Closed-Form Modeling and Control of Spacecraft Swarms in Eccentric Orbits

Spacecraft formation-flying and swarm missions have received great attention in the space sector over the past few decades. Some modern multi-satellite missions require the use of an eccentric orbit to access varying altitudes, new ground tracks, and a lower perturbation environment. The use of a distributed mission architecture becomes more challenging in eccentric orbits due to the increased complexity of relative motion dynamics, and important mission considerations such as passive safety, differential perturbation modeling, and efficient impulsive control become more difficult to resolve in closed-form. This paper introduces a new state representation denoted Eccentric Relative Orbit Elements (EROE) to address these issues. The EROE provide an insightful geometric tie into relative position and velocity in eccentric orbits, revealing closed-form expressions for passive safety. Leveraging the fact that EROE are functions of orbit elements, state transition matrices including differential J2, solar radiation pressure, and third body perturbations are also presented. Finally, this paper maps the EROE state onto an existing impulsive control methodology to compute maneuver schemes in closed-form. Using the advantages provided by the chosen state representation, design and maintenance strategies are proposed for swarms, all of which require little computational effort. These results are applied to the mission design and simulation of a conceptual three-spacecraft swarm mission denoted the Mars Gravity Experiment. Simulation results demonstrate over two orders of magnitude improved positional accuracy over short time periods when using the STMs provided in this paper compared to Yamanaka-Ankersen, and delta-v budgeting is accurate within 0.5% of nonlinear simulation.