minral ZPETC x z z. 98 ZPETC 3. 84z z2 2. 2z z PI.6.4 k px = 69k ix = 28k p = 7k i = 位移 /mm 位移 /mm (a)x 轴 (b)y 轴 X Y ZPETC H Fig. 4 Position rsponss of X axis and Y axis using diffrnt controllrs with load 跟踪误差 /mm 跟踪误差 /mm (a)x 轴 (b)y 轴 G ccc PID k p = k i = k d = 42 6 轮廓误差 /mm 6 Fig WANG Guangan ZHANG RunxiaoSHUAI Mit al. Contour rror controllr in NC machin tools J. Machin Tool & Hdraulics J. Fig. 5 Tracking rrors of X axis and Y axis using diffrnt controllrs with load WANG BoLIANG YingchunDONG Shn. Fuzz contour control stratg for th ultra prcision machin toolj. Journal of 4 5 PI Harbin Institut of Tchnolog ZPETC H J 法向交叉耦合控制的轮廓误差非交叉耦合控制的轮廓误差 Contour rrors of XY tabl using diffrnt controllrs with paramtr mismatch of two axs 6 ZPETC H 4 XY H ZPETC ZPETC H. J.

6 QU YonginZHAO XimiGUO Qingding. Cross coupling control of contour rrors basd on zro phas rror tracking controllr J. China Mchanical Enginring SUN DongSHAO XiaoinFENG Gang. A modlfr crosscoupld control for position snchronization of multiaxis motions J. IEEE Transactions on Control Sstms Tchnolog CHENG MingangLEE Chngchin. Motion controllr dsign for contourfollowing tasks basd on raltim contour rror stimation J. IEEE Trans. on Industrial Elctronics H J LAN YipngGUO Qingding. Robust H control for linar motor srvo sstm J. Transactions of China Elctrotchnical Socit DOB ZPETC J ZHAO XimiGUO Qingding. Linar srvo robust tracking control basd on DOB and ZPETC to improv th contour machining prcision J. Transactions of China Elctrotchnical Socit YEH ShShiuhHSU PauLo. Estimation of th contouring rror vctor for th crosscoupld control dsignj. IEEE /ASME Trans. on Mchatronics

Fig. 1 Th priodic signal gnrator / 1 Tds T d 2 K q s T d 2 r 2 Fig. 2 Th block diagram of th sstm basd on 3 th plugin tp rptitiv cont

Fig. 1 Th priodic signal gnrator / 1 Tds T d 2 K q s T d 2 r 2 Fig. 2 Th block diagram of th sstm basd on 3 th plugin tp rptitiv cont 17 2 213 2 ELECTRI C MACHINES AND CONTROL Vol. 17 No. 2 Fb. 213 1 2 1 2 2 1. 1187 2 1116 : 针对直线电机驱动的凸轮加工随动系统实际加工应用中, 要求系统既要对周期性输入信号具有跟踪能力, 又要对周期性扰动具有抑制能力对这一问题, 根据基于内模原理的重复控制理论设计实用性强的插入型重复控制器, 来消除由周期性输入的基波和谐波所引起的误差,