【Paper】2023_Distributed adaptive fixed-time formation control for UAV-USV heterogeneous multi-agent

Liu H, Weng P, Tian X, et al. Distributed adaptive fixed-time formation control for UAV-USV heterogeneous multi-agent systems[J]. Ocean Engineering, 2023, 267: 113240.

1. Introduction

2. Preliminaries and problem statement

2.1. Graph theory

2.2. Problem formulation

Consider one quadrotor as the UAV system. The dynamics model of the UAV is as follows:

$$\begin{array}{rl}{\ddot{x}}_a& =(\cos \phi \sin \theta \cos \xi +\sin \phi \sin \xi )u_1/m_a-{\rho }_x{\dot{x}}_a/m_a\\ {\ddot{y}}_a& =(\cos \phi \sin \theta \sin \xi -\sin \phi \cos \xi )u_1/m_a-{\rho }_y{\dot{y}}_a/m_a\\ {\ddot{z}}_a& =(\cos \phi \sin \theta )u_1/m_a-{\rho }_z{\dot{z}}_a/m_a-g\\ \ddot{\phi }& =\dot{\theta }\dot{\xi }(I_y-I_z)/I_x-\dot{\theta }\Lambda I_r/I_x+u_2/I_x-{\rho }_{\phi }\dot{\phi }/I_x\\ \ddot{\theta }& =\dot{\phi }\dot{\xi }(I_z-I_x)/I_y-\dot{\phi }\Lambda I_r/I_y+u_3/I_y-{\rho }_{\theta }\dot{\theta }/I_y\\ \ddot{\xi }& =\dot{\phi }\dot{\theta }(I_x-I_y)/I_z+u_4/I_z-{\rho }_{\xi }\dot{\xi }/I_z\end{array}\text{\hspace{1em}(1)}$$

where
${\chi }_a=[x_a,y_a,z_a{]}^{\text{T}}$ denote the position variable,
${\omega }_a=[\phi ,\theta ,\xi {]}^{\text{T}}$ denote angle variable,
$u_1,u_2,u_3,u_4$ denote control thrust, three control torques,
$I_x,I_y,I_z$ denote the moments of inertia,
${\rho }_x,{\rho }_y,{\rho }_z,{\rho }_{\phi },{\rho }_{\theta },{\rho }_{\xi }$ represent the aerodynamic damping coefficients,
$m_a$ is the mass,
$g$ is the acceleration of gravity,
$I_r$ is the rotor blade of inertia and
$\Lambda$ is the rotor angular.

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For the leader quadrotor, the attitude system is stable in (Tayebi and McGilvray, 2006). Thus, the researched UAV control problem in this paper concentrates primarily on position systems. Considering external disturbances, parametric uncertainties and input saturation, the UAV three degrees of freedom model can be redefined as follows:

$$\begin{array}{rl}{\ddot{\chi }}_a(t)& =A\text{sat}(u_a)+f({\chi }_a)+{\Delta }_{a}f({\chi }_a)+d({\chi }_a,t)\end{array}\text{\hspace{1em}(2)}$$

where
$\text{sat}(u_a)=[\text{sat}(u_{xa}),\text{sat}(u_{ya}),\text{sat}(u_{za}){]}^{\text{T}}\in {\mathbb{R}}^3$ denote the actual control input vector,
$u_a(t)=[u_{xa}(t),u_{ya}(t),u_{za}(t){]}^{\text{T}}\in {\mathbb{R}}^3$
${\Delta }_{a}f({\chi }_a)=[{\Delta }_a{f}_x,{\Delta }_a{f}_y,{\Delta }_a{f}_z{]}^{\text{T}}\in {\mathbb{R}}^3$ denote the model uncertainties,
$d(\cdot )=[d_x(t),d_y(t),d_z(t){]}^{\text{T}}\in {\mathbb{R}}^3$ denote the external disturbances.

$$\{\begin{array}{rl}{u}_{xa}& =(\cos \phi \sin \theta \cos \xi +\sin \phi \sin \xi )u_1\\ u_{ya}& =(\cos \phi \sin \theta \sin \xi -\sin \phi \cos \xi )u_1\\ u_{za}& =(\cos \phi \cos \theta )u_1\end{array}\text{\hspace{1em}(3)}$$

Let
$$\{\begin{array}{rl}{\chi }_{l,1}& ={\chi }_a\\ {\chi }_{l,2}& ={\dot{\chi }}_a\end{array}\text{\hspace{1em}(5)}$$

Then leader UAV dynamic model can be rewritten as:
Let
$$\{\begin{array}{rl}{\dot{\chi }}_{l,1}& ={\chi }_{l,2}\\ {\dot{\chi }}_{l,2}& =A\text{sat}(u_a)+f({\chi }_{l,1})+F_a({\chi }_{l,1},t)\end{array}\text{\hspace{1em}(6)}$$

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Consider $N$ surface vessels as the USV systems. The dynamics of the $k$th USV are described below:

$$\{\begin{array}{rl}{\dot{\eta }}_{s,k}& =R({\Psi }_{s,k})v_{s,k}\\ {\bar{M}}_k{\dot{v}}_{s,k}+{\bar{C}}_k{v}_{s,k}+{\bar{D}}_k{v}_{s,k}& =\text{sat}(u_{s,k})+R^{\text{T}}({\Psi }_{s,k}){\delta }_k(t)\end{array}\text{\hspace{1em}(7)}$$

where
${\eta }_{s,k}=[x_{s,k},y_{s,k},{\Psi }_{s,k}{]}^{\text{T}}$ denote the position variable,
$v_{s,k}=[v_{xs,k},v_{ys,k},v_{\Psi s,k}{]}^{\text{T}}$ denote the velocity variable,
$\text{sat}(u_{sk})=[\text{sat}(u_{xsk}),\text{sat}(u_{ysk}),\text{sat}(u_{\Psi sk}){]}^{\text{T}}$ represent the actual control input vector,
$u_{sk}(t)=[u_{xsk},u_{ysk},u_{\Psi sk}{]}^{\text{T}}$ represent the desired control inputs vector of the $k$th USV,
${\delta }_k(t)=[{\delta }_{xk},{\delta }_{yk},{\delta }_{\Psi k}{]}^{\text{T}}$ denote external disturbances of the $k$th USV,
$R({\Psi }_{s,k})$ is a rotation matrix
${\bar{M}}_k$ denotes the inertia matrix,
${\bar{D}}_k$ denotes the damping matrix,
${\bar{C}}_k$ denotes the Coriolis and centripetal matrix.

Let
$$\{\begin{array}{rl}{\chi }_{sk,1}& ={\eta }_{s,k}\\ {\chi }_{sk,2}& ={\dot{\eta }}_{s,k}\end{array}\text{\hspace{1em}(9)}$$

Then leader UAV dynamic model can be rewritten as:
Let
$$\{\begin{array}{rl}{\dot{\chi }}_{s,1}& ={\chi }_{s,2}\\ {\dot{\chi }}_{s,2}& =M_s^{-1}(R_s\text{sat}(u_s)-C_s{\chi }_{s,2}-D_s{\chi }_{s,2})+F_s\end{array}\text{\hspace{1em}(10)}$$

$$\{\begin{array}{rl}\left[\begin{array}{c}{\dot{\chi }}_{s1,1}\\ {\dot{\chi }}_{s2,1}\\ {\dot{\chi }}_{s3,1}\\ {\dot{\chi }}_{s4,1}\end{array}\right]& =\left[\begin{array}{c}{\chi }_{s1,2}\\ {\chi }_{s2,2}\\ {\chi }_{s3,2}\\ {\chi }_{s4,2}\end{array}\right]\\ \left[\begin{array}{c}{\dot{\chi }}_{s1,2}\\ {\dot{\chi }}_{s2,2}\\ {\dot{\chi }}_{s3,2}\\ {\dot{\chi }}_{s4,2}\end{array}\right]& =M_s^{-1}(R_s\text{sat}(u_s)-C_s\left[\begin{array}{c}{\chi }_{s1,2}\\ {\chi }_{s2,2}\\ {\chi }_{s3,2}\\ {\chi }_{s4,2}\end{array}\right]-D_s\left[\begin{array}{c}{\chi }_{s1,2}\\ {\chi }_{s2,2}\\ {\chi }_{s3,2}\\ {\chi }_{s4,2}\end{array}\right])+F_s\end{array}\text{\hspace{1em}(10)}$$

2.3. Radial basis function neural networks

$$\Theta (L)=w^{\ast \text{T}}S(L)+\epsilon (L)\text{\hspace{1em}(11)}$$

$$S(L)=\exp (-\frac{\Vert L-\mu {\Vert }^2}{2h^2})\text{\hspace{1em}(12)}$$

3. Main results

3.1. Fixed-time control for the leader UAV

The trajectory tracking error of the UAV can be given as follows:
$$\{\begin{array}{rl}{e}_{l,1}& ={\chi }_{l,1}-{\chi }_{ad}\\ e_{l,2}& ={\chi }_{l,2}-{\dot{\chi }}_{ad}\end{array}\text{\hspace{1em}(13)}$$

$$\{\begin{array}{rl}\left[\begin{array}{c}{e}_{xa,1}\\ e_{ya,1}\\ e_{za,1}\end{array}\right]& =\left[\begin{array}{c}{\chi }_{xa,1}\\ {\chi }_{ya,1}\\ {\chi }_{za,1}\end{array}\right]-{\chi }_{ad}\\ \left[\begin{array}{c}{e}_{xa,2}\\ e_{ya,2}\\ e_{za,2}\end{array}\right]& =\left[\begin{array}{c}{\chi }_{xa,2}\\ {\chi }_{ya,2}\\ {\chi }_{za,2}\end{array}\right]-{\dot{\chi }}_{ad}\end{array}\text{\hspace{1em}(13)}$$

where
${\chi }_{ad}$ is the UAV desired position,
${\dot{\chi }}_{ad}$ is the UAV desired velocity,
${\chi }_{l,1}$ is the UAV position,
${\dot{\chi }}_{l,2}$ is the UAV velocity.
$e_{l,1}=[e_{xa,1},e_{ya,1},e_{za,1}{]}^{\text{T}}$,
$e_{l,2}=[e_{xa,2},e_{ya,2},e_{za,2}{]}^{\text{T}}$.

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Now, a fixed-time nonsingular fast terminal sliding mode surface of the leader UAV is defined as follows:

$$s_l=e_{l,2}+g_1{H}_l^{{\lambda }_1}(e_{l,1})+g_2{\text{Sig}}^{{\lambda }_2}(e_{l,1})\text{\hspace{1em}(14)}$$

$$s_l=\left[\begin{array}{c}{e}_{xa,2}\\ e_{ya,2}\\ e_{za,2}\end{array}\right]+\left[\begin{array}{ccc}{g}_{11}& & \\ & g_{12}& \\ & & g_{13}\end{array}\right]\left[\begin{array}{c}{h}_l^{{\lambda }_1}(e_{xa,1})\\ h_l^{{\lambda }_1}(e_{ya,1})\\ h_l^{{\lambda }_1}(e_{za,1})\end{array}\right]+\left[\begin{array}{ccc}{g}_{21}& & \\ & g_{22}& \\ & & g_{23}\end{array}\right]\left[\begin{array}{c}{\text{Sig}}^{{\lambda }_2}(e_{xa,1})\\ {\text{Sig}}^{{\lambda }_2}(e_{ya,1})\\ {\text{Sig}}^{{\lambda }_2}(e_{za,1})\end{array}\right]\text{\hspace{1em}(14)}$$

where
$H_l^{{\lambda }_1}(e_{l,1})=[h_l^{{\lambda }_1}(e_{xa,1}),h_l^{{\lambda }_1}(e_{ya,1}),h_l^{{\lambda }_1}(e_{za,1}){]}^{\text{T}}$,
${\text{Sig}}^{{\lambda }_2}(e_{l,1})=[{\text{Sig}}^{{\lambda }_2}(e_{xa,1}),{\text{Sig}}^{{\lambda }_2}(e_{ya,1}),{\text{Sig}}^{{\lambda }_2}(e_{za,1}){]}^{\text{T}}$,
g 1 = diag { g 11 , g 12 , g 13 } \red{g_1} = \text{diag} \{g_{11}, g_{12}, g_{13}\} g1​=diag{ g11​,g12​,g13​},
g 2 = diag { g 21 , g 22 , g 23 } \red{g_2} = \text{diag} \{g_{21}, g_{22}, g_{23}\} g2​=diag{ g21​,g22​,g23​},
$p_{ia},i=x,y,z$ is a designed positive,
${\lambda }_1,{\lambda }_2$ are positive constants.

in which:
$$\begin{array}{rl}{h}_l^{{\lambda }_1}(e_{ia,1})& =\{\begin{array}{r}|e_{ia,1}{|}^{{\lambda }_1}\text{sign}(e_{ia,1}),|e_{ia,1}|\ge p_{ia}\\ p_{ia}^{{\lambda }_1-1}{e}_{ia,1},|e_{ia,1}|<p_{ia}\end{array}\\ {\text{Sig}}^{{\lambda }_2}(e_{ia,1})& =|e_{ia,1}{|}^{{\lambda }_2}\text{sign}(e_{ia,1}),i=x,y,z\end{array}\text{\hspace{1em}(15)}$$

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$$u_a=A^{-1}\left[-f({\chi }_{l1})+{\ddot{\chi }}_{ad}-g_1{\vartheta }_1{e}_{l,2}-g_2{\vartheta }_2{e}_{l,2}-(k_1+\sigma )\text{sign}(s_l)-k_2{s}_l{s}_l^{\text{T}}{s}_l\right]\text{\hspace{1em}(20)}$$

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3.2. Fixed-time control for the follower USVs

In view of assumption 5 and the USV model Equation (27), the follower USVs control law can be given as follows:

$$\begin{array}{rl}{u}_s& =R_s^{-1}[C_s{\chi }_{s,2}+D_s{\chi }_{s,2}-M_s(-{\dot{\beta }}_{s,2}-{\dot{\chi }}_{l,2}+((L+B{)}^{-1}\otimes I_3)\\ & (b_1{\Phi }_1{e}_{s,2}+b_2{\Phi }_2{e}_{s,2}+(r_{1s}+\tau )\text{sign}(s_p)+r_{2s}{s}_p{s}_p^{\text{T}}{s}_p))]\end{array}\text{\hspace{1em}(29)}$$

$$\{\begin{array}{rl}{\dot{\chi }}_{s,1}& ={\chi }_{s,2}\\ {\dot{\chi }}_{s,2}& =M_s^{-1}(R_s\text{sat}(u_s)-C_s{\chi }_{s,2}-D_s{\chi }_{s,2})+F_s\\ & =M_s^{-1}(R_s{u}_s-C_s{\chi }_{s,2}-D_s{\chi }_{s,2})+F_s\\ & =M_s^{-1}(C_s{\chi }_{s,2}+D_s{\chi }_{s,2}-M_s(-{\dot{\beta }}_{s,2}-{\dot{\chi }}_{l,2}+((L+B{)}^{-1}\otimes I_3)(b_1{\Phi }_1{e}_{s,2}+b_2{\Phi }_2{e}_{s,2}+(r_{1s}+\tau )\text{sign}(s_p)+r_{2s}{s}_p{s}_p^{\text{T}}{s}_p))-C_s{\chi }_{s,2}-D_s{\chi }_{s,2})+F_s\\ & =M_s^{-1}(-M_s(-{\dot{\beta }}_{s,2}-{\dot{\chi }}_{l,2}+((L+B{)}^{-1}\otimes I_3)(b_1{\Phi }_1{e}_{s,2}+b_2{\Phi }_2{e}_{s,2}+(r_{1s}+\tau )\text{sign}(s_p)+r_{2s}{s}_p{s}_p^{\text{T}}{s}_p)))+F_s\\ & =(-(-{\dot{\beta }}_{s,2}-{\dot{\chi }}_{l,2}+((L+B{)}^{-1}\otimes I_3)(b_1{\Phi }_1{e}_{s,2}+b_2{\Phi }_2{e}_{s,2}+(r_{1s}+\tau )\text{sign}(s_p)+r_{2s}{s}_p{s}_p^{\text{T}}{s}_p)))+F_s\\ & =({\dot{\beta }}_{s,2}+{\dot{\chi }}_{l,2}-((L+B{)}^{-1}\otimes I_3)(b_1{\Phi }_1{e}_{s,2}+b_2{\Phi }_2{e}_{s,2}+(r_{1s}+\tau )\text{sign}(s_p)+r_{2s}{s}_p{s}_p^{\text{T}}{s}_p))+F_s\end{array}\text{\hspace{1em}(10)}$$

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main_UAV.m

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程序名字：main_UAV_USV.m

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