  This paper first formalises a new observability concept, called weak regular
observability, that is adapted to Fast Moving Horizon Estimation where one aims
to estimate the state of a nonlinear system efficiently on rolling time windows
in the case of small initial error. Additionally, sufficient conditions of weak
regular observability are provided in a problem of Simultaneous Localisation
and Mapping (SLAM) for different measurement models. In particular it is shown
that following circular trajectories leads to weak regular observability in a
second order 2D SLAM problem with several possible types of sensors.
