Minimization of Delta-v for Satellite Swarm Maintenance Using a Virtual Chief

Abstract

This article presents a novel methodology for minimizing delta-v costs associated with relative orbit-keeping of a satellite swarm. The minimization is achieved through the selection of a virtual chief spacecraft for the reference of the swarm. By casting the minimization in relative orbital elements (ROE), the proposed approach leverages state-transition matrices (STMs) and analytical representations of delta-v usage. The STMs and analytical expressions are used to represent the swarm delta-v usage as a function of the virtual chief spacecraft parameters. The minimization of the resulting swarm delta-v function is manipulated to take the form of a convex program that is applicable to various delta-v formulations, ROE sets, and dynamic environments, generalizing approaches used for binary satellite systems to larger swarms. The novel quadratically constrained, linear objective convex program can be solved efficiently with available algorithms, such as interior point methods or commercial solvers. Furthermore, a new analytical delta-v lower bound for 6 degrees-of-freedom reconfigurations of quasi-nonsingular ROEs is developed. By leveraging the novel delta-v lower bound in the optimization, analytical solutions to the optimization for various orbit environments that include the perturbing effects of J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or drag are produced. For the problems that do not have an analytical solution, geometric insight is provided to reduce computational effort in solving the optimization. The presented solutions and insights are validated through a convex optimization solver, demonstrating the accuracy and value of the analysis. Additionally, a high-fidelity simulation demonstrates fuel savings in a realistic mission scenario.