• Optics and Precision Engineering
  • Vol. 29, Issue 12, 2891 (2021)
Hua-yang SAI1,2, Zhen-bang XU1,3,*, Shuai HE1, En-yang ZHANG1, and Chao QIN1
Author Affiliations
  • 1CAS Key Laboratory of On-orbit Manufacturing and Integration for Space Optics System, Changchun Institute of Optics, Fine Mechanics and Physics, CAS, Changchun30033, China
  • 2University of Chinese Academy of Sciences, Beijing100049, China
  • 3Materials and Optoelectronics Research Center, University of Chinese Academy of Sciences, Beijing100049, China
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    DOI: 10.37188/OPE.20212912.2891 Cite this Article
    Hua-yang SAI, Zhen-bang XU, Shuai HE, En-yang ZHANG, Chao QIN. Predefined-time sliding mode control for rigid spacecraft[J]. Optics and Precision Engineering, 2021, 29(12): 2891 Copy Citation Text show less

    Abstract

    To minimize system uncertainty and external disturbance in attitude tracking control for rigid spacecraft, a predefined-time sliding mode controller (PTSMC) is proposed. First, the spacecraft attitude tracking system is developed with quaternion parameterization, and the predefined time sliding surface is designed using an error quaternion and error angular velocity. Then, considering the uncertainties and external disturbances of the spacecraft system, a PTSMC with a non conservative upper bound is designed, and the noise of the system is reduced using boundary layer technology. Finally, by designing the Lyapunov function, the predefined-time stability of the proposed controller and the non conservative upper bound of the system convergence are demonstrated. The simulation results show that using the proposed approach, the attitude tracking accuracy of rigid spacecraft can reach 1.5×10-6 rad, and the angular velocity tracking accuracy can reach 2×10-6 rad/s. Compared with the existing predefined time control and non singular terminal sliding mode control, the upper bound of the stabilization time of the proposed control is more non conservative and has higher tracking accuracy and robustness. The effectiveness of the control scheme is further illustrated by the attitude tracking experiment of the 3 DOF airborne platform. The angle tracking error is less than 0.1 rad, and the position tracking error is less than 0.2 m.
    q˙v=12(q4I3+qv×)ωq˙4=-12qvTω(1)

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    a×=0-a3a2a30-a1-a2a10  a=[a1,a2,a3] R3(2)

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    q˙dv=12(qdv×+qd4I3)ωdq˙d4=-12qdvTωd(3)

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    e˙v=12(e4I3+ev×)ωee˙4=-12evTωe(4)

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    ωe=ω-Cωd(5)

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    C=(e42-evTev)I3+2evevT-2e4ev×(6)

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    Jω˙+ω×Jω=u+d(7)

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    e˙v=12(e4I3+ev×)ωeω˙e=ωe×Cωd-J-1[(Cωd)×Jω+ωe×Jω-u-d](8)

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    J=J0+ΔJ(9)

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    ω˙e=ωe×Cωd-J0-1[(Cωd)×J0ω+ωe×J0ω-u-ρ](10)

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    ρ=d-ΔJωe=ΔJωe×Cωd-(Cωd)×ΔJω-ωd×ΔJω.(11)

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    ρb0+b1ω+b2ω2(12)

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    x˙=f(x;ρ),x(0)=x0(13)

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    T(x0)Tc   x0Rn(14)

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    V˙(x)-γTc(αV(x)p+βV(x)q)k(15)

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    γ(ρ)=Γ(mp)Γ(mq)αkΓ(k)(q-p)αβmp(16)

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    sa=ωe2+γ22Tc12(αevp+βevq)(17)

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    s=ωe+sa12(18)

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    u=u1+u2u1=J0|ωe|η-a|sa|12+|ωe|-J0η-J0ωe×Cωd+(Cωd)×J0ω+ωe×J0ωu2=-k0sign(s)η=γTc2(α|s|p+β|s|q)ksign(s)a=γ22Tc2(αp|ev|p-1+βq|ev|q-1)ωek0=b0+b1ω+b2ω2(19)

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    ω˙e=f(ω)+b(ω)u+J0-1ρ(20)

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    s˙=ω˙e+|ωe|ω˙e|sa|12+γ22Tc12(αp|ev|p-1+βq|ev|q-1)ωe|sa|12(21)

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    s˙=f(ω)+b(ω)u+J0-1ρ+|ωe|(f(ω)+b(ω)u+J0-1ρ)|sa|12+γ22Tc12(αp|ev|p-1+βq|evq-1|)ωe|sa|12(22)

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    s˙=|ωe|η-a|sa|12+|ωe|-η-J0-1k0sign(s)+J0-1ρ+|ωe||ωe|η-a|sa|12+|ωe|-η-J0-1k0sign(s)+J0-1ρ|sa|12+γ22Tc12(αp|ωe|p-1+βq|ωe|q-1)ωe|sa|12(23)

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    s˙=-η+J0-1ρ-J0-1k0sign(s)+|ωe|(J0-1ρ-J0-1k0sign(s)|sa|12(24)

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    s˙-η=-γTc2(α|s|p+β|s|q)ksign(s)(25)

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    V˙1=s˙sign(s)-γTc2(α|s|p+β|s|q)k=-γTc2(αV1(s)p+βV1(s)q)k(26)

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    ωe=-γTc1(α|ev|p+β|ev|q)ksign(ev)(27)

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    V˙2(ev)=-γTc1(αV2(ev)p+βV2(ev)q)k(28)

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    a=γ22Tc2(αp|ev|p-1+βq|ev|q-1)  |ωe|>εεI3×1   |ωe|ε(29)

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    u=-γ12(q1+p1|x1|p1)|x1|q1-1exp(|x1|p1)sign(σ)-γ2exp(α2|σ|p2)σβ2q2-ksign(σ)σ=x2+x22+2γ12exp(|x1|p1)x1q112(30)

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    u=u0+u1u0=ω×J0ω-J0(ωe×Cωd-Cω˙d)-12λJ0L(ev)(e4I3+ev×)ωeu1=-K1S-K2sigα0(S)-(b0+b1ω+b2ω2)sign(S)S=ωe+λf(ev)s(x)=|x|rsign(x)                  |x|δax+b|x|r0 sign(x)   |x|<δf(ev)=[s(e1),s(e2),s(ee)]TL(ev)=diag(l(e1),l(e2),l(e3))sigα0(ζ)=[|ζ|α0sign(ζ1),...,|ζn|α0sign(ζn)]T(31)

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    u=-J0kpev-J0kdev-J0ωe×Cωd-J0-1(Cωd)×J0ω-J0-1ωe×J0ω(32)

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    Hua-yang SAI, Zhen-bang XU, Shuai HE, En-yang ZHANG, Chao QIN. Predefined-time sliding mode control for rigid spacecraft[J]. Optics and Precision Engineering, 2021, 29(12): 2891
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