T-ESKF: Transformed Error-State Kalman Filter for Consistent Visual-Inertial Navigation

TL;DR

This work tackles the persistence of estimation inconsistency in visual-inertial navigation caused by observability-mismatch in the ESKF. It introduces T-ESKF, a consistent VINS estimator built on a linear time-varying transformation of the error-state that yields a state-independent unobservable subspace, preserving correct observability across linearization points. The authors derive an efficient covariance propagation mechanism that leverages the transformation to keep the IMU-core computations fixed-size, enabling scalable operation with many landmarks. They analytically prove observability properties and demonstrate competitive accuracy and strong consistency in Monte-Carlo simulations and real-world datasets (e.g., EuRoC, TUM-VI), with code publicly available. The results indicate that transforming the error-state provides a principled path to mitigating observability-induced inconsistencies while preserving computational efficiency for practical VINS deployments.

Abstract

This paper presents a novel approach to address the inconsistency problem caused by observability mismatch in visual-inertial navigation systems (VINS). The key idea involves applying a linear time-varying transformation to the error-state within the Error-State Kalman Filter (ESKF). This transformation ensures that \textrr{the unobservable subspace of the transformed error-state system} becomes independent of the state, thereby preserving the correct observability of the transformed system against variations in linearization points. We introduce the Transformed ESKF (T-ESKF), a consistent VINS estimator that performs state estimation using the transformed error-state system. Furthermore, we develop an efficient propagation technique to accelerate the covariance propagation based on the transformation relationship between the transition and accumulated matrices of T-ESKF and ESKF. We validate the proposed method through extensive simulations and experiments, demonstrating better (or competitive at least) performance compared to state-of-the-art methods. The code is available at github.com/HITCSC/T-ESKF.

Figures (15)

• Figure 1: The unobservable subspace of the original system depends on the state, while that of the transformed system is state-independent. The dimension of ESKF unobservable subspace is reduced by one (with the global rotation in ESKF becoming erroneously observable). In contrast, the unobservable subspace of T-ESKF remains invariant to changes in linearization points.
• Figure 2: Pipeline of T-ESKF. It propagates and updates the covariance estimates in the transformed space. The state estimate is propagated by integrating \ref{['equ:ode1']} and updated using the state correction derived from the transformed system.
• Figure 3: Simulated trajectories (black dashed lines) with green and red stars marking the starting and ending points, respectively, and estimated trajectories (solid lines).
• Figure 4: The NEES of 1000 Monte-Carlo simulation runs. The upper three subfigures show the average NEES over time for 1000 runs. The lower three subfigures depict the frequency distribution of the NEES for 1000 runs throughout the simulation time, with the black dashed lines representing the theoretical value of the chi-square distribution. The lines of T-ESKF and RI-EKF coincide.
• Figure 5: Orientation and position RMSE of 100 Monte-Carlo runs with various measurement noises on Udel-Gore.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6