Satellite tracking using a second-order stochastic nonlinear filter

Tarun Kumar Rawat^* and Harish Parthasarathy

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodelled stochastic acceleration, is not assumed in the particle dynamics. The small, unmodelled stochastic acceleration produces an additional random force on a particle. Estimation algorithms of a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. In particular, this paper examines the effect of the stochastic acceleration on the motion of the orbiting satellite, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means, and conditional variances for estimating the state of the satellite. By linearizing the stochastic differential equations about the mean of the state vector using first-order approximation, the mean trajectory of the resulting first-order approximated stochastic differential model does not preserve the perturbation effect felt by the orbiting satellite; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the perturbed model is examined on the basis of the second-order approximations of the system nonlinearity. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation `between the observation' and a functional difference equation for the conditional probability density `at the observation'. The effectiveness of the second-order nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting satellite and the signal-noise ratio. The Kolmogorov forward equation, however, is not appropriate for numerical simulations, since it is the equation for the evolution of the conditional probability density. Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting satellite. Even these equations are not appropriate for the numerical simulations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher-order moments. Hence we consider the approximations to these moment evolution equations. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.

Keywords: Satellite tracking, stochastic differential equation, Brownian-motion process, Fokker-Planck Kolmogorov equation, mean, variance.