Trigonometric Moments of a Generalized von Mises Distribution in 2-D Range-Only Tracking

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.