  In this paper, we first review the theory of symmetry-preserving observers
and we mention some recent results. Then, we apply the theory to Extended
Kalman Filter-based Simultaneous Localization and Mapping (EKF SLAM). It allows
to derive a new (symmetry-preserving) Extended Kalman Filter for the non-linear
SLAM problem that possesses convergence properties. We also prove a special
choice of the gains ensures global exponential convergence.
