# Research on Optimal Landing Trajectory Planning Method between an UAV and a Moving Vessel
## 2 Approach Landing Trajectory Estimation between Drone and Vessel
\begin{array}{l}x^{\prime}_{\text{C2}}=x^{\prime}_{A3}+Rsin\gamma_{ BT}\\ z^{\prime}_{\text{C2}}=z^{\prime}_{A3}+Rcos\gamma_{BT}\end{array} \right. \tag{12}\]  
In addition, according to Equation (7), the angle \(\varphi_{A2}\) can be obtained, and then the coordinates of the point \(\mathbf{A}_{1}\) and \(\mathbf{A}_{2}\) are obtained as follows:  
\[\left\{ \begin{array}{l}x^{\prime}_{A1}=x^{\prime}_{C1}-Rsin\varphi_{A2}\\ z^{\prime}_{A1}=z^{\prime}_{C1}+Rcos\varphi_{A2}\\ x^{\prime}_{A2}=x^{\prime}_{C2}-Rsin\varphi_{A2}\\ z^{\prime}_{A2}=z^{\prime}_{C2}+Rcos\varphi_{A2}\end{array} \right. \tag{13}\]  
The difficulty in solving Equations (4)-(13) is that the value \(t_{3}\) is indeterminate. To this end, this paper proposes an iterative method to determine the values of \(t_{3}\). For the approximation of \(t_{3}\), we can use:  
\[t_{3}(i=1)=\frac{\sqrt{\left(x_{B0}-x_{A0}\right)^{2}+\left(z_{B0}-z_{A0} \right)^{2}}}{\left(V_{A}\pm V_{B}\right)}+\frac{\pi R}{V_{A}} \tag{14}\]  
where if the UAV and the vessel are facing each other, the symbol \"\(+\)\" is selected, but if not, the symbol \"\(-\)\" is selected. In formula (14), the angle between the UAV flight speed vector and the ship's navigation speed vector cannot be equal to 90\({}^{\circ}\). The error \(\Delta\)t of a single approximation does not exceed the following value:  
\[\Delta\text{t}=\frac{\pi R}{V_{A}} \tag{15}\]  
Within the interval \([t_{3}(i=1)-\Delta t,\ \ t_{3}(i=1)+\Delta t]\), the value \(t_{3}\) is searched and determined by the binary method. The search termination condition is:  
\[\left|A_{2\text{II}}(t_{3}(i))-A_{2\text{III}}(t_{3}(i))\right|<\varepsilon \tag{16}\]  
where \(\varepsilon\)--sets the coordinate error of point \(A_{2}\) in the track segment \(L_{\text{II}}-A_{2\text{III}}(t_{3}(i))\) and \(L_{\text{III}}-A_{2\text{III}}(t_{3}(i))\). The value \(\varepsilon\) is determined by the accuracy of the actual landing trajectory approach, and in this paper is selected as \(\varepsilon\leq 10\) m. \(A_{2\text{II}}\) refers to the point at the end of the left side trajectory of point \(A_{2}\) (the second trajectory); \(A_{2\text{III}}\) refers to the point at the end of the right side trajectory of point \(A_{2}\) (the third trajectory). When the vessel is cruising at a constant speed, at \(\varphi_{A2}(t_{3}(i))\) according to Equation (7), the coordinates of the point \(A_{2\text{II}}\) are:  
\[\left\{ \begin{array}{l}x^{\prime}_{2\text{II}}=x^{\prime}_{C1}-Rsin\varphi_ {A2}(t_{3}(i))+L_{\text{II}}cos\varphi_{A2}(t_{3}(i))\\ z^{\prime}_{2\text{III}}=z^{\prime}_{C1}+Rcos\varphi_{A2}(t_{3}(i))+L_{\text{ II}}sin\varphi_{A2}(t_{3}(i))\end{array} \right. \tag{17}\]  
With the coordinates of the point \(A_{2\text{III}}\) being:  
\[\left\{ \begin{array}{l}x^{\prime}_{2\text{III}}=x^{\prime}_{C2}-Rsin\varphi_ {A2}(t_{3}(i))\\ z^{\prime}_{2\text{III}}=z^{\prime}_{C2}+Rcos\varphi_{A2}(t_{3}(i))\end{array} \right. \tag{18}\]  
After the value \(t_{3}\) is determined, the total length \(L\) of the approach landing trajectory is determined by the sum of the length of the four-part trajectory segments:  
\[L=L_{\text{I}}+L_{\text{II}}+L_{\text{III}}+L_{\text{IV}} \tag{19}\]  
By calculating the length \(L\) of the shortest approach landing trajectory in real time, according to Equation (19), we can control the likelihood of the UAV returning to the landing device on the vessel. If the remaining fuel amount of the aircraft is less than the fuel amount \(M_{KP}\) required to return to the vessel, then the UAV must perform a return maneuver. Among them, \(M_{KP}\) is determined according to Equation (20). \[M_{KP}=Lm_{L} \tag{20}\]  
where \(m_{L}\) refers to the amount of fuel consumed by the drone unit path. It is worth noting that, if \(L_{\text{II}}\geq L_{\text{I}}+L_{\text{III}}+L_{\text{IV}}\), the approximate calculation Formula (21) can be used to replace the exact calculation formula of the landing trajectory approach \(L\) (14).