• Chinese Journal of Ship Research
  • Vol. 20, Issue 1, 263 (2025)
Qiang GUO1, Jiaqi WANG1, Xianku ZHANG2, and Daocheng MA2
Author Affiliations
  • 1School of Electrical and Control Engineering, Xi'an University of Science and Technology, Xi'an 710054, China
  • 2Navigation College, Dalian Maritime University, Dalian 116026, China
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    DOI: 10.19693/j.issn.1673-3185.04240 Cite this Article
    Qiang GUO, Jiaqi WANG, Xianku ZHANG, Daocheng MA. BLF-based adaptive path following control for unmanned surface vehicles under shallow water effects[J]. Chinese Journal of Ship Research, 2025, 20(1): 263 Copy Citation Text show less

    Abstract

    Objective

    This study investigates how to effectively address path-dependent constraints during the path-following of unmanned surface vessels in complex waterways while ensuring navigation safety and stability.

    Method

    First, performance and feasibility constraints are established for the vessel's navigation based on the precision and safety requirements of autonomous ships in shallow waters. Next, to address the issues of the path parameter representation and convergence requirements of the controller, a barrier Lyapunov function (BLF) combined with a fixed-time convergence strategy is applied. A path-dependent controller capable of converging within a fixed time is then designed, and radial basis function neural networks (RBFNN) and adaptive robust terms are used to handle nonlinearities and environmental disturbances. Finally, the intelligent unmanned surface vehicle model "Dazhi" is used to simulate shallow water effects, and the controller's performance is analyzed through simulations.

    Results

    The simulation results show that the path tracking error converges rapidly to the desired region without violating the constraints. Compared to the unconstrained case, the controller demonstrates clear advantages in convergence speed and precision, verifying its effectiveness and robustness.

    Conlusions

    The proposed control strategy is innovative and significant in addressing path-dependent constraints for ship navigation, ensures precise path tracking within a fixed time, and has significant theoretical and practical application value. Future research may further optimize the control strategy to address more complex water environments and higher-precision path tracking tasks.

    $ \left\{ x˙=ucos(ψ)vsin(ψ)y˙=usin(ψ) + vcos(ψ)ψ˙=r \right. $(1)

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    $ \left\{ mΔu˙=XuΔu+Xu˙Δu˙mv˙+mV0r+mxcr˙=Yνv+Yrr+Yv˙v˙+Yr˙r˙+YδδIzzr˙+mxcv˙+mxcV0r=Nνv+Nrr+Nv˙v˙+Nr˙r˙+Nδδ \right. $(2)

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    $ \dot r = {f_r}(\upsilon ) + \frac{{{N_\delta }}}{{{I_{{\textit{z}}{\textit{z}}}}}}\delta + {d_{wr}} $(3)

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    $ \left\{ xe=x(t)xd(ϖ(t))ye=y(t)yd(ϖ(t))ψe=ψψrze(t,ϖ(t))=xe2+ye2 \right. $(4)

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    $ {\psi _{\text{r}}} \triangleq 0.5[1 - {{\rm{sign}}} ({x_{\rm{e}}})]{\rm{sign}}({y_{\rm{e}}}){\text{π}} + \arctan\left(\frac{{{y_{\text{e}}}}}{{{x_{\text{e}}}}}\right) $(5)

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    $ {{\textit{z}}_{\text{e}}}(t,\varpi (t)) < {M_{\text{H}}}(\varpi (t)) $(6)

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    $ \left| {{\psi _{\text{e}}}} \right| < {M_\psi } $(7)

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    $ \left\{ VZe=12ηz2Vψ=12ηψ2 \right. $(8)

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    $ \left\{ ηz=MHze(MHze)(zeλ)ηψ=Mψ2ψeMψ2ψe2 \right. $(9)

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    $ \left\{ α1(x)V(x)α2(x)V˙(x)γ1Vp(x)γ2Vq(x)+Ψ,t0 \right. $(10)

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    $ \left\{ {\mathop {\lim x(t)}\limits_{t \to T} \left| {V(x(t)) \leqslant \min \left\{ {\gamma _1^{ - \tfrac{1}{p}}{{\left(\frac{\varPsi }{{1 - \varsigma }}\right)}^{\tfrac{1}{p}}},\gamma _2^{ - \tfrac{1}{q}}{{\left(\frac{\varPsi }{{1 - \varsigma }}\right)}^{\tfrac{1}{q}}}} \right\}} \right.} \right\} $(11)

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    $ T \leqslant {T_{\max }} = \frac{1}{{\varsigma {\gamma _1}(1 - p)}} + \frac{1}{{\varsigma {\gamma _2}(q - 1)}} $(12)

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    $ {\left| {\textit{z}} \right|^\mu }{\left| \zeta \right|^\theta } \leqslant \frac{\mu }{{\mu + \theta }}\iota {\left| {\textit{z}} \right|^{\mu + \theta }} + \frac{\theta }{{\mu + \theta }}{\iota ^{\tfrac{{ - \mu }}{\theta }}}{\left| \zeta \right|^{\mu + \theta }} $(13)

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    $ 0 \leqslant \left| \omega \right| - \omega \tanh \left(\frac{\omega }{\sigma }\right) \leqslant 0.2785\sigma $(14)

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    $ \chi {(\omega - \chi )^q} \leqslant \frac{q}{{q + 1}}({\omega ^{q + 1}} - {\chi ^{q + 1}}) $(15)

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    $ F{\text{(}}{\boldsymbol{X}}{\text{)}} = {{\boldsymbol{W}}^{*{\text{T}}}}\varphi {\text{(}}{\boldsymbol{X}}{\text{) + }}\varepsilon $(16)

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    $ {\varphi _i}\left( {{{\boldsymbol{X}}}} \right) = {\text{exp}}\left( { - \frac{{{{\left( {{\boldsymbol{X}} - {{\boldsymbol{\mu}} _i}} \right)}^{\text{T}}}\left( {{\boldsymbol{X}} - {{\boldsymbol{\mu }}_i}} \right)}}{{\sigma _i^2}}} \right),\;\;i = 1,2,...,l $(17)

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    $ \left\{ η˙z=ηzze(ucos(ψψr)vsin(ψψr))ηzzeϖ˙(cosψrdxddϖ+sinψrdyddϖ)+ϖ˙ηzMHdMHdϖη˙ψ=dηψdψerdηψdψeuzesin(ψψr)dηψdψevzecos(ψψr)+dηψdψeϖ˙(1zecosψrdxddϖ1zesinψrdyddϖ) \right. $(18)

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    $ αu=Kze(12)2ηz3ηzzeKze(12)341ηzze1ηzSu1ηzezeudηzMHdMHdϖ+2usin2(ψψr2)+vsin(ψψr)+ud(cosψrdxddϖ+sinψrdyddϖ) $(19)

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    $ αr=Kψ(12)2ηψ3dηψdψeKψ(12)341dηψdψe1ηψSψ+uzesin(ψψr)+vzecos(ψψr)ud(1zecosψrdxddϖ1zesinψrdyddϖ)$(20)

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    $ {S}_{\ell }=\left\{ (η2)34,|η|ε0j=12aj(η2)j(ε02)j+34,\right. $(21)

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    $ V˙1Kze(12ηz2)2Kze(12ηz2)34Kψ(12ηψ2)2Kψ(12ηψ2)34+zϖ[ηzηzze(cosψrdxddϖ+sinψrdyddϖ)+ηzeηzMHdMHdϖ]ηzeηzzeueηψdηψdψwre+ηψdηψdψezϖ(1zecosψrdxddϖ1zesinψrdyddϖ)+εz+εψ $(22)

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    $ τrβ˙r+βr=αr,βr(0)=αr(0) $(23)

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    $ {\dot r_{\text{e}}} = {\dot \beta _r} - {f_r}(\upsilon ) - \frac{{{F_r}}}{{{m_r}}}\delta - {d_{wr}} $(24)

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    $ {F_r}({X_r}) = - {f_r}(\upsilon ) + \frac{1}{2}{r_{e} } $(25)

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    $ {F_r}({X_r}) = {W_r}^{{\text{*T}}}{\varphi _r}{\text{(}}{X_r}{\text{) + }}{\varepsilon _r} $(26)

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    $ reFr(Xr)=re[WrTφ(Xr)+εr]|re|[Wrφ(Xr)+εr]12ai2re2θrφrT(Xr)φ(Xr)+ar22+re22+εr $(27)

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    $ {V_3} = \frac{1}{2}r_{\text{e}}^2 + \frac{1}{{2{\mu _{\theta r}}}}\tilde \theta _r^{\text{T}}{\tilde \theta _r} + \frac{1}{{2{\mu _{\xi r}}}}\tilde \xi _r^2 $(28)

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    $ ατr=Kr(12)2re2+Kr(12)341reSr+tanh(reχr)ξ^r+β˙+12ar2re2θ^rφrT(Xr)φ(Xr) $(29)

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    $ \left\{ θ^˙r=μθr2ar2re2φrT(Xr)φ(Xr)σθr1θ^rσθr2θ^r3ξ^˙r=μξrretanh(reχr)σξr1ξ^rσξr2ξ^r3 \right. $(30)

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    $ {S}_{r}=\left\{ (re2)34,|re|εr0j=12aj(re2)j(εr02)j+34,\right. $(31)

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    $ \left\{ σθθ~rTθ^rσθθ~r22+σθθr22σξξ~rξ^rσξξ~r22+σξξr22 \right. $(32)

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    $ \left\{ (θ~r22μθr)pθ~r22μθr+(1p)pp1p(ξr22μξr)pξr22μξr+(1p)pp1p\right. $(33)

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    $ \left\{ θ~rθ^r334(θr4θ~r4)ξ~rξ^r334(ξr4ξ~r4) \right. $(34)

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    $ V˙=V˙1+V˙2Kze(12ηz2)2Kze(12ηz2)34Kψ(12ηψ2)2Kψ(12ηψ2)34Kr(12re2)2Kr(12re2)343σθr2μθr(θr22μθr)2σθr1(θr22μθr)34 $()

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    $ 3σξr2μξr(ξ~r22μξr)2σξr1(ξ~r22μξr)34+14σθr1+3σθr24μθrθr4+σθrθ~r22μθr+14σξr1+3σξr24μξrξr4+σξrξr22μξr+ar22+0.2785χrξr+ηzηzMHdMHdϖ+zϖ[ηzηzze(cosψrdxddϖ+sinψrdyddϖ)+ηψdηψdψe(1zecosψrdxddϖ1zesinψrdyddϖ)]+ε¯ $(35)

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    $ ϖ¨=Kϖ(12)2zϖ3Kϖ(12)341zϖSω+duddϖϖ˙+ηzeηzze(cosψrdxddϖ+sinψrdyddϖ)ηzeηzMHdMHdϖηψdηψdψe(1zecosψrdxddϖ1zesinψrdyddϖ)$(36)

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    $ {S}_{\varpi }=\left\{ (zϖ2)34,|zϖ|εϖ0j=12aj(zϖ2)j(εϖ02)j+34,\right. $(37)

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    $ V˙Kze(12ηz2)2Kze(12ηz2)34Kψ(12ηψ2)2Kψ(12ηψ2)34Kr(12re2)2Kr(12re2)34Kϖ(12zϖ2)2Kϖ(12zϖ2)343σθr2μθr(θ~r22μθr)2σθr1(θ~r22μθr)343σξr2μξr(ξ~r22μξr)2σξr1(ξ~r22μξr)34+3σθr24μθrθr4+σθrθ~r22μθr+14σθr1+14σξr1+3σξr24μξrξr4+σξrξr22μξr+ar22+0.2785χrξr+ε¯ $(38)

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    $ \dot V(x) \leqslant - {\gamma _1}{V^p}(x) - {\gamma _2}{V^q}(x) + \varPsi $(39)

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    $ {\gamma _1} = \min \{ {K_{{\textit{z}}{\text{e}}}},{K_\psi },{K_{\text{r}}},{K_\varpi },{\sigma _{\theta u1}},{\sigma _{\theta r1}},{\sigma _{\xi r1}}\} $()

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    $ γ2=min{Kze,Kψ,Kr,Kϖ,3σθu2μθu,3σθr2μθr,3σξu2μξu,3σξr2μξr}$()

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    $ Ψ=14σθr1+3σθr24μθrθr4+σθrθ~r22μθr+14σξr1+3σξr24μξrξr4+σξrξr22μξr+ar22+0.2785χrξr+ε¯ $(40)

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    Qiang GUO, Jiaqi WANG, Xianku ZHANG, Daocheng MA. BLF-based adaptive path following control for unmanned surface vehicles under shallow water effects[J]. Chinese Journal of Ship Research, 2025, 20(1): 263
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