# Convergence and Consistency Analysis for A 3D Invariant-EKF SLAM

## TL;DR

This paper analyzes the convergence and invariance properties of an RI-EKF SLAM algorithm, proving its convergence without restrictive assumptions and demonstrating its superior performance over other EKF variants in 3D SLAM tasks.

## Contribution

It provides the first convergence proof for RI-EKF SLAM without assuming Jacobian evaluation at ground truth and highlights its invariance and improved consistency.

## Key findings

- RI-EKF is invariant under stochastic rigid body transformations.
- Proven convergence of RI-EKF without restrictive Jacobian assumptions.
- Simulation shows RI-EKF outperforms other EKF-based SLAM methods.

## Abstract

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.