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Time-delayed Dynamic Mode Decomposition for families of periodic trajectories in Cislunar Space

Sriram Narayanan, Mohamed Naveed Gul Mohamed, Indranil Nayak, Suman Chakravorty, Mrinal Kumar

Abstract

In recent years, the development of the Lunar Gateway and Artemis missions has renewed interest in lunar exploration, including both manned and unmanned missions. This interest necessitates accurate initial orbit determination (IOD) and orbit prediction (OP) in this domain, which faces significant challenges such as severe nonlinearity, sensitivity to initial conditions, large state-space volume, and sparse, faint, and unreliable measurements. This paper explores the capability of data-driven Koopman operator-based approximations for OP in these scenarios. Three stable periodic trajectories from distinct cislunar families are analyzed. The analysis includes theoretical justification for using a linear time-invariant system as the data-driven surrogate. This theoretical framework is supported by experimental validation. Furthermore, the accuracy is assessed by comparing the spectral content captured to period estimates derived from the fast Fourier transform (FFT) and Poincare-like sections.

Time-delayed Dynamic Mode Decomposition for families of periodic trajectories in Cislunar Space

Abstract

In recent years, the development of the Lunar Gateway and Artemis missions has renewed interest in lunar exploration, including both manned and unmanned missions. This interest necessitates accurate initial orbit determination (IOD) and orbit prediction (OP) in this domain, which faces significant challenges such as severe nonlinearity, sensitivity to initial conditions, large state-space volume, and sparse, faint, and unreliable measurements. This paper explores the capability of data-driven Koopman operator-based approximations for OP in these scenarios. Three stable periodic trajectories from distinct cislunar families are analyzed. The analysis includes theoretical justification for using a linear time-invariant system as the data-driven surrogate. This theoretical framework is supported by experimental validation. Furthermore, the accuracy is assessed by comparing the spectral content captured to period estimates derived from the fast Fourier transform (FFT) and Poincare-like sections.

Paper Structure

This paper contains 12 sections, 21 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Mathematical Background
  4. Cislunar Dynamics
  5. Koopman Operator Theory
  6. Modeling Periodic Trajectories: Auto-regression and Hankel DMD
  7. Results
  8. L1 - Halo Orbit
  9. L1/L2 - Butterfly Orbit
  10. Resonant (2:1) Orbit
  11. Spectral Comparison
  12. Conclusion & Future Work

Figures (8)

  • Figure 1: L1 - Halo orbit
  • Figure 2: L1/L2 - Butterfly orbit
  • Figure 3: Resonant (2:1) orbit
  • Figure 4: Spectral comparison for each orbit type: (a) L1 Halo orbit (b) L1/L2 Butterfly orbit (c) Resonant (2:1) orbit
  • Figure 5: Eigenvalues of $\Tilde{\mathbf{A}}$ matrix for each orbit type: (a) L1 Halo orbit (b) L1/L2 Butterfly orbit (c) Resonant (2:1) orbit
  • ...and 3 more figures