Robust Closed-form Framework for Drag-Propulsive Control of Formation Flight

Distributed Space Systems (DSS) can accomplish many objectives that would be difficult or impossible with a single monolithic spacecraft, but often use spacecraft with limited propulsion and computing power. To conserve propellant, DSS spacecraft can leverage environmental forces to produce low cost relative forces, such as atmospheric drag in Low Earth Orbit (LEO). Atmospheric drag can be controlled on-board spacecraft by modulating differential attitude to create differential drag with little to no propulsion. This work maximizes this strategy by developing robust, closed-form algorithms for optimal "hybrid" control, defined here as complementing the magnitude and directionality deficiencies of differential drag with the superior control authority of propulsive maneuvers within the same control window. The use of drag introduces a significant source of dynamic uncertainty to the control problem, primarily from inaccurate atmospheric density modelling. This paper develops a robust control architecture to reject dynamic modeling errors and recover final state accuracy performance. The architecture periodically re-solves the remaining drag maneuver plan in a diminishing horizon policy, using the current state estimate to reject operational and navigation errors while retaining the cost savings offered by differential drag. The approach is built upon reachable set theory, a control analysis technique that provides conditions for optimality for Linear Time Variant (LTV) systems with norm-like cost functions and optimally decouples hybrid control into separate propulsive and differential drag sub-problems. The optimal decoupling allows the architecture to solve control for hybrid or drag-only maneuvering. The optimality conditions are leveraged to prove the asymptotic Lyapunov stability of ideal reachable set-based control solutions and derive error bounds on this stability. A covariance analysis on the primary sources of error in hybrid control determines that the dominating source of uncertainty is the atmospheric density model. This conclusion allows the bounds on Lya-punov stability to be completely met by simply conditioning the atmospheric density value used in the control solutions. The derived Lyapunov stability bounds and performance of the algorithm are demonstrated experimentally in high fidelity full-force orbital simulations of the upcoming Space Weather Reconfigurable Multiscale Experiment (SWARM-EX) mission. The overall approach used by this work is not specific to DSS and can be extrapolated to any LTV control system with norm-like cost.