# Research on Optimal Landing Trajectory Planning Method between an UAV and a Moving Vessel
## 2 Approach Landing Trajectory Estimation between Drone and Vessel  
A typical approach landing trajectory between a common UAV and a vessel is shown in Figure 1. At the time t = 0 (starting of the approach maneuver), the UAV is at the point \(\mathbf{A}_{0}\), with coordinates \(\left( \begin{array}{cc}x_{A0}&y_{A0}&z_{A0}\end{array} \right)\), speed \(\mathbf{V}_{A}(t)\), and speed direction expressed as \(\mathbf{\varphi}_{A0}\)\(\left( \begin{array}{cc}\gamma_{A0}&\beta_{A0}\end{array} \right)\). Similarly, at the time t = 0, the vessel carrying the land-based device is located at a point \(\mathbf{B}_{0}\), with coordinates \(\left( \begin{array}{cc}x_{B0}&y_{B0}&z_{B0}\end{array} \right)\), speed \(\mathbf{V}_{B}(t)\), and speed direction of \(\mathbf{\varphi}_{B0}\)\(\left( \begin{array}{cc}\gamma_{B0}&\beta_{B0}\end{array} \right)\). It is assumed that during the UAV landing approach, the mode of the UAV motion velocity vector is \(\mathbf{V}_{A}(t)\), the mode of the vessel motion velocity vector is \(\mathbf{V}_{B}(t)\), and the vessel motion velocity direction \(\mathbf{\varphi}_{B0}\) are known constants. For the UAV landing time \(t_{B}\), the predicted position of the vessel is \(\mathbf{B}_{T}\Big{(} \begin{array}{cc}x_{BT}&y_{BT}&z_{BT}\end{array} \Big{)}\). At this time, the mode of the vessel's motion velocity vector is constant \(\mathbf{V}_{B}\) with its direction \(\mathbf{\varphi}_{B0}\Big{(} \begin{array}{cc}\gamma_{B0}&\delta_{B0}\end{array}  \Big{)}\). Assuming that the approach landing trajectory is in the plane \(\Big{(} \begin{array}{cc}\mathbf{A}_{0}&\mathbf{B}_{0}&\mathbf{V}_{B}\end{array} \Big{)}\), and the origin of the initial coordinate system is selected at the point \(\mathbf{B}_{0}\), the coordinates of the point in the new obtained coordinate system are:  
\[\left\{ \begin{array}{l}x_{1}^{\prime}=x-x_{B0}\\ y_{1}^{\prime}=y-y_{B0}\\ z_{1}^{\prime}=z-z_{B0}\end{array} \right. \tag{1}\]  
where \(x^{\prime}\), \(y^{\prime}\), \(z^{\prime}\), are the coordinates of the points in the new coordinate system, and \(x\), \(y\)\(z\) are the coordinates of the points in the original coordinate system. In order for the ship's velocity vector to lie in the horizontal plane, the resulting coordinate system needs to be rotated by \(-\delta_{B0}\) along the axis \(x^{\prime}\). The new point coordinates in the resulting coordinate system are:  
\[\left\{ \begin{array}{c}x_{2}^{\prime}=x_{1}^{\prime}\\ y_{2}^{\prime}=-z_{1}^{\prime}sin(-\delta_{B0})+y_{1}^{\prime}cos(-\delta_{B0}) \\ z_{2}^{\prime}=-z_{1}^{\prime}cos(-\delta_{B0})+y_{1}^{\prime}sin(-\delta_{B0}) \end{array} \right. \tag{2}\]  
Finally, we rotate the obtained coordinate system along the axis \(z^{\prime}\) by an angle:  
\[\theta=-\text{arctan}\frac{\left(y_{2B0}^{\prime}-y_{2A0}^{\prime}\right)}{ \left(x_{2B0}^{\prime}-x_{2A0}^{\prime}\right)}\]  
The point coordinates in the resulting new coordinate system are:  
\[\left\{ \begin{array}{c}x^{\prime}=x_{2}^{\prime}cos\theta-y_{2}^{\prime}sin \theta\\ y^{\prime}=x_{2}^{\prime}sin\theta+y_{2}^{\prime}cos\theta\\ z^{\prime}=z_{2}^{\prime}\end{array} \right. \tag{3}\]  
The complete approach trajectory from point \(\mathbf{A}_{0}\) to point \(\mathbf{B}_{T}\) in the resulting coordinate system will be in the plane \(X^{\prime}Z^{\prime}\) (as shown in Figure 2). Figure 1: Approach trajectory between an UAV and a vessel. The case shown in Figure 2 occurs when the pitch angle of both the UAV and the vessel is numerically close to each other. For example, when the UAV flies at a constant altitude before starting the approach landing decision. The sum direction of the axis \(X^{\prime}\) and axis \(Z^{\prime}\) and the positive direction of the angle in the plane \(X^{\prime}Z^{\prime}\) are given in Figure 2.  
In general, a landing trajectory approach consists of four feature segments. The first segment corresponds to the UAV landing turn, from the initial UAV position \(\mathbf{A}_{0}\) to point \(\mathbf{A}_{1}\) at the end of the turn.