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

• * Corresponding author

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.

Mathematics Subject Classification: Primary: 93B27.

 Citation: • • Figure 1.  Fiber bundle, projection, base space and orbits

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

Figure 3.  Section

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

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

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

Table 1.  Summary of Structure

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

Table 2.  Summary of Structure

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

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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Tables(3)

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