Post-Maneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions

This paper describes the use of polynomial chaos expansions to approximate the probability of a collision between two satellites after at least one performs a translation maneuver. Polynomial chaos provides a computationally efficient means to generate an approximate solution to a stochastic differential equation without introducing any assumptions on the a posteriori distribution. The stochastic solution then allows for orbit state uncertainty propagation. For the maneuvering spacecraft in the presented scenarios, the polynomial chaos expansion is sparse, allowing for the use of compressive sampling methods to improve solution tractability. This paper first demonstrates the use of these techniques for possible intra-formation collisions for the Magnetospheric Multiscale mission. The techniques are then applied to a potential collision with debris in low Earth orbit. Results demonstrate that these polynomial chaos-based methods provide a Monte Carlo-like estimate of the collision probability, including adjustments for a spacecraft shape model, with only minutes of computation cost required for scenarios with a probability of collision as low as 10−6. A graphics processing unit (GPU) implementation of the polynomial chaos expansion analysis further reduces the computation time for the scenarios presented.