Polynomial superlevel set approximation of swept-volume for computing collision probability in space encounters

Computing long-term collision probability in space encounters is usually based on integration of a multivariate Gaussian distribution over the volume of initial conditions which generate collisions in the considered time interval. As this collision set is very difficult to determine analytically, for practical computation various simplifications are made in the literature. We present a new method for computing the collision probability based on two steps. Firstly, a higher-order outer-approximation of the swept-volume by a polynomial superlevel set is obtained as an optimal solution of a polynomial optimization problem. This has the advantage of providing approximate closed-form descriptions of the collision-prone states which can then be effectively used for long-term and repeated conjunctions analysis. Numerically, a hierarchy of linear matrix inequality problems is solved, which provide approximations (i) of increasing accuracy and (ii) convergent in volume to the original set. Secondly, once such a polynomial representation is computed, a high-order quadrature scheme for volumes implicitly defined by a polynomial superlevel sets is employed. Finally, the method is illustrated on a practical numerical example.