• Chinese Journal of Ship Research
  • Vol. 20, Issue 1, 326 (2025)
Jiwen ZHANG, Bo XU, Chuyan WANG, and Zhaoyang WANG
Author Affiliations
  • College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China
  • show less
    DOI: 10.19693/j.issn.1673-3185.04045 Cite this Article
    Jiwen ZHANG, Bo XU, Chuyan WANG, Zhaoyang WANG. A neural dynamics model prediction-based adaptive control system for AUV formation control[J]. Chinese Journal of Ship Research, 2025, 20(1): 326 Copy Citation Text show less

    Abstract

    Objectives

    This paper seeks to provide a solution for the formation control issue that arises when autonomous underwater vehicles (AUVs) are subjected to interference from obstacles and complex ocean currents.

    Methods

    To tackle the issue of AUV hysteresis resulting from an overly rapid predicted convergence speed during dynamic obstacle avoidance, a multi-AUV formation adaptive control method (NDP-ABS) based on brain dynamics model prediction is created. Active and inhibitory sources are created to solve the local optimization problem of potential field methods. When paired with optimal control, dynamic obstacle avoidance, formation control, and predicted tracking are accomplished. Second, a nonlinear adaptive backstepping method is used to design the AUV expected tracking controller, which resolves the interference of shallow ocean current disturbances and nonlinear factors on the AUV expected tracking control in consideration of unknown nonlinear factors and ocean current disturbances introduced in the control law of the NDP process. Finally, Lyapunov theory is used to demonstrate the system's stability.

    Results

    The anti-interference and obstacle avoidance performance of the NDP-ABS system are tested in six sets of comparative simulation tests, and the results confirm its efficacy.

    Conclusions

    The NDP-ABS formation scheme offers several benefits, including cheap obstacle avoidance costs, robust resistance to interference from ocean currents, high stability, and clear advantages in the non-explicit formation control of multiple AUVs.

    $ {\boldsymbol{M}}\dot {\boldsymbol{v}} + {\boldsymbol{C}}({\boldsymbol{v}}){\boldsymbol{v}} + {\boldsymbol{D}}({\boldsymbol{v}}){\boldsymbol{v}} + {\boldsymbol{g}}(\eta ) = {\boldsymbol{F}} $(1)

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    $ \left[ (m\boldsymbolXu˙)u˙(mYv˙)v˙(mZw˙)w˙(IyMq˙)q˙(IzNr˙)r˙ \right] = \left[ FXFYFZFMFN \right] + {{\boldsymbol{F}}_{{\mathrm{b}}v}} + \left[ ΔuΔvΔwΔqΔr \right] $(2)

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    $ F_{\mathrm{bv}}=\left[\boldsymbolXuuu2+\boldsymbolXvv\boldsymbolv2+\boldsymbolXwww2+\boldsymbolXqqq2Yuvuv(mYur)ur+Yv|v|v|v|mzgq2(mZuq)uq+Zuwuw+Zw|w|w|w|Mq|q|q|q|MuquqMuwuwNuvuv+Nv|v|v|v|+Nurur\right] $(3)

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    $ \left\{ \boldsymbolx˙1=\boldsymbolx2+Δ(\boldsymbolx1)\boldsymbolx˙2=\boldsymbolMi1(\boldsymbolF\boldsymbolC(\boldsymbolx2)\boldsymbolx2D(\boldsymbolx2)\boldsymbolx2)+Δ(\boldsymbolx2)\right. $(4)

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    $ \left\{ uCR=MC1cos(kCxωCt)sin(kCyωCt)\boldsymbolvCR=MC2cos(kCyωCt)sin(kCxωCt)wCR=MC3sin(kCxωCt)sin(kCyωCt)\right. $(5)

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    $ \dot{\boldsymbol{z}}_{\rm{b}}^{\rm{i}}=\left[\boldsymbol{\omega}\right]\times\boldsymbol{z}_{\rm{b}}^{\rm{i}}+(\boldsymbol{v}^{\rm{b}}-\boldsymbol{R}(\theta_{\rm{b}},\varphi_{\rm{b}})\boldsymbol{v}_{\rm{d}}^{\rm{i}}) $(6)

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    $ \left\{ θF=arctan(z˙dx˙d2+y˙d2)φF=arctan(y˙dx˙d)\right. $(7)

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    $ {\boldsymbol{R}}({\theta _{\rm{b}}},{\varphi _{\rm{b}}}) = \left[ cosφbcosθbsinφbcosφbsinθbsinφbcosθbcosφbsinφbsinθbsinθb0cosθb \right] $(8)

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    $ \left\{ x˙b=rybq\boldsymbolzb+ucosφbcosθb+vsinφbwcosφbsinθby˙b=rxb+vusinφbcosθbvcosφbwcosφbsinθbz˙b=qxb+w+usinθbwcosθbφ˙b=r/cosθφ˙Fθ˙b=qθ˙F \right. $(9)

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    $ \frac{\mathrm{d}y_i}{\mathrm{d}t}=-Ay_i+(B-y_i)In_i^{\text{p}}(t)-(D+y_i)In_i^{\text{n}}(t) $(10)

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    $ \left\{Inip(t)=[T]++j=1KωijS[yj]+Inin(t)=[T]+j=1KωijH[yj]\right. $(11)

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    $ T=\left\{ BD0\right.$(12)

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    $ dyidt=Ayi+B(Byi)jS=1KS(ωijS)i,jSD(D+yi)jH=1KH(ωijH)i,jH $(13)

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    $ JiS=tktk+Δt{JiH(t)+di+xiAT2+di1,,dii1,dii+1,,d1NAT2}dt $(14)

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    $ J_i^{{H}}(t) = \sum\limits_{o = 1}^{{N_2}} {\mathop {\lim }\limits_{t \to \infty } \left\{ {\frac{\varUpsilon }{{{{\rm{e}}^{{d_{i,{{o}}}}}} - 1}} - \frac{\varUpsilon }{{{{\rm{e}}^{{R_{{H}}}}} - 1}}} \right\}} = 0 $(15)

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    $ \mathop {\lim }\limits_{t \to \infty } \left\| {{\boldsymbol{p}}_{{s}}^{{i}} - {\boldsymbol{p}}_{{AT}}^{{i}}} \right\|_{{AT}}^2 = 0 $(16)

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    $ \mathop {\lim }\limits_{t \to \infty } \left\| {{\boldsymbol{p}}_{{s}}^{{i}} - {\boldsymbol{p}}_{{s}}^j} \right\|_{\rm{FM}}^2 = d_{ij}^{\rm{FM}} $(17)

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    $ \boldsymbol{p}_{HP}=\left[\boldsymbolpHP(k)\boldsymbolpHP(k+1|k)...\boldsymbolpHP(k+sp|k+sp1)\right] $(18)

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    $ \left\{ \boldsymbolx˙1=\boldsymbolx2+Δ(\boldsymbolx1)\boldsymbolx˙2=\boldsymbolM1\boldsymbolF\boldsymbolΘ\boldsymbolf(\boldsymbolx2)+Δ(\boldsymbolx2) \right. $(19)

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    $ {\lambda _{\min }}({\boldsymbol{A}}){\left\| {\boldsymbol{z}} \right\|^2} \leqslant {{\boldsymbol{z}}^{\rm{T}}}{\boldsymbol{A}}{\boldsymbol{z}} \leqslant {\lambda _{\max }}({\boldsymbol{A}}){\left\| {\boldsymbol{z}} \right\|^2} $()

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    $ \left\{ \boldsymbolz1=\boldsymbolx1\boldsymbolxd\boldsymbolz2=\boldsymbolx2\boldsymbolα\boldsymbolx˙d\right. $(20)

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    $ \left\{ \boldsymbolz˙1=\boldsymbolx˙1\boldsymbolx˙1d=\boldsymbolz2+\boldsymbolα+Δ(\boldsymbolx1)\boldsymbolα=k1\boldsymbolz114ε12\boldsymbolz1V˙1=\boldsymbolz1T\boldsymbolz˙1=\boldsymbolz1T(\boldsymbolz2+\boldsymbolα+Δ(\boldsymbolx1)) \right. $(21)

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    $ \left\{ V˙1=\boldsymbolz1T\boldsymbolz2k1\boldsymbolz1214ε12\boldsymbolz12+\boldsymbolz1TΔ(\boldsymbolx1)|\boldsymbolz1TΔ(\boldsymbolx1)|14ε12\boldsymbolz12+ε1\boldsymbolβ(\boldsymbolx1)2V˙1\boldsymbolz1T\boldsymbolz2k1\boldsymbolz12+ε1\boldsymbolβ(\boldsymbolx1)2 \right. $(22)

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    $ \boldsymbolz˙2=\boldsymbolx˙2\boldsymbolα˙\boldsymbolx¨d=\boldsymbolM1\boldsymbolF\boldsymbolΘ\boldsymbolf(\boldsymbolx2)+Δ(\boldsymbolx2)\boldsymbolx¨d(\boldsymbolα\boldsymbolx1\boldsymbolx˙1+\boldsymbolα\boldsymbolxd\boldsymbolx˙d+\boldsymbolα\boldsymbolx1Δ(\boldsymbolx1)) $(23)

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    $ \left\{ V2=12\boldsymbolz12+12\boldsymbolz22+12\boldsymbolθ~T\boldsymbolΓ1\boldsymbolθ~V˙2=V˙1+\boldsymbolz2T\boldsymbolz˙2\boldsymbolθ~T\boldsymbolΓ1\boldsymbolθ^\right. $(24)

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    $ V˙2\boldsymbolz1T\boldsymbolz2k1\boldsymbolz12+ε1\boldsymbolβ(\boldsymbolx1)2+\boldsymbolz2Tu\boldsymbolz2T\boldsymbolx¨d+\boldsymbolz2T(\boldsymbolα1\boldsymbolx1\boldsymbolx˙1+\boldsymbolα1\boldsymbolxd\boldsymbolx˙d+\boldsymbolα1\boldsymbolx1Δ(\boldsymbolx1))\boldsymbolz2T\boldsymbolΘ\boldsymbolf(\boldsymbolx2)+\boldsymbolz2TΔ(\boldsymbolx2)\boldsymbolθ~T\boldsymbolΓ1\boldsymbolθ^˙ $(25)

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    $ \left\{ |\boldsymbolz2TΔ(\boldsymbolx2)|\boldsymbolz224ε22+ε2\boldsymbolβ(\boldsymbolx2)2|\boldsymbolz2T\boldsymbolα1\boldsymbolx1Δ(\boldsymbolx1)|\boldsymbolz224ε32(\boldsymbolα1\boldsymbolx1)2+ε2\boldsymbolβ(\boldsymbolx2)2\right. $(26)

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    $ \left\{ \boldsymbolu=\boldsymbolz1k2\boldsymbolz2\boldsymbolΘ^\boldsymbolf(\boldsymbolx1)14ε22\boldsymbolz2+\boldsymbolx¨d+(\boldsymbolα\boldsymbolx1\boldsymbolx˙1+\boldsymbolα\boldsymbolxd\boldsymbolx˙d)+14ε32(\boldsymbolα\boldsymbolx1)2\boldsymbolz2\boldsymbolθ^˙=\boldsymbolΓ\boldsymbolf(\boldsymbolx2)\boldsymbolz2q\boldsymbolΓ(\boldsymbolθ^\boldsymbolθ0) \right. $(27)

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    $ \dot {\boldsymbol{x}} = {\boldsymbol{f}}({\boldsymbol{x}},{\boldsymbol{u}}),{\boldsymbol{f}}(0,0) = 0,{\boldsymbol{x}}\in {{\bf{R}}^n},\boldsymbol{u} \in {{\bf{R}}^m} $()

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    $ \dot{V}(\boldsymbol{x})\leqslant-\xi V(\boldsymbol{x})+M_1+M_2 $()

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    $ V(t) \leqslant ({M_1} + {M_2})/\xi + \left[ {V(0) - ({M_1} + {M_2})/\xi } \right]{{\rm{e}}^{ - \xi t}} $()

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    $ V˙2k1\boldsymbolz12k2\boldsymbolz22+ε1\boldsymbolβ(\boldsymbolx1)2+ε2\boldsymbolβ(\boldsymbolx2)2ε3\boldsymbolβ(\boldsymbolx1)2q\boldsymbolθ~T(\boldsymbolθ^\boldsymbolθ0) $(28)

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    $ q \cdot {\tilde {\boldsymbol{\theta}} ^{\rm{T}}}(\hat {\boldsymbol{\theta}} - {{\boldsymbol{\theta}} _0}) \leqslant - \frac{1}{2}q{\left\| {\tilde {\boldsymbol{\theta}} } \right\|^2} + \frac{1}{2}q{\left\| {{\boldsymbol{\theta}} - {{\boldsymbol{\theta}} _0}} \right\|^2} $(29)

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    $ \left\{ M1=q\boldsymbolθ\boldsymbolθ02/2M2=ε1\boldsymbolβ(\boldsymbolx1)2+ε2\boldsymbolβ(\boldsymbolx2)2ε3\boldsymbolβ(\boldsymbolx1)2 \right. $(30)

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    $ {\dot V_2} \leqslant - {k_1}{\left\| {{{\boldsymbol{z}}_1}} \right\|^2} - {k_2}{\left\| {{{\boldsymbol{z}}_2}} \right\|^2} - \frac{1}{2}q{\left\| {\tilde {\boldsymbol{\theta}} } \right\|^2} + {M_1} + {M_2} $(31)

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    $ \left\{ V2γ1Vk1\boldsymbolz12+k2\boldsymbolz22+q\boldsymbolθ~2/2γ2Vk1\boldsymbolz12k2\boldsymbolz22q\boldsymbolθ~2/2γ2V2/γ1\right. $(32)

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    $ {V_2}(t) \leqslant \frac{{{\gamma _1}({M_1} + {M_2})}}{{{\gamma _2}}} + \left[ {{V_2}(0) - \frac{{{\gamma _1}({M_1} + {M_2})}}{{{\gamma _2}}}} \right]{{\text{e}}^{ - \tfrac{{{\gamma _2}}}{{{\gamma _1}}}t}} $(33)

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    $ \left\{ \boldsymbolz12/2V2(t)limt|\boldsymbolx1\boldsymbolx1d|limt2γ1(M1+M2)/γ2 \right. $(34)

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    $ \frac{{{\gamma _1}({M_1} + {M_2})}}{{{\gamma _2}}} = \frac{{\max \{ 1,1/\lambda ({\gamma _2})\} }}{{\min \{ 2{k_i},q\} }}({M_1} + {M_2}) $(35)

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    $ \frac{{({M_1} + {M_2})}}{{2{\gamma _2}}} \geqslant \frac{{{M_1} + {M_2}}}{{2{k_i}}} \geqslant \frac{{({M_1} + {M_2})}}{q} $(36)

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    $ ({M_1} + {M_2})/2{\gamma _2} = ({M_1} + {M_2})/q $(37)

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    $ \left\{ γ1(M1+M2)γ2(M1+M2)2γ2q2\boldsymbolθ\boldsymbolθ02q2γ1(M1+M2)/γ2\boldsymbolθ\boldsymbolθ02 \right. $(38)

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    $ \left\{ \boldsymbolxATi=[5+sin(Δθ)]cos(πt/5000)5yATi=[5+sin(Δθ)]sin(πt/5000)\boldsymbolzATi=t/1000+cos(Δθ)Δθ=(i1)2π/3\right. $(39)

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    $ \left\{ Δ(\boldsymbolx1)=0.1cos(\boldsymbolx1)Δ(\boldsymbolx2)=0.3sin2(\boldsymbolx2)\right. $(40)

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    Jiwen ZHANG, Bo XU, Chuyan WANG, Zhaoyang WANG. A neural dynamics model prediction-based adaptive control system for AUV formation control[J]. Chinese Journal of Ship Research, 2025, 20(1): 326
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