Alternating Direction Implicit Method-Based Nonlinear Filtering for Relative Orbit Estimation

Abstract

Nonlinear filtering theory, based upon the Fokker-Planck equation and Bayes’ rule, has been around since the 1970s. This method does not require linearized dynamics and/or measurement models, and also the assumption of the Gaussian process is not necessary. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with only a few exceptions, and normally the solution is obtained through numerical approaches. In this paper, this type of nonlinear filtering method is reinvestigated and applied to a relative orbit estimation problem. The time evolution of the state conditional probability density function between measurements is obtained by solving the associated Fokker-Planck equation numerically using the numerical scheme called the alternating direction implicit method. To further mitigate the computational load, an enhanced moving domain technique is employed and the probability distribution function is partitioned.