# Research on Optimal Landing Trajectory Planning Method between an UAV and a Moving Vessel
## 2 Approach Landing Trajectory Estimation between Drone and Vessel
If the inertia of the UAV control system in the horizontal plane is neglected, then the maneuvering process can be considered to be done along the center of the circle \(\mathbf{C}_{1}\Big{(} \begin{array}{cc}x^{\prime}_{C1}&y^{\prime}_{C1}\end{array}  \Big{)}\), with a minimum allowable turning radius R (in plane \(X^{\prime}Z^{\prime}\)). At this time:  
\[\left\{ \begin{array}{l}x^{\prime}_{C1}=x^{\prime}_{A0}+Rsin\gamma_{A0}\\ z^{\prime}_{C1}=z^{\prime}_{A0}-Rcos\gamma_{A0}\end{array} \right. \tag{4}\]  
The length of the first track is:  
\[L_{\rm I}=\varphi_{\rm I}{\rm R} \tag{5}\]  
of which the angular distance between \(\mathbf{A}_{0}\) and \(\mathbf{A}_{1}\) is \(\varphi_{\rm I}\), and R is the turning radius of the UAV. On this trajectory, the value of R is determined by the lateral overload allowable value \(n_{z}\)[18]. For example, for \(V_{A}=\left|\mathbf{V}_{A}(t)\right|=60\) m/s, \(n_{z}=1\), we can get:  
\[{\rm R}=\frac{V_{A}^{2}}{n_{z}g}\approx 360\ {\rm m}\]  
In this case, the angular velocity of turn is:  
\[\omega_{y}=\frac{V_{A}}{R}=\frac{60}{360}=\frac{1}{6}\ {\rm rad}/s\Big{(} \omega_{y}=10^{\circ}/s\Big{)}\]  
The second segment of the landing trajectory approach corresponds to a straight line segment, between the first maneuver completion point \(\mathbf{A}_{1}\) and the second maneuver start point \(\mathbf{A}_{2}\). The UAV enters the vessels navigation direction. The length of the trajectory is equal to the distance between the center of the first and second turns of the UAV, namely:  
\[L_{\rm II}=\sqrt{\left(x_{C2}-x_{C1}\right)^{2}+\left(z_{C2}-z_{C1}\right)^{2}} \tag{6}\]  
On this trajectory, the direction angle \(\varphi_{A2}\) of the UAV's motion trajectory is calculated by:  
\[\varphi_{A2}=\left\{\begin{array}{l}arctan\Big{(}\frac{z_{C2}-z_{C1}}{x_{C 2}-z_{C1}}\Big{)},\  
In the third segment, along the center of the circle \(\mathbf{C}_{2}\), the arc of radius \(\mathbf{R}\) is flying from the point \(\mathbf{A}_{2}\) to the point \(\mathbf{A}_{3}\). At the point \(\mathbf{A}_{3}\), the UAV direction lies in the same direction of the ship carrying the landing device (ship). The length of the path is:  
\[L_{\text{III}}=\varphi_{\text{III}}R \tag{8}\]  
of which \(\varphi_{\text{III}}\) refers to the angular distance between \(\mathbf{A}_{2}\) and \(\mathbf{A}_{3}\). In the fourth section of the landing trajectory approach, the UAV moves along the direction of the vessel, making an approximate linear motion between the second maneuver completion point \(\mathbf{A}_{3}\) and the landing end point \(\mathbf{B}_{T}\). The length \(L_{\text{IV}}\) of the section is a constant value, about 200 to 300 m. The length of this passage is selected in advance based on the conditions that accurately guide the UAV to the landing device. On this path segment, a compensation of the calculated landing error, as well as a reduction in the UAV flight speed to the minimum required for landing, is achieved. Assuming that \(t_{3}\) is the time when the UAV ends its turn (return to point \(\mathbf{A}_{3}\)), then, in general, the following equation can be obtained:  
\[\mathbf{A}_{3}=\mathbf{B}_{0}+\int\limits_{0}^{t_{3}}\mathbf{V}_{\text{B}}(t)dt \tag{9}\]  
For a specific uniform linear motion of the ship in the plane, we can get:  
\[\left\{ \begin{array}{l}x^{\prime}_{A3}=x^{\prime}_{B0}+V_{B}t_{3}cos\gamma_{ B0}\\ z^{\prime}_{A3}=z^{\prime}_{B0}+V_{B}t_{3}sin\gamma_{B0}\end{array} \right. \tag{10}\]  
where the coordinates of the end point \(\mathbf{B}_{T}\), the UAV and the vessel are connected at this point:  
\[\left\{ \begin{array}{l}x^{\prime}_{BT}=x^{\prime}_{B0}+(V_{B}t_{3}+L_{\text {IV}})cos\gamma_{B0}\\ z^{\prime}_{BT}=z^{\prime}_{B0}+(V_{B}t_{3}+L_{\text{IV}})sin\gamma_{B0}\end{array} \right. \tag{11}\]  
After determining the coordinates of the point \(\mathbf{A}_{3}\), the coordinates of the turning circle center \(\mathbf{C}_{2}\) of the third segment of the landing trajectory can be obtained with the following relationship:  
\[\left\{