• Optics and Precision Engineering
  • Vol. 30, Issue 24, 3139 (2022)
Yi LIU, Xiaoteng MA, Zongqiang FENG, Jiantao YAO*, and Yongsheng ZHAO
Author Affiliations
  • Laboratory of Parallel Robotics and Mechatronic Systems in Hebei Province, Yanshan University, Qinhuangdao066004, China
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    DOI: 10.37188/OPE.20223024.3139 Cite this Article
    Yi LIU, Xiaoteng MA, Zongqiang FENG, Jiantao YAO, Yongsheng ZHAO. Stiffness prediction-neural network based error compensation for attitude adjustment platform[J]. Optics and Precision Engineering, 2022, 30(24): 3139 Copy Citation Text show less

    Abstract

    The impact of deformation error on the end positioning accuracy of high-precision attitude adjustment equipment cannot be ignored. To improve the accuracy of a 2RRPU/2RPU/U two-axis parallel attitude platform, an error compensation model is proposed based on a stiffness model to predict the error trend and a neural network algorithm to improve the prediction accuracy. The theoretical stiffness model is first established based on the full Jacobi and elastic deformation matrices of the attitude-adjusting platform. The validity of the prediction of the loaded deformation trend is verified by comparing it with the prediction by the Ansys data stiffness model. Then, a Simulink-Adams-Ansys-OPC-based simulation environment is built, and the platform full attitude simulation data is collected under random load. Next, the attitude and drive error trends are predicted based on the stiffness model and velocity Jacobi matrix, and the mapping from end error to drive compensation is realized based on the velocity Jacobi. The accuracy of the error prediction is further improved by using a neural network algorithm. The simulation results show that the attitude accuracy of the platform is improved by 9% after adopting the error compensation model, which verifies the effectiveness of the “stiffness prediction-neural network” model in the improvement of the platform attitude accuracy.
    vp1a=v+ω×ep1(1)

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    l˙1=Q1T·vp1a(2)

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    l˙1=[Q1T(ep1×Q1)T]vω=[J1]vω(3)

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    fr1C1·v+(fr1C1×d1)·ω=0(4)

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    0=[C1   0]vω=[J1rf]vω(5)

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    l˙2=[Q2T(ep2×Q2)T]vω=[J2]vω(6)

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    fr2C2·v+(fr2C×2d2)·ω=0(7)

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    0=[fr2C2    0]vw=[J2rf]vw(8)

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    MrCc2·ω=0(9)

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    0=[0    MrCc2]vw=[J2rm]vw(10)

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    fUiCUi·v+(fUiCUi×dUi)·ω=0(11)

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    0=[CU1   0]vω=[JUi]vω(12)

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    MUZCUZm·ω=0(13)

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    0=[0   MUZCUZm]vω=[JUMZ]vω(14)

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    l˙10l˙20000009×1=J1J1rfJ2J2rfJ2rmJUXJUYJUZJUMZ9×6·vω6×1=[Ja]9×6vω6×1(15)

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    Δλ11=F1(l1-l1g)E1A1(16)

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    Δλ12=F1l1gE1gA1g(17)

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    F1p=F1Q1·RR(18)

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    Δλ13=F1pl1RE1RA1R(19)

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    F1v=F1(Q1×R)R(20)

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    Δλ14=F1vl1R33E1RI1R(21)

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    ΔλR=Δλ132+Δλ142(22)

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    Δλ1=Δλ11+Δλ12+ΔλR(23)

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    k1=F1Δλ1+F1Δλ11+Δλ12+ΔλR(24)

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    k1=1(l1-l1g)E1A1+l1gE1gA1g+Q1·Rl1RRE1RA1R2+Q1×R(l1R)33E1RI1RR2Δλ1(25)

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    Δλ1rf=F1rl133E1I1(26)

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    M1a=F1r(S2×ep1)S2(27)

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    M1ap=M1a×Q1(28)

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    Δλ1am=M1apl122E1I1(29)

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    Δλ1r=(Δλ1rf)2+(Δλ1am)2(30)

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    k1r=FrΔλ1r(31)

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    k1r=1l133E1I12+(S2×ep1)×Q1l122S2E1I12(32)

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    Δλ2p=F2(l2-l2g)E2A2(33)

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    Δλ2g=F2l2gE2gA2g(34)

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    Δλ2=Δλ2p+Δλ2g(35)

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    k2=F2Δλ2(36)

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    k2=1(l2-l2g)E2A2+l2gE2gA2g(37)

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    Δλ2r=F2rtl233E2I2(38)

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    M2a=F2r×ep2(39)

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    M2a=M2a×Q2(40)

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    Δλ2a=M2rvl222E2I2(41)

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    Δλ2rf=(Δλ2r)2+(Δλ2a)2(42)

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    k2rf=1Δλ2rf(43)

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    k2rf=1l233E2I22+S1×ep2×Q2l222E2I22(44)

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    M2r=M2r·(ep2×Ua1)ep2(45)

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    M2rv=M2r×Q2(46)

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    Δλ2rmv=M2rvl222E2I2(47)

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    k2rm=1Δλ2rmv(48)

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    k2rm=1(ep2×S1)×Q2l222E2I2ep2(49)

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    ΔλUX=FUXlU33EUIU(50)

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    kUX=FUXΔλUX=3EUIUlU3(51)

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    ΔλUY=FUYlU33EUIU(52)

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    kUY=FUYΔλUY=3EUIUlU3(53)

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    ΔλUZ=FUZlUEUAU(54)

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    kUZ=FUZΔλUZ=EUAUlU(55)

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    MUZ=MUrm(Rx×Ry)(56)

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    MUrmz=MUrm×Z(57)

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    ΔλUrm=MUrmzlU22EUAU(58)

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    kUrmz=MUrmΔλUrm(59)

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    kUrmz=2EUIU(Rx×Ry)×ZlU2(60)

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    Kp=diga(k1,k1r,k2,k2rf,k2rm,kUx,kUy,kUz,kUmz)(61)

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    τ9×1=[Kp]9×9δq9×1(62)

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    τT·δq=FT·D(63)

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    D=[J]δq(64)

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    F=Kδp(65)

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    δrδθ=[Kα,β]FeMe(66)

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    δθ=δαωα+δβωβ(67)

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    δα=δθxδβ=δθycosα(68)

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    δl1δl2=[J]δαδβ(69)

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    Yi LIU, Xiaoteng MA, Zongqiang FENG, Jiantao YAO, Yongsheng ZHAO. Stiffness prediction-neural network based error compensation for attitude adjustment platform[J]. Optics and Precision Engineering, 2022, 30(24): 3139
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