## 2 Problem Description
### Geometrical Model of Speed-Varying Collision Avoidance for Super-Large Vessels  
The geometric model of vessel speed-varying collision avoidance plots the dynamic information of the vessel and the incoming vessel obtained by radar or AIS in the same chart. Plane geometry drawing is used to obtain relevant parameters, determine whether there is a CR, and provide a reference for drivers to take corresponding vessel maneuvering. Let the speed and heading of this vessel be \(v_{0}\) and \(C_{0}\), respectively, the speed and heading of the target vessel be \(v_{1}\) and \(C_{1}\), respectively, the true bearing of the target vessel be \(T_{B}\), the distance be \(D\), the speed \(v_{01}\), and relative heading \(C_{01}\) of this vessel relative to the target vessel. Then, the \(d_{CPA}\) and \(t_{CPA}\) between vessels can be calculated from the above parameters [28]. The expressions are as follows:  
\[d_{CPA}=f_{d_{CPA}}(v_{0},\ C_{0},\ v_{1},\ C_{1},\ D,\ T_{B}) \tag{1}\]  
\[t_{CPA}=f_{t_{CPA}}(v_{0},\ C_{0},\ v_{1},\ C_{1},\ D,\ T_{B},\ v_{01}) \tag{2}\]  
The vessel slows down to the predetermined speed, requiring a deceleration stroke (\(D_{s}\)) and stroke time (\(T_{s}\)) before slowing down to the predetermined speed. The \(d_{CPA}\) value of each target vessel is affected by \(D_{s}\) and \(T_{s}\). \(T_{s}\) refers to the time required for a vessel to move from the point of loss of control until its inertia disappears. Its size can be calculated according to Equation (3):  
\[T_{s}=-T_{st}(v_{t}/\ v_{0}) \tag{3}\]\(D_{s}\) refers to the distance traveled by vessel in the direction of vessel speed during the time of impact \(T_{s}\), which can be estimated by Equation (4):  
\[D_{s}=v_{0}T_{st}\Big{(}1-e^{-T_{s}/\ T_{st}}\Big{)} \tag{4}\]  
This vessel maintains its course and takes deceleration and collision avoidance operations, assuming the target vessel maintains its course and speed. During the deceleration process, the \(d_{CPA}\) values between vessels will continue to change. When the deceleration reaches the predetermined speed \(v_{0N}\), the \(d_{CPA}\) will no longer change, and the position of the target vessel relative to the vessel at this time is obtained \(\left(x_{1},\ y_{1}\right)\). \[\left\{ \begin{array}{l}x_{1}=DsinB+v_{1}T_{s}sinC_{1}+\ D_{s}sin(C_{0}+180^{ \circ})\\ y_{1}=DcosB+v_{1}T_{s}cosC_{1}+D_{s}cos(C_{0}+180^{\circ})\end{array} \right. \tag{5}\]  
Then, we can obtain the new distance \(D_{N}\) of the target vessel  
\[D_{N}=\sqrt{\left(x_{1}\right)^{2}+\ \left(y_{1}\right)^{2}} \tag{6}\]  
The velocity components of the vessel's speed on the \(x\) and \(y\) axes are:  
\[\left\{ \begin{array}{l}v_{x_{0}}=v_{0N}sinC_{0}\\ v_{y_{0}}=v_{0N}cosC_{0}\end{array} \right. \tag{7}\]  
The velocity components of the target vessel's speed on the \(x\) and \(y\) axes are:  
\[\left\{ \begin{array}{l}v_{x_{1}}=v_{1}sinC_{1}\\ v_{y_{1}}=v_{1}cosC_{1}\end{array} \right. \tag{8}\]  
The velocity components of the relative speed of two vessels on the \(x\) and \(y\) axes are:  
\[\left\{ \begin{array}{l}v_{x_{01}}=v_{x_{1}}-v_{x_{0}}\\ v_{y_{01}}=v_{y_{1}}-v_{y_{0}}\end{array} \right. \tag{9}\]  
The relative speed of the two vessels \(v_{01N}\) is:  
\[v_{01N}=\sqrt{v_{x_{01}}^{2}+{v_{y_{01}}}^{2}} \tag{10}\]  
The relative heading of the two vessels \(C_{01N}\) is:  
\[C_{01N}=\left\{ \begin{array}{l l}arctan\big{(}v_{y_{01}}/v_{x_{01}}\big{)} &v_{x_{01}}\geq 0,v_{y_{01}}\geq 0\\ 90^{\circ}&v_{x_{01}}\geq 0,v_{y_{01}}=0\\ 180^{\circ}+arctan\big{(}v_{y_{01}}/v_{x_{01}}\big{)}&v_{y_{01}}<0\\ 270^{\circ}&v_{x_{01}}<0,v_{y_{01}}=0\\ 360^{\circ}+arctan\big{(}v_{y_{01}}/v_{x_{01}}\big{)}&v_{x_{01}}<0,v_{y_{01}}< 0\end{array} \right. \tag{11}\]  
The true orientation of the target vessel \(T_{BN}\) is:  
\[T_{BN}=\left\{ \begin{array}{l l}arctan(x_{1}/y_{1})&y_{1}>0\\ 180^{\circ}+arctan(x_{1}/y_{1})&y_{1}<0\\ 0^{\circ}&y_{1}=0,x_{1}=0\\ 90^{\circ}&y_{1}=0,x_{1}>0\\ 270^{\circ}&y_{1}=0,x_{1}<0\end{array} \right. \tag{12}\]\(d_{CPA1}\) and \(t_{CPA1}\) after deceleration are denoted as:  
\[\left\{