Fourier Series Quadratic Bézier Approximation for Gradient Vector Field Trajectory Following

This work investigated precise arrival-time control for uncrewed aerial vehicles with turnrate– constrained dynamics. Trajectory planning and following with Bézier curves allowed for time-controlled interceptions by providing continuous curved paths that meet arrival time requirements and did not violate fixed-wing aircraft turning constraints. Bézier curves were parametric and required approximation of original routes for Gradient Vector Field (GVF)-based trajectory following to overcome the discretization problem inherent in waypointbased tracking. The specific problem of this work was identifying a length and arrival time controllable trajectory function that was partially differentiable using Fourier series as a curve fitting basis function. Arrival time deviations and tracking error experiments were conducted in two-dimensional kinematic airplane simulations using GVF tracking and the non-linear guidance law for comparison. Fourier series routes for convergence towards the route resulted in improved path tracking and arrival time adhesion when compared to the NGL and polynomial regression approximation of the Bézier route.