# American Institute of Mathematical Sciences

June  2021, 13(2): 247-271. doi: 10.3934/jgm.2021015

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

 1 Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra, 400076, India 2 Department of Mechanical Engineering, University of Peradeniya, KY20400, Sri Lanka 3 School of Postgraduate Studies, Sri Lanka Technological Campus, Padukka, CO 10500, Sri Lanka 4 Systems and Control Engineering Group, Indian Institute of Technology Bombay, Mumbai, Maharashtra, 400076, India

* Corresponding author

Received  September 2019 Revised  June 2021 Published  June 2021 Early access  June 2021

Fund Project: The authors would like to thank the Indian Institute of Technology Bombay and the Sri Lanka Technological Campus, Padukka, for their support both logistical and financial.

The article presents a bundle framework for nonlinear observer design on a manifold with a 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 ${\mathop{\mathbb{SO}(3)}}$ on $\mathbb{R}^3 \setminus \{0\}$ and the simultaneous localisation and mapping (SLAM) problem.

Citation: Anant A. Joshi, D. H. S. Maithripala, Ravi N. Banavar. A bundle framework for observer design on smooth manifolds with symmetry. Journal of Geometric Mechanics, 2021, 13 (2) : 247-271. doi: 10.3934/jgm.2021015
##### References:

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##### References:
Fiber bundle, projection, base space and orbits
Isotropy subgroup of $p$, $G_p \subset G$
Section
Horizontal and vertical space decomposition at any arbitrary point $p \in P$
Action of $\gamma_{\sigma_P}$
Figure for the proof of Lemma 3.1. (Arrows indicate vectors)
Summary of Structure
 $P = \mathbb{R}^3 \setminus \{0\}$ $G = {\mathop{\mathbb{SO}(3)}}$ $\mathcal{Y} = \mathbb{S}^2\times \mathbb{S}^2$
 $P = \mathbb{R}^3 \setminus \{0\}$ $G = {\mathop{\mathbb{SO}(3)}}$ $\mathcal{Y} = \mathbb{S}^2\times \mathbb{S}^2$
Summary of Structure
 $P = \mathbb{R}^3 \setminus \{0\}$ $G = {\mathop{\mathbb{SO}(3)}}$ $\phi(g,p) = gp$
 $P = \mathbb{R}^3 \setminus \{0\}$ $G = {\mathop{\mathbb{SO}(3)}}$ $\phi(g,p) = gp$
Summary of Structure
 $P ={\mathop{\mathbb{SE}(3)}} \times \mathbb{E}$ $G = {\mathop{\mathbb{SE}(3)}}$ $\mathcal{Y} = \mathbb{E}$
 $P ={\mathop{\mathbb{SE}(3)}} \times \mathbb{E}$ $G = {\mathop{\mathbb{SE}(3)}}$ $\mathcal{Y} = \mathbb{E}$
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