Bézier Curves for Smooth Entry into Elliptic Orbits

Abstract

This paper describes a path planning approach for smooth entry of an aerospace vehicle (ASV) into a 3D elliptic orbit. The generated path represents a polynomial Bezier curve connecting a given position of the ASV with the orbital entry point. Recursive and non-recursive analytical formulas for the last k intermediate control points determining Ck smooth Bezier path are derived. To evaluate the performance of the proposed approach, a practical measure of path smoothness such as path length, average curvature, and maximum curvature is introduced to choose the best entry point on the orbit and the corresponding Bezier path. The simulation results demonstrate that minimizing the maximum curvature yields a path that optimizes the proposed smoothness measure.