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Uncertainty Propagation and Filtering via the Koopman Operator in Astrodynamics

Simone Servadio, William Parker, Richard Linares

TL;DR

The paper addresses uncertainty propagation in strongly nonlinear astrodynamics by introducing a Koopman Operator (KO) framework that propagates central moments of the state distribution to arbitrary order. It combines a Galerkin-based computation of KO eigenfunctions using Legendre polynomials with a KO-based filter (KOF) that treats measurements as polynomial observables, enabling closed-form moment propagation and Kalman-like updates without numerical integration. The methodology is demonstrated on CRTBP problems (Earth–Moon halo and Sun–Earth Lyapunov orbits), where KO accurately tracks means and higher-order moments and exhibits superior consistency compared with EKF, IKF, and UKF in Monte Carlo tests. This approach offers a robust, global tool for uncertainty quantification and state estimation in highly nonlinear orbital dynamics with limited measurements.

Abstract

The Koopman Operator (KO) provides an analytical solution of dynamical systems in terms of orthogonal polynomials. This work exploits this representation to include the propagation of uncertainties, where the polynomials are modified to work with stochastic variables. Thus, a new uncertainty quantification technique is proposed, where 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, where measurements are considered as additional observables in the KO mathematics. Numerical simulations in astrodynamics assess the accuracy and performance of the new methodologies.

Uncertainty Propagation and Filtering via the Koopman Operator in Astrodynamics

TL;DR

The paper addresses uncertainty propagation in strongly nonlinear astrodynamics by introducing a Koopman Operator (KO) framework that propagates central moments of the state distribution to arbitrary order. It combines a Galerkin-based computation of KO eigenfunctions using Legendre polynomials with a KO-based filter (KOF) that treats measurements as polynomial observables, enabling closed-form moment propagation and Kalman-like updates without numerical integration. The methodology is demonstrated on CRTBP problems (Earth–Moon halo and Sun–Earth Lyapunov orbits), where KO accurately tracks means and higher-order moments and exhibits superior consistency compared with EKF, IKF, and UKF in Monte Carlo tests. This approach offers a robust, global tool for uncertainty quantification and state estimation in highly nonlinear orbital dynamics with limited measurements.

Abstract

The Koopman Operator (KO) provides an analytical solution of dynamical systems in terms of orthogonal polynomials. This work exploits this representation to include the propagation of uncertainties, where the polynomials are modified to work with stochastic variables. Thus, a new uncertainty quantification technique is proposed, where 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, where measurements are considered as additional observables in the KO mathematics. Numerical simulations in astrodynamics assess the accuracy and performance of the new methodologies.
Paper Structure (11 sections, 52 equations, 5 figures)

This paper contains 11 sections, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Koopman Operator Theory
  3. Computing the Koopman Eigenfunctions via Galerkin Method
  4. Koopman modes
  5. Uncertainty Propagation with the Koopman Operator
  6. The Koopman Operator Filter
  7. The Circular Restricted Three Body Problem
  8. Hamiltonian in Normal Form
  9. Prediction of Central Moments
  10. Orbit Determination
  11. Conclusions

Figures (5)

  • Figure 1: Monte Carlo propagation of $10^4$ orbits for $t_f = 2$ given the initial PDF $\mathcal{N} ({\bf{x}}_{0},{\bf P}_0)$. Projection on the $(x,y)$ plane.
  • Figure 2: Prediction of raw moments of the state PDF for a Halo orbit orbiting around $\mathcal{L}_1$ in the earth-Moon system.
  • Figure 3: Prediction of central moments of the state PDF for a Halo orbit orbiting around $\mathcal{L}_1$ in the earth-Moon system.
  • Figure 4: 500 runs Monte Carlo analysis of the KOF for a Lyapunov orbit around $\mathcal{L}_1$.
  • Figure 5: Position and Velocity standard deviation analysis among EKF, IKF, UKF, and KOF.