## 2 Modeling of USV
### Kinetic Equation  
To study the planar kinetic equation, the following standards are used: \(m\) is vehicle's mass, \(x_{g}\) is the distance from the barycenter of USV to the body-fixed frame, \(I_{z}\) is the vehicle's moment of inertia about the \(z\)-axis, \(X_{u}\), \(Y_{v}\), \(Y_{r}\), \(N_{v}\) and \(N_{r}\) are the linear damping terms, \(X_{\dot{u}}\), \(Y_{\dot{v}}\), \(Y_{r}\), \(N_{\dot{v}}\) and \(N_{r}\) are the hydrodynamic added mass terms. The planar kinetic equation of three DOFs is established by Lagrange mechanics theory. \[M\dot{v}+C(v)v+D(v)v=\tau+\tau_{b} \tag{4}\]  
where \(\tau=[\tau_{u},\tau_{v},\tau_{r}]^{T}\), \(\tau_{u}\), \(\tau_{v}\) and \(\tau_{r}\) represent the control forces and moment in all directions, \(\tau_{b}=[b_{u},b_{v},b_{r}]^{T}\), \(b_{u}\), \(b_{v}\), \(b_{r}\) denote the external disturbances caused by wind, wave and currents, \(M\) is the inertia matrix, \(C(v)\) is the Coriolis and centripetal matrix and \(D(v)\) is hydrodynamic damping matrix. \[M=\left[ \begin{array}{ccc}m-X_{\dot{u}}&0&0\\ 0&m-Y_{\dot{v}}&mx_{g}-Y_{r}\\ 0&mx_{g}-Y_{r}&I_{z}-N_{r}\end{array} \right] \tag{5}\]  
\[C(v)=\left[ \begin{array}{ccc}0&0&a_{13}\\ 0&0&a_{23}\\ a_{31}&a_{32}&0\end{array}   
When the propulsion angle is \(\delta\), the vector thrusts in different directions are:  
\[\left\{ \begin{array}{l}\tau_{u}=T\cos\delta\\ \tau_{v}=T\text{sin}\delta\\ \tau_{r}=LT\text{sin}\delta\end{array} \right. \tag{8}\]  
\(T\) is the thrust of the thruster, which is a function related to the rotating speed of the propeller. \(L\) is the length from the center of rotation to the fulcrum of the propulsor. The effective attack angle of the pod \(\alpha_{R}\) is a small value with the unit \"rad\". Therefore, we can draw \(\sin\alpha_{R}=\alpha_{R}=\delta\), where the propulsion angle \(\delta=[-0.5236\) rad, \(0.5236\) rad] [27]. Hence, if \(\delta\) is a small value and \(L>1\), then \(\tau_{v}\approx 0\). **Assumption 1**. : _For a typical pod, it can be rotated \(360^{\circ}\). However, in order to reduce the difficulty of manipulating USV, its rotatable angle is often limited to a certain range, such as \(\pm 30^{\circ}\)._  
Define \(m_{11}=m-X_{\delta}\), \(m_{22}=m-Y_{v}\), \(m_{23}=mx_{g}-Y_{r}\), \(m_{32}=mx_{g}-N_{\phi}\), \(m_{33}=I_{z}-N_{r}\), \(d_{11}=-X_{u}\), \(d_{22}=-Y_{v}\), \(d_{23}=-Y_{r}\), \(d_{32}=-N_{v}\), \(d_{33}=-N_{r}\), \(X_{p}=\tau_{u}\) and \(N_{p}=\tau_{r}\). Assuming that USV is symmetrical and the barycenter of it coincides with the center of the body-fixed frame, that is to say, \(x_{g}=0\), \(Y_{r}=0\), \(N_{\phi}=0\), \(Y_{r}=0\) and \(N_{v}=0\), then (4) can be simplified as (9). \[\left\{ \begin{array}{l}\dot{u}=f_{u}+\frac{1}{m_{11}}\tau_{u}+\frac{1}{m_{11 }}b_{u}\\ \dot{v}=f_{v}+\frac{1}{m_{22}}b_{v}\\ \dot{r}=f_{r}+\frac{1}{m_{33}}\tau_{r}+\frac{1}{m_{33}}b_{r}\end{array} \right. \tag{9}\]  
where:  
\[\left\{ \begin{array}{l}f_{u}=\frac{m_{22}}{m_{11}}\
u r-\frac{d_{11}}{m_{11 }}u\\ f_{v}=-\frac{m_{11}}{m_{22}}\
u ur-\frac{d_{22}}{m_{22}}v\\ f_{r}=\frac{m_{11}-m_{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r\end{array} \right. \tag{10}\]  
**Remark 1**. : _From the modeling principle, many assumptions and simplifications are used; from the actual navigation, the operating conditions are constantly changing. Therefore, in essence, \(f_{u}\), \(f_{v}\) and \(f_{r}\) are uncertain dynamic functions:_  
**Remark 2**. : _The sway \(v\) has been proven in [28] to be passive-bounded stable._  
**Assumption 2**. : _The external disturbances accord with the following assumptions: \(|b_{u}|\leq b_{u\max}\), \(|b_{v}|\leq b_{v\max}\) and \(|b_{r}|\leq b_{r\max}\), where \(b_{u\max}\), \(b_{v\max}\) and \(b_{r\max}\) are unknown positive constants._  
**Assumption 3**.