Position Extrema in Keplerian Relative Motion: A Gröbner Basis Approach

Abstract

This paper analyzes the relative motion between two spacecraft in orbit. Specifically, the paper provides bounds for relative spacecraft position-based measures which impact spacecraft formation-flight mission design and analysis. Previous efforts have provided bounds for the separation distance between two spacecraft. This paper presents a methodology for bounding the local vertical, horizontal, and cross track components of the relative position vector in a spacecraft centered, rotating reference frame. Three metrics are derived and a methodology for bounding them is presented. The solution of the extremal equations for the metrics is formulated as an affine variety and obtained using a Gröbner basis reduction. No approximations are utilized and the only assumption is that the two spacecraft are in bound Keplerian orbits. Numerical examples are included to demonstrate the efficacy of the method. The metrics have utility to the mission designer of formation flight architectures, with relevance to Earth observation constellations.