## 2 Methodology
### Improved Velocity Obstacle  
The VO method defines a collection of velocity sets in which a collision occurs if the velocity falls inside one of them. Assuming that two ships, \(V_{O}\) and \(V_{T}\), represent the velocity of the OS and the TS, respectively. \(V_{OT}\) is the relative velocity of OS caused by TS, then this can be expressed in the following way:  
\[\overrightarrow{V_{OT}}=\overrightarrow{V_{O}}-\overrightarrow{V_{T}} \tag{10}\]  
As displayed in Figure 4, the white circle region, the TS's ship domain, defines all of the TS's possible positions when a collision occurs [28]. Once \(V_{OT}\) falls into the Relative collision cone (RCC), the collision between ships exists. On the contrary, there is no risk of collision existing between ships. The absolute collision cone (ACC, purple region) is presented by RCC along \(V_{T}\), which implies that the \(V_{O}\) dropping into ACC is equivalent to \(V_{OT}\) falling into RCC. In other words, the collision between ships exists when \(V_{O}\) falls into the ACC, which can be shown as Equation (11) considering that the VO algorithm is time-varying:  
\[\overrightarrow{V_{o}}^{(t)}\in\mathrm{VO}^{(t)}=\mathrm{RCC}\otimes \overrightarrow{V_{T}}^{(t)} \tag{11}\]  
where operation \(\otimes\) is the Minkowski addition. However, since it ignores the characteristics of ship manoeuvring and the particular navigation environment, typical VO cannot be \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Telegraph** & **RPM (r/min)** & **Speed (knot)** \\ \hline \multirow{4}{*}{Ahead} & Nav. ahead & 92 & 14 \\  & Full ahead & 85 & 12 \\  & Half ahead & 70 & 10.5 \\  & Slow ahead & 48 & 5 \\  & Dead slow ahead & 35 & 3 \\ \hline \multirow{4}{*}{Astern} & Stop & 0 & 0 \\ \cline{2-4}  & Dead slow astern & \(-\)35 & \(-\)3 \\ \cline{1-1}  & Slow astern & \(-\)48 & \(-\)5 \\ \cline{1-1}  & Half astern & \(-\)70 & \(-\)10.5 \\ \cline{1-1}  & Full astern & \(-\)85 & \(-\)12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Propulsion particulars about RPM and speed of ship “Huayang dream”. directly applied to ship collision avoidance even though it is very effective and simple in identifying the presence of a collision. Therefore, an IVO is presented, which combines typical VO, MMG, and COLREGs. The ship's movement is non-linear due to the large inertia of its properties, especially during significant changes in course and speed [34]. Therefore, an IVO model based on the consideration of the ship-manoeuvring motion characteristics is proposed to quantify the VO set in the encounter situation, which includes telegraph and course being able to be manoeuvred by the ship. The typical VO set is shown as the red region in Figure 5a, which is linear and appropriate for dynamic collision avoidance methods under ideal conditions. The outer and inner of the circle denote the maximum and minimum speeds of the ship, respectively, which are based on the manoeuvrability parameter of the propeller. In practice, the maximum and minimum speeds are maintained by the telegraph of \"Full. ahead\" (\(F\)) of the ship and \"Dead slow ahead\" (\(D\)), respectively, and the other speed for collision avoidance can be controlled by the \"Half. ahead\" (\(H\)) and \"Slow. ahead\" (\(S\)). Except for these, other telegraphs are seldom used to avoid collisions based on good seamanship since their slow speeds make it impossible to maintain the course. Therefore, an IVO algorithm shown in Figure 5b is provided combined with the four telegraphs and a typical VO, which can be used to avoid collision based on the ship's manoeuvrability. Four sectors denote the VO set of the different telegraphs, which can be expressed as followed:  
\[\left\{ \begin{array}{l}VO_{F}=ACC\oplus T_{F}\\ VO_{H}=ACC\oplus T_{H}\\ VO_{S}=ACC\oplus T_{S}\\ VO_{D}=ACC\oplus T_{D}\end{array} \right. \tag{12}\]  
where \(\oplus\) shows mathematical expression. \(T_{F}\), \(T_{H}\), \(T_{S}\), and \(T_{D}\) denote telegraph \(F\), \(H\), \(S\), and \(D\), respectively.