A Comparison of Parametric and Non-Parametric Machine Learning Approaches for the Uncertain Lambert Problem

Abstract

The uncertain Lambert problem has important applications in Space Situational Awareness (SSA). While formulating the solution to this problem, it is of great interest to characterize the uncertainty associated with the solution as a function of position vector uncertainties at initial and final times. Previous work in this respect has concentrated on deriving a stochastic framework that exploits dynamical system theory in conjunction with non-product quadrature methods to compute higher order sensitivity matrices for accurately characterizing the uncertainty associated with Lambert problem solution. While deep learning tools have gained tremendous attention in various fields such as physics, biology, and manufacturing, existing tools for regression and classification do not capture model uncertainty. In comparison, Bayesian-based models offer a solid and robust mathematically grounded framework to reason about model uncertainty, but usually come with a prohibitive computational cost. In aerospace systems, representing model uncertainty is of crucial importance. The objective of this work will be to conduct a detailed comparison between classical dynamical system based approaches with recent advances in Machine Learning (ML) to characterize the uncertainty associated with the Lambert problem solution. In particular, we will consider parametric ML approaches such as multi-layered neural networks and a non-parametric Bayesian approach known as Gaussian Process Regression to learn a surrogate model representing the Lambert problem solution in the neighborhood of the nominal solution. Numerical experiments will be conducted to assess the relative merits of each of the methods considered in terms of accuracy of representing the uncertainty associated with the Lambert problem solution as well as numerical efficiency.