## 2 Methodology
### Improved MMG Model
The function of the thrust derating coefficient by the ship moving can be expressed as follows:  
\[\big{(}1-t_{p}\big{)}=\big{(}1-t_{p0}\big{)}+f \tag{6}\]  
where the \(f\) is the function of the effect of ship motion, which is determined as follows:  
\[\left\{ \begin{array}{ll}f=k_{t}\beta_{R}\\ k_{t}=0.00023(\gamma_{A}\cdot L/D_{\rho})-0.028\\ \gamma_{A}=(B/d)\{1.3(1-C_{b})-3.1l_{cb}\}\\ l_{cb}=x_{c}/L\cdot 100\\ \beta_{R}=\beta-l_{R}\frac{\
u}{\
u}\end{array} \right. \tag{7}\]  
where \(x_{c}\) is the vertical coordinate of buoyancy. \(\beta_{R}\) denotes the drift angle at the rudder, and \(\beta\) is the drift angle at the center of the ship, which can be determined by \(\beta=\text{atan}(-u/\
u)\). \(B\) and \(d\) denote the ship's width and the ship draft, respectively. \(L\) denotes the ship's length and \(D_{\rho}\) is the diameter of the propeller. \(\gamma_{A}\) and \(l_{cb}\) are the rectification coefficients of the hull and floating centre coefficient, respectively. \(C_{b}\) and \(l_{R}\) denote the block coefficient of the ship and drift angle coefficient. The force of the propeller, mainly from the thrust of the propeller, is expressed as follows:  
\[T=\rho(NP)^{2}D_{p}^{2}K_{T} \tag{8}\]  
where \(K_{T}\) is the factor of thrust; \(\rho\) is the density of water; \(D_{p}\) represents the diameter of the propeller; and \(NP\) denotes the real-time rotation rate of the propeller, which uses to keep and control the ship speed. There are two different styles of ship manoeuvre at sea: first is manoeuvring in the manner of only altering course while maintaining a constant revolution per minute (RPM), which nearly invariably occurs in open water. The second is manoevring in the manner of changing speed by reducing or increasing the revolution of the main engine, which almost always occurs in narrow water or high-density traffic seas. In previous studies, the ship's speed was rarely taken into account in the ship motion model, with just the ship's course being taken into account. Moreover, the study using ship models that are affected by changes in direction and speed almost exclusively focuses on small boats. The difference of characteristics of propellers between tiny ships and large ships is huge. Furthermore, the ship in the variable-speed process has more influencing components than the ship in the variable-course motion process, and the interaction between each factor is more sophisticated. Furthermore, minor speed changes have little effect on the ship's collision avoidance in practice, which is why captains and pilots prefer to manage the telegraph rather than using a small variation of the main engine revolution to modify the speed. Telegraphs are used to control the ship's speed, which includes ten different presupposed specific NPs. Propulsion particulars about a bulk carrier called \"Huayang dream\" are illustrated in Table 1, where the gap of RPM between different telegraphs is relatively large in order to accommodate a large vessel speed change. Those telegraphs are applied to adjust the speed to manoeuvre the ship to avoid collision by OOW and pilots in practice. If \(NP_{1}\) is the initial rotation rate of the propeller of a telegraph at the time \(t_{1}\), and \(NP_{2}\) is the final rotation rate of the propeller of a telegraph at the time \(t_{2}\), the rotation rate of the propeller at a time \(t\) can be determined by the following:  
\[NP^{t}=\left\{ \begin{array}{ll}NP_{2},&t>t_{2}\\ NP_{1},&t<t_{1}\\ NP_{1}-(t-t_{1})\times k,&t_{1}<t<t_{2},\;\mathrm{NP}_{1}>\!\!NP_{2}\\ NP_{1}+(t-t_{1})\times k,&t_{1}<t<t_{2},\;\mathrm{NP}_{1}<NP_{2}\end{array} \right. \tag{9}\]  
where \(k\) is the meaning of a change of rotation rate, which was deduced to be around 0.25 rotation per second based on the researcher's experience as the ship's captain/officers and the model projection. The Runge-Kutta method and improved MMG should be used to deduce the future state of the ship by combining the initial state of OS, such as speed, course, and position.