Bayesian Cramér-Rao Lower Bounds for Magnetic Field-based Train Localization

In this paper, the theoretically achievable accuracy of magnetic field-based localization in railway environments is analyzed. The analysis is based on the Bayesian Cramér-Rao lower bound (BCRLB) that bounds the mean squared error of an estimator from below. The derivation of the BCRLB for magnetic field-based localization is not straightforward because the magnetic field cannot be described by an analytical equation but must be derived from measurements. In this paper we show how the BCRLB can be calculated by fitting a Gaussian process (GP) to magnetometer measurements to obtain an analytical expression of the magnetic field along a railway line. The proposed GP-based BCRLB is evaluated with the magnetic field of a 1 km long track segment. Furthermore, a comparison between the bound and the estimation error of a particle filter shows the sub-optimality of the particle filter for magnetic railway localization.