Nonlinear Uncertainty Propagation for Perturbed Two-Body Orbits

The main objective of this paper is to present the development of the computational methodology, based on the Gaussian mixture model, that enables accurate propagation of the probability density function through the mathematical models for orbit propagation. The key idea is to approximate the density function associated with orbit states by a sum of Gaussian kernels. The unscented transformation is used to propagate each Gaussian kernel locally through nonlinear orbit dynamical models. Furthermore, a convex optimization problem is formulated by forcing the Gaussian mixture model approximation to satisfy the Kolmogorov equation at every time instant to solve for the amplitudes of Gaussian kernels. Finally, a Bayesian framework is used on the Gaussian mixture model to assimilate observational data with model forecasts. This methodology effectively decouples a large uncertainty propagation problem into many small problems. A major advantage of the proposed approach is that it does not require the knowledge of system dynamics and the measurement model explicitly. The simulation results are presented to illustrate the effectiveness of the proposed ideas.