# An Equivariant Observer Design for Visual Localisation and Mapping

## TL;DR

This paper introduces a novel equivariant observer framework for visual SLAM that unifies pose and landmark estimation, ensuring stability and robustness in non-linear visual localization tasks.

## Contribution

It formulates visual SLAM as a continuous-time equivariant observer on a symmetry group, providing a new stability-guaranteed approach for pose and landmark estimation.

## Key findings

- Observer error system is almost globally asymptotically stable.
- Exponential stability in the large is achieved.
- Decoupled Riccati-gains improve landmark estimation stability.

## Abstract

This paper builds on recent work on Simultaneous Localisation and Mapping (SLAM) in the non-linear observer community, by framing the visual localisation and mapping problem as a continuous-time equivariant observer design problem on the symmetry group of a kinematic system. The state-space is a quotient of the robot pose expressed on SE(3) and multiple copies of real projective space, used to represent both points in space and bearings in a single unified framework. An observer with decoupled Riccati-gains for each landmark is derived and we show that its error system is almost globally asymptotically stable and exponentially stable in-the-large.