Relative Position Estimation Using Fokker-Planck and Bayes' Formula

Abstract

The nonlinear filtering theory using the Fokker-Planck equation and Bayes’ rule has been around since 1970s. Theoretically, this type of estimation method outperforms the widely used Extended Kalman filters because the method does not require linearized dynamics and 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 FokkerPlanck equation with the exception of a few special cases, and they usually have to be evaluated by numerical methods. In this paper the Fokker-Planck equation and Bayes’ rule based estimation method is investigated and applied to a relative orbital position estimation problem. The time evolution of the state probability density function between measurements is obtained by solving the Fokker-Planck equation numerically using the conventional finite difference method. The Bayes’ rule is also numerically evaluated to update the state probability density function obtained from the Fokker-Planck equation according to measurements. So far this estimation method has not been employed much in practice because of the high computational cost needed in solving the Fokker-Planck equation numerically. To decrease computational cost the moving domain concept is employed to reduce the domain of integration while solving the Fokker-Planck equation. Simulation results show that the accuracy of the estimation is improved as compared with the Extended Kalman filter and the computational cost is significantly lower with the propose moving grid scheme than without it.