OPTIMIZATION OF FLIGHT TRAJECTORIES OF UNMANNED AERIAL VEHICLES WITH GIVEN FINAL CONDITIONS

The article considers the urgent problem of optimizing the flight trajectory of unmanned aerial vehicles (UAVs) with given final conditions. An approach to constructing the optimal UAV trajectory along successive sections using approximation by a fifth-degree polynomial is presented. A mathematical model is developed that takes into account the features of using local inertial coordinate systems on each section of the trajectory. A motion optimization method is proposed based on minimizing the quadratic functional characterizing both the accuracy of reaching specified points and the energy costs for control. The solution to the optimization problem is obtained using analytical design of the optimal controller. Analytical expressions are given for determining the optimal lateral acceleration of the UAV, ensuring the required positioning accuracy with minimal energy costs. Features of coordinate transformation between trajectory sections to ensure continuity of motion are considered. A practical example of calculating the optimal UAV trajectory with given initial conditions is given.

1. Alqudsi Y. UAV swarms: research, challenges, and future directions / Y. Alqudsi, M. Makaraci // Journal of Engineering and Applied Science. – 2025. – Vol. 72. – Р. 12. https://doi.org/10.1186/s44147-025-00582-3.

2. UAV Path planning trends from 2000 to 2024: A bibliometric analysis and visualization / Q. Wu et al / Drones. – 2025. – Vol. 9. – Р.128. https://doi.org/10.3390/drones9020128.

3. Online decision making and path planning framework for safe operation of unmanned aerial vehicles in urban scenarios / M. Quinones-Grueiro et al // International Journal of Prognostics and Health Management. – 2021. – Vol. 12. – Р. 3. https://doi.org/10.36001/ijphm.2021.v12i3.2953.

4. A spherical vector-based adaptive evolutionary particle swarm optimization for UAV path planning under threat conditions / Y. Liu et al // Scientific Reports. – 2025. – Vol. 15. – Р. 2116. https://doi.org/10.1038/s41598-025-85912-4.

5. Performance analysis of hybrid optimization approach for UAV path planning control using FOPID-TID controller and HAOAROA algorithm / N. Basil et al // Scientific Reports. – 2025. – Vol. 15. – Р. 4840. https://doi.org/10.1038/s41598-025-86803-4.

6. Rgg-pso+: Random geometric graphs based particle swarm optimization method for UAV path planning / Y. Liu et al // International Journal of Computational Intelligence Systems. – 2024. – Vol. 17. – Р. 127. https://doi.org/10.1007/s44196-024-00511-x.

7. Zhao Y. Survey on computational-intelligence-based UAV path planning / Y. Zhao, Z. Zheng, Y. Liu // Knowledge-Based Systems. – 2018. – Vol. 158. – Р. 54-64. https://doi.org/10.1016/j.knosys.2018.05.033.

8. Aljalaud F. Optimizing Autonomous Multi-UAV Path Planning for Inspection Missions: A Comparative Study of Genetic and Stochastic Hill Climbing Algorithms / F. Aljalaud, Y. Alohali // Energies. – 2025. – Vol.18(1). – Р. 50. https://doi.org/10.3390/en18010050.

9. Enhanced Multi‐UAV Path Planning in Complex Environments With Voronoi‐Based Obstacle Modelling and Q‐Learning / W. Su et al // International Journal of Aerospace Engineering. – 2024. – Vol. 1. – Р. 5114696. https://doi.org/10.1155/2024/5114696.

10. Hybrid PSO-HSA and PSO-GA algorithm for 3D path planning in autonomous UAVs / B. Abhishek et al // SN Applied Sciences. – 2020. – Vol. 2. – Р. 1-16. https://doi.org/10.1007/s42452-020-03498-0.

11. A Comprehensive Review of Energy-Efficient Techniques for UAV-Assisted Industrial Wireless Networks / Yijia Zhang et al // Energies. – 2024. – Vol. 17(18). – Р. 4737. https://doi.org/10.3390/en17184737.

12. Pounds P. Modelling and control of a large quadrotor robot / P. Pounds, R. Mahony, P. Corke // Control Engineering Practice. – 2010. – Vol. 18. – Р. 691-699. https://doi.org/10.1016/j.conengprac.2010.02.008.