Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem

This paper addresses the problem of relative position control of spacecraft formation flying (SFF) utilizing the framework of the circular restricted three-body problem (CR3BP) with the Sun and Earth as the primary gravitational bodies. Particularly, the results are not confined to the close vicinity of the collinear Lagrangian libration points. Rather, a linearization is performed relative to an arbitrary non-Keplerian reference trajectory, so that linear time-varying differential equations result. It is rigorously proved that the open-loop linearized SFF dynamics is unstable but controllable. Uncontrolled formations with bounded relative separation that constitute the stable subspace of the dynamics are found using the dual system method. This constitutes a powerful observation, since bounded formations in three-body trajectories have been found thus far only at the vicinity of the Lagrangian equilibrium points. Assuming ideal state measurements, a time-varying continuous linear-quadratic control law is subsequently developed. A complete internal disturbance model is used, rendering a robust disturbance rejection performance. An illustrative example is used to show that the propulsion for deep-space formationkeeping requires a considerable dynamic range. It is concluded that plasma electric propulsion having a micro-thrusting capability is a most suitable means for deep-space formationkeeping.