  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.
