## 2 Kinematics and Dynamic Models of USV  
A USV moves in six degrees of freedom (6-DOF) with three translation displacements to define the location and three angular displacements to define the orientation. These motions are often described in two types of reference frame, namely the inertial and body-fixed frames. The motions in heavy, roll,  
Figure 1: Concept of non-strict trajectory tracking.  
and pitch are neglected. The kinematics and dynamics of the USV without external environmental disturbances are described as follows [26]:  
\[\dot{\mathbf{\eta}}=\mathbf{J}(\psi)\mathbf{\
u}, \tag{1}\]  
\[\dot{\mathbf{M}}\mathbf{\
u}+\mathbf{D}(\mathbf{\
u})\mathbf{\
u}+\mathbf{C}(\mathbf{\
u})\mathbf{\
u}=\mathbf{\tau}, \tag{2}\]  
where  
\[\mathbf{J}(\psi)=\left[\begin{array}{ccc}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{array}\right],\mathbf{M}=\left[\begin{array}{ccc}m_{11}&0&0\\ 0&m_{22}&m_{23}\\ 0&m_{32}&m_{33}\end{array}\right],\]  
\[\mathbf{C}(\mathbf{\
u})=\left[\begin{array}{ccc}0&0&C_{13}(\mathbf{\
u})\\ 0&0&C_{23}(\mathbf{\
u})\\ -C_{13}(\mathbf{\
u})&-C_{23}(\mathbf{\
u})&0\end{array}\right],\mathbf{D}(\mathbf{\
u})= \left[\begin{array}{ccc}d_{11}(\mathbf{\
u})&0&0\\ 0&d_{22}(\mathbf{\
u})&d_{23}(\mathbf{\
u})\\ 0&d_{32}(\mathbf{\
u})&d_{33}(\mathbf{\
u})\end{array}\right],\]  
\[m_{11}=m-X_{ii},\,m_{22}=m-Y_{\dot{\
u}},\,m_{23}=mx_{g}-Y_{\dot{\
u}},\,m_{3 2}=mx_{g}-N_{\dot{\
u}},\,m_{33}=I_{z}-N_{\dot{\
u}},\,Y_{\dot{\
u}}=N_{\dot{ \
u}}.\]  
\[C_{13}(\mathbf{\
u})=-(m-Y_{\dot{\
u}})v-(mx_{g}-Y_{\dot{\
u}})r,\,C_{23}(\mathbf{\
u} )=(m-X_{\dot{u}})u,\]  
\[d_{11}(\mathbf{\
u})=-X_{u}-X_{|u|u|}|u|-X_{uuuu}u^{2},\,d_{22}(\mathbf{\
u})=-Y_{\dot {\
u}}-Y_{|v|v|}|v|-Y_{|r|v|}|r,\]  
\[d_{23}(\mathbf{\
u})=-Y_{\dot{\
u}}-Y_{|v|r|}|v|-Y_{|r|r}|r|,\,d_{32}(\mathbf{\
u})=- N_{\dot{\
u}}-N_{|v|v|}|v|-N_{|r|v|}|r|,\]  
\[d_{33}(\mathbf{\
u})=-N_{\dot{\
u}}-N_{|v|r|}|v|-N_{|r|r}|r|.\]  
The system inertia matrix \(\mathbf{M}=\mathbf{M}^{\mathbf{T}}>0\); \(\mathbf{C}(\mathbf{\
u})=-\mathbf{C}^{\mathbf{T}}(\mathbf{\
u})\,\,\forall\,\,\mathbf{\
u}\in\mathbb{ R}^{3}\) is a skew-symmetric matrix of Coriolis and centripetal terms; \(\mathbf{D}(\mathbf{\
u})\) is the hydrodynamic damping matrix and it is strictly positive such that \(\mathbf{D}(\mathbf{\
u})>0\,\,\forall\,\,\mathbf{\
u}\in\mathbb{R}^{3}\).  
Here \(\mathbf{\eta}=\left[x,\,y,\,\,\mathbf{\psi}\right]^{T}\) denotes the position \((x,y)\) in the heading angle \(\psi\) of the USV in the earth-fixed frame; \(\mathbf{\
u}=\left[u,\,v,\,r\right]^{T}\) present the velocities of the USV in the body-fixed frame (surge velocity: \(u\), sway velocity: \(v\); and yaw velocity: \(r\)); \(\mathbf{\tau}=\left[\tau_{u},\,\,\tau_{v},\tau_{r}\right]^{T}\) is the control input of the USV (surge force: \(\tau_{u}\), sway force: \(\tau_{\dot{\
u}}\); and yaw moment: \(\tau_{\dot{\
u}}\)); \(m\) is the mass of the USV; \(X_{ii}\), \(Y_{\dot{\
u}}\), \(Y_{\dot{\
u}}\), and \(N_{\dot{\
u}}\) are the added masses; \(x_{g}\) is the \(X_{\dot{\
u}}\)-coordinate of the USV center of gravity in the body-fixed frame; and \(I_{z}\) is the inertia with respect to the vertical axis. The coefficients \(X_{u}\), \(X_{|u|u}\), \(X_{uuuu}\), \(Y_{\dot{\
u}}\), \(Y_{|v|}\), \(Y_{\dot{\
u}|v}\), \(Y_{\dot{\
u}|r}\), \(Y_{|r|r}\), \(N_{\dot{\
u}|v}\), \(N_{|\dot{\
u}|v}\), \(N_{\dot{\
u}|r}\), \(N_{r}\), \(N_{