Consider a space mission aiming to rendezvous with and capture a non-cooperative, possibly tumbling, space object. While ground observation campaigns can estimate the angular velocity of the target, its accurate rotational state remains unknown until observations are made during the mission itself. Therefore, during mission planning every possible rotational state must be considered subject to a maximum angular velocity which limits the burden on the spacecraft design. We assume that to mitigate the risks of failing to capture the target and of generating more debris through breakup, the target must be captured along a particular axis and without relative rotation. This motivates the technique of motion synchronization, in which the chaser makes its final approach to the target along a single axis in the latter’s body-fixed frame. The aim of this study is to explore the parameter space of rotational motions that the tumbling body may make, obtaining conditions under which a motion synchronization trajectory exists.
We approach the problem from two directions. First, we attempt to find and optimize motion synchronization trajectories for the entire parameter space subject to path constraints. Second, we provide a mathematical description of the complementary set of parameters for which no such trajectory was found, giving mathematical arguments that motion synchronization trajectories respecting the path constraints do not exist for these cases. In this way we obtain a complete description of the parameter space: for any combination of parameters describing the tumbling motion, we either find a motion synchronization trajectory or prove that none exists. Averaging over the parameter space gives a statistical measure of success, the probability that motion synchronization can be performed. We apply the methodology to a notional case based on realistic path constraints such as illumination and ground station visibility.
In addition, the mathematical description forms a heuristic for deciding whether motion synchronization is possible for any particular rotational state of the target body. Since this heuristic is in agreement with the trajectory optimization results, it yields a method of rapid estimation of the impact of spacecraft and mission design changes on the motion synchronization phase. We conclude the study by discussing the effect of changing several such design parameters of the notional mission.