## 2 Problem Formulation  
As shown in Figure 1, a group of USVs consists of \(N\) USVs, and the model of the \(i\)th USV is given as [35]  
\[\dot{x}_{i} =u_{i}\cos(\psi_{i})-v_{i}\sin(\psi_{i}), \tag{1}\] \[\dot{y}_{i} =u_{i}\sin(\psi_{i})+v_{i}\cos(\psi_{i}),\] (2) \[\dot{\psi}_{i} =r_{i}, \tag{3}\]  
and  
\[\dot{u}_{i} =\sigma_{i,1}+\frac{1}{m_{i,11}}\tau_{i,u}, \tag{4}\] \[\dot{v}_{i} =\sigma_{i,2},\] (5) \[\dot{r}_{i} =\sigma_{i,3}+\frac{1}{m_{i,33}}\tau_{i,r}, \tag{6}\]  
where \((x_{i},y_{i})^{T}\in\mathbb{R}^{2}\) is the position in the Earth-fixed frame (\(X_{E}O_{E}Y_{E}\)); \(\psi_{i}\) is the yaw angle in \(X_{E}O_{E}Y_{E}\); \(u_{i}\) is the surge velocity in the body-fixed frame (\(X_{B}O_{B}Y_{B}\)); \(v_{i}\) is the sway velocity in \(X_{B}O_{B}Y_{B}\); \(r_{i}\) is the yaw rate in \(X_{B}O_{B}Y_{B}\); \(m_{i,11}\) and \(m_{i,33}\) denote the inertia mass; uncertainties \(\sigma_{i,1}\), \(\sigma_{i,2}\) and \(\sigma_{i,3}\) contain Coriolis parameters, hydrodynamic damping parameters and time-varying disturbances, respectively; and \(\tau_{i,u}\) and \(\tau_{i,r}\) are input signals. To facilitate observer design and analysis, the \(i\)th USV model is written as  
\[\dot{\eta}_{i} =R_{i}v_{i} \tag{7}\] \[\dot{v}_{i} =M_{i}^{-1}\tau_{i}+\sigma_{i}, \tag{8}\] \begin{table}
\begin{tabular}{l l} \hline \(\alpha_{i,h}\) & Auxiliary variables \\ \(\omega_{i,h}\) & Errors caused by the first-order filter \\ \(\omega_{\tau_{1}},\omega_{\tau_{2}}\) & Control laws at the kinetic level \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: Earth-fixed frame and body-fixed frame. where \(\eta_{i}=[x_{i},y_{i},\psi_{i}]^{T}\), \(\
u_{i}=[u_{i},v_{i},r_{i}]^{T}\), \(\tau_{i}=[\tau_{i,\mu},0,\tau_{i,\
u}]^{T}\), \(M_{i}=\text{diag}\{m_{i,11},m_{i,22},m_{i,33}\}\), \(m_{i,22}\) is the inertia mass. \(\sigma_{i}=[\sigma_{i,1},\sigma_{i,2},\sigma_{i,3}]^{T}\), and  
\[R_{i}=\left[ \begin{array}{ccc}\cos(\psi_{i})&-\sin(\psi_{i})&0\\ \sin(\psi_{i})&\cos(\psi_{i})&0\\ 0&0&1\end{array} \right].\]  
The mismatch function between input without saturation and input with saturation is described as \(\varpi_{i}=[\varpi_{i,\mu},0,\varpi_{i,r}]^{T}=\tau_{ic}-\tau_{i}\), where \(\tau_{ic}=[\tau_{i,uc},0,\tau_{i,rc}]^{T}\); \(\tau_{i,uc}\) and \(\tau_{i,rc}\) are the surge force and yaw moment calculated by the proposed controller, respectively; and \(\tau_{i}=[\tau_{i,\mu},0,\tau_{i,r}]^{T}\). Then, the saturated control is given by \(\tau_{i}=\tau_{ic}-\varpi_{i}\). **Assumption 1**. : _The unknown function \(\sigma_{i}\) satisfies \(\varphi_{i}\leq\sigma_{i}^{*}\), where \(\sigma_{i}^{*}\) is a positive constant._  
\(\sigma_{i}\) _is a vector containing velocity, Coriolis parameters, hydrodynamic damping parameters and time-varying disturbances. It is natural to assume that their derivatives are bounded. Control inputs to drive USVs are bounded; therefore, Assumption 1 is reasonable._  
**Assumption 2**. : _The velocity of the ith USV is bounded such that \(\|\
u_{i}\|\leq\
u_{i}^{*}\), \(\
u_{i}^{*}\) is a positive constant._  
The control inputs to drive USVs are bounded, and the energy of external disturbances is limited. Therefore, Assumption 2 is reasonable. **Assumption 3**. : _The desired distance (\(\rho_{ij,d}\)) and angle (\(\lambda_{ij,d}\)) are bounded._  
Assumption 3 means that the desired value is reasonable. The following definitions are posited. As shown in Figure 2, the subscript \(j\) is the index of the leader (\(j\)). \(\eta_{j}=[x_{j},y_{j},\psi_{j}]^{T}\) is used to denote the position and yaw of the leader (\(j\)). \(\
u_{j}=[u_{j},v_{j},r_{j}]^{T}\) is the velocity and yaw rate of the leader (\(j\)). Figure 2: Leader–follower structure.