## 2 System Modeling
### Problem Formulation and Control Objective  
First, the position tracking errors \(x_{e}\) and \(y_{e}\) are defined as follows:  
\[ \begin{cases}x_{e}=x-x_{t}\\ y_{e}=y-y_{t}\quad,\end{cases}  \tag{3}\]  
where \(x_{t}\) and \(y_{t}\) are the time-varying positions with respect to the earth-fixed frame. The derivative of the position tracking error can be expressed by  
\[ \begin{cases}\dot{x}_{e}=\dot{x}-\dot{x}_{t}=\dot{x}_{d}-\dot{x}_{t}+(\dot{x}- \dot{x}_{d})\\ \dot{y}_{e}=\dot{y}-\dot{y}_{t}=\dot{y}_{d}-\dot{y}_{t}+(\dot{y}-\dot{y}_{d}) \quad,\end{cases}  \tag{4}\]  
where \(\dot{x}_{d}\) and \(\dot{y}_{d}\) are designed desired velocities in the earth-fixed frame. **Proposition 1**[34]. : _System (4) will be input-to-state stable if the desired velocities \(\dot{x}_{d}\) and \(\dot{y}_{d}\) are chosen as:_  
\[ \begin{cases}\dot{x}_{d}=-k_{x}\text{tanh}(x_{e})+\dot{x}_{t}\\ \dot{y}_{d}=-k_{y}\text{tanh}(y_{e})+\dot{y}_{t}\quad,\end{cases}  \tag{5}\]  
_where \(k_{x}>0\) and \(k_{y}>0\). Note that the choice of controller gains \(k_{x}\) and \(k_{y}\) will determine how fast the trajectories converge to zero and can be tuned on the basis of the system's desired output behavior._  
Based on Proposition 1, \(x_{e}\) and \(y_{e}\) will be ultimately bounded while \(\dot{x}-\dot{x}_{d}\) and \(\dot{y}-\dot{y}_{d}\) converge to zero. Then, the system control object is converted to design \(T\) and \(\tau_{r}\) in system (2) to guarantee \(\dot{x}-\dot{x}_{d}\) and \(\dot{y}-\dot{y}_{d}\) converge to zero. That means that the vehicle needs to stabilize to the desired velocities \(u_{d}\) and \(v_{d}\) in the body-fixed frame. \[ \begin{bmatrix}u_{d}\\ v_{d}\end{bmatrix} = \begin{bmatrix}\cos(\psi)&\sin(\psi)\\ -\sin(\psi)&\cos(\psi)\end{bmatrix}\begin{bmatrix}\dot{x}_{d}\\ \dot{y}_{d}\end{bmatrix} . \tag{6}\]  
The sway speed \(v_{d}\) cannot be directly controlled due to the under-actuated characteristic of the system. Define \(u_{d}=\sqrt{\dot{x}_{d}^{2}+\dot{y}_{d}^{2}-v^{2}}\) and \(\gamma_{d}=\arctan(\dot{y}_{d}/\dot{x}_{d})\). Then, the control requirement is transformed to make surge speed \(u\) and the trajectory angle \(\gamma=\psi+\beta\) to the desired ones, and \(\beta=\arctan(v/u)\), \(\dot{x}_{d}\
eq 0\). The tracking errors \(u_{e}\) and \(\gamma_{e}\) are obtained as follows:  
\[ \begin{cases}\gamma_{e}=\psi+\beta-\gamma_{d}\\ u_{e}=u-u_{d}\end{cases} . \tag{7}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Terms** & **Description** \\ \hline \(x,y\) & position of the AUV in the earth-fixed frame \\ \(\psi\) & yaw angle \\ \(u\) & surge velocity \\ \(v\) & sway velocity \\ \(r\) & yaw angular velocity \\ \(T\) & surge force \\ \(\tau_{r}\) & yaw torque \\ \(m\) & mass terms \\ \(d_{1},d_{2}\) & unknown disturbances \\ \(X_{u},\,Y_{v},\,N_{r}\) & hydrodynamic coefficients of the AUV model \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The model parameters. The derivatives of \(\gamma_{e}\) and \(u_{e}\) are  
\[ \begin{cases}\dot{\gamma}_{e}=r+\dot{\beta}-\dot{\gamma}_{d}\\ \dot{u}_{e}=M_{1}(X_{u}u+a_{23}vr+T+d_{1})-\dot{u}_{d}\\ \end{cases} . \tag{8}\]  
The trajectory tracking control problem can be reformulated in the new mathematical model as follows:  
\[ \begin{cases}\dot{\gamma}_{e}=r+\dot{\beta}-\dot{\gamma}_{d}\\ \dot{u}_{e}=M_{1}(X_{u}u+a_{23}vr+T+d_{1})-\dot{u}_{d}\\ \dot{v}=M_{2}(Y_{v}v+a_{13}ur)\\ \dot{r}=M_{3}(N_{r}r+a_{12}uv+\tau_{r}+d_{2})\\ \end{cases} . \tag{9}\]  
**Remark 1**. : _As an illustration, the position trajectory tracking control for \(x_{t}\) and \(y_{t}\) is transformed into the stabilization problem of \(u_{e}\) and \(\gamma_{e}\). It could decouple the tracking error subsystem and avoid the complex form caused by the iterative solution controller of the Lyapunov function for \(x_{e}\) and \(y_{e}\). The control objective is to design the control law of \(T\) and \(\tau_{r}\)._  
The following assumption holds, considering the AUV's physical limitation during the trajectory tracking control. **Assumption 1**.