Modern astronautics problems require methods of tractable uncertainty quantification for problems with data in a variety of mathematical spaces. Polynomial Chaos Expansions (PCEs) enable tractable uncertainty propagation, sensitivity analysis, and provide a surrogate model to efficiently solve stochastic optimization problems. Existing PCE methods are mostly isolated to bases defined via tensor products of univariate functions over subdomains of the real line or integers. The goal of this work is to incorporate random vectors on the n-dimensional sphere, thereby extending the use of PCEs to problems that include spacecraft attitude uncertainty. Random inputs with probability densities on the n-sphere are generally correlated. When directional random inputs are independent, products of univariate functions fail to produce an orthogonal basis. Basis functions must preserve the periodic response of the system produced by the underlying structure of the domain. This paper presents an approach to generating an orthogonal basis with respect to a density function on the n-sphere by combining hyperspherical harmonics with an orthogonalization procedure based on the raw moments of the harmonic functions. For highly concentrated densities on the unit sphere, the spherical cap harmonics provide a more numerically stable solution while preserving periodicity. Numeric errors in the proposed procedure are presented for multiple cases. Performance of the PCEs is quantified when propagating uncertainty for a highly eccentric orbit with a random translational maneuver error, and a second case based on rigid-body dynamics with the attitude state parameterized as a quaternion.
Incorporating Directional Uncertainties into Polynomial Chaos Expansions for Astronautics Problems
J Astronaut Sci
2023-06-21
Article (Journal)
Electronic Resource
English
Uncertainty quantification , Directional statistics , Polynomial chaos , Maneuver execution errors , Attitude propagation Engineering , Aerospace Technology and Astronautics , Mathematical Applications in the Physical Sciences , Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics)
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