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
The Unified State Model is a method for expressing orbits using a set of seven elements. The elements consist of a quaternion and three parameters based on the velocity hodograph. A complete derivation of the original model is given in addition to two proposed modifications. Both modifications reduce the number of state elements from seven to six by replacing the quaternion with either modified Rodrigues parameters or the Exponential Map. Numerical simulations comparing the original Unified State Model, the Unified State Model with modified Rodrigues parameters, and the Unified State Model with Exponential Map, with the traditional Cartesian coordinates have been carried out. The Unified State Model and its derivatives outperform the Cartesian coordinates for all orbit cases in terms of accuracy and computational speed, except for highly eccentric perturbed orbits. The performance of the Unified State Model is exceptionally better for the case of orbits with continuous lowthrust propulsion with CPU simulation time being an order of magnitude lower than for the simulation using Cartesian coordinates. This makes the Unified State Model an excellent state propagator for mission optimizations.
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