American Institute of Mathematical Sciences

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

A bundle framework for observer design on smooth manifolds with symmetry

Anant A. Joshi 1, , D. H. S. Maithripala 2,3, and  Ravi N. Banavar 4,,

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.

Keywords: Lie group symmetry, observer design, bundle structure, fibre bundle, principal bundle.
Mathematics Subject Classification: Primary: 93B27.
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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show all references

References:
[1]

D. Auroux and S. Bonnabel, Symmetry-based observers for some water-tank problems, IEEE Transactions on Automatic Control, 56 (2011), 1046-1058.  doi: 10.1109/TAC.2010.2067291.  Google Scholar

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[15]

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[16]

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[17]

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[19]

N. G. Branko Ristic Sanjeev Arulampalam, Beyond the Kalman Filter, Wiley-IEEE, 2004. Google Scholar

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C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, I. Reid and J. J. Leonard, Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age, IEEE Transactions on Robotics, 32 (2016), 1309-1332.  doi: 10.1109/TRO.2016.2624754.  Google Scholar

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P. Coote, J. Trumpf, R. Mahony and J. C. Willems, Near-optimal deterministic filtering on the unit circle, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009, 5490–5495. doi: 10.1109/CDC.2009.5399999.  Google Scholar

[23]

J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems, 2nd edition, Chapman & Hall/CRC, 2012.  Google Scholar

[24]

J. L. Crassidis, F. L. Markley and Y. Cheng, Survey of nonlinear attitude estimation methods, Journal of Guidance, Control, and Dynamics, 30 (2007), 12-28.  doi: 10.2514/1.22452.  Google Scholar

[25]

G. Dissanayake, S. Huang, Z. Wang and R. Ranasinghe, A review of recent developments in simultaneous localization and mapping, 2011 6th International Conference on Industrial and Information Systems, ICIIS 2011 - Conference Proceedings. doi: 10.1109/ICIINFS.2011.6038117.  Google Scholar

[26]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Springer-Verlag Berlin Heidelberg, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[27]

H. Durrant-Whyte and T. Bailey, Simultaneous localization and mapping: part i, IEEE Robotics Automation Magazine, 13 (2006), 99-110.  doi: 10.1109/MRA.2006.1638022.  Google Scholar

[28]

M.-D. Hua, M. Zamani, J. Trumpf, R. Mahony and T. Hamel, Observer design on the special euclidean group se(3), in 2011 50th IEEE Conference on Decision and Control and European Control Conference, 2011, 8169–8175. doi: 10.1109/CDC.2011.6160453.  Google Scholar

[29]

S. J. Julier and J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE, 92 (2004), 401-422.  doi: 10.1109/JPROC.2003.823141.  Google Scholar

[30]

S. J. Julier, J. K. Uhlmann and H. F. Durrant-Whyte, A new approach for filtering nonlinear systems, in Proceedings of 1995 American Control Conference - ACC'95, vol. 3, 1995, 1628–1632. Google Scholar

[31]

R. E. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar

[32]

R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, Journal of Basic Engineering, 83 (1961), 95-108.  doi: 10.1115/1.3658902.  Google Scholar

[33]

A. Khosravian, T. Chin, I. Reid and R. Mahony, A discrete-time attitude observer on so(3) for vision and gps fusion, in 2017 IEEE International Conference on Robotics and Automation (ICRA), 2017, 5688–5695. doi: 10.1109/ICRA.2017.7989669.  Google Scholar

[34]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 1, Interscience Publishers, 1963.  Google Scholar

[35]

C. Lageman, J. Trumpf and R. Mahony, Gradient-like observers for invariant dynamics on a lie group, IEEE Transactions on Automatic Control, 55 (2010), 367-377.  doi: 10.1109/TAC.2009.2034937.  Google Scholar

[36]

F. Le Bras, T. Hamel, R. Mahony and C. Samson, Observer design for position and velocity bias estimation from a single direction output, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 7648–7653. doi: 10.1109/CDC.2015.7403428.  Google Scholar

[37]

K. W. Lee, W. S. Wijesoma and J. I. Guzman, On the observability and observability analysis of slam, in 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006, 3569–3574. doi: 10.1109/IROS.2006.281646.  Google Scholar

[38]

E. J. Lefferts, F. L. Markley and M. D. Shuster, Kalman filtering for spacecraft attitude estimation, Journal of Guidance, Control, and Dynamics, 5 (1982), 417-429.  doi: 10.2514/3.56190.  Google Scholar

[39]

D. G. Luenberger, Observing the state of a linear system, IEEE Transactions on Military Electronics, 8 (1964), 74-80.  doi: 10.1109/TME.1964.4323124.  Google Scholar

[40]

Ba-Ngu Vo Mahendra Mallick Vikram Krishnamurthy (ed.), Integrated Tracking, Classification, and Sensor Management, Artech House, 2012. Google Scholar

[41]

R. Mahony and T. Hamel, A geometric nonlinear observer for simultaneous localisation and mapping, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), 2017, 2408–2415. doi: 10.1109/CDC.2017.8264002.  Google Scholar

[42]

R. Mahony, T. Hamel and J.-M. Pflimlin, Nonlinear complementary filters on the special orthogonal group, IEEE Transactions on Automatic Control, 53 (2008), 1203-1218.  doi: 10.1109/TAC.2008.923738.  Google Scholar

[43]

R. Mahony, T. Hamel, J. Trumpf and C. Lageman, Nonlinear attitude observers on so(3) for complementary and compatible measurements: A theoretical study, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009, 6407–6412. doi: 10.1109/CDC.2009.5399821.  Google Scholar

[44]

R. Mahony, T. Hamel, P. Morin and E. Malis, Nonlinear complementary filters on the special linear group, International Journal of Control, 85 (2012), 1557-1573.  doi: 10.1080/00207179.2012.693951.  Google Scholar

[45]

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