Semi-analytical formation-flying control in J2-perturbed eccentric orbits

Abstract

This paper presents a methodology for quasi-fuel-optimal control of the relative motion in Earth-orbiting spacecraft formations in J2-perturbed eccentric orbits using impulsive maneuvers. A closed-form expression describing the inverse relative dynamics with multiple impulsive maneuvers over a defined time window is derived, enabling a semi-analytical approach for obtaining quasi-optimal solutions under a path-minimization assumption. The combined guidance and control formulation, as described by the closed-form expression, is parameterized in terms of quasi non-singular mean relative orbital elements (ROEs) and leverages a linearized J2-perturbed relative motion model for state prediction using a state transition matrix (STM). The proposed method extends prior ROE-based semi-analytical impulsive-control formulations to the case of J2-perturbed eccentric orbits. This approach provides geometric insights into secular and long-period effects within the finite reconfiguration windows and is applicable to orbits with arbitrary eccentricity.