# A DDPG-Based USV Path-Planning Algorithm
## 2 Methodology
### Motion Model for Unmanned Surface Vehicles  
In this study, an underactuated USV is employed as the subject of research for path planning. The position of the USV within the Earth's coordinate system is depicted by the position state vector \(\eta=[x\;y\;\psi]^{T}\), where \((x,y)\) signifies the location, and \(\psi\) denotes the orientation angle. The horizontal velocity of the USV is represented by the velocity vector \(v=[u\;v\;r]^{T}\), in which \(u,v\), and \(r\) correspond to the forward velocity, lateral velocity, and yaw rate, respectively. To create a simplified model of the USV that is conducive to the design of a path planning system, the following assumptions have been adopted:  
1. The USV exhibits six degrees of freedom: heave, sway, yaw, roll, pitch, and surge. The last three have a minimal influence on the USV and are thus disregarded. 2. The USV's mass is uniformly distributed, and it is symmetric with respect to the OXZ plane; that is, \(I_{xy}=I_{yz}=0\). 3. The origin of the body coordinate system is located at the center of the USV. 4. An underactuated ship refers to one that cannot generate thrust in the lateral direction (or has transverse thrusters that are inoperative for other reasons). It has a longitudinal propeller only. Therefore, the thrust vector \(\tau\) can be expressed as \([\tau_{u}\;0\;\tau_{r}]^{r}\), where \(\tau_{u}\) is the main propulsion thrust, and \(\tau_{r}\) is the turning moment induced by the rudder. As no force is applied in the lateral direction, the value is zero. 5. Higher-order hydrodynamic terms are disregarded. Based on these simplifications, the nonlinear three-degree-of-freedom model for the USV on the horizontal plane can be derived as  
\[\dot{\mathbf{\eta}}=\mathbf{R}(\mathbf{\eta})\
u \tag{9}\]  
\[\mathbf{M}\dot{w}=\mathbf{\tau}+\mathbf{\tau}_{w}-\mathbf{C}(\mathbf{v})v-\mathbf{D}(\mathbf{v})v \tag{10}\]  
Here, the transformation matrix \(\mathbf{R}(\mathbf{\eta})\) is  
\[\mathbf{R}(\mathbf{\eta})= \begin{bmatrix}\cos(\psi)&-\sin(\psi)&0\\ \sin(\psi)&\cos(\psi)&0\\ 0&0&1\end{bmatrix}  \tag{11}\]  
\(\mathbf{M}\) is the inertia matrix (including added mass):  
\[\mathbf{M}= \begin{bmatrix}m_{11}&0&0\\ 0&m_{22}&m_{23}\\ 0&m_{32}&m_{33}\end{bmatrix} = \begin{bmatrix}m-X_{i}&0&0\\ 0&m-Y_{i}&-Y_{i}\\ 0&-Y_{i}&I_{2}-N_{i}\end{bmatrix}  \tag{12}\]\(\mathbf{C}(\mathbf{v})\) is the Coriolis centripetal moment matrix:  
\[\mathbf{C}(\mathbf{v})= \begin{bmatrix}0&0&C_{13}\\ 0&0&C_{23}\\ -C_{13}&-C_{23}&0\end{bmatrix} , \tag{13}\]  
where \(C_{13}=-m_{22}v-m_{23}v\) and \(C_{23}=m_{11}u\). The Coriolis centripetal moment matrix, represented as \(\mathbf{C}(\mathbf{v})\), is formulated as  
\[\mathbf{C}(\mathbf{v})= \begin{bmatrix}0&0&C_{13}\\ 0&0&C_{23}\\ -C_{13}&-C_{23}&0\end{bmatrix} ,C_{13}=-(m_{22}+m_{23})v,C_{23}=u\cdot m_{1}(1) \tag{14}\]  
The damping matrix, denoted by \(\mathbf{D}(\mathbf{v})\), is expressed as  
\[\mathbf{D}(\mathbf{v})= \begin{bmatrix}d_{11}&0&0\\ 0&d_{22}&d_{23}\\ 0&d_{32}&d_{33}\end{bmatrix} = \begin{bmatrix}-X_{u}&0&0\\ 0&-Y_{v}&-Y_{r}\\ 0&-N_{v}&-N_{r}\end{bmatrix}  \tag{15}\]  
Here, the values of \(m_{11}\), \(m_{22}\), and \(m_{33}\) correspond to \(m-x_{u}\), \(m-Y_{v}\), and \(I_{z}-N_{r}\), respectively. \(d_{11}\), \(d_{22}\), and \(d_{33}\) signify \(-X_{u}\), \(-Y_{v}\), and \(-N_{r}\) in the inertial parameters. \(X\), \(Y\), and \(N\) stand for the hydrodynamic derivatives. The Ocean disturbance is represented by \(\tau_{uv}= \begin{bmatrix}\tau_{uu}&\tau_{uvr}&\tau_{uvr}\end{bmatrix} \). The symbols and their meanings of the Motion Model for USV are detailed in Table 2, the USV Symbol Definition Table. A simplified model of USV is shown in Figure 2. Parameters of unmanned surface vehicles are shown in Table 3.