# An Efficient Trajectory Planning Method for High-Speed Interception of Invasive Drones
## 6 Performance Verification and Error Analysis
### Multi Target Trajectory Comparison  
This section designs a scenario for unmanned aerial vehicles to approach multiple targets, mainly to verify the applicability of the algorithm proposed in this paper in anti-drone swarm task scenarios. There are multiple target drones distributed in an area of 150 m \(\times\) 150 m \(\times\) 50 m, assuming that the positions of all targets are known and can be accurately tracked and predicted. Our drones start from the origin and approach all target drones in sequence. It is required that the relative velocities of both sides be zero when reaching the target position, in order to facilitate the payload operation of our drones. Figure 10: Simulation results of our method under high-speed conditions. The calculation takes 0.004 s.  
Thanks to the high computational efficiency of the closed form solution of the optimal trajectory, which can evaluate a large number of trajectories in a short time, this paper uses numerical methods to solve the shortest completion time of a single target trajectory that meets the control input constraints, and combines genetic algorithms to optimize the task sequence of multiple objectives. Figure 11 shows the trajectory planning results and physical simulation effects of the comparison method. The maximum planning speed of this method exceeded 30 m/s, which is far beyond the performance limit of drones considering the air resistance in the actual physical environment. This phenomenon of exceeding the capability range is also reflected in the large error between the simulated state trajectory and the expected state trajectory, which is unacceptable. Figure 12 elucidates the fluctuations in relative distance and speed between the drone, controlled by the contrastive method planning results, and multiple targets during actual flight. The findings suggest that there is a notable discrepancy between the simulated flight outcomes and the preset expected values, whether considering relative distance or speed. This significant deviation undoubtedly fails to satisfy the stringent demands of the drone's payload for its operational environment. Figure 13 provides a three-dimensional visualization of the flight trajectory comparison between the drone and its target. It is evident that there exists a notable discrepancy between the actual flight path of the drone and the anticipated trajectory, which compromises precise control and task execution. Figure 14 illustrates the trajectory planning outcomes and the physical simulation effects of the method proposed in this study. It is important to note that the maximum speed governed by this method does not surpass 20 m/s, thereby ensuring that the planned speed remains within the drone's performance capabilities. Furthermore, both the position and velocity trajectory tracking results are highly satisfactory. Figure 11: Simulation results of the approach task for five targets (comparative method). The task took 23.29 s.  
Figure 15 offers a comprehensive record of the fluctuations in relative distance and speed between the drone, controlled by the planning outcomes of this paper's method, and multiple targets during actual flight. The data reveals that at the terminal moment, both the relative distance and speed approximate zero. This suggests that the proposed method is capable of adequately fulfilling the operational requirements of the drone's payload. Figure 16 provides a visual representation of the high degree of congruence between the actual flight trajectory of the drone and the anticipated trajectory, nearly achieving a state of superimposition. It is noteworthy that the task order of the two methods diverges when executing multi-objective close-proximity tasks. This divergence originates from the disparate outcomes of single-task optimal trajectory planning between the two methodologies, resulting in marked differences in the optimization results of combined multi-objective tasks. Figure 12: The relative distance and speed of the target in the comparison method.