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Deep Learning Based Dynamics Identification and Linearization of Orbital Problems using Koopman Theory

George Nehma, Madhur Tiwari, Manasvi Lingam

TL;DR

The paper addresses the challenge of obtaining a globally valid linear representation for nonlinear orbital dynamics described by the Two-Body Problem (2BP) and Circular Restricted Three-Body Problem (CR3BP) to enable efficient control and state estimation. It develops a data-driven Koopman operator framework powered by a deep neural network that lifts states to a high-dimensional observable space and learns a finite-dimensional linear operator that propagates these observables, enabling a linear time-invariant (LTI) formulation for both 2BP variants and CR3BP around the L1 point. The approach demonstrates global linearization across circular, elliptical, and perturbed 2BP scenarios and extends to CR3BP, with invariants such as radius, angular momentum, and the Jacobi constant being preserved to a meaningful degree. The framework offers a fast, offline-trained baseline for control and estimation that generalizes to different central bodies and perturbations, with potential online adaptation to maintain accuracy during long-duration missions and formation flying.

Abstract

The study of the Two-Body and Circular Restricted Three-Body Problems in the field of aerospace engineering and sciences is deeply important because they help describe the motion of both celestial and artificial satellites. With the growing demand for satellites and satellite formation flying, fast and efficient control of these systems is becoming ever more important. Global linearization of these systems allows engineers to employ methods of control in order to achieve these desired results. We propose a data-driven framework for simultaneous system identification and global linearization of the Circular, Elliptical and Perturbed Two-Body Problem as well as the Circular Restricted Three-Body Problem around the L1 Lagrange point via deep learning-based Koopman Theory, i.e., a framework that can identify the underlying dynamics and globally linearize it into a linear time-invariant (LTI) system. The linear Koopman operator is discovered through purely data-driven training of a Deep Neural Network with a custom architecture. This paper displays the ability of the Koopman operator to generalize to various other Two-Body systems without the need for retraining. We also demonstrate the capability of the same architecture to be utilized to accurately learn a Koopman operator that approximates the Circular Restricted Three-Body Problem.

Deep Learning Based Dynamics Identification and Linearization of Orbital Problems using Koopman Theory

TL;DR

The paper addresses the challenge of obtaining a globally valid linear representation for nonlinear orbital dynamics described by the Two-Body Problem (2BP) and Circular Restricted Three-Body Problem (CR3BP) to enable efficient control and state estimation. It develops a data-driven Koopman operator framework powered by a deep neural network that lifts states to a high-dimensional observable space and learns a finite-dimensional linear operator that propagates these observables, enabling a linear time-invariant (LTI) formulation for both 2BP variants and CR3BP around the L1 point. The approach demonstrates global linearization across circular, elliptical, and perturbed 2BP scenarios and extends to CR3BP, with invariants such as radius, angular momentum, and the Jacobi constant being preserved to a meaningful degree. The framework offers a fast, offline-trained baseline for control and estimation that generalizes to different central bodies and perturbations, with potential online adaptation to maintain accuracy during long-duration missions and formation flying.

Abstract

The study of the Two-Body and Circular Restricted Three-Body Problems in the field of aerospace engineering and sciences is deeply important because they help describe the motion of both celestial and artificial satellites. With the growing demand for satellites and satellite formation flying, fast and efficient control of these systems is becoming ever more important. Global linearization of these systems allows engineers to employ methods of control in order to achieve these desired results. We propose a data-driven framework for simultaneous system identification and global linearization of the Circular, Elliptical and Perturbed Two-Body Problem as well as the Circular Restricted Three-Body Problem around the L1 Lagrange point via deep learning-based Koopman Theory, i.e., a framework that can identify the underlying dynamics and globally linearize it into a linear time-invariant (LTI) system. The linear Koopman operator is discovered through purely data-driven training of a Deep Neural Network with a custom architecture. This paper displays the ability of the Koopman operator to generalize to various other Two-Body systems without the need for retraining. We also demonstrate the capability of the same architecture to be utilized to accurately learn a Koopman operator that approximates the Circular Restricted Three-Body Problem.
Paper Structure (26 sections, 45 equations, 28 figures, 2 tables, 2 algorithms)

This paper contains 26 sections, 45 equations, 28 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. NONLINEAR DYNAMICS AND THE KOOPMAN OPERATOR
  3. Koopman Operator Theory
  4. Koopman Operator
  5. Dynamic Mode Decomposition and its Extensions
  6. Koopman Approximation Algorithm
  7. Orbital Dynamics
  8. Circular Restricted Three-Body Dynamics
  9. NEURAL NETWORK ARCHITECTURE
  10. Deep Neural Network Structure and Loss Function Definition
  11. MODEL ACCURACY METRICS
  12. Global vs Local Error
  13. Two-Body Circular Metrics
  14. CR3BP Jacobi Constant
  15. SIMULATION AND RESULTS
  16. ...and 11 more sections

Figures (28)

  • Figure 1: Deep learning framework including the prediction and correction steps. Tiwari
  • Figure 2: Four position and velocity states of the Earth orbits over one orbital period in comparison to the full nonlinear dynamics.
  • Figure 3: Global error in position states over one orbital period around Earth.
  • Figure 4: Local error in position states over one orbital period around Earth.
  • Figure 5: Four position and velocity states of the Earth orbits over ten orbital periods in comparison to the full nonlinear dynamics.
  • ...and 23 more figures