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
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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.

[2]

T. Bailey and H. Durrant-Whyte, Simultaneous localization and mapping (slam): Part II, IEEE Robotics Automation Magazine, 13 (2006), 108-117.  doi: 10.1109/MRA.2006.1678144.

[3]

G. Baldwin, R. Mahony and J. Trumpf, A nonlinear observer for 6 dof pose estimation from inertial and bearing measurements, in 2009 IEEE International Conference on Robotics and Automation, 2009, 2237–2242. doi: 10.1109/ROBOT.2009.5152242.

[4]

G. Baldwin, R. Mahony, J. Trumpf, T. Hamel and T. Cheviron, Complementary filter design on the special euclidean group se(3), in 2007 European Control Conference (ECC), 2007, 3763–3770. doi: 10.23919/ECC.2007.7068746.

[5]

A. Barrau and S. Bonnabel, Intrinsic filtering on lie groups with applications to attitude estimation, IEEE Transactions on Automatic Control, 60 (2015), 436-449.  doi: 10.1109/TAC.2014.2342911.

[6]

A. Barrau and S. Bonnabel, The invariant extended kalman filter as a stable observer, IEEE Transactions on Automatic Control, 62 (2017), 1797-1812.  doi: 10.1109/TAC.2016.2594085.

[7]

A. Barrau and S. Bonnabel, Three examples of the stability properties of the invariant extended kalman filter, IFAC-PapersOnLine, 50 (2017), 431 – 437, 20th IFAC World Congress. doi: 10.1016/j.ifacol.2017.08.061.

[8]

A. Barrau and S. Bonnabel, Invariant kalman filtering, Annual Review of Control, Robotics, and Autonomous Systems, 1 (2018), 237-257.  doi: 10.1146/annurev-control-060117-105010.

[9]

A. N. Bishop, P. N. Pathirana and A. V. Savkin, Target tracking with range and bearing measurements via robust linear filtering, in 2007 3rd International Conference on Intelligent Sensors, Sensor Networks and Information, 2007,131–135. doi: 10.1109/ISSNIP.2007.4496832.

[10]

A. M. BlochDong Eui ChangN. E. Leonard and J. E. Marsden, Controlled lagrangians and the stabilization of mechanical systems. ii. potential shaping, IEEE Transactions on Automatic Control, 46 (2001), 1556-1571.  doi: 10.1109/9.956051.

[11]

A. M. BlochN. E. Leonard and J. E. Marsden, Controlled lagrangians and the stabilization of mechanical systems. i. the first matching theorem, IEEE Transactions on Automatic Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.

[12]

A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-3017-3.

[13]

S. Bonnabel, Left-invariant extended kalman filter and attitude estimation, in 2007 46th IEEE Conference on Decision and Control, 2007, 1027–1032. doi: 10.1109/CDC.2007.4434662.

[14]

S. BonnabelP. Martin and P. Rouchon, Symmetry-preserving observers, IEEE Transactions on Automatic Control, 53 (2008), 2514-2526.  doi: 10.1109/TAC.2008.2006929.

[15]

S. BonnabelP. Martin and P. Rouchon, Non-linear symmetry-preserving observers on lie groups, IEEE Transactions on Automatic Control, 54 (2009), 1709-1713.  doi: 10.1109/TAC.2009.2020646.

[16]

S. Bonnabel, P. Martin, P. Rouchon and E. Salaün, A separation principle on lie groups, IFAC Proceedings Volumes, 44 (2011), 8004 – 8009, 18th IFAC World Congress. doi: 10.3182/20110828-6-IT-1002.03353.

[17]

S. Bonnable, P. Martin and E. Salaün, Invariant extended kalman filter: theory and application to a velocity-aided attitude estimation problem, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 2009, 1297–1304. doi: 10.1109/CDC.2009.5400372.

[18]

G. Bourmaud, R. Mégret, A. Giremus and Y. Berthoumieu, Discrete extended kalman filter on lie groups, in 21st European Signal Processing Conference (EUSIPCO 2013), 2013, 1–5.

[19]

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

[20]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag New York, 2005. doi: 10.1007/978-1-4899-7276-7.

[21]

C. CadenaL. CarloneH. CarrilloY. LatifD. ScaramuzzaJ. NeiraI. 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.

[22]

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.

[23]

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

[24]

J. L. CrassidisF. 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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[35]

C. LagemanJ. 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.

[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.

[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.

[38]

E. J. LeffertsF. 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.

[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.

[40]

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

[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.

[42]

R. MahonyT. 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.

[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.

[44]

R. MahonyT. HamelP. 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.

[45]

R. Mahony, J. Trumpf and T. Hamel, Observers for kinematic systems with symmetry, IFAC Proceedings Volumes, 46 (2013), 617–633, 9th IFAC Symposium on Nonlinear Control Systems. doi: 10.3182/20130904-3-FR-2041.00212.

[46]

M. MallickY. Bar-ShalomT. Kirubarajan and M. Moreland, An improved single-point track initiation using gmti measurements, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 2697-2714.  doi: 10.1109/TAES.2015.140599.

[47]

F. L. Markley, Attitude error representations for kalman filtering, Journal of Guidance, Control, and Dynamics, 26 (2003), 311-317.  doi: 10.2514/2.5048.

[48]

F. L. Markley and J. L. Crassidis, Fundamentals of Spacecraft Attitude Determination and Control, 1st edition, Springer-Verlag New York, 2014. doi: 10.1007/978-1-4939-0802-8.

[49]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag New York, 1999. doi: 10.1007/978-0-387-21792-5.

[50] J. Milnor, Morse Theory, vol. 51 of Annals of Mathematics Studies, Princeton University Press, 1963.  doi: 10.1515/9781400881802.
[51]

A. SacconJ. TrumpfR. Mahony and A. P. Aguiar, Second-order-optimal minimum-energy filters on lie groups, IEEE Transactions on Automatic Control, 61 (2016), 2906-2919.  doi: 10.1109/TAC.2015.2506662.

[52]

J. Thienel and R. M. Sanner, A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise, IEEE Transactions on Automatic Control, 48 (2003), 2011-2015.  doi: 10.1109/TAC.2003.819289.

[53]

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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)
Figure 7.  Radar
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 $
$ 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 $
$ 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} $
$ P ={\mathop{\mathbb{SE}(3)}} \times \mathbb{E} $ $ G = {\mathop{\mathbb{SE}(3)}} $ $ \mathcal{Y} = \mathbb{E} $
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