Combining Differential Algebra and Subset Simulation for Efficient Long-Term Collision Risk Assessment

Abstract

This paper presents a methodology to enhance the GSOC Collision Avoidance System (CAS) for better assessing collision risk during longterm close approaches. The technique employs Differential Algebra (DA) to model the non-linear relative motion via high-order Taylor polynomial expansions, with accuracy maintained with Automatic Domain Splitting (ADS) that subdivides the initial uncertainty set. By sampling the initial relative conditions and evaluating the polynomial expansions over time, the method reduces traditional Monte-Carlo (MC) integration to a one-dimensional polynomial evaluation. Computational efficiency is further improved by integrating the Subset Simulation (SS) algorithm, an estimation technique that drastically reduces the required samples by reformulating a rare event as a sequence of more probable conditional events. The methodology’s performance is evaluated, using representative synthetic benchmark scenarios from the literature as well as a real long-term encounter.