An adaptive Monte Carlo method for uncertainty forecasting in perturbed two-body dynamics

A key objective of space situational awareness is to provide an accurate characterization of state uncertainty of space objects (SOs) in order to support predictive analytics, such as assessments of the probability of collision with other SOs. Due to a large number of SOs in comparison with the limited number of sensors available for tracking them, physics based uncertainty forecasting is often required for long periods, during which state uncertainty can become highly non-Gaussian. In this paper, we consider the problem of uncertainty forecasting in perturbed two-body dynamics via adaptive Monte Carlo simulations (MCS). While widely used, the standard Monte Carlo approach is limited by two main challenges: L) its slow rate of convergence, which is especially detrimental to its application in complex systems where each simulation is computationally expensive; and ii.) ambiguity in the ensemble's capability to accurately characterize the true state-pdf over time with a fixed ensemble size, given the fact that the propagated state uncertainty is time-varying and unknown in advance. We employ a newly developed adaptive Monte Carlo method that aims to characterize the propagated state-pdf within user defined accuracy bounds in the perturbed two-body dynamics. In the adaptive Monte Carlo algorithm, the transient performance of MCS is quantified in terms of the standard deviation of its approximation error, which is estimated via bootstrapping. A particle addition scheme is activated when the approximation error exceeds the user-defined upper bound at the current time and new particles are introduced sequentially, one at a time. These are selected through a two-layered algorithm that (a.) enforces optimal inter-particle projective distance, and (b.) minimizes sample discrepancy with respect to the true initial state-pdf in terms of the space-filling and non-collapsing properties. On the other hand, particles are removed (halted) for future propagation when current MCS error is estimated to be lower than the user defined threshold, in the interest of reducing computational load. Particle removal is dependent on their own evolved probability-density values which are obtained by solving the corresponding stochastic Liouville equation (SLE) numerically. As a result, the proposed approach is expected to generate a minimal ensemble of particles to characterize the propagated state-pdf within user defined accuracy bounds at all future times. Numerical simulations are shown to demonstrate the effectiveness of the proposed forecasting method.