An approach for nonlinear uncertainty propagation: Application to orbital mechanics

An approach for nonlinear propagation of orbit uncertainties is discussed while making use of the Fokker-Planck-Kolmogorov Equation (FPKE). The central idea is to replace evolution of initial conditions for a dynamical system with the evolution of a probability density function (pdf) for state variables. The transition pdf corresponding to dynamical system state vector is approximated by using a finite Gaussian mixture model. The mean and covariance of different components of the Gaussian mixture model are propagated through the use of an Unscented Kalman Filter (UKF). Furthermore, the unknown amplitudes corresponding to different components of the Gaussian mixture model are found by minimizing the FPKE error over the entire volume of interest. This leads to a convex quadratic minimization problem guaranteed to have a unique solution. The two-body problem model with non-conservative atmospheric drag forces and initial uncertainty will be used to show the efficacy of the ideas developed in this paper.