Spacecraft Uncertainty Propagation Using Gaussian Mixture Models and Polynomial Chaos Expansions

Abstract

Polynomial Chaos Expansion (PCE) and Gaussian Mixture Models (GMMs) are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. PCE models uncertainty by performing an expansion using orthogonal polynomials (OPs). The accuracy of PCE for a given problem can be improved by increasing the order of the OP expansion. The number of terms in the OP expansion increases factorially with dimensionality of the problem, thereby reducing the effectiveness of the PCE approach for problems of moderately high dimensionality. This paper shows a combination of GMM and PCE, GMM-PC as an alternative form of the multi-element PCE. GMM-PC reduces the overall order required to reach a desired accuracy. The initial distribution is converted to a GMM, and PCE is used to propagate the state uncertainty represented by each of the elements through the nonlinear dynamics. Splitting the initial distribution into a GMM 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 GMM-PC. The resulting distributions are shown to

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