# A Closed-Form Method for Simultaneous Target Localization and UAV Trajectory Optimization
## 4 Simulations, Experiments and Results Analysis
### Simulations
The trajectory optimization result is shown in Figure 7.  
From the optimized trajectory, we can see that similar to the constant velocity scenario, the optimized trajectory of UAV tends to form a circle around the target with the target as the center. The comparison of localization performances for the proposed method and the referenced method is shown in Figure 8. We can see that the optimized trajectory by the proposed method in this paper can still achieve a better localization performance than that of the referenced methods. Figure 6: Root mean squared error (RMSE) errors of target localization for the moving target with constant velocity. Figure 7: Trajectory optimization result with the constant turn model. (**a**) Absolute trajectory. (**b**) Relative trajectory with target as the origin. In the scenario with the constant acceleration motion model, the initializations and constraints were the same as the above scenarios; and the acceleration of the target were set as 0.135m/s\({}^{2}\), 0.260 m/s\({}^{2}\) in the \(X^{g}\), \(Y^{g}\) directions, respectively. The trajectory optimization result is shown in Figure 9. From the result, we can see that the optimal trajectory was similar to the other two scenarios. The localization performance is shown in Figure 10. The RMSE comparisons of the proposed method and the referenced methods showed that using the predicted information matrix could also better predict the target position and improve the effectiveness of the trajectory for minimizing the localization uncertainty of the target with an acceleration motion model. All three simulation results coincided with the phenomenon that the optimal path for a moving sensor to maximize a stationary target's observability is a circle around it. Furthermore, how large the circle is determined by the UAV's limit and the UAV's allowable turn rate, which is essentially determined by the UAV's acceleration. In fact, the circle around the target exists because the allowable minimum speed is not zero. According to the simulation setting where the relative UAV height from the target is known, the optimal position to minimize the target position estimate uncertainty is just above the target location. The algorithm always tries to find a waypoint nearest to the target at each optimization step while satisfying the UAV maneuverability requirement. Figure 8: RMSE errors of target localization for the moving target with constant turn rate. Figure 5.0 Trajectory optimization result with the constant acceleration motion model. (**a**) Absolute trajectory. (**b**) Relative trajectory with target as the origin. From the simulation results, it can be seen that the simultaneous target localization and UAV trajectory optimization algorithm based on EIF is applicable to the common motion models, and can achieve an obvious improvement on the localization performance compared with the existing methods. The underlying reason includes two aspects. First, the proposed algorithm exploits the EIF to consistently estimate the target state by predicting the target state based on the target dynamic model with the previous states and correcting it with the latest obtained measurement. Second, the consistent estimate of the target position was utilized to optimize the next UAV's waypoint, which makes the waypoint more effective for suppressing the target estimate uncertainty. However, the other two methods, which either use the stochastic model or cannot consistently estimate the target state, are likely to degrade the localization performance.