Open Access

ARTICLE

Orthogonal Probability Approximation for Highly Accurate and Efficient Orbit Uncertainty Propagation

Pugazhenthi Sivasankar^1,*, Austin B. Probe², Tarek A. Elgohary¹
1 Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA
2 Emergent Space Technologies, Austin, TX 78752, USA
* Corresponding Author: Pugazhenthi Sivasankar. Email:
(This article belongs to the Special Issue: Advances in Methods of Computational Modeling in Engineering Sciences, a Special Issue in Memory of Professor Satya Atluri)

Digital Engineering and Digital Twin https://doi.org/10.32604/dedt.2024.052805

Received 16 April 2024; Accepted 30 September 2024; Published online 22 October 2024

Abstract

In Space Situational Awareness (SSA), accurate and efficient uncertainty quantification and propagation are essential for various applications, such as conjunction analysis, track correlation, and orbit prediction. The propagation of the probability density function (PDF) in nonlinear systems results in non-Gaussian distributions, which are difficult to approximate. Furthermore, the computational cost of approximating the PDF increases exponentially with the number of random variables, a phenomenon known as the curse of dimensionality. To address these challenges, the Orthogonal Probability Approximation (OPA) method is presented for high-fidelity uncertainty propagation and PDF approximation in nonlinear dynamical systems. The method leverages Liouville’s theorem, sampled PDF values at specific evaluation nodes, and Chebyshev polynomial basis functions to approximate the PDF at the desired time of interest. A grid filtering approach is employed to enhance computational efficiency, and scaling of the PDF values is utilized for non-conservative systems. Numerical results are presented for a linear oscillator, a Duffing oscillator with and without damping, and a planar orbit problem. Validation is performed using linear error propagation theory and Monte Carlo simulations. The results demonstrate that OPA is at least eight times more computationally efficient in approximating the PDF of dynamical systems compared to conventional Monte Carlo methods.

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Keywords

Numerical approximation; uncertainty propagation; astrodynamics; Chebyshev polynomials

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