Quaternion kinematics for the error-state Kalman filter

TL;DR

The work provides a comprehensive, convention-aware treatment of quaternions and 3D rotations, detailing their algebra, exponential/logarithmic maps, and the relationship to the rotation group SO(3). It then builds a rigorous error-state Kalman filter (ESKF) framework for IMU-driven systems, deriving continuous- and discrete-time kinematics, Jacobians, perturbations, and integration schemes, with both local and global angular-error formulations. A major focus is on precise, practitioner-friendly formulations for fusing IMU data with complementary sensors (GPS/vision), including SLERP, isoclinic rotations, and careful handling of perturbations, noise, and reset operations to minimize drift. The result is a practical toolkit for robust attitude and navigation estimation in real-world inertial sensing applications. The guidance aims to support reliable integration of high-rate IMU data with lower-rate absolute measurements, ensuring consistency, stability, and accuracy in pose estimation.

Abstract

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).

Figures (14)

• Figure 1: Rotation of a vector ${\bf x}$, by an angle $\phi$, around the axis ${\bf u}$. See text for details.
• Figure 2: Exponential maps of the rotation matrix.
• Figure 3: Exponential maps of the quaternion.
• Figure 4: Double cover of the rotation manifold. Left: the quaternion ${\bf q}$ in the unit 3-sphere defines an angle $\theta$ with the identity quaternion ${\bf q}_1$. Center: the resulting 3D rotation ${\bf x}'={\bf q}\otimes{\bf x}\otimes{\bf q}^*$ has double angle $\phi$ than that of the original quaternion. Right: Superposing the 4D and 3D rotation planes, observe how one turn of the quaternion ${\bf q}$ over the 3-sphere (red) represents two turns of the rotated vector ${\bf x}$ in 3D space (blue).
• Figure 5: Rotation composition. In ${\mathbb R}^2$, we would simply do $\theta_{\mathcal{A}\mathcal{C}} = \theta_{\mathcal{A}\mathcal{B}}+\theta_{\mathcal{B}\mathcal{C}}$, with an operation 'sum' that is commutative. In ${\mathbb R}^3$ composition satisfies ${\bf q}_{\mathcal{A}\mathcal{C}} = {\bf q}_{\mathcal{A}\mathcal{B}}\otimes{\bf q}_{\mathcal{B}\mathcal{C}}$ and, in matrix form, ${\bf R}_{\mathcal{A}\mathcal{C}} = {\bf R}_{\mathcal{A}\mathcal{B}}\,{\bf R}_{\mathcal{B}\mathcal{C}}$. These operators are not commutative and one must respect the order strictly ---a proper notation helps: 'AB' chains with 'BC' to create 'AC'.