## 3 Super-Large Vessel Collision Avoidance Model
### CRI Objective Function Model
The ship domain is a water area around the vessel that exists to avoid collision accidents and ensure the safety of vessel navigation, and its size is affected by a variety of factors, such as the size of the vessel, speed and angle of encounter, and visibility of the water [32]. Since the inception of the elliptical ship domain concept established by [PERSON] [33], numerous experts and scholars worldwide have conducted relevant research in the ship domain model, establishing various ship domain models of different shapes and applying them to domains such as maritime traffic risk assessment, collision avoidance, and water-way traffic planning. These models include the sector model for open water proposed by [PERSON] [34], the circular model with center offset proposed by [PERSON] et al. [35], the elliptical model proposed by [PERSON] [36] and [PERSON] et al. [37], and the azimuthal octagonal model proposed by [PERSON] et al. [38]. [PERSON] et al. [39] classified these into three types of ship domain model, such as circular, elliptical, and polygonal, based on which corresponding mathematical models were established for simulation and analysis. The results indicate that the DAVIS and PIETRZYKOWSKI models are suitable for risk assessment in maritime traffic, while the FUJII, GOODWIN, COLDWELL, and KIJIMA models are suitable for collision avoidance. In practical applications, the use of the GOODWIN or KIJIMA models is safer than use of the COLDWELL and FUJII models. Although the GOODWIN model considers the different safe encounter distances of vessels in different encounter scenarios, it is a static model related only to the length of the vessel. In contrast, the KIJIMA model is influenced by multiple parameters, such as vessel length, width, course, and speed, and is a dynamic domain model that comprehensively considers vessel scale and encounter situations, better reflecting the characteristic differences of the vessel's safe encounter distances in different scenarios. The KIJIMA model consists of two parts: watching and blocking areas. The watching area is a section of the vessel in the forward direction, which is used to detect the obstacles in front of the vessel. The length of the watching area can be set according to the actual situation, which usually depends on the vessel's speed and reaction time and other factors. The blocking area refers to some area around the ship, which describes a situation in which the vessel is blocked during movement. The blocking area is calculated based on information such as the location and size of obstacles detected in the watching area. Figure 2 shows that the KIJIMA model is an approximate ellipse shape, consisting of two differently shaped semi-ellipses at the top and bottom. They share the half-axis \(S_{b}\) in the forward transverse direction of the vessel, while the lengths of the half-axes \(R_{bf}\) and \(R_{bf}\) in the bow and transom directions are different. In addition, the lengths of all the semi-axes change  
were obtained based on the KIJIMA model. This work is based on the ellipse origin to edge distance equation:  
\[d=\sqrt{a^{2}b^{2}/(a^{2}sinr^{2}+a^{2}cosr^{2})} \tag{28}\]  
where \(a\) is the long half-axis of the ellipse, \(b\) is the short half-axis of the ellipse, and \(r\) is the angle with the long half-axis \(a\). Derive the formula for \(d_{1}\):  
\[d_{1}=\left\{ \begin{array}{ll}\sqrt{R_{bf}^{2}S_{b}^{2}/\left(R_{bf}^{2}sinr _{1}^{2}+S_{b}^{2}cosr_{1}^{2}\right)}&y\geq 0\\ \sqrt{R_{ba}^{2}S_{b}^{2}/\left(R_{ba}^{2}sinr_{2}^{2}+S_{b}^{2}cosr_{2}^{2} \right)}&y<0\end{array} \right. \tag{29}\]  
where \((x,y)\) are the coordinates of the closest point of approach in the coordinate system established in the ship domain; \(r_{1},\ r_{2}\) are the angles between the line from the vessel to the closest point of approach and \(R_{bf}\), \(R_{ba}\), respectively. \[r_{t_{CPA}}= \begin{cases}1&0\leq t_{CPA}\leq t_{1}\\ \left(\frac{t_{2}-t_{CPA}}{t_{2}-t_{1}}\right)^{2}&t_{1}<t_{CPA}\leq\ t_{2}\\ 0&t_{CPA}\geq t_{2}\end{cases}  \tag{30}\]  
\[t_{1}=