## 3 Super-Large Vessel Collision Avoidance Model  
### CRI Objective Function Model  
CRI is a fuzzy concept that measures the degree of collision risk between vessels. The CRI \(\in\) [0,1], specifically, CRI = 0 indicates that there is no CR between vessels, and the vessel can continue sailing in its current state; 0 \(<\) CRI \(<\) 1 indicates that there is a CR between vessels [29]. By setting a safety threshold, it can be divided into several levels, such as \"low risk\", \"moderate risk\", and \"high risk\", and appropriate avoidance operations can be undertaken based on this. CRI = 1 indicates that the vessel has entered an emergency, and relying solely on the vessel's avoidance operation will not avoid collision with the target vessel. The main way to determine whether there is a CR between vessels is to comprehensively analyze the distance \(D\) and relative orientation between the two vessels \(\theta\), vessel speed ratio \(K\), nearest encounter distance \(d_{CPA}\), recent meeting time \(t_{CPA}\), [PERSON] et al. [30] used the above five elements as basic evaluation parameters and quantitatively analyzed the vessel risk model using the fuzzy comprehensive evaluation method based on fuzzy mathematics. Considering the five factors that affect the risk of vessel collision, establish a target factor set:  
\[U=\{D,\ \theta,\ K,\ d_{CPA},\ t_{CPA}\} \tag{14}\]  
Establish a comment set:  
\[V=\{u_{1},\ u_{2}\} \tag{15}\]  
where \(u_{1}\) indicates danger; \(u_{2}\) indicates security. \(W\) is the weight matrix of influencing factors:  
\[W\ =\left(w_{D},\ w_{\theta},\ w_{K},\ w_{d_{CPA}},\ w_{t_{CPA}}\right)=(0.14, \ 0.10,\ 0.08,\ 0.36,\ 0.32) \tag{16}\]  
Establish an evaluation matrix:  
\[r=\left( \begin{array}{cc}u_{D}&1-u_{D}\\ u_{\theta}&1-u_{\theta}\\ u_{K}&1-u_{K}\\ u_{d_{CPA}}&1-u_{d_{CPA}}\\ u_{t_{CPA}}&1-u_{t_{CPA}}\end{array} \right) \tag{17}\]  
where \(r_{D}\), \(r_{\theta}\), \(r_{K}\), \(r_{d_{CPA}}\) and \(r_{t_{CPA}}\) represent risk membership for each influencing factor; the range of values is [0,1]. \[u_{D}= \begin{cases}1&0\leq D\leq D_{1}\\ \left(\frac{D_{2}-D}{D_{2}-D_{1}}\right)^{2}&D_{1}<D\leq D_{2}\\ 0&D>D_{2}\end{cases}  \tag{18}\]  
\[u_{\theta}=1/2\bigg{(}cos(\theta-19)+\sqrt{400/289\ +cos^{2}(\theta-19)}\bigg{)} -5/17 \tag{19}\]  
\[u_{K}=1/\Big{(}1\ +\ k_{w}/K\sqrt{K^{2}+1\ +2 KsinC}\Big{)} \tag{20}\]  
where \(C\) is the collision angle between two vessels, \(k_{w}\) is a constant taken as 2, \(D_{1}=K_{1}K_{2}K_{3}D_{LA}\), and \(D_{LA}\) is the distance of the last-minute action, whose value is generally taken as 12 times the vessel's length [31]. \(D_{2}=K_{1}K_{2}K_{3}R\); the value of \(K_{1}\) depends on the visibility, the value of \(K_{2}\) depends on the waters in which the vessel is navigating, and the value of \(K_{3}\) depends on the human factor. \(R\) is the radius of the dynamic boundary area of this vessel. \[R=1.7 cos(\theta-19)+\sqrt{4.4+2.89 cos^{2}(\theta-19)} \tag{21}\]  
\[r_{d_{CPA}}=\left\{ \begin{array}{ll}1&0\leq d_{CPA}\leq d_{1}\\ 1/2-1/2 sin(180^{\circ}/(d_{2}-d_{1})(d_{CPA}-(d_{2}-d_{1})/2))&d_{1}\leq d_{CPA }\leq d_{2}\\ 0&d_{CPA}\geq d_{2}\end{array} \right. \tag{22}\]  
where \(d_{1}\) is the vessel's safe encounter distance and \(d_{2}\) is the absolute safe encounter distance; generally, take \(d_{2}=2d_{1}\). In the traditional method of determining the safe encounter distance of vessels, a fixed threshold is mainly set. There may be a greater CR when the distance between vessels is less than this determined threshold, but the accuracy of this method can be limited due to the complexity and dynamics of vessel movements in ports. Incorporating a ship domain model can improve the accuracy of encounter distance calculations by more accurately simulating vessel motion and behavior and considering factors such as vessel size, shape, draft, speed, and maneuvering.