## 2 USV Dynamic Models  
### Reference Frames and Motion Equations  
To describe the maneuvering motion of a USV, a geographic reference frame and a body-fixed reference frame are established as shown in Figure 1. The geographic reference frame is fixed to the Earth, pointing North-East-down; the body-fixed reference frame is fixed to the hull, with the origin at the center of gravity of the USV and pointing forward-right-down.  
Considering the horizontal motions of a USV, namely, surge, sway, and yaw motions, while neglecting roll, pitch, and heave motions, a simplified three degree of freedom vectorial model for a USV can be expressed as [15]:  
\[\dot{\boldsymbol{\eta}}=\boldsymbol{J}(\boldsymbol{\eta})\boldsymbol{\
u} \tag{1}\]  
\[\boldsymbol{M}\dot{\boldsymbol{\
u}}+\boldsymbol{C}(\boldsymbol{\
u}) \boldsymbol{\
u}+\boldsymbol{D}(\boldsymbol{\
u})\boldsymbol{\
u}=\boldsymbol {\tau}+\boldsymbol{\tau}_{\text{wind}}+\boldsymbol{\tau}_{\text{wave}} \tag{2}\]  
where \(\boldsymbol{\eta}=\begin{bmatrix}x&y&\psi\end{bmatrix}^{\text{T}}\) is the pose vector in the geographic reference frame, containing north position (\(x\)), east position (\(y\)), and yaw angle (\(z\)). \(\boldsymbol{\
u}=\begin{bmatrix}u&v&r\end{bmatrix}^{\text{T}}\) is the velocity vector in the body-fixed reference frame, containing surge velocity (\(u\)), sway velocity (\(v\)), and yaw rate (\(r\)). \(\boldsymbol{M}\) is the mass matrix, \(\boldsymbol{C}(\boldsymbol{\
u})\) is the Coriolis matrix, \(\boldsymbol{D}(\boldsymbol{\
u})\) is the drag matrix, \(\boldsymbol{\tau}\) is the vector of forces and moment generated by thrusters, and \(\boldsymbol{\tau}_{\text{wind}}\) and \(\boldsymbol{\tau}_{\text{wave}}\) are the vectors of forces and moment caused by wind and waves, respectively.  
Figure 1: Geographic and body-fixed reference frame.  
Assuming that the hull's weight is evenly distributed and the hull itself is symmetrical both laterally and longitudinally, \(\mathbf{J}(\mathbf{\eta})\), \(\mathbf{M}\), \(\mathbf{C}(\mathbf{\
u})\), and \(\mathbf{D}(\mathbf{\
u})\) can be expressed, respectively, as follows:  
\[\mathbf{J}(\mathbf{\eta})=\begin{bmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix} \tag{3}\]  
\[\mathbf{M}=\begin{bmatrix}m-X_{\text{\tiny{H}}}&0&0\\ 0&m-Y_{\text{\tiny{0}}}&0\\ 0&0&I_{\text{\tiny{2}}}-N_{\text{\tiny{j}}}\end{bmatrix} \tag{4}\]  
\[\mathbf{C}(\mathbf{\
u})=\begin{bmatrix}0&0&-\left(m-Y_{\text{\tiny{0}}}\right)v\\ 0&0&\left(m-X_{\text{\tiny{H}}}\right)u\\ \left(m-Y_{\text{\tiny{0}}}\right)v&-\left(m-X_{\text{\tiny{H}}}\right)u&0 \end{bmatrix} \tag{5}\]  
\[\mathbf{D}(\mathbf{\
u})=-\begin{bmatrix}X_{u}+X_{u|u|}\mid u\mid&0&0\\ 0&Y_{\text{\tiny{P}}}+Y_{\text{\tiny{$\
u$}}|\mathbf{\
u}|}\mid v\mid&0\\ 0&0&N_{\text{\tiny{r}}}+N_{\text{\tiny{$\
u$}}|\mathbf{\
u}|}\mid r\mid\end{bmatrix} \tag{6}\]  
where \(m\) is the mass, \(I_{\text{\tiny{2}}}\) is the moment of inertia about the \(z_{\text{\tiny{b}}}\) axis, \(X_{\text{\tiny{H}}}\), \(Y_{\text{\tiny{0}}}\), and \(N_{\text{\tiny{r}}}\) represent the hydrodynamic added mass, \(X_{u}\), \(Y_{\text{\tiny{0}}}\), and \(N_{\text{\tiny{r}}}\) are the linear damping coefficients, and \(X_{u|u|}\), \(Y_{\text{\tiny{$\
u$}}|\mathbf{\
u}|}\), and \(N_{\text{\tiny{$\
u$}}|\mathbf{\
u}|}\) are the quadratic damping coefficients.