Segmented UAV Trajectory Optimization via Fifth-Order Polynomial Approximation and Analytical Optimal Control

This study aims to optimize the flight trajectory of Unmanned Aerial Vehicles (UAVs) with given final conditions. An approach for constructing the optimal UAV trajectory through successive sections using a fifth-degree polynomial approximation is presented. A mathematical model is developed that considers the features of using local inertial coordinate systems on each trajectory section. A motion optimization method is proposed based on minimizing the quadratic functional that characterizes both the accuracy of reaching specified points and the energy cost for control. The solution to the optimization problem is obtained using the analytical design of the optimal controller. The analytical synthesis is novel and yields closed-form expressions for the optimal control on each segment, ensuring minimum-energy trajectories that meet the specified waypoints exactly. The proposed approach is validated through a practical example and MATLAB/Simulink simulation, demonstrating accurate and stable path-following by a UAV.