# Research on a Support System for Automatic Ship Navigation in Fairway
## 3 Mathematical Model of Ship Motion
### Ship Dynamic Model  
To solve the problems of ship maneuvering or ship control systems, the mathematical model of the ship needs to be described first. This model has to represent the precise characteristics of the ship in the real world. Ship motions, such as surge, sway and yaw are enough to express the dynamic characteristics of the ship on the Cartesian coordinate system. The Maneuvering Modeling Group (MMG) model has been employed in this research to show ship motion during the berthing process, in which the hydrodynamic components (forces and moments acting on the ship) were divided into separate modules, such as the hull, rudder, propeller, tugboat and thruster. According to MMG model, the maneuvering equation of the ship is shown as follows:  
\[ \begin{gathered}(m+m_{x})\dot{u}-\big{(}m+m_{y}\big{)}vr=X_{H}+X_ {P}+X_{R}+X_{W},\\ \big{(}m+m_{y}\big{)}\dot{v}+(m+m_{x})ur=Y_{H}+Y_{P}+Y_{R}+Y_{W}+ Y_{Thr},\\ (I_{zz}+J_{zz})\dot{r}=N_{H}+N_{R}+N_{W}+N_{Thr},\end{gathered}  \tag{2}\]  
where \(u,v,\) and \(r\) are the ship's velocity in the surge, sway and yaw directions, respectively; \(X_{H}\), \(Y_{H}\), and \(N_{H}\) are the hydrodynamic forces and moments acting on the ship's hull. They are defined by the Kijima equations [33], employed as in Equation (3):  
\[ \begin{gathered} X_{H}=\frac{\rho}{2}LdU^{2}\bigg{(}X_{\beta r}^{ \prime}\sin\beta+X_{uu}^{\prime}\cos^{2}\beta\frac{rL}{U}\bigg{)}\\ \Upsilon_{H}=\frac{\rho}{2}LdU^{2}\bigg{(}Y_{\beta}^{\prime} \beta+Y_{r}^{\prime}\frac{rL}{U}+Y_{\beta\beta}^{\prime}\beta|\beta|+Y_{rr}^{ \prime}\frac{rL}{U}\bigg{|}\frac{rL}{U}\bigg{|}+Y_{\beta\beta r}^{\prime}\beta ^{2}\frac{rL}{U}+Y_{\beta rr}^{\prime}\beta\bigg{(}\frac{rL}{U}\bigg{)}^{2} \bigg{)}\\ N_{H}=\frac{\rho}{2}L^{2}dU^{2}\bigg{(}N_{\beta}^{\prime}\beta+N_{ r}^{\prime}\frac{rL}{U}+N_{\beta\beta}^{\prime}\beta|\beta|+N_{rr}^{\prime} \frac{rL}{U}\bigg{|}\frac{rL}{U}\bigg{|}+N_{\beta\beta r}^{\prime}\beta^{2} \frac{rL}{U}+N_{\beta rr}^{\prime}\beta\bigg{(}\frac{rL}{U}\bigg{)}^{2} \bigg{)},\end{gathered}  \tag{3}\]  
Figure 2: Ship dynamic in Cartesian reference frame. where \(X^{\prime}_{\beta r},X^{\prime}_{uu},\ldots,N^{\prime}_{\beta r\beta r},\text{ and }N^{\prime}_{\beta rr}\) are hydrodynamic coefficients which are assumed as the method described by the authors [33]. The longitudinal force of the propeller is determined as follows:  
\[X_{P}=(1-t_{P})T \tag{4}\]  
\[T=\rho D^{4}_{p}n^{2}K_{T}(J)\]  
The hydrodynamic forces and moment by the rudder are expressed as:  
\[X_{R}=-(1-t_{R})F_{N}\sin\delta\]  
\[Y_{R}=-(1+a_{H})F_{N}\cos\delta \tag{5}\]  
\[N_{R}=-(x_{R}+a_{H}x_{H})F_{N}\cos\delta,\]  
where  
\[F_{N}=\frac{\rho}{2}f_{\alpha}(\Lambda)A_{R}U^{2}_{R}\sin\alpha_{R}. \tag{6}\]  
Considering the dynamic model of the bow thruster on ship maneuvering, the method of [PERSON] et al. [34] was proposed. This model was also employed in the research of [PERSON] and [PERSON] [28]. The expression is shown as follows:  
\[T=512\cdot\left(\frac{w_{0}^{2}\cdot L^{3}\cdot H}{M_{0}^{2}}\right), \tag{7}\]  
where \(T\) is the total lateral thrust of impeller and surface forces; _wo_ and \(L\) are the rate of turn and length on the load water line; \(H\) is the ship's draft; _Mo_ is the rotation rate constant. \(T_{f}\) and \(T_{S}\) are the outputs from the bow and stern thrusters, respectively, making the forces in the _y-axis_ and moments in the _z-axis_ yielded by the side thrusters expressed as:  
\[\left\{ \begin{array}{c}Y_{Thr}=T_{f}+T_{S}\\ N_{Thr}=T_{f}x_{b}+T_{S}x_{s}\end{array} \right.. \tag{8}\]  
In this research, the thruster is only used in bow positioning and the tugboat is made fast astern. Thus, the components \(T_{S}\) and \(Tug_{S}\) become zero in Equations (7) and (8), respectively. In this study, in order to predict the hydrodynamic coefficients, the training ship (named SAENURI) of Mokpo National Maritime University was employed as the model ship. The particulars of this ship are shown in Table 1. Hydrodynamic coefficients predicted by the Kijima model are presented in Table 2.