  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 (IEKF). Then, under a
simple compatibility assumption between the error and the transformation group,
we prove 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.
