Utilizing the Finite Fourier Series to Generate Quadrotor Trajectories Through Multiple Waypoints

Motion planning is critical for ensuring precise and efficient operations of unmanned aerial vehicles (UAVs). While polynomial parameterization has been the prevailing approach, its limitations in handling complex trajectory requirements have motivated the exploration of alternative methods. This paper introduces a finite Fourier series (FFS)-based trajectory parameterization for UAV motion planning, highlighting its unique capability to produce piecewise infinitely differentiable trajectories. The proposed approach addresses the challenges of fixed-time minimum-snap trajectory optimization by formulating the problem as a quadratic programming (QP) problem, with an analytical solution derived for unconstrained cases. Additionally, we compare the FFS-based parameterization with the polynomial-based minimum-snap algorithm, demonstrating comparable performance across several representative trajectories while uncovering key differences in higher-order derivatives. Experimental validation of the FFS-based parameterization using an in-house quadrotor confirms the practical applicability of the FFS-based minimum-snap trajectories. The results indicate that the proposed FFS-based parameterization offers new possibilities for motion planning, especially for scenarios requiring smooth and higher-order derivative continuity at the expense of minor increase in computational cost.