Initialization of Particle Filter and Posterior Cramér-Rao Bound for Bearings-Only Tracking in Modified Polar Coordinate System

We here address the classical bearings-only tracking problem (BOT) for a single object, which belongs to the general class of non linear filtering problems. Recently, algorithms based on sequential Monte Carlo methods (particle filtering) have been proposed. However, initializing particle filtering is often the main difficulty, especially if the state is only partially observed (BOT). To remedy for this problem, the problem is immersed in a modified polar coordinate (MP) framework. This approach leads us to consider an original formulation of the BOT problem within the MP system. In particular, it is shown that this problem is relevant to a more general class of problems: non-linear filtering with unknown state covariance. Inside this particular framework, particle filters can be quite convinently initialized by using only observed bearings (optimization problem). The whole algorithm performs quite satisfactorily, avoiding the need of a strong prior about target location and velocity. Simulation results illustrate the benefits of this approach. The Posterior Cramér-Rao Bound (PCRB) provides a lower bound on the mean square error. Original PCRB approximations for the «partial» state target (the observable components) are derived. It is well-known that the "usual" PCRB is (very) over-optimistic. Relaxing the asymptotic unbiasness hypothesis, a new bound is derived, both for partial or complete state vectors, which presents a good agreement with estimated MSE from simulated data