Stochastic Convergence of Sobol-Based Mahalanobis Shell Sampling Collision Probability Computation

Abstract

Sample-based computation of the joint-time probability of collision motivates developing the Mahalanobis Shell Sampling (MSS) algorithm, which samples nondegenerate normal random variables, enabling rare event simulation without unduly penalizing sample size. The MSS method has unbiased estimators in sample mean and covariance, and it may achieve arbitrary precision when approximating probability measures. For Clohessy–Wiltshire relative orbital dynamics, computational MSS exponential rates of error convergence (in the mean-square-error sense) are shown to improve by one order of magnitude (for sample mean and covariance) over Monte Carlo; when reproducing the instantaneous probability of collision, MSS has a comparable mean-square-error convergence rate performance to Monte Carlo.