**1. Introduction**

Engineering is an imperfect science. Noisy measurements from sensors in state estimation [1,2], a constantly changing environment in guidance [3–5], and improperly actuated controls [6] are all major sources of error. The more these sources of error are understood, the better the final product will be. Ideally, every variable with some sort of uncertainty associated with it would be completely and analytically described with its probability density function (pdf). Unfortunately, even if this is feasible for the initialization of a random variable, its evolution through time rarely yields a pdf with an analytic form. If the pdf cannot be given in analytic form, then approximations and assumptions must be made.

In many cases, a random variable is quantified using only its first two moments—as with the unscented transform [7]—and a further assumption is that the distribution is Gaussian. In cases where the variable's uncertainty is relatively small and the dynamics governing its evolution are not highly nonlinear, this is not necessarily a poor assumption. In these cases, the higher order moments are highly dependent on the first two moments; i.e., there is a minimal amount of unique information in the higher order moments. In contrast, if either the uncertainty is large or the dynamics become highly nonlinear, the higher order moments become less dependent on the first two moments and contain larger amounts of unique information. In this case, the error associated with using only the first two moments becomes significant [8,9].

One method of quantifying uncertainty that does not require an assumption of the random variable's pdf is polynomial chaos expansion (PCE) [10–14]. PCE characterizes a random variable as a coordinate in a polynomial vector space. Useful deterministic information about the random variable lies in this coordinate, including the moments of the random variable [15]. The expression of the coordinate depends on the basis in which it is expressed. In the case of PCE, the bases are made up of polynomials that are chosen based on the assumed density of the random variable; however, any random variable can be represented using any basis [16]. It is strongly noted that assuming the density of the random variable simply eases computation; with enough computing power, any random variable can be quantified with any basis [17]. The most common basis polynomials are those that are orthogonal with respect to common pdfs, such as the Hermite-Gaussian and Jacobi-beta polynomial-pdf pairs.

Polynomial chaos has been used to quantify and propagate the uncertainty in nonlinear systems including fluid dynamics [18–21], orbital mechanics in multiple element spaces [22], and has been expanded to facilitate Gaussian mixture models [23]. While polynomial chaos has been well-studied for variables that exist in the *n*-dimensional space of real numbers ( R*n*), many variables do not lie in this field. The variables that are of concern in this paper are angular variables. When estimating an angular random variable, such as the true anomaly of a body in orbit or vehicle attitude, a common approach is to assume that the variable is approximately equal to its projection on R*n* and use methods designed for variables on that space. When the uncertainty is very small, this approximation is relatively effective; however, as the uncertainty increases, this approximation becomes invalid. Recently, work on directional statistics has been incorporated into state estimation, providing a method for estimating angular random variables directly on the *n*-dimensional special orthogonal group (S*n*) [24–26]. Recent work [27] has shown that polynomial chaos can be used to estimate the mean and variance of one-dimensional angular random variables and the set of equinoctial orbital elements elements [28]. In neither case was a correlation estimated that included an angular state.

The sections of this paper include a detailed discussion of orthogonal polynomials in Section 2, including those that are orthogonal on the unit circle and two of the more common circular pdfs. In Section 3 an overview of polynomial chaos is given as well as the extension that has been made to include angular random variables, including the correlation between two angles. Finally, in Section 4, numerical results are presented comparing the correlation between two angular random variables calculated using Monte Carlo and PCE.
