Papers and Reading

Papers and Reading Path¶

This page collects and categorizes key papers on invariant and equivariant filtering.

0. Surveys and Tutorials¶

• The Manifold Kalman Filter Hierarchy: Equivariant Filters (2026)
  Frank Dellaert, Rohan Bansal
  A comprehensive overview of the Equivariant Filter theory, design, and code implementation in the GTSAM framework.

• Progress in symmetry preserving robot perception and control through geometry and learning (2022)
  Maani Ghaffari, Ray Zhang, Minghan Zhu, Chien Erh Lin, Tzu-Yuan Lin, Sangli Teng, Tingjun Li, Tianyi Liu and Jingwei Song
  Comprehensive overview of how symmetry is used in perception, state estimation, and control.

• SE(3)-Equivariant Robot Learning and Control: A Tutorial Survey (2025)
  Joohwan Seo, Soochul Yoo, Junwoo Chang, Hyunseok An, Hyunwoo Ryu, Soomi Lee, Arvind Kruthiventy, Jongeun Choi, Roberto Horowitz
  A recent tutorial unifying mathematical notation for group equivariant deep learning and geometric control.

• A Micro Lie Theory for State Estimation in Robotics (2018)
  Joan Solà, Jeremie Deray, Dinesh Atchuthan
  A primer on Lie group theory from the perspective of roboticists, not just physicists and mathematicians.

1. Foundations of Invariant and Equivariant Filtering¶

1.1 Core IEKF Theory¶

• Non-linear symmetry-preserving observers on Lie groups (2009)
  Silvère Bonnabel, Philippe Martin, Pierre Rouchon
  Direct successor to the 2008 “Symmetry-Preserving Observers” paper, specialized to systems on Lie groups. Establishes the autonomous error-state property that underpins later IEKF and EqF theory.

• Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem (2009)
  Silvère Bonnabel, Philippe Martin, Erwan Salaün
  The seminal paper introducing the IEKF, showing improved consistency for attitude estimation.

• Non-linear state error based extended Kalman filters with applications to navigation (2013)
  Axel Barrau
  Comprehensive PhD thesis on IEKF theory and applications.

• The Invariant Extended Kalman Filter as a Stable Observer (2017)
  Axel Barrau, Silvère Bonnabel
  Rigorous stability analysis of the IEKF, establishing conditions for local asymptotic stability.

• The Difference between the Left and Right Invariant Extended Kalman Filter (2025)
  Yixiao Ge, Giulio Delama, Martin Scheiber, Alessandro Fornasier, Pieter van Goor, Stephan Weiss, Robert Mahony
  A recent clarification paper proving that left and right IEKFs are mathematically equivalent when the reset step is correctly implemented, debunking the myth that handedness must match measurement physics.

1.2 Symmetry-Preserving and Geometric Observers¶

• Symmetry-preserving observers (2004)
  S. Bonnabel, Ph. Martin, P. Rouchon
  Early work on observer design that preserves system symmetries.

• Nonlinear Complementary Filters on the Special Orthogonal Group (2008)
  Robert Mahony, Tarek Hamel, Jean-Michel Pflimlin
  Nonlinear complementary filter on SO(3) that strongly influenced later invariant and equivariant attitude observers.

• Observer Design on the Special Euclidean Group SE(3) (2011)
  Minh-Duc Hua, Mohammad Zamani, Jochen Trumpf, Robert Mahony, Tarek Hamel
  Early work on observer design for pose estimation on SE(3).

2. Equivariant Filters (EqF) and Unified Viewpoints¶

2.1 Equivariant Filter Design¶

• Equivariant Filter (EqF) (2023)
  Pieter van Goor, Tarek Hamel, Robert Mahony
  Presents the full Equivariant Filter framework for systems on homogeneous spaces. Develops the lifted system, equivariant error, and origin-linearized filter design in their most general form.

• Equivariant Filter (EqF): A General Filter Design for Systems on Homogeneous Spaces (2020)
  Pieter van Goor, Tarek Hamel, Robert Mahony
  The original paper that introduces the Equivariant Filter design methodology. Establishes the core idea of designing a Kalman-style filter directly on a homogeneous space using a transitive Lie-group symmetry, and is the foundational reference for the entire EqF line of work.

• Equivariant Filter Design for Kinematic Systems on Lie Groups (2021)
  Robert Mahony, Jochen Trumpf
  Framework for embedding kinematic systems on Lie groups into equivariant systems.

• Equivariant Systems Theory and Observer Design (2020)
  Robert Mahony, Tarek Hamel, Jochen Trumpf
  Technical companion to the EqF papers that develops the underlying systems-theoretic foundation: equivariant lifts, the relationship between symmetry groups and homogeneous state spaces, and the structural properties that enable EqF and IEKF design.

2.2 Advanced EqF Architectures¶

• Equivariant Filter Design for Inertial Navigation Systems with Input Measurement Biases (2022)
  Alessandro Fornasier, Yonhon Ng, Robert Mahony, Stephan Weiss
  The foundational bias-aware EqF for inertial navigation systems. Introduces the symmetry group that natively includes IMU biases as part of the state, rather than tacking them on as in standard IEKF approaches.

• Revisiting Multi-GNSS Navigation for UAVs – An Equivariant Filtering Approach (2023)
  Martin Scheiber, Alessandro Fornasier, Christian Brommer, Stephan Weiss
  Applies the equivariant filter framework to the IMU + multi-GNSS sensor fusion problem common in outdoor UAV navigation, demonstrating EqF advantages over the multiplicative EKF in terms of consistency, convergence rate, and robustness to poor initial estimates.

• MSCEqF: A Multi State Constraint Equivariant Filter for Vision-Aided Inertial Navigation (2024)
  Alessandro Fornasier, Pieter van Goor, Eren Allak, Robert Mahony, Stephan Weiss
  Extends the MSCKF (Multi-State Constraint Kalman Filter) to the equivariant framework, allowing for consistent VIO with onboard calibration.

• An Equivariant Approach to Robust State Estimation for the ArduPilot Autopilot System (2024)
  Alessandro Fornasier, Yixiao Ge, Pieter van Goor, Martin Scheiber, Andrew Tridgell, Robert Mahony
  A novel EqF formulation exploiting Semi-Direct-Bias symmetry for multi-sensor fusion, specifically designed to handle GNSS outliers and shifts in open-source autopilot software.

• Equivariant Symmetries for Aided Inertial Navigation (2024)
  Alessandro Fornasier
  Dissertation that advances the understanding of equivariant symmetries in the context of inertial navigation systems.

• Overcoming Bias: Equivariant Filter Design for Biased Attitude Estimation with Online Calibration (2022)
  Alessandro Fornasier, Yonhon Ng, Christian Brommer, Christoph Böhm, Robert Mahony, Stephan Weiss
  Paper that introduces a new generic formulation for a gyroscope aided attitude estimator, taking advantage of states in a single equivariant geometric structure.

3. Applications and Advanced Topics¶

3.1 Legged Robotics & Contact Estimation¶

• Contact-Aided Invariant Extended Kalman Filtering for Legged Robot State Estimation (2018)
  Ross Hartley, Maani Ghaffari, Ryan M. Eustice, Jessy W. Grizzle
  The “killer app” for InEKF, using contact constraints as invariant measurements.

• Adaptive Invariant Extended Kalman Filter for Legged Robot State Estimation (2025)
  Kyung-Hwan Kim, DongHyun Ahn, Dong-hyun Lee, JuYoung Yoon, Dong Jin Hyun
  Proposes an adaptive mechanism to improve proprioceptive state estimation and handle slip more robustly than standard InEKF.

• Legged Robot State Estimation With Invariant Extended Kalman Filter Using Neural Measurement Network (2024)
  Donghoon Youm, Hyunsik Oh, Suyoung Choi, Hyeongjun Kim, Jemin Hwangbo
  A hybrid approach integrating a Neural Measurement Network (NMN) with an InEKF to bridge the sim-to-real gap in terrain-aware state estimation.

3.2 Visual-Inertial Odometry (VIO)¶

• Equivariant Filter Design for Range-only SLAM (2025)
  Yixiao Ge, Arthur Pearce, Pieter van Goor, Robert Mahony
  Designs an EqF for range-only SLAM, and constructs a symmetry group acting jointly on the robot pose and the landmark positions, producing a filter with better consistency and convergence properties than EKF baselines for beacon-based localization.

• Equivariant Filter for Feature-Based Homography Estimation for General Camera Motion (2023)
  Tarek Bouazza, Katrina Ashton, Pieter van Goor, Tarek Hamel
  Applies the EqF methodology to homography and planar-scene structure estimation under arbitrary camera motion, using only camera-velocity measurements and direct point-feature correspondences.

• Invariant Kalman Filtering for Visual-Inertial SLAM (2018)
  Martin Brossard, Silvère Bonnabel, Axel Barrau
  Application of IEKF to visual-inertial odometry.

• EqVIO: An Equivariant Filter for Visual-Inertial Odometry (2023)
  Pieter van Goor, Robert Mahony
  The definitive journal paper on EqVIO, demonstrating superior consistency over standard VIO (like VINS-Mono) by utilizing an equivariant output approximation.

3.3 Other Multi-sensor Fusion¶

• Eq-LIO: Equivariant Filter for Tightly Coupled LiDAR-Inertial Odometry (2024)
  Anbo Tao, Yarong Luo, Chunxi Xia, Chi Guo, Xingxing Li
  An EqF-based tightly coupled LiDAR-inertial odometry system. Uses a semi-direct product symmetry group that jointly couples IMU bias, navigation state, and LiDAR extrinsic calibration into a single equivariant structure, suppressing linearization error and improving robustness.

4. General Equivariance¶

Disclaimer: There are many more works in the field of general equivariance than we can list here. This part will not be exhaustive, but rather an interesting pointer into that literature!

4.1 General Knowledge¶

• Geometric deep learning: going beyond Euclidean data (2017)
  Michael M. Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, Pierre Vandergheynst
  The foundational survey of geometric deep learning, introducing a unified perspective on neural networks operating on non-Euclidean domains (graphs, manifolds, groups). Provides the broader context in which group-equivariant networks and, by extension, modern equivariant filtering sit.

4.1 Equivariant Diffusion Policies (Robot Learning)¶

• Equivariant Diffusion Policy (2024)
  Dian Wang, Stephen Hart, David Surovik, Tarik Kelestemur, Haojie Huang, Haibo Zhao, Mark Yeatman, Jiuguang Wang, Robin Walters, Robert Platt
  Introduces a diffusion policy that leverages domain symmetries (like SO(2)) to drastically improve sample efficiency in imitation learning.

• EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning (2024)
  Jingyun Yang, Zi-ang Cao, Congyue Deng, Rika Antonova, Shuran Song, Jeannette Bohg
  Extends equivariance to the SIM(3) group (translation, rotation, and scale), allowing robots to learn manipulation tasks that generalize to objects of different sizes and poses.

• Spherical Diffusion Policy (2025)
  Xupeng Zhu, Fan Wang, Robin Walters, Jane Shi
  A state-of-the-art policy that embeds the diffusion process in spherical Fourier space to achieve exact SE(3) equivariance for complex 3D manipulation.

• ET-SEED: Efficient Trajectory-Level SE(3) Equivariant Diffusion Policy (2025)
  Chenrui Tie, Yue Chen, Ruihai Wu, Boxuan Dong, Zeyi Li, Chongkai Gao, Hao Dong
  Introduction of a SE(3)-equivariant diffusion policy that generates robot trajectories more efficiently and generalizes better with less data.

• Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation (2023)
  Hyunwoo Ryu, Jiwoo Kim, Hyunseok An, Junwoo Chang, Joohwan Seo, Taehan Kim, Yubin Kim, Chaewon Hwang, Jongeun Choi, Roberto Horowitz
  Bi-equivariant denoising generative modeling on SE(3) for visual robotic manipulation.

• A Practical Guide for Incorporating Symmetry in Diffusion Policy (2025)
  Dian Wang, Boce Hu, Shuran Song, Robin Walters, Robert Platt
  Practical guidance on incorporating symmetry into diffusion policies without requiring fully equivariant architectures.

For software and practical implementations, see the Examples and Implementations section of this site and the GTSAM EKF variants documentation.