Graduation Semester and Year
Spring 2026
Language
English
Document Type
Thesis
Degree Name
Master of Science in Aerospace Engineering
Department
Mechanical and Aerospace Engineering
First Advisor
Kamesh Subbarao
Second Advisor
Julio César Benavides
Third Advisor
Animesh Chakravarthy
Abstract
Uncertainties, that are inherent to dynamic models, can be associated with state initial conditions, force modelling errors, navigation and actuation errors. In system modelling stochastic differential equations are used to represent dynamic phenomena with uncertainties, for which the solutions are probability density functions of quantities of interest characterizing the realization of the stochastic processes. In Polynomial Chaos Expansion (PCE) propagation, these solutions are represented as weighted sums of multivariate spectral polynomials that are functions of the input random variables. Generalized polynomial chaos expansion (gPC) is an extension to the original homogenous PCE which projects the random solution onto a basis of polynomials which are orthogonal to the probability density function of the input random variables, reducing the problem of finding the solutions of SDEs to computing the coefficients of the expansion. This problem can be solved by applying a class of non-intrusive techniques known as stochastic collocation methods which combines the strength of sampling-based Monte Carlo technique and intrusive Galerkin methods and thus can be used as a ‘black box’ simulator. This approach, however, poses a major challenge since it requires implementation of an efficient quadrature technique to obtain the collocation nodes from a joint PDF representing the initial mixed distributions of state and parametric uncertainties. Several previously developed quadrature techniques suffer from either the curse of dimensionality (e.g. Gaussian quadrature tensor grid rules) or being incapable of handling mixed distribution types or being intrusive. For this purpose, a mixed sparse grid quadrature rule is adopted that carries out the problem non-intrusively by the means of repetitive simulation generated from random sampling as done so in Monte Carlo or Latin hypercube and can incorporate independent random variables governed by different distribution types. This efficient quadrature technique, when incorporated into the gPC framework, drastically reduces the computation cost and increases efficiency in the nonlinear propagation of orbital uncertainty.
The propagated uncertainty is then used to analyze close conjunction events between earth bound satellites that are subject to state and drag related mixed (Gaussian normal and uniform) uncertainties at the orbit determination (OD) epoch. Orbital conjunction probability is computed using semi-analytical 2-dimensional conjunction plane analysis techniques and sampling-based MC techniques for various stochastic initial conditions at the OD. For all cases, MSG-gPC based semi-analytical conjunction probability results show close agreement with the Monte Carlo reference truth. Finally, the MSG-gPC propagation-based conjunction probability computation algorithm has been tested using two archived conjunction events obtained from NASA’s Conjunction Assessment and Risk Analysis (CARA) software repository and verified.
Keywords
Uncertainty Quantification, Orbital Mechanics, Optimal Estimation Techniques, Polynomial Chaos, Orbital Debris Environment, Conjunction Assessment, Stochastic Differential Equations, Collision Probability Estimation, Monte Carlo Simulation, Covariance Propagation Techniques
Disciplines
Aerospace Engineering | Applied Mathematics | Astrophysics and Astronomy | Aviation | Mathematics | Operations Research, Systems Engineering and Industrial Engineering | Physics | Statistics and Probability
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Karim, Monalisa, "Uncertainty Quantification, Propagation & Conjunction Assessment in Orbital Mechanics using Generalized Polynomial Chaos Expansion & 2-Dimensional Conjunction Plane Analysis Techniques" (2026). Mechanical and Aerospace Engineering Theses. 4.
https://mavmatrix.uta.edu/mechaerospace_theses2/4
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Aerospace Engineering Commons, Applied Mathematics Commons, Astrophysics and Astronomy Commons, Aviation Commons, Mathematics Commons, Operations Research, Systems Engineering and Industrial Engineering Commons, Physics Commons, Statistics and Probability Commons