A polynomial-based Monte Carlo approach for estimating long-term collision probabilities

Abstract

This paper introduces a versatile approach for computing the risk of collision specifically tailored for scenarios featuring low relative encounter velocities, but with potential applicability across a wide range of situations. The technique employs Differential Algebra (DA) to express the non-linear dynamical flow of the initial distribution in the primary-secondary objects relative motion through high-order Taylor polynomials. The entire initial uncertainty set is subdivided into subsets through Automatic Domain Splitting (ADS) techniques to control the accuracy of the Taylor expansions. The methodology samples the initial conditions of the relative state and evaluates the polynomial expansions for each sample while retaining their temporal dependency. The classical numerical integration of the initial statistics over the set of conditions for which a collision occurs is thus reduced to an evaluation of mono-dimensional time polynomials. Specifically, samples reaching a relative distance below a critical value are identified along with the time at which this occurs. The approach is tested against a Monte Carlo (MC) simulation for various literature test cases, yielding accurate results and a consistent gain in computational time. Specifically, the time reduction can reach up to 99.9 %, depending on the dynamics and the number of samples employed in the analysis, while the estimated final Probability of Collision (Pc), for long-term scenarios, deviates from the reference by no more than 0.018 %.