Closed-form Posterior Cramér-rao Bound for Bearings-only Tracking Thomas Bréhard and Jean-pierre Le Cadre

We here address the classical bearings-only tracking problem (BOT) for a single object, which belongs to the general class of nonlinear filtering problems. Recently, algorithms based on sequential Monte Carlo methods (particle filtering) have been proposed. As far as performance analysis is concerned, the Posterior Cramér-Rao Bound (PCRB) provides a lower bound on the mean square error. Classically, under a technical assumption named ”asymptotic unbiasedness assumption”, the PCRB is given by the inverse Fisher Information Matrix (FIM). The latter is computed using Tichavský’s recursive formula via Monte Carlo methods. In this paper, two major problems are studied. First, we show that the ”asymptotic unbiasedness assumption” can be replaced by an assumption which is more meaningful. Second, an exact algorithm to compute the PCRB is derived via Tichavský’s recursive formula without using Monte-Carlo methods. This result is based on a new coordinates system named Logarithmic Polar Coordinates (LPC) system. Simulation results illustrate that PCRB can now be computed accurately and quickly, making it suitable for sensor management applications. Key-words: bearings-only tracking, sequential Monte Carlo methods, posterior Cramér-Rao bound, performance analysis, sensor management.