## 3 Methodology
### Conversion Model of Anchor Position  
The anchor position refers to a position where the ship is relatively stable by grasping the bottom material while anchoring \(A_{2}\) is shown in Figures 1 and 6. In the horizontal direction of sea bottom, there is a certain displacement \(L_{0}\) of the anchor from the dropping anchor position \(A_{1}\) that is instructed to let go of the anchor to the anchor position \(A_{2}\), as shown in Figure 1. Reasons for the formation of \(L_{0}\) are relatively complex and hard to calculate, which are related to the ship's maneuverability, type, tonnage, speed, loading situation, anchor type and other factors, as well as external wind and waves, water depth and bottom material.  
Figure 6: Schematic diagram of the position deviation of ship position sensors. (**a**) The position deviation of ship position sensors when anchoring. (**b**) The position deviation of ship position sensors after anchoring.  
There is a certain time interval from the issued order of anchoring to the actual settlement of the anchor. Since the ship moves with a certain speed, a certain displacement of the anchor in the horizontal plane of the ship's movement is detectable. The operation of anchoring by the windlass takes longer than gravity anchoring, and will generate more horizontal displacement.
2. There is a certain horizontal displacement during the process of the anchor falling to the bottom, inserting into the bottom material and stabilizing to the anchor position.
3. The anchor and anchor chain are affected by wind and waves during the anchoring operation, resulting in a certain displacement of the anchor in the horizontal direction.
4. During the reversing process of the ship, a large horizontal force upon the ship is generated, and the anchor is dragged, thus presenting a certain horizontal displacement.  
Therefore, the vector model of the anchoring motion is summarized in Equation (10).  
\[\overset{\rightarrow}{L}_{0}=\overset{\rightarrow}{V}*t+\overset{ \rightarrow}{L}_{A}+\overset{\rightarrow}{L}_{W}+\overset{\rightarrow}{L}_ {P} \tag{10}\]  
In Equation (10), \(\overset{\rightarrow}{L}_{0}\) is the anchoring motion vector; \(\overset{\rightarrow}{V}\) represents the ship vector speed; \(t\) the ship anchoring time, that is, the time from the moment when the order is issued to the time when the anchor claw penetrates into the bottom material; \(\overset{\rightarrow}{L}_{A}\) is the anchor dragging motion vector; \(\overset{\rightarrow}{L}_{W}\) is the horizontal motion vector generated by the wind, waves and currents; and \(\overset{\rightarrow}{L}_{P}\) represents the anchor dragging vector due to the reverse horizontal force. The anchoring motion vector model involves many factors and it is difficult to quantify them. Then, in order to find the anchor position, as shown in Figure 6b, the position sensor coordinates of \(A_{3}(\varphi_{3},\lambda_{3})\), namely, the ship position after anchoring, can be calculated by Equation (11).  
\[\left\{\begin{array}{l}\varphi_{3}^{\prime}\approx\varphi_{3}+\frac{Z^{ \prime}}{1852}\times\cos\theta^{\prime}\\ \lambda_{3}^{\prime}\approx\lambda_{3}+\frac{Z^{\prime}}{1852}\times\sin \theta^{\prime}\end{array}\right. \tag{11}\]  
where, \(Z^{\prime}\) is the distance between the anchor position of the ship and the position actually recorded by the ship during anchor dropping, and \(\theta^{\prime}\) is the included angle between the connecting line from the anchor position to the ship position in the coordinate system and the abscissa.