Spacecraft Uncertainty Propagation Using Gaussian Mixture Models and Polynomial Chaos Expansions

Polynomial chaos expansion and Gaussian mixture models are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. Polynomial chaos expansion models uncertainty by performing an expansion using orthogonal polynomials. The accuracy of polynomial chaos expansion for a given problem can be improved by increasing the order of the orthogonal polynomial expansion. The number of terms in the orthogonal polynomial expansion increases factorially with dimensionality of the problem, thereby reducing the effectiveness of the polynomial chaos expansion approach for problems of moderately high dimensionality. This paper shows a combination of Gaussian mixture model and polynomial chaos expansion, Gaussian mixture model–polynomial chaos expansion as an alternative form of the multi-element polynomial chaos expansion. Gaussian mixture model–polynomial chaos expansion reduces the overall order required to reach a desired accuracy. The initial distribution is converted to a Gaussian mixture model, and polynomial chaos expansion is used to propagate the state uncertainty represented by each of the elements through the nonlinear dynamics. Splitting the initial distribution into a Gaussian mixture model reduces the size of the covariance associated with each new element, thereby reducing the domain of approximation and allowing for lower-order polynomials to be used. Several spacecraft uncertainty propagation examples are shown using Gaussian mixture model–polynomial chaos expansion. The resulting distributions are shown to efficiently capture the full shape of the true non-Gaussian distribution.