Calculating collision probability for long-term satellite encounters through the reachable domain method

Satellite encounters during close operations, such as rendezvous, formation, and cluster flights, are typical long-term encounters. The collision probability in such an encounter is a primary safety concern. In this study, a parametric method is proposed to compute the long-term collision probability for close satellite operations with initial state uncertainty. Random relative state errors resulting from system uncertainty lead to possible deviated trajectories with respect to the nominal one. To describe such a random event meaningfully, each deviated trajectory sample should be mapped to a unique and time-independent element in a random variable (RV) space. In this study, the RV space was identified as the transformed state space at a fixed initial time. The physical dimensions of both satellites were characterized by a combined hard-body sphere. Transforming the combined hard-body sphere into the RV space yielded a derived ellipsoid, which evolved over time and swept out a derived collision volume. The derived collision volume was solved using the reachable domain method. Finally, the collision probability was computed by integrating a probability density function over the derived collision volume. The results of the proposed method were compared with those of a nonparametric computation-intensive Monte Carlo method. The relative difference between the two results was found to be < 0.6%, verifying the accuracy of the proposed method.