• Chinese Journal of Ship Research
  • Vol. 20, Issue 1, 278 (2025)
Shijie LI1,2, Jiawei HE2, Jialun LIU1,3,4, Taixu LIU2, and Chengqi XU2
Author Affiliations
  • 1State Key Laboratory of Maritime Technology and Safety, Wuhan 430063, China
  • 2School of Transportation and Logistics Engineering, Wuhan University of Technology, Wuhan 430063, China
  • 3East Lake Laboratory, Hubei Province, Wuhan, 420202, China
  • 4National Engineering Research Center for Water Transport Safety, Wuhan 430063, China
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    DOI: 10.19693/j.issn.1673-3185.03888 Cite this Article
    Shijie LI, Jiawei HE, Jialun LIU, Taixu LIU, Chengqi XU. Ship heading and trajectory control method based on L1-GPR[J]. Chinese Journal of Ship Research, 2025, 20(1): 278 Copy Citation Text show less

    Abstract

    Objectives

    Intelligent ships at sea are influenced by environmental interference, and the influence of the uncertainty of model parameters leads to the problem of low ship motion control accuracy, so it is necessary to improve the resistance of the ship's control algorithm to such interference.

    Methods

    Based on an L1 adaptive control algorithm and Gaussian process regression (GPR) model, an L1 adaptive controller combined with a GPR model controller for underactuated ship path tracking control is proposed, and the control law is derived using the Lyapunov control function. The L1 adaptive controller is a new technique that considers both robustness and fast adaptivity. The closed-loop control system has proven to be consistently globally asymptotically stable, while the GPR model is used to model sudden disturbances and environmental disturbances during ship navigation, and achieve the rapid elimination of the effects of such disturbances in combination with the adaptive law.

    Results

    The simulation results show that adding the GPR model reduces the average rudder amplitude by 14.9%, average absolute heading error by 23.2%, and maximum absolute heading error by 12.1%. The effects of environmental disturbances can be cancelled out faster and a stable state reached more rapidly than in cases without added disturbances.

    Conclusions

    The proposed L1-GPR adaptive controller can effectively resist various disturbances during navigation.

    $ \dot {\boldsymbol{\eta}} = {{\boldsymbol{T}}}(\varphi ){\boldsymbol{v}} $(1)

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    $ {{\boldsymbol{T}}}(\varphi ) = \left[ cosφsinφ0sinφcosφ0001 \right] $(2)

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    $ {T_1}{T_2}\ddot r + ({T_1} + {T_2})\dot r + r = K(\delta + {\delta _{\text{c}}}) + K{T_3}{\delta _{\text{c}}} + {f_{{\text{ci}}}} $(3)

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    $ \left\{ (xlosx)2+(ylosy)2=R02(xlosxn)/(ylosyn)=(xn+1xlos)/(yn+1ylos) \right. $(4)

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    $ φlos(n)=arctan(Δy(n)/Δx(n))=arctan((ylosy)/(xlosx)) $(5)

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    $ {\hat f_{\Delta {\text{ci}}}} ~ {\rm{GP}}(m(\xi ),{k_i}(\xi ,\xi ')) $(6)

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    $ {k_i}(\xi ,\xi ') = \sigma _{f{i}}^2\exp \left(\sum\limits_{j = 1}^6 ( {\xi _j} - \xi _{{j}}')^2/( - 2l_{ij}^2)\right) $(7)

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    $ f_{{{\text{ci}}}}^k({\xi ^ * }) ~ {\rm N}(\mu _{i}^k({\xi ^ * }),{var} _{i}^k({\xi ^ * })) $(8)

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    $ \left\{ μik(ξ)=(\boldsymbolαik)\boldsymbolkik(ξ)varik(ξ)=ki(ξ)(\boldsymbolkik)T\boldsymbolCik\boldsymbolkik \right. $(9)

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    $ \left\{ T1T2=ηT1T2T1+T2=σ(T1+T2)r=βrK(δ+δr)=ρK(δ+δr) \right. $(10)

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    $ \left\{ ηT1T2r¨+σ(T1+T2)r˙+βr=ρKδ+fciφ˙=r \right. $(11)

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    $ \left\{ φ~=φ^φr~=r^r \right. $(12)

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    $ \left\{ φ~˙ideal=Γ1φ~r~˙ideal=Γ2r~ \right. $(13)

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    $ M(x(k),k) = \{ {\hat f_{\rm{ci}}}(x(k),k),{\hat e_f}(x(k),k)\} $(14)

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    $ f^ci(x(k),k)=uk(x(k))e^f(x(k),k)=ef,k(x(k)) $(15)

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    $ \left\{ φ^˙=Γ1φ~+rr^˙=Γ1r~+(T1+T2)1(ρ^σKδβ^σrη^σT1T2r¨+f^ci) \right. $(16)

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    $ \left\{ φ~˙=Γ1φ~+rr~˙=Γ2r~+(T1+T2)1(ρ~σKδβ~σrη~σT1T2r¨+f~ci) \right. $(17)

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    $ Vpred=1/2((1/γρσ)ρ~σ2+(1/γβσ)β~σ2+(1/γησ)η~σ2+(1/γfciσ)f~ciTf~ci)+1/2φ~+1/2(T1+T2)r~2$(18)

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    $ V˙pred=((1/γρσ)ρ~σρ^˙σ+(1/γβσ)β~σβ^˙σ+(1/γησ)η~ση^˙σ+(1/γfciσ)f~cif^˙ci)+φ~φ~˙+r~(T1+T2)r~˙=(1/γρσ)ρ~σρ^˙σ+(1/γβσ)β~σβ^˙σ+(1/γησ)η~ση^˙σ+(1/γfciσ)f~cif^˙ciΓ1φ~2Γ2(T1+T2)r~2+r~(ρ~σKδβ~σrη~σT1T2r¨+f~ci)=ρ~σ((1/γρσ)ρ^˙σ+r~Kδ)+β~σ((1/γβσ)β^˙σr~r)+η~σ((1/γησ)η^˙σr~T1T2r¨)+f~ci((1/γfciσ)f^˙ci+r~)Γ1φ~2(T1+T2)Γ2r~2 $(19)

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    $ \left\{ ρ^˙σ=γρσr~Kδβ^˙σ=γβσr~rη^˙σ=γησr~T1T2r¨f^˙ci=γfciσr~ \right. $(20)

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    $ \left\{ ρ^˙σ = γησProj(ρσ,r~Kδ)β^˙σ=γβσProj(βσ,r~r)η^˙σ=γησProj(ησ,r~T1T2r¨)f^˙ci=γfciσProj(fci,r~) \right. $(21)

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    $ {\dot V_{\rm{pred}}} = - {\varGamma _1}{\tilde \varphi ^2} - ({T_1} + {T_2}){\varGamma _2}{\tilde r^2} \leqslant 0 $(22)

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    $ \left\{ z1=φφdz2=rα \right. $(23)

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    $ {V_1} = 1/2{\textit{z}}_1^2 $(24)

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    $ {\dot V_1} = {{\textit{z}}_1}{\dot {\textit{z}}_1} = {{\textit{z}}_1}(r - {\dot \varphi _{\rm{d}}}) = {{\textit{z}}_1}{{\textit{z}}_2} + {{\textit{z}}_1}(\alpha - {\dot \varphi _{\rm{d}}}) $(25)

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    $ {\dot V_1} = - {K_1}{\textit{z}}^2_1 + {{\textit{z}}_1}{{\textit{z}}_2} $(26)

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    $ T{{\dot {\textit{z}}}_2} = T\dot r - T\dot \alpha = {{\hat \rho }_\sigma }K\delta - {{\hat \beta }_\sigma }r - {{\hat \eta }_\sigma }{T_1}{T_2}\ddot r + {{\hat f}_{\rm{ci}}} - T\dot \alpha $(27)

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    $ {V_2} = 1/2{{\textit{z}}_2}{T^{ - 1}}{{\textit{z}}_2} + {V_1} $(28)

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    $ V˙2=z2T1z˙2+V˙1=z2(ρ^σKδβ^σrη^σT1T2r¨+f^ciTα˙)z1k1z1+z1z2$(29)

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    $ V˙2==z2(ρ^σKδβ^σrβ^σαη^σT1T2r¨+f^ciTα˙+z1)K1z12 $(30)

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    $ {{\hat \rho }_\sigma }K\delta = {{\hat \beta }_\sigma }r + {{\hat \beta }_\sigma }\alpha + {{\hat \eta }_\sigma }{T_1}{T_2}\ddot r - {{\hat f}_{\rm{ci}}} + T\dot \alpha - {{\textit{z}}_1} - {K_2}{{\textit{z}}_2} $(31)

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    $ {\dot V_2} = - {K_2}{{\textit{z}}_2}^2 - {K_1}{{\textit{z}}_1^2} \leqslant 0 $(32)

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    $ Vctrl=Vpred+V2=1/2((1/γρσ)ρ~σ2+(1/γβσ)β~σ2+(1/γησ)η~σ2+(1/γfciσ)f~ciTf~ci)+1/2φ~+1/2(T1+T2)r~2+z2T1z˙2+1/2z1 $(33)

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    $ V˙ctrl=V˙pred+V˙2=Γ1φ~2(T1+T2)Γ2r~2K2z22K1z120$(34)

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    $ \delta _{\rm{ctl}} = C(s)\delta = {\omega ^2}\delta /({s^2} + 2\zeta \omega s + {\omega ^2}) $(35)

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