Trigonometric Moments of a Generalized von Mises Distribution in 2-D Range-Only Tracking
Abstract
A 2D range-only tracking scenario is non-trivial due to two main reasons.
First, when the states to be estimated are in Cartesian coordinates, the
uncertainty region is multi-modal. The second reason is that the probability
density function of azimuth conditioned on range takes the form of a
generalized von Mises distribution, which is hard to tackle. Even in the case
of implementing a uni-modal Kalman filter, one needs expectations of
trigonometric functions of conditional bearing density, which are not available
in the current literature. We prove that the trigonometric moments (circular
moments) of the azimuth density conditioned on range can be computed as an
infinite series, which can be sufficiently approximated by relatively few terms
in summation. The solution can also be generalized to any order of the moments.
This important result can provide an accurate depiction of the conditional
azimuth density in 2D range-only tracking geometries. We also present a simple
optimization problem that results in deterministic samples of conditional
azimuth density from the knowledge of its circular moments leading to an
accurate filtering solution. The results are shown in a two-dimensional
simulation, where the range-only sensor platform maneuvers to make the system
observable. The results prove that the method is feasible in such applications.