# Quaternion kinematics for the error-state Kalman filter

arXiv: https://arxiv.org/abs/1711.02508 · HTML: https://mlsanity.com/paper/1711.02508/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2017.

## Built on by (most cited first, 30 of 30)

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- [EDI: ESKF-based Disjoint Initialization for Visual-Inertial SLAM Systems](https://mlsanity.com/paper/2308.02670/) (2023)
- [Continuous Pose for Monocular Cameras in Neural Implicit Representation](https://mlsanity.com/paper/2311.17119/) (2023)
- [Legged Robot State Estimation With Invariant Extended Kalman Filter Using Neural Measurement Network](https://mlsanity.com/paper/2402.00366/) (2024)
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- [MagShield: Towards Better Robustness in Sparse Inertial Motion Capture Under Magnetic Disturbances](https://mlsanity.com/paper/2506.22907/) (2025)
- [Sleep Posture One-Shot Learning Framework Using Kinematic Data Augmentation: In-Silico and In-Vivo Case Studies](https://mlsanity.com/paper/2205.10778/) (2022)
- [Multi-Visual-Inertial System: Analysis, Calibration and Estimation](https://mlsanity.com/paper/2308.05303/) (2023)
- [Enhancing Visual Inertial SLAM with Magnetic Measurements](https://mlsanity.com/paper/2409.09904/) (2024)
- [Reinforcement Learning with Lie Group Orientations for Robotics](https://mlsanity.com/paper/2409.11935/) (2024)
- [SSPINNpose: A Self-Supervised PINN for Inertial Pose and Dynamics Estimation](https://mlsanity.com/paper/2506.11786/) (2025)
- [An Improved Multi-State Constraint Kalman Filter for Visual-Inertial Odometry](https://mlsanity.com/paper/2210.08117/) (2022)
- [Locking On: Leveraging Dynamic Vehicle-Imposed Motion Constraints to Improve Visual Localization](https://mlsanity.com/paper/2306.17529/) (2023)
- [High-accuracy Vision-Based Attitude Estimation System for Air-Bearing Spacecraft Simulators](https://mlsanity.com/paper/2312.08146/) (2023)
- [Multi-Robot Relative Pose Estimation in SE(2) with Observability Analysis: A Comparison of Extended Kalman Filtering and Robust Pose Graph Optimization](https://mlsanity.com/paper/2401.15313/) (2024)
- [Sky-GVIO: an enhanced GNSS/INS/Vision navigation with FCN-based sky-segmentation in urban canyon](https://mlsanity.com/paper/2404.11070/) (2024)
- [Tightly-Coupled, Speed-aided Monocular Visual-Inertial Localization in Topological Map](https://mlsanity.com/paper/2411.05497/) (2024)
- [KN-LIO: Geometric Kinematics and Neural Field Coupled LiDAR-Inertial Odometry](https://mlsanity.com/paper/2501.04263/) (2025)
- [HDVIO2.0: Wind and Disturbance Estimation with Hybrid Dynamics VIO](https://mlsanity.com/paper/2504.00969/) (2025)
- [Aleatoric Uncertainty from AI-based 6D Object Pose Predictors for Object-relative State Estimation](https://mlsanity.com/paper/2509.01583/) (2025)
- [Adaptive Covariance and Quaternion-Focused Hybrid Error-State EKF/UKF for Visual-Inertial Odometry](https://mlsanity.com/paper/2512.17505/) (2025)
- [Observability Conditions and Filter Design for Visual Pose Estimation via Dual Quaternions](https://mlsanity.com/paper/2605.02054/) (2026)
- [SubSplat: High-Resolution Pixel-aligned 3DGS via Sub-pixel Gaussian Reparameterization](https://mlsanity.com/paper/2607.20813/) (2026)
- [QuacamFM: Quaternion-Constrained Flow Matching for Camera Pose Estimation](https://mlsanity.com/paper/2609.32455/) (2026)

## Abstract (arXiv)

This article is an exhaustive revision of concepts and formulas related to quaternions and rotations in 3D space, and their proper use in estimation engines such as the error-state Kalman filter. The paper includes an in-depth study of the rotation group and its Lie structure, with formulations using both quaternions and rotation matrices. It makes special attention in the definition of rotation perturbations, derivatives and integrals. It provides numerous intuitions and geometrical interpretations to help the reader grasp the inner mechanisms of 3D rotation. The whole material is used to devise precise formulations for error-state Kalman filters suited for real applications using integration of signals from an inertial measurement unit (IMU).