Exploiting Symmetries to Design EKFs With Consistency Properties for Navigation and SLAM

The extended Kalman filter (EKF) is both the historical algorithm for multi-sensor fusion and still state of the art in numerous industrial applications. However, it may prove inconsistent in the presence of unobservability under a group of transformations. In this paper, we first build an alternative EKF based on an alternative nonlinear state error. This EKF is intimately related to the theory of the invariant EKF. Then, under a simple compatibility assumption between the error and the transformation group, we prove that the linearized model of the alternative EKF automatically captures the unobservable directions, and many desirable properties of the linear case then directly follow. This provides a novel fundamental result in filtering theory. We apply the theory to multi-sensor fusion for navigation, when all the sensors are attached to the vehicle and do not have access to absolute information, as typically occurs in GPS-denied environments. In the context of simultaneous localization and mapping (SLAM), Monte Carlo runs and comparisons to OC-EKF, robocentric EKF, and optimization-based smoothing algorithms (iSAM) illustrate the results. The proposed EKF is also proved to outperform standard EKF and to achieve comparable performance to iSAM on a publicly available real dataset for multi-robot SLAM.