Posterior Cramér-Rao bound and suboptimal filtering for IMU/GNSS based cooperative train localization

The optimal estimator for the hidden state of nonlinear systems is often not known or it is computational unfeasible. In this situation suboptimal algorithms must be used. An important performance metric for these algorithms is the difference of their root mean square error (RMSE) compared to the RMSE of the optimal estimator. If the optimal estimator is unknown it is useful to have a lower bound for the RMSE. Such a bound is defined by the posterior Cramér-Rao lower bound (PCRB) which is also valid for biased estimators. In this paper a version of the PCRB for nonlinear systems considering known inputs is applied to analyze the performance of an extended Kalman filter (EKF) and unscented Kalman filter (UKF) for GNSS/IMU based cooperative train localization. The analysis is realized by performing a simulation study for different track and satellite geometries. After an initial phase with larger errors both, the EKF and UKF are able to estimate the bias on the pseudoranges and the receiver clocks and they attain the PCRB.