## 3 Autonomous Collision Avoidance Method
### Dynamic Feasible Manoeuvring Interval  
The feasible manoeuvring decision interval, which contains all the decision-making for collision avoidance of ships, can be provided by the IVO algorithm based on the improved MMG. If both ships maintain their original course and speed, as shown in Figure 6, a risk of collision between ships is possible. The \(C_{0}\) and \(C_{1}\) are the minimal course changes of the feasible manoeuvring decision to the port and starboard side respectively at the present  
Figure 5: The diagram of the IVO method: (**a**) Traditional VO set; (**b**) Non-linear VO set.  
timing. The moving target will enter (not enter) the OS's domain if the angle is lower (greater) than this.  
However, under the constraint of COLRGEs, some decisions should be discarded in marine practice. Figure 7 illustrates collision avoidance decision-making under the COLREGs constraint in a variety of encounter scenarios. The dotted line indicates the side of the course followed by the COLREGs, with Figure 7a indicating an overtaking situation, Figure 7b indicating a head-on situation, and Figure 7c indicating a cross situation. The course is followed by the rules illustrated by the dotted line. Thus, the dynamically feasible manoeuvring interval of OS is offered for preventing collision action based on the MMG model, COLREGs, and IVO algorithm. The interval made at the time \(t\) is described as follows, which includes all feasible ship-manoeuvring decisions:  
\[D^{t}=\left[\begin{array}{c}(C_{1}^{t},T_{F}^{t}),\ldots,(C_{m}^{t},T_{F}^{t })\\ (C_{1}^{t},T_{1}^{t}),\ldots,(C_{m}^{t},T_{H}^{t})\\ (C_{1}^{t},T_{5}^{t}),\ldots,(C_{p}^{t},T_{S}^{t})\\ (C_{1}^{t},T_{D}^{t}),\ldots,(C_{q}^{t},T_{S}^{t})\end{array}\right] \tag{16}\]  
where \(C\) means the altering course which \(C\ \in\ (-90,\ 90)\) in the overtaking situation and \(\Delta C\in(0,\ 90)\) in the head-on and cross situation. \(D^{t}\) denotes OS's dynamic feasible manoeuvring interval at a time \(t\). \(m,n,p\) and \(q\) denotes the total number of the sets at the telegraph of \(F\), \(H\), \(S\) and \(D\), respectively.  
Furthermore, combined with the model of CRI, the dynamic feasible manoeuvring intervals are still appropriate for collision avoidance in the multi-ship scenario. There are generally two types of collision avoidance for TSs in the multi-ship scenario. One is that OS should take measures to avoid collision for one TS where the value of CRI is greater than the threshold. Although, a variety of TSs are in danger for OS, where the value of CRI of some TSs is too low to take collision avoidance at this moment. Therefore, in this situation, OS just avoids collision for the one TS at the same time. The other is that OS should take measures to avoid collision for two or more ships at once where their values of CRI are all  
Figure 6: Feasible manoeuvring decision interval.  
greater than the threshold. Therefore, the dynamically feasible manoeuvring interval of TSs is the intersection of the manoeuvring interval of each TS, which is shown as Equation (17):  
\[D^{t}=D_{1}^{t}\cap\ D_{2}^{t}\cdots D_{i}^{t}\cdots D_{m-1}^{t}\cap\ D_{m}^{t} \tag{17}\]  
where \(m\) denotes the total number of TSs with the higher value of CRI and \(D_{i}^{m}\) denotes the dynamically feasible manoeuvring interval of \(TS_{i}\).