A bundle framework for observer design on smooth manifolds with symmetry

Joshi Anant A.; Maithripala D. H. S.; Banavar Ravi N.

doi:10.3934/jgm.2021015

Article Contents

2021, Volume 13, Issue 2: 247-271. Doi: 10.3934/jgm.2021015

This issue Previous Article Matched pair analysis of the Vlasov plasma Next Article Erratum: Constraint algorithm for singular field theories in the

$ k $

-cosymplectic framework

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
^* Corresponding author

Received: September 2019

Revised: June 2021

Early access: June 2021

Published: June 2021

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..

Abstract / Introduction Full Text(HTML) Figure(7) / Table(3) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93B27.

Citation:

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Figure 1. Fiber bundle, projection, base space and orbits

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Figure 2. Isotropy subgroup of $ p $, $ G_p \subset G $

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Figure 3. Section

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Figure 4. Horizontal and vertical space decomposition at any arbitrary point $ p \in P $

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Figure 5. Action of $ \gamma_{\sigma_P} $

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Figure 6. Figure for the proof of Lemma 3.1. (Arrows indicate vectors)

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Figure 7. Radar

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Table 1. Summary of Structure

$ P = \mathbb{R}^3 \setminus \{0\} $ $ G = {\mathop{\mathbb{SO}(3)}} $ $ \mathcal{Y} = \mathbb{S}^2\times \mathbb{S}^2 $

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Table 2. Summary of Structure

$ P = \mathbb{R}^3 \setminus \{0\} $ $ G = {\mathop{\mathbb{SO}(3)}} $ $ \phi(g,p) = gp $

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Table 3. Summary of Structure

$ P ={\mathop{\mathbb{SE}(3)}} \times \mathbb{E} $ $ G = {\mathop{\mathbb{SE}(3)}} $ $ \mathcal{Y} = \mathbb{E} $

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