# Fast Path Planning for Autonomous Ships in Restricted Waters
## 2 Materials and Methods
### Environment Modeling
#### 2.1.2 Ship Dynamic Properties Modeling  
Practical path planning and CA algorithms require precise knowledge about the dynamic properties of a ship. Because the steering of a ship can be regarded as a rigid-body motion on the horizontal plane, a three-degree-of-freedom dynamic model regarding surge, sway and yaw is selected in this paper [12]. The ship maneuvering motion in the earth-fixed and body-fixed coordinate systems is shown in Figure 5, and the relevant equations are as follows:  
\[\begin{array}{l}\text{surge}:\ m(\dot{u}-vr-x_{G}r^{2})=\mathbf{X}\\ \text{sway}:\ m(\dot{v}+ur+x_{G}\dot{r})=\mathbf{Y}\\ \text{yaw}:\ l_{2}\dot{r}+mx_{G}(ur+\dot{v})=\mathbf{N}\end{array}, \tag{6}\]  
where \(m\) is the mass of the ship, \(u\) and \(v\) are the surge speed and the sway speed, respectively, \(\dot{u}\) and \(\dot{v}\) are the corresponding accelerations, \(U\) is the resultant velocity of \(u\) and \(v\), \(\psi\), \(r\) and \(\dot{r}\) represent the yaw angle, yaw rate and yaw acceleration, respectively, and \(\psi\) is measured in the inertial frame (\(\dot{\psi}\equiv r\)). It is assumed that the vessel is symmetric with respect to the centerline, so the center of gravity of the vessel is in the position (\(x_{G}\),0). \(I_{x}\) is the yaw moment of inertia of the vessel, \(\mathbf{X}\) and \(\mathbf{Y}\) denote the hydrodynamic forces acting along the axes \(x\) and \(y\), respectively, \(\mathbf{N}\) is the yaw hydrodynamic moment, and \({}_{\delta}\) is the rudder angle.  
It can be assumed that \(\mathbf{X}\), \(\mathbf{Y}\), and \(\mathbf{N}\) are functions of various state variables and control variable \(\delta\). Therefore, \(\mathbf{X}\), \(\mathbf{Y}\) and \(\mathbf{N}\) can be written as follows:  
\[[\mathbf{X}\,\mathbf{Y}\mathbf{N}]^{T}=f(u,v,r,\dot{u},\dot{v},\dot{r},d), \tag{7}\]  
The use of a third-order truncated Taylor series of the \(\mathbf{X}\), \(\mathbf{Y}\) and \(\mathbf{N}\) functions will lead to a set of non-linear maneuvering equations for ship motion in the horizontal plane. This is sufficient to show the maneuvering behavior of the vessel. The non-linear model of Mariner class vessels has been discussed by [PERSON] [19]. In this paper, we choose this model and consider the saturations in the rudder mechanics. The main characteristics of this vessel are described in Table 4. The time required for the ship to turn 90\({}^{\circ}\) with maximum rudder angle is 150 s [20].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spatial Objects** & **Features** & **Distance (nm)** & **Parameter** \\ \hline point & wrecks, reefs & \textgreater{}2 & \(\beta<1\) \\  & aids to navigation & \textgreater{}0.5 & \(\beta<10\) \\  & safety contours & & \\ curve & channel limits & \textgreater{}1 & \(\gamma<3\) \\  & channel limits & \textgreater{}0.2 & \(\gamma<10\) \\ surface & islands, shoals & \textgreater{}3 & \(\gamma<0.5\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Suggested configuration of the parameters \(\beta\) and \(\gamma\) for different spatial objects.  
TSs are assumed to keep their motion parameters or change their course at any point configured by operators. A low-level PD (proportional derivative) controller is incorporated with the governing equations of ship maneuvering:  
\[\delta=-K_{p}(\psi-\psi_{desired})+K_{d}r, \tag{8}\]  
where \(\delta\) is the input control variable, \(K_{p}\) is the proportional constant, and \(K_{d}\) is the derivative constant, \(\psi\) is the instantaneous yaw angle, i.e., course angle, \(\psi_{desired}\) is the desired course, and \(r\) is the yaw rate [21; 22].