## 3 Mathematical Modelling of Ship Power Positioning Systems
### Build a Mathematical Model of the Ship's Motion  
The ship power positioning system, which is mainly studied in this paper, focuses on the positioning control of the ship on the sea surface, so the effects of vertical oscillation, transverse motions, and longitudinal rocking are neglected, and only the three degrees of freedom of longitudinal oscillation, transverse oscillation, and bow rocking are considered. The ship's motion in the horizontal plane is superimposed by the ship's low-frequency and high-frequency motion. The low-frequency motion affects the real position change of the ship, which is the main research object. The simplified three-degree-of-freedom ship low-frequency motion model is obtained [22]:  
\[\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{y}\end{bmatrix}=\begin{bmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}u\\ v\\ r\end{bmatrix}\Rightarrow\dot{\eta}=R(\psi)\
u, \tag{1}\]  
\[M\dot{v}+D(v)v=\tau+d \tag{2}\]  
Figure 1: Ship planar motion variables.  
where \(M\) is the mass matrix and \(D\) is the damping matrix, \(M=\begin{bmatrix}m-X_{u}&0&0\\ 0&m-Y_{\hat{v}}&0\\ 0&0&I_{z}-N_{r}\end{bmatrix}\),  
\(D(v)=\begin{bmatrix}-X_{u}&0&0\\ 0&-Y_{v}&0\\ 0&0&-N_{r}\end{bmatrix}\). \(X_{u},X_{u},Y_{v},Y_{v},Y_{r},Y_{r},N_{v},N_{v},N_{r},N_{\hat{r}}\) is the hydrodynamic derivative, \(m\) is the mass of the ship, \(I_{z}\) is the rotational moment of inertia of the ship, \(\tau\) is the total thrust of the propellers, and \(d\) is the ambient force of the ocean. Due to the extremely complex and irregular geometry of the attached ship including oars and rudders, it is unlikely that the hydrodynamic theoretical method can be used to carry out the computation of this series of hydrodynamic conductors. Most of the cases are used in the experimental ship modeling, and the formulae used for the estimation of the hydrodynamic conductors of the linear fluid are adopted here. The experimental data of the ship modeling, which was sorted out by [PERSON], are shown below [1; 23; 24; 25; 26]:  
\[I_{z}=\Big{(}1-C_{b}^{4.5}\Big{)}\Big{(}\frac{D}{24g}\Big{)}\Big{(}L^{2}+B^{2} \Big{)} \tag{3}\]  
\[X^{\prime}_{u}=-SC_{t}\Big{(}\frac{1}{Ld_{m}0.5\rho V^{2}L^{2}}\Big{)} \tag{4}\]  
\[X^{\prime}_{u}=-\frac{1}{100}\Big{[}0.398+11.97C_{b}\Big{(}1+3.73\frac{d_{m}}{ B}\Big{)}-2.89C_{b}\frac{L}{B}\Big{(}1+1.13\frac{d_{m}}{B}\Big{)}+0.175C_{b} \Big{(}\frac{L}{B}\Big{)}^{2}\Big{(}1+0.541\frac{d_{m}}{B}\Big{)}-1.107\frac{ L}{B}\frac{d_{m}}{B}\Big{]}\Big{(}\frac{m}{0.5\rho L^{3}}\Big{)} \tag{5}\]  
\[Y_{\hat{v}}=-\Bigg{[}1+\frac{0.16C_{b}B}{d_{m}}-5.1\Big{(}\frac{B}{L}\Big{)}^ {2}\Bigg{]}.\pi\Big{(}\frac{d_{m}}{L}\Big{)}^{2} \tag{6}\]  
\[Y^{\prime}_{r}=-\Bigg{[}\frac{0.67B}{L}-0.0033\bigg{(}\frac{B}{d_{m}}\Big{)}^ {2}\Bigg{]}.\pi\Big{(}\frac{d_{m}}{L}\Big{)}^{2} \tag{7}\]  
\[N^{\prime}_{i}=-\bigg{(}\frac{1.1B}{L}-\frac{0.41B}{d_{m}}\bigg{)}.\pi\bigg{(} \frac{d_{m}}{L}\bigg{)}^{2} \tag{8}\]  
\[N^{\prime}_{r}=-\bigg{(}\frac{1}{12}+\frac{0.017C_{b}B}{d_{m}}-\frac{0.33B}{L }\bigg{)}.\pi\Big{(}\frac{d_{m}}{L}\Big{)}^{2} \tag{9}\]  
\[Y^{\prime}_{v}=-\bigg{(}1+\frac{0.40C_{b}B}{d_{m}}\bigg{)}.\pi\bigg{(}\frac{d_ {m}}{L}\bigg{)}^{2} \tag{10}\]  
\[Y^{\prime}_{r}=-\bigg{(}-\frac{1}{2}+\frac{2.2B}{L}-\frac{0.080B}{d_{m}}\bigg{)}.\pi\bigg{(}\frac{d_{m}}{L}\bigg{)}^{2} \tag{11}\]  
\[N^{\prime}_{v}=-\Big{(}\frac{1}{2}+\frac{2.4d_{m}}{L}\Big{)}.\pi\Big{(}\frac{ d_{m}}{L}\Big{)}^{2} \tag{12}\]  
\[N^{\prime}_{r}=-\bigg{(}\frac{1}{4}+\frac{0.039B}{d_{m}}-\frac{0.56B}{L}\bigg{)}.\pi\bigg{(}\frac{d_{m}}{L}\bigg{)}^{2}, \tag{13}\]  
where \(L\) is the hull waterline length, \(d_{m}\) is the average draft depth; \(B\) is the ship's breadth; and \(C\) is the hull squareness coefficient [27].