awesome invariant-and-equivariant filter-and-observer

Paper list for invariant and equivariant filter and observer. Work-in-progress.

Feel free to suggest relevant papers in the following format.

**Non-linear state error based extended Kalman filters with applications to navigation**  
Axel Barrau, PhD thesis, 2015 [paper](https://pastel.archives-ouvertes.fr/tel-01344622)   

Acknowledgement: I would like to thank [Silvère Bonnabel][http://www.silvere-bonnabel.com/Publications.html], [Tarek Hamel][https://dblp.uni-trier.de/pid/41/1354.html], [Jochen Trumpf][http://users.cecs.anu.edu.au/~trumpf/], [Pieter van Goor][https://pvangoor.github.io/], for paper suggestions!

model-based nonlinear estimation methods.

Invariant Observer

1. **Nonlinear State Estimation and Modeling of a Helicopter UAV **

   Martin Barczyk, PhD thesis, 2012

   Note: employing a rotation matrix representation for the state manifold to obtain designs amenable to global stability analysis, obtaining a direct nonlinear design for gains of the AHRS observer, modifying the previously-proposed Invariant EKF systematic method for computing gains, and culminating in simulation and experimental validation of the observers.

2. **Features of Invariant Extended Kalman Filter Applied to Unmanned Aerial Vehicle Navigation **

   Nak Yong Ko, Wonkeun Youn, In Ho Choi, Gyeongsub Song and Tae Sik Kim. sensors 2018. [paper][https://www.mdpi.com/1424-8220/18/9/2855]

3. **Optimal Invariant Observers Theory for Nonlinear State Estimation **

   Jean-Philippe Condomines, Cédric Seren, Gautier Hattenberger. [paper][http://recherche.enac.fr/~jean-philippe.condomines/wp-content/uploads/2015/12/IUKF16_book_chapter.pdf]

4. **Nonlinear Kalman Filtering for Multi-Sensor Navigation of Unmanned Aerial Vehicles **

   Jean-Philippe Condomines, Book 2018.

   Note: Invariant-UKF and $\pi$-IUKF. These two methods apply the general framework of invariant observers to the nonlinear estimation of the state of a dynamic system using an Unscented Kalman Filter (UKF) method, from the more general class of Sigma Point (SP) nonlinear filtering algorithms.

5. Augmented invariant-EKF designs for simultaneous state and disturbance estimation

   Kevin Coleman, Master thesis, 2020 [paper][https://www.proquest.com/docview/2493158725/D51863A194EF4C31PQ/1?accountid=15157]

   Note: study Invariant-EKF designs for invariant systems with disturbances. Three different IEKF designs are presented for a unicycle robot under linear disturbances.

Invariant Filter

1. Non-linear state error based extended Kalman filters with applications to navigation
   Axel Barrau, PhD thesis, 2015 paper

   Note: Using the fact that the Jacobians are state-independent, it can be shown that the IEKF is a locally asymptotically stable observer, no matter the trajectory.

2. Smoothing algorithms for navigation, localisation and mapping based on high-grade inertial sensors

   Paul Chauchat , PhD thesis, 2020 paper

3. Deep learning, Inertial Measurements Units, and Odometry: Some Modern Prototyping Techniques for Navigation Based on Multi-Sensor Fusion

Martin Brossard , PhD thesis, 2020 paper

4. Invariant smoothing on Lie groups

   P. Chauchat, A. Barrau, and S. Bonnabel. IROS 2018. [paper][https://ieeexplore.ieee.org/document/8594068]

   Note: leverage ideas from the IEKF framework, where the Jacobians are state independent, to create an IEKF-based SWF (ISWF), and demonstrate its robustness relative to IEKFs and traditional batch-estimation frameworks.

5. Invariant Sliding Window Filtering for Attitude and Bias Estimation

   Alex Walsh;Jonathan Arsenault;James Richard Forbes. 2019 [paper][https://ieeexplore.ieee.org/document/8814702/]

   Note: further study the ISWF and apply it to non-group-affine process models.

6. An Invariant Extended $H_{\infty}$ Filter

   M. Lavoie, J. Arsenault, and J. R. Forbes. CDC2019 [paper][https://ieeexplore.ieee.org/document/9029289/]

7. **The Invariant Rauch-Tung-Striebel Smoother **

   Niels van der Laan , Mitchell Cohen , Jonathan Arsenault , and James Richard Forbes. IEEE Robotics and automation letters. 2020 [paper][https://ieeexplore.ieee.org/document/9126191]

8. Robust Linearly Constrained Invariant Filtering for a Class of Mismatched Nonlinear Systems

   Paul Chauchat, Jordi Vilà-Valls, Eric Chaumette. IEEE Control Systems Letters 2021 [paper][https://ieeexplore.ieee.org/document/9373652]

   Note: using linear constraints is an effective way to robustify the KF. InEKF should be all the more sensitive to a possible system model mismatch.

Theory

1. **Kalman filtering with a class of geometric state equality constraints **

   P. Chauchat, A. Barrau, S. Bonnabel. CDC 2017. [paper][https://ieeexplore.ieee.org/document/8264033]

2. Extended Kalman Filtering With Nonlinear Equality Constraints: A Geometric Approach

   Axel Barrau, Silvère Bonnabel. IEEE Transactions on Automatic Control 2020. [paper][https://ieeexplore.ieee.org/document/8765615]

3. **Linear observed systems on groups **

   Axel Barrau, Silvère Bonnabel. Systems and Control Letters 2019. [paper][https://www.sciencedirect.com/science/article/pii/S0167691119300805]

4. Equivariant Filter Design for Kinematic Systems on Lie Groups

   Robert Mahony, Jochen Trumpf. MTNS2020 [paper][https://arxiv.org/abs/2004.00828]

   Note: invariant system --> group affine system --> equivariant system

5. Equivariant Systems Theory and Observer Design

   Robert Mahony, Tarek Hamel, Jochen Trumpf. [paper][https://arxiv.org/abs/2006.08276]

   Note: A discussion of invariant errors for systems with homogeneous state that motivates an observer/filter architecture with the observer state posed on the symmetry group.

6. A bundle framework for observer design on smooth manifolds with symmetry

   Anant A. Joshi, D.H.S. Maithripala, Ravi N. Banavar. [paper][https://arxiv.org/abs/1907.09234]

   Note: the special case when the group action is free is given special emphasis

Equivariant Filter

1. **A Geometric Nonlinear Observer for Simultaneous Localisation and Mapping **

   Robert Mahony, Tarek Hamel. CDC 2017

2. Attitude Observation for Second Order Attitude Kinematics
   Yonhon Ng, Pieter van Goor, Robert Mahony, Tarek Hamel. CDC 2019 paper
   Note:

3. Equivariant Systems Theory and Observer Design for Second Order Kinematic Systems on Matrix Lie Groups
   Yonhon Ng, Pieter van Goor, Tarek Hamel, Robert Mahony. CDC 2020 paper
   Note:

4. A Geometric Observer Design for Visual Localization and Mapping

   Pieter van Goor, Robert Mahony, Tarek Hamel, Jochen Trumpf. CDC 2019 [paper][https://arxiv.org/pdf/1904.02452.pdf]

5. An Observer Design for Visual Simultaneous Localisation and Mapping with Output Equivariance

   Pieter van Goor, Robert Mahony, Tarek Hamel, Jochen Trumpf. IFAC 2020 [paper][https://arxiv.org/abs/2005.14347]

6. Constructive Observer Design for Visual Simultaneous Localisation and Mapping

   Pieter van Goor, Robert Mahony, Tarek Hamel, Jochen Trumpf. Submitted to Automatica [paper][https://arxiv.org/abs/2006.05053]

7. Equivariant Filter (EqF): A General Filter Design for Systems on Homogeneous Spaces
   Pieter van Goor, Tarek Hamel and Robert Mahony CDC 2020 [paper][https://ieeexplore.ieee.org/abstract/document/9303813] [presentation][https://pvangoor.github.io/talks/2021/05/07/cdc2020_talk.html] [YouTube][https://www.youtube.com/watch?v=AwlDJU_3nuc]

8. **EQUIVARIANT FILTER (EqF) **
   Pieter van Goor, Tarek Hamel and Robert Mahony paper

   Note: In cases where the system output is also equivariant the EqF leads to linearised dynamics with a constant output matrix.

9. **Equivariant Visual Odometry in the Wild **

   Robert Mahony, Pieter van Goor, Mina Henein, Ryan Pike, Jun Zhang and Yonhon Ng. CDC 2020

10. An Equivariant Filter for Visual Inertial Odometry
    Pieter van Goor, Robert Mahony ICRA 2021 paper [Code][https://github.com/pvangoor/eqf_vio] [presentation][https://pvangoor.github.io/talks/2021/03/17/ardupilot_vio.html] [YouTube][https://www.youtube.com/watch?v=vLZdBKRjRi4]

    Note: The Equivariant Filter for Visual Inertial Odometry.

Application

Inertial-integrated navigation

1. **Contact-Aided Invariant Extended Kalman Filtering for Legged Robot State Estimation **

   [code][GitHub - UMich-BipedLab/Contact-Aided-Invariant-EKF: Example code for contact-aided invariant extended Kalman filtering.]

2. **Contact-Aided Invariant Extended Kalman Filtering for Robot State Estimation **

   Ross Hartley , Maani Ghaffari , Ryan M Eustice and Jessy W Grizzle. IJRR 2020.

3. **$SE_2(3)$ based Extended Kalman Filtering and Smoothing Framework for Inertial-Integrated Navigation **

   Yarong Luo, Chi Guo, Jingnan Liu. [paper][https://arxiv.org/abs/2102.12897]

4. Equivariant Filtering Framework for Inertial-Integrated Navigation

   Yarong Luo, Chi Guo, Jingnan Liu. [paper][https://arxiv.org/abs/2103.14873]

5. **Legged Robot State Estimation in Slippery Environments Using Invariant Extended Kalman Filter with Velocity Update **

   Sangli Teng, Mark Wilfried Mueller, Koushil Sreenath. [paper][https://arxiv.org/abs/2104.04238]

SLAM

1. Practical Considerations and Extensions of the Invariant Extended Kalman Filtering Framework

   Jonathan Arsenault. Master thesis, 2019.

   Note：Invariant Filtering in Continuous Time and Discrete Time.

2. **Toward Invariant Visual-Inertial State Estimation using Information Sparsification **

   Shih-Chieh (Jerry) Hsiung. Master thesis, 2018.

3. **Associating Uncertainty to Extended Poses for on Lie Group IMU Preintegration with Rotating Earth **

   M. Brossard, A. Barrau, P. Chauchat, S. Bonnabel. IEEE Transactions on Robotics 2021. [paper][https://arxiv.org/abs/2007.14097v2]

4. **A Mathematical Framework for IMU Error Propagation with Applications to Preintegration **

   Martin Brossard, Axel Barrau, Paul Chauchat, and Silvere Bonnabel. ICRA 2020. [paper][https://ieeexplore.ieee.org/document/9197492]

5. **Consistent EKF-based visual-inertial odometry on matrix Lie group **

6. **Consistent EKF-based visual-inertial navigation using points and lines **

7. **Observability analysis and consistency improvements for visual-inertial odometry on the matrix Lie group of extended poses **

8. **Observability Analysis of IMU Intrinsic Parameters in Stereo Visual-Inertial Odometry **

9. **Consistent Right-Invariant Fixed-Lag Smoother with Application to Visual Inertial SLAM **