An homogeneous space geometry for simultaneous localisation and mapping

Simultaneous Localisation and Mapping (SLAM) is the archetypal chicken and egg problem: Localisation of a robot with respect to a map requires an estimate of the map, while mapping an environment from data acquired by a robot requires an estimate of the robot localisation. The nonlinearity and co-dependence of the SLAM problem has made it an ongoing research problem for more than thirty years. The present paper details recent advances in understanding the SLAM problem, specifically the existence of an underlying geometry and symmetry structure that provides significant insight into the difficulties that have plagued many SLAM algorithms. To demonstrate the power of the geometric insight we derive a constant gain observer for the SLAM problem that; that does not depend on linearisation, has globally asymptotically stable error dynamics, is very robust, and operates in dynamic environments (estimating the landmark velocities as states in the observer).