An adaptive Gaussian sum filtering approach for orbit uncertainty estimation

An approach for nonlinear filtering when measurements are sparse is discussed which makes use of the Fokker-Planck-Kolmogorov Equation (FPKE). The central idea is to replace the evolution of initial state estimates for a dynamical system with the evolution of a probability density function (pdf) for state variables. The transition pdf corresponding to a dynamical system state vector is approximated by using a finite Gaussian mixture model. The mean and covariance of each Gaussian mixture model component are propagated through the use of an Unscented Kalman Filter (UKF), and the unknown amplitudes 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 with non-conservative atmospheric drag forces and initial condition uncertainty will be used to show the efficacy of the ideas developed in this paper.