Using a Virtual Chief to Minimize Delta-v for Satellite Swarm Maintenance in Eccentric Orbits

Abstract

This work presents an approach to minimize swarm-keeping delta-v costs through the use of a virtual chief spacecraft. Current approaches in minimizing delta-v swarm-keeping costs are either computationally expensive or restricted to certain orbits and swarm geometries. To reduce the delta-v costs, a virtual chief spacecraft is used as the reference that the deputy spacecraft control their orbit with respect to. The relation between the virtual chief spacecraft parameters and swarm-keeping delta-v consumption is constructed by leveraging relative orbital elements (ROE) to form a convex program. Specifically, the choice of the virtual chief spacecraft is formulated as a quadratically constrained, linear program where the independent variables are the virtual chief spacecraft properties. To apply the optimization to the quasi-nonsingular ROE set, a novel delta-v lower-bound that captures the cost of 6 degree-of-freedom reconfigurations in eccentric orbits is required. This new delta-v lower-bound is derived and proven using primer-vector theory. While the quadratically constrained, linear program can be solved using available convex software, the computational effort is reduced by examining the structure of the optimization to find analytical solutions. To that end, the authors explore the optimal virtual chief spacecraft for swarm-keeping in orbits perturbed by $J^{2}$ and solar radiation pressure. Analytical solutions are provided if possible. If not possible, guidance on reducing the computational effort is presented. The analytical solutions are validated using CVX, a convex optimization solver. Furthermore, the solutions are further validated for a realistic swarm geometry in a high-fidelity simulation of a representative mission scenario. The delta-v required by the swarm spacecraft to maintain the desired geometry is compared for both optimal and non-optimal virtual chief spacecraft properties to demonstrate delta-v savings on the order of 15 to 20% across the swarm.