Closed-form Posterior Cramér-Rao Bound for Active Measurement Scheduling

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, the PCRB is given by the inverse Fisher Information Matrix (FIM). The latter is computed using Tichavský’s recursive formula via Monte Carlo methods. Recently an exact algorithm to compute the PCRB via Tichavský’s recursive formula without using Monte-Carlo methods have been derived by Bréhard et al. This result is based on a new coordinates system named Logarithmic Polar Coordinates (LPC) system. This paper illustrates that PCRB can now be computed accurately and quickly, making it suitable for active measurement scheduling. NOTATION Xt: is the target state in the Cartesian coordinates system, Yt: is the target state in the LPC system, ny: size of the target state (ny = 4), <: inequality R < S means that R − S is a positive semidefinite matrix, Idn: n × n identity matrix, 0n×m : n × m matrix composed of zero element, ⊗: Kronecker product, X: denotes the transpose of matrix X . δ: Dirac delta function, ∆: Laplacian operator, ∇: gradient operator,