Application of the Finite Fourier Series for Smooth Motion Planning of Quadrotors

Abstract

Motion planning and control system design are crucial in the development of unmanned aerial vehicles (UAVs) for various applications. To maximize the mission performance, trajectory design is often performed to minimize motor output and power consumption along the entire path while avoiding obstacles. Polynomial parameterizations have traditionally been used to express UAV trajectories due to their simplicity and effectiveness of implementation. In this work, an alternative parameterization based on Finite Fourier Series (FFS) is investigated. The results of the FFS-based parameterization are compared with the state-of-the-art polynomial-based trajectory generation algorithms. One unique feature of the FFS parameterization is the theoretical piece-wise infinite differentiability of multi-segment trajectories. For fixed-time minimum-snap motion-planning problems using FFS parameterization, it is 1) shown that motion-planning problems can be formulated as quadratic programming (QP) problems, and 2) derived an analytic solution to an unconstrained QP problem. Leveraging the analytic solution, formulation, and solution of time-allocated minimum-snap multi-segment trajectories is also presented. The practical limitations of the FFS method are also discussed in detail. Finally, formulation and numerical comparison of the minimum-snap, multi-segment trajectories with time-allocation are also presented. Performance (with respect to the computation time, power required, and number of iterations) of the FFS and polynomial parameterizations are compared using five representative trajectories. The results show the comparable performance of the two parameterizations with important differences between the two on high-level derivatives. An in-house quadrotor is used with a six-degree-of-freedom (6DoF) cascaded PID flight control logic to experimentally validate the tracking of the generated smooth trajectories.