# A Closed-Form Method for Simultaneous Target Localization and UAV Trajectory Optimization
## 4 Simulations, Experiments and Results Analysis
### Simulations  
The simulations were conducted based on the full measurement of the UAV target localization system (i.e., the measurement contains the attitude angles of the UAV and camera). Utilizing (6), the required measurement errors of the azimuth and elevation angle of LoS were computed. The measurement noises were both set as zero mean Gaussian, of which the symbol (SYM) and standard deviations (STD) are listed in Table 1.  
For the simulations, three scenarios were designed: the first scenario had with a moving target with a constant velocity model; the second scenario had the constant turn rate model; and the third had a constant acceleration motion model. For simplicity, the flight heights were all fixed at 600 m. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Roll Angle** & **Pitch Angle** & **Yaw Angle** & **Azimuth** & **Elevation** \\ \hline SYM & \(\phi\) & \(\theta\) & \(\varphi\) & \(\alpha\) & \(\beta\) \\ STD (degree) & 0.8 & 0.8 & 1.2 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement noise setting. In the scenario with the constant velocity model, the initial point of the target was set as the origin of \(C\)s, the velocity was set as 25 m/s and 12 m/s in the two axes of the horizontal plane; the initial position of the UAV was set as (\(-1000\) m, 0 m, 600 m), and the minimum and maximum velocity were set as 70 m/s and 100 m/s, respectively, and the maximum turn rate was 12 degree/s. The time interval of measurement was one second and the tracking lasts 100 s. Using the above trajectory optimization algorithm, the trajectory result is obtained as Figure 5. To learn the relation between the target and the optimized trajectory, the trajectory optimization result relative to the target position is shown in Figure 5 (i.e., the optimized trajectory when treating the target position as the origin). From the trajectory result, we can see that the optimal trajectory tends to form circles around the target, where the target position is the circle center. To illustrate the localization performance, 100 Monte Carlo trails were conducted. The root mean squared error (RMSE) results are shown in Figure 6, where the localization method in [1] (EKF based on stochastic model) and the method similar to the referenced method in [17] (notated by STLUTO-IDEAL) are also presented. The two referenced methods are also simultaneous target localization and trajectory optimization strategies, among which the first one assumes target motion as a stochastic model, and the second one does not take into account the process noise. In the simulations, for the first method, the process noise covariance was set as \(27.7\times 27.7\times I_{2\times 2}\) because the maximum velocity value was 27.7m/s; for the second method, the MLE was used to estimate the target position. From the comparison, we could see that the simultaneous target localization and UAV trajectory optimization based EIF (notated by STLUTO-EIF) had overall lower errors than the two referenced methods. The method of EKF based on the stochastic model had an obviously large error than the other two methods, of which the localization error tended to diverge at \(t\approx 45\) s. The underlying reason is that the EKF-based method predicts the position badly using the stochastic model. The method without considering the process noise had an obvious better performance than the EKF-based method, but it has a large error than the proposed method. This shows that the consideration of process noise can improve the effectiveness of the optimized trajectory of the UAV. Figure 5: Trajectory optimization result with the constant velocity motion model. (**a**) Absolute trajectory. (**b**) Relative trajectory with target as the origin. For the second scenario, the target motion model was changed accordingly. The initializations of the target state and UAV, and the constraints settings were the same as scenario 1; the turn rate of the target was set as 2 degree/s.