  In this paper, we investigate the convergence and consistency properties of
an Invariant-Extended Kalman Filter (RI-EKF) based Simultaneous Localization
and Mapping (SLAM) algorithm. Basic convergence properties of this algorithm
are proven. These proofs do not require the restrictive assumption that the
Jacobians of the motion and observation models need to be evaluated at the
ground truth. It is also shown that the output of RI-EKF is invariant under any
stochastic rigid body transformation in contrast to $\mathbb{SO}(3)$ based EKF
SLAM algorithm ($\mathbb{SO}(3)$-EKF) that is only invariant under
deterministic rigid body transformation. Implications of these invariance
properties on the consistency of the estimator are also discussed. Monte Carlo
simulation results demonstrate that RI-EKF outperforms $\mathbb{SO}(3)$-EKF,
Robocentric-EKF and the "First Estimates Jacobian" EKF, for 3D point feature
based SLAM.
