State Forecasting in an Estimation Framework with Surrogate Sensor Modeling

1. Problem Statement

The paper addresses the critical challenge of state estimation and orbit prediction for Resident Space Objects (RSOs) in the context of Space Situational Awareness (SSA). Key difficulties include:

• Partial Observability: Observational data (range, bearing, elevation) is often intermittent, incomplete, or noisy due to sensor limitations and orbital geometry.
• Model Uncertainty: Traditional physics-based models (e.g., Keplerian two-body) often fail to capture complex perturbative forces (atmospheric drag, solar radiation pressure, $J_2$ effects) or the specific intent of maneuvering objects.
• Long-Horizon Forecasting: Unlike robotic control, space orbit prediction requires forecasting over long durations where errors accumulate rapidly.
• The Gap: Pure data-driven methods (Machine Learning) struggle with extrapolation beyond training data, while pure physics-based models lack robustness against unmodeled dynamics. The paper seeks a framework to fuse these approaches to extract meaningful state information from partial measurements.

2. Methodology

The authors propose a decoupled, iterative estimation framework that integrates a simplified reference dynamics model with a data-driven surrogate measurement model within an Extended Kalman Filter (EKF) structure.

A. Core Components

1. Reference Dynamics Model (Placeholder):
   • A simplified, physics-based model (e.g., Keplerian two-body) used to propagate state estimates forward in time.
   • It does not need to be highly accurate or account for all perturbations; its role is to provide a baseline state trajectory and initial conditions.
2. Surrogate Measurement Model (Data-Driven):
   • Built using Hankel-Dynamic Mode Decomposition (Hankel-DMD), a technique rooted in Koopman operator theory.
   • Instead of modeling the full system dynamics, the surrogate learns the evolution of measurements ($z(k)$) directly from historical data.
   • It uses time-delay embedding (Hankel matrix reconstruction) to capture spectral features and ergodic attractors of nonlinear systems.
   • Role: It predicts what the measurements should be at time $k+1$ when direct observations are missing or noisy.

B. The Fusion Process (EKF Framework)

The framework operates in a standard EKF loop but with a specific division of labor:

• Prediction Step: The reference dynamics model propagates the state estimate $\hat{x}^+(k)$ to $\hat{x}^-(k+1)$.
• Update Step:
  • Instead of relying solely on a physical measurement function $h(\hat{x})$, the filter incorporates the surrogate model's prediction of the measurement.
  • The EKF fuses the reference model's state prediction with the surrogate's measurement forecast to correct the state estimate.
  • This allows the system to "hallucinate" accurate measurement trends based on learned dynamics when sensors are blind, correcting the drift of the simplified reference model.

3. Key Contributions

• Decoupled Modeling Strategy: The paper introduces a novel approach where the data-driven model is trained only on measurements, while the physics model handles state propagation. This avoids the need to assign physical meaning to the surrogate's output, simplifying integration.
• Hankel-DMD for Surrogate Modeling: The application of Hankel-DMD to learn the temporal evolution of sensor data (range, azimuth, elevation) rather than state vectors, enabling effective extrapolation in partial observability scenarios.
• Consistency Analysis: The authors provide a mathematical proof showing that the Hankel-DMD operator is unbiased ($E[A] = A$) but demonstrate that the estimation error variance grows over time due to the inability to perfectly extrapolate. This highlights the necessity of periodic retraining.
• Comprehensive Validation: The framework is tested across a wide spectrum of complexities, from simple harmonic oscillators to high-fidelity orbital mechanics (ISS, Molniya orbits, and Cislunar Halo orbits).

4. Results and Case Studies

The framework was evaluated using normalized 2-norm errors against three baselines: (i) Reference model alone, (ii) The proposed DMD-EKF fusion, and (iii) A hypothetical "perfect measurement" scenario.

• Simple Pendulum: The DMD-EKF successfully tracked the damped pendulum despite the reference model ignoring damping. However, it diverged after ~200 seconds as the autoregressive model failed to capture dissipative effects, proving the limits of one-shot learning.
• International Space Station (ISS):
  • Noise-Free: The framework matched the performance of the hypothetical perfect-measurement case for various dynamics (Keplerian, $J_2$, Drag, SGP4).
  • Noisy Observations: The framework effectively filtered 5% sensor noise, keeping estimates within $3\sigma$ bounds, whereas the reference model alone diverged.
  • Nonlinear Observables: Successfully handled range, azimuth, and elevation measurements, leveraging the periodic nature of the orbit to maintain accuracy.
• Cislunar Environment (L1 Halo Orbit): Demonstrated robustness in the complex Circular Restricted Three-Body Problem (CR3BP) with 1% Gaussian noise, successfully filtering noise and maintaining stability where the reference model alone showed divergence.
• Limitations Identified:
  • Molniya-3-50: The framework failed (2 orders of magnitude worse error) because the high eccentricity and long orbital period (5 days) caused perturbations to drift the spectral content beyond the surrogate's extrapolation capability.
  • Van der Pol Oscillator: When the placeholder model's parameters (damping $\mu$) were significantly different from the true system, the fusion failed. This indicates the reference model must be a "sufficiently good" approximation of the true dynamics.

5. Significance and Conclusion

This work offers a practical solution for Space Situational Awareness where sensor data is sparse and system dynamics are complex.

• Robustness: It significantly improves state estimation accuracy compared to using simplified physics models alone, particularly in noisy or intermittent data environments.
• Flexibility: The framework is adaptable to various filtering algorithms (EKF, UKF, PF) and is not limited to orbital mechanics; it applies to any system requiring forecasting with partial observations (e.g., robotics, weather).
• Future Directions: The authors emphasize that while the current "one-shot" learning approach works for short horizons, long-term forecasting requires periodic retraining of the surrogate model as new measurements become available to correct for spectral drift and maintain consistency.

In summary, the paper validates a hybrid architecture that leverages the generalization capability of physics-based models and the pattern recognition power of data-driven surrogates to solve the inverse problem of state estimation under partial observability.