## 2 Background Description
### Mathematical Model of the AUV  
The AUV design in this study is a six-DOF (Degree of Freedom) navigation in water; moreover, we also took into account the actual application situation and the type of sensor equipped, as well as the fact that the AUV actually navigates at the same depth. Thus, we simplified the entire control model such that the AUV formation moves on the horizontal plane (Figure 1). The designed AUV has two horizontal channels to realize the change in the head direction, so the AUV in the plane space is fully driven, and the kinematics and dynamics formulas [25] are as follows:  
\[\left\{ \begin{array}{l}\dot{\eta}=J(\psi)v\\ M\dot{v}=\Gamma(\eta,v)+\tau+d-\Delta_{f}\end{array} \right., \tag{1}\]  
where \(\eta\) is a state vector for the AUV, and \(\eta=[x,y,\psi]^{\mathrm{T}}\), where \(x\) is the portrait position, \(y\) is the lateral position, and \(\psi\) is the bow direction angle. Additionally, \(v=[u,v,r]^{\mathrm{T}}\), where \(u\) is the speed along X, \(v\) is the speed along Y, and \(r\) is the bow angular velocity. Finally, \(\tau=[\tau_{1},\tau_{2},\tau_{3}]^{\mathrm{T}}\), where \(\tau\) is the control input vector to be designed, \(\tau_{1}\) is the forward force, \(\tau_{2}\) is the transverse force, and \(\tau_{3}\) is the yawing moment. \(J(\varphi)\) is the bow's rotation matrix, defined as follows:  
\[J(\psi)= \begin{pmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{pmatrix}  \tag{2}\]  
\[\dot{J}(\psi)=J(\psi)S(r) \tag{3}\]  
\[J^{\mathrm{T}}(\psi)S(r)I(\psi)=J(\psi)S(r)J^{\mathrm{T}}(\psi)=S(r) \tag{4}\]  
\[S(r)= \begin{bmatrix}0&-r&0\\ r&0&0\\ 0&0&0\end{bmatrix}  \tag{5}\]  
Let  
\[d=M\mathrm{J}^{\mathrm{T}}\delta, \tag{6}\]  
\[\delta=[\delta_{1},\delta_{2},\delta_{3}], \tag{7}\]  
Figure 1: The autonomous underwater vehicle (AUV) coordinate diagram. \[\vartheta=[\vartheta_{1},\vartheta_{2},\vartheta_{3}]\text{ is uncertainty of the model}\] \[\Gamma(\eta,v)=-C(v)v-Dv, \tag{8}\]  
where \(C(v)\) is the Coriolis force and centripetal force matrix, \(D\) is the damping coefficient matrix, and \(M\) is the inertia coefficient matrix. The port and starboard structure of the AUV is symmetrical, so \(Y_{i}=N_{v}\), and the geometric center of the AUV coincides with its center of gravity; thus, \(x_{g}=0\):  
\[M= \begin{bmatrix}m-X_{u}&0&0\\ 0&m-Y_{v}&mxg-Y_{r}\\ 0&mxg-N_{v}&I_{z}-N_{r}\end{bmatrix} = \begin{bmatrix}m_{11}&0&0\\ 0&m_{22}&0\\ 0&0&m_{33}\end{bmatrix} , \tag{9}\]  
\[C(v)= \begin{bmatrix}0&0&-m_{22}v-m_{23}r\\ 0&0&m_{11}u\\ m_{22}v+m_{23}r&-m_{11}u&0\end{bmatrix} , \tag{10}\]  
\[D=- \begin{bmatrix}X_{u}&0&0\\ 0&Y_{v}&Y_{r}\\ 0&N_{v}&N_{r}\end{bmatrix} = \begin{bmatrix}d_{11}&0&0\\ 0&d_{22}&d_{23}\\ 0&d_{32}&d_{33}\end{bmatrix} , \tag{11}\]  
where \(m\) is the AUV mass, \(I_{z}\) is the moment of inertia, \(x_{g}\) is the distance between the AUV geometric center D and the center of gravity G, and the symbols \(X_{*}\), \(Y_{*}\), and \(Z_{*}\) represent the corresponding hydrodynamic derivatives. A multi-AUV system has \(N\) AUVs. The interaction topology of the multi-AUV system can be detailed by a directed graph \(G\). The \(i\)th AUV can be represented by node \(q_{i}\) in \(G\), and the interaction channel from the \(i\)th AUV to the \(j\)th AUV can be expressed as \(e_{ij}\), \(i,j\in\{i=1,2,\ldots N\}\). Thus, the directed graph \(G\) obtains a spanning tree. For these AUVs, the inner loop can decide the attitude, and the outer loop can decide position with desired velocity of the system (Figure 2). The assumed operation scenario carried out in this study is shown in Figure 3. The AUVs in the formation all move in the fixed-depth layer, so the coordination of AUVs in the depth direction is not considered. The multi-AUV system can de designed as follows:  
\[ \begin{cases}\dot{x}_{i}(t)=v_{i}(t)\\ \dot{v}_{i}(t)=u_{i}(t)^{\prime}\end{cases}