Uncertainty Estimation in Orbit Determination: A Comparison of Machine Learning, Monte Carlo and Least Squares Approaches

Abstract

The field of Space Situational Awareness (SSA) has become increasingly important in recent years due to the rapid rise in active satellites and the accumulation of space debris in Earth orbit. Accurate orbit determination (OD) and, more importantly, reliable estimates of uncertainty are essential for planning collision avoidance manoeuvres and preserving a safe orbital environment. Over time, machine learning (ML) has also seen increasing use in this area, as its algorithms hold the potential to improve classical OD methods by leveraging measurement data.

Scorsoglio et al. (2023) demonstrated that a specialized type of neural network, known as a Physics-Informed Extreme Learning Machine (PIELM), can perform rapid orbit determination without requiring an initial guess of the state vector. By incorporating the governing differential equations, PIELMs reduce the “black box” nature typically associated with standard neural networks. However, estimating realistic prediction uncertainties remains an open challenge for nonlinear systems, particularly in contexts where Bayesian approaches cannot be directly applied.

In this study, we investigate and compare uncertainty quantification methods in orbit determination by analysing the behaviour of the covariance matrix across different estimation frameworks. Specifically, we examine the classical covariance propagation using the state transition matrix as used in the weighted least squares (WLS) method, a Monte Carlo simulation-based approach employing a standard orbital propagator, and strategies to assess the uncertainty associated with OD results obtained via a PIELM. The comparative analysis aims to assess the fidelity and characteristics of uncertainty estimates produced by each method. All computations are carried out within the AI4POD (Artificial Intelligence for Precise Orbit Determination) framework.

Acknowledgements

The project Artificial Intelligence for Precise Orbit Determination (AI4POD) is funded by Deutsches Zentrum für Luft- und Raumfahrt, Bonn-Oberkassel, under grant 50LZ2308.

References

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[2] Scorsoglio, A., Ghilardi, L. & Furfaro, R. A Physic-Informed Neural Network Approach to Orbit Determination. J Astronaut Sci 70, 25 (2023). https://doi.org/10.1007/s40295-023-00392-w

[3] Liu, Xu, Wen Yao, Wei Peng, und Weien Zhou. „Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data“. Neurocomputing 549 (September 2023): 126425. https://doi.org/10.1016/j.neucom.2023.126425.

[4]  Aigner, B., Dallinger, F., Andert, T., and Pätzold, M.: Integrating Machine Learning algorithms into Orbit Determination: The AI4POD Framework, Europlanet Science Congress 2024, Berlin, Germany, 8–13 Sep 2024, EPSC2024-521, https://doi.org/10.5194/epsc2024-521, 2024.

[5]  Dallinger, F., Aigner, B., Andert, T., and Pätzold, M.: Physics Informed Neural Networks as addition to classical Precise Orbit Determination, Europlanet Science Congress 2024, Berlin, Germany, 8–13 Sep 2024, EPSC2024-514, https://doi.org/10.5194/epsc2024-514, 2024.