Closed-form Posterior Cramér-Rao Bound for a Manoeuvring Target in the Bearings-Only Tracking Context Using Best-Fitting Gaussian Distribution

In this paper, we investigate the problem of the computation of the posterior Cramer-Rao bound (PCRB) in the context of bearings-only tracking (BOT) for a manoeuvring target. The PCRB provides a lower bound on the mean square error. In a recent paper, Hernandez et al have proposed a new approach named best-fitting Gaussian (BFG) model to calculate the bound for jump Markov linear filtering problems with a linear measurement equation. Thanks to the linear property of the measurement equation, an exact formula for the PCRB associated to the BFG model can be obtained via a classical Riccati-like recursion. However, in the BOT framework, the measurement equation is non linear so that we do not have a closed-form formula. Consequently, the BFG-PCRB must be approximated using Monte-Carlo methods. This implies a high computational burden. We show in this paper that the BFG model associated to the BOT problem can be computed exactly using another coordinate system named log polar coordinate (LPC) system