Uncertainty Propagation and Filtering via the Koopman Operator in Astrodynamics

Abstract

The Koopman Operator (KO) theory is applied to generate an analytical solution of dynamical systems. The approach proposed in this work exploits a novel derivation of the KO with orthogonal polynomials to represent and propagate uncertainties, where the polynomials are modified to work with stochastic variables. Thus, a new uncertainty quantification technique is proposed in which the KO solution is expanded to include the prediction of central moments, up to an arbitrary order. The propagation of uncertainties is then expanded to develop a new filtering algorithm, for which the measurements are considered as additional observables in the KO mathematics. The uncertainty propagation technique is tested by predicting the state probability density function of a spacecraft in a halo orbit. The performance of the technique is assessed with a Monte Carlo analysis and proven to obtain accurate estimates for the state covariance, skewness, and kurtosis. The novel filtering methodology is then applied to an orbit determination application regarding a Lyapunov orbit, where an analysis on the filter accuracy and consistency shows that the new KO filter outperforms other common estimators.