# Research on Optimal Landing Trajectory Planning Method between an UAV and a Moving Vessel
## 3 Initial Error Estimation of an UAV Autonomous Guidance System
### UAV Motion Simulation Based on Optimal Approach Landing Trajectory  
The dynamic structure of the UAV attitude control system in the horizontal and vertical planes is shown in Figure 3. The values \(T_{1}\), \(T_{2}\), \(T_{3}\), and \(T_{4}\) correspond to the time constant of the steering gear system and the aircraft; the \(\xi_{1}\) and \(\xi_{2}\) are the damping coefficients; with \(k_{1}\), \(k_{2}\), \(k_{\gamma}\), \(k_{\delta}\), \(k_{\dot{g}}\), the track azimuth \(\gamma_{A}\) and the track dip angle \(\delta_{A}\) and its control loop transfer coefficient of rate of change. In the coordinate converter \(F_{1}\), the calculation formulas of the track azimuth \(\gamma_{tr}\) and the track dip angle \(\delta_{tr}\) of the UAV flight speed vector at the \(i\) approach landing trajectory point are:  
\[\left\{ \begin{array}{c}\gamma_{tr}(i)=\arctan\!\left(\frac{z_{tr}(i+1)-z_{tr }(i)}{x_{tr}(i+1)-z_{tr}(i)}\right)\\ \delta_{tr}(i)=\arctan\!\left(\frac{y_{tr}(i+1)-y_{tr}(i)}{\sqrt{\left(x_{tr }(i+1)-x_{tr}(i)\right)^{2}+\left(z_{tr}(i+1)-z_{tr}(i)\right)^{2}}}\right) \end{array} \right. \tag{29}\]  
\[S_{A}(i)=\sqrt{\left(x_{tr}(i+1)-x_{tr}(i)\right)^{2}+\left(y_{tr}(i+1)-y_{tr }(i)\right)^{2}+\left(z_{tr}(i+1)-z_{tr}(i)\right)^{2}} \tag{30}\]  
The integration step is defined as:  
\[\Delta\mathrm{T}=\frac{S_{A}(i)}{V_{A}(i)} \tag{31}\]  
In the transverse plane, the finite difference forms of the equation of motion of the UAV are:  
Figure 3: Structurally-dynamic scheme of UAV control system for angle in horizontal and vertical planes. \[\left\{ \begin{array}{c}\mathrm{U}_{1\gamma}(i)=k_{\gamma}(\gamma_{tr}(i)-\gamma_{A }(i-1))-k_{\gamma}U_{3\gamma}(i-1)\\ U_{2\gamma}(i)=U_{2\gamma}(i-1)+\frac{\left(k_{1}U_{1\gamma}(i-1)-U_{2\gamma}(i -1)\right)\Delta T}{T_{1}}\\ U^{\prime}_{3\gamma}(i)=U^{\prime}_{3\gamma}(i-1)+\frac{\left(l_{2\gamma}(i-1) -2T_{2}\xi_{1}l^{\prime}_{3\gamma}(i-1)-U_{3\gamma}(i-1)\right)\Delta T}{T_{2 }^{2}}\\ U_{3\gamma}(i)=U_{3\gamma}(i-1)+U^{\prime}_{3\gamma}(i-1)\Delta T\\ \gamma_{A}(i)=\gamma_{A}(i-1)+U^{\prime}_{3\gamma}(i-1)\Delta T\end{array} \right. \tag{32}\]  
In the longitudinal plane, the finite difference forms of the equation of the UAV motion are:  
\[\left\{ \begin{array}{c}\mathrm{U}_{18}(i)=k_{8}(\delta_{tr}(i)-\delta_{A}(i- 1))-k_{\gamma}U_{3\delta}(i-1)\\ U_{28}(i)=U_{28}(i-1)+\frac{\left(k_{2}U_{18}(i-1)-U_{2\gamma}(i-1)\right) \Delta T}{T_{3}}\\ U^{\prime}_{3\delta}(i)=U^{\prime}_{3\delta}(i-1)+\frac{\left(l_{2\delta}(i-1 )-2T_{2}\xi_{2}U^{\prime}_{3\delta}(i-1)-U_{3\delta}(i-1)\right)\Delta T}{T_{ 4}^{2}}\\ U_{3\delta}(i)=U_{3\delta}(i-1)+U^{\prime}_{3\delta}(i-1)\Delta T\\ \delta_{A}(i)=\delta_{A}(i-1)+U^{\prime}_{3\delta}(i-1)\Delta T\end{array} \right. \tag{33}\]  
After processing of the track azimuth \(\gamma_{A}\) and track dip angle \(\delta_{A}\) of the UAV flight speed vector by the coordinate converter \(F_{2}\), the finite difference forms of the UAV's centroid coordinates are as follows:  
\[\left\{ \begin{array}{c}x_{A}(i)=x_{A}(i-1)+S_{A}(i)cos\gamma_{A}(i)\\ y_{A}(i)=y_{A}(i-1)+S_{A}(i)cos\delta_{A}(i)\\ z_{A}(i)=z_{A}(i-1)+S_{A}(i)sin\gamma_{A}(i)\end{array} \right. \tag{34}\]  
In the design of the control system, the transmission azimuth loop transfer coefficient \(\left(k_{\gamma},\;\;k_{\gamma}\right)\) and the track pitch loop transfer coefficient \(\left(k_{\delta},\;\;k_{\delta}\right)\) must be carefully selected to ensure the stability margin and the quality of the transition process. Since the inertia of the control object is large, there is a distinct dynamic error.