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Matched pair analysis of the Vlasov plasma
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 |
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.
References:
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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. Bloch, Dong Eui Chang, N. 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. Bloch, N. 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. |
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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. Bonnabel, P. Martin and P. Rouchon,
Symmetry-preserving observers, IEEE Transactions on Automatic Control, 53 (2008), 2514-2526.
doi: 10.1109/TAC.2008.2006929. |
[15] |
S. Bonnabel, P. 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. |
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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. |
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N. G. Branko Ristic Sanjeev Arulampalam, Beyond the Kalman Filter, Wiley-IEEE, 2004. |
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F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag New York, 2005.
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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. |
[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. |
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J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems, 2nd edition, Chapman & Hall/CRC, 2012. |
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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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Mallick, Y. Bar-Shalom, T. 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. Saccon, J. Trumpf, R. 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.
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[53] |
J. Trumpf, R. Mahony and T. Hamel, On the structure of kinematic systems with complete symmetry, in 2018 IEEE Conference on Decision and Control (CDC), 2018, 1276–1280.
doi: 10.1109/CDC.2018.8619718. |
[54] |
J. F. Vasconcelos, R. Cunha, C. Silvestre and P. Oliveira, Landmark based nonlinear observer for rigid body attitude and position estimation, in 2007 46th IEEE Conference on Decision and Control, 2007, 1033–1038.
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[55] |
M. Wang and A. Tayebi, Geometric nonlinear observer design for slam on a matrix lie group, in 2018 IEEE Conference on Decision and Control (CDC), 2018, 1488–1493.
doi: 10.1109/CDC.2018.8619501. |
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D. S. Watkins, Fundamentals of Matrix Computations, 3rd edition, John Wiley & Sons, Inc., Hoboken, New Jersey, 2010. |
[57] |
M. Zamani, J. Trumpf and R. Mahony,
Near-optimal deterministic filtering on the rotation group, IEEE Transactions on Automatic Control, 56 (2011), 1411-1414.
doi: 10.1109/TAC.2011.2109436. |
[58] |
M. Zamani, J. Trumpf and R. Mahony,
Minimum-energy filtering for attitude estimation, IEEE Transactions on Automatic Control, 58 (2013), 2917-2921.
doi: 10.1109/TAC.2013.2259092. |
[59] |
D. E. Zlotnik and J. R. Forbes,
Gradient-based observer for simultaneous localization and mapping, IEEE Transactions on Automatic Control, 63 (2018), 4338-4344.
doi: 10.1109/TAC.2018.2829467. |
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. Bloch, Dong Eui Chang, N. 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. Bloch, N. 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. Bonnabel, P. Martin and P. Rouchon,
Symmetry-preserving observers, IEEE Transactions on Automatic Control, 53 (2008), 2514-2526.
doi: 10.1109/TAC.2008.2006929. |
[15] |
S. Bonnabel, P. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Mallick, Y. Bar-Shalom, T. 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. Saccon, J. Trumpf, R. Mahony and A. P. Aguiar,
Second-order-optimal minimum-energy filters on lie groups, IEEE Transactions on Automatic Control, 61 (2016), 2906-2919.
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