# A bundle framework for observer design on smooth manifolds with symmetry

## TL;DR

This paper introduces a geometric bundle framework for nonlinear observer design on manifolds with symmetry, decomposing the problem into orbit and quotient space designs, with applications to Lie groups and SLAM.

## Contribution

It develops a unified geometric approach for observer design on manifolds with symmetry, especially focusing on free group actions and gradient-based methods on Lie groups.

## Key findings

- Framework applied to SO(3) actions on R^3
- Decomposition simplifies observer design tasks
- Illustrated with SLAM and well-known examples

## Abstract

The article presents a bundle framework for nonlinear observer design on a manifold with a Lie group action. The group action on the manifold decomposes the manifold to a quotient structure and an orbit space, and the problem of observer design for the entire system gets decomposed to a design over the orbit (the group space) and a design over the quotient space. The emphasis throughout the article is on presenting an overarching geometric structure; the special case when the group action is free is given special emphasis. Gradient based observer design on a Lie group is given explicit attention. The concepts developed are illustrated by applying them on well known examples, which include the action of $\mathbb{SO}(3)$ on $\mathbb{R}^3 \setminus \{0\}$ and the simultaneous localisation and mapping (SLAM) problem.