# Stochastic Expansions Including Random Inputs on the Unit Circle

Stochastic expansion-based methods of uncertainty quantification, such as polynomial chaos and separated representations, require basis functions orthogonal with respect to the density of random inputs. Many modern engineering problems employ stochastic circular quantities, which are defined on the unit circle in the complex plane and characterized by probability density functions on this periodic domain. Hence, stochastic expansions with circular data require corresponding orthogonal polynomials on the unit circle to allow for their use in uncertainty quantification. Rogers-Szego polynomials enable uncertainty quantification for random inputs described by the wrapped normal density. For the general case, this paper presents a framework for numerically generating orthogonal polynomials as a function of the distribution's characteristic function and demonstrates their use with the von Mises density. The resulting stochastic expansions allow for estimating statistics describing the posterior density using the expansion coefficients. Results demonstrate the exponential convergence of these stochastic expansions and apply the proposed methods to propagating orbit-state uncertainty with equinoctial elements. The astrodynamics application of the theory improves robustness and accuracy when compared to approximating angular quantities as variables on the real line.

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