derive error-state kalman filter

This work addresses the limitations of conventional local navigation algorithms, which neglect Earth’s curvature, rotation, and gravity variations, thereby failing to meet the demands of high-precision state estimation at a global scale. By leveraging Lie group symmetry and invariant Kalman filtering theory, the paper systematically derives and unifies the global navigation dynamics for four classes of error-state Kalman filters—including standard, left-invariant, and right-invariant formulations. It presents the first comprehensive comparison of ESKF equations under different error-state representations, clarifying their respective applicability conditions and performance characteristics in global scenarios. The proposed framework accommodates complex sensor configurations and dynamic environments, delivering a directly implementable, high-precision, and robust state estimation algorithm that advances the practical deployment of trajectory-independent error propagation theory.

This work addresses the challenge that nonlinear Kalman filters—such as the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF)—often struggle to balance robustness and accuracy due to a lack of systematic design principles. To this end, the paper introduces a covariance compensation framework that quantifies the deviation from EKF’s covariance prediction and establishes design criteria for performance improvement. It presents, for the first time, the concept of covariance compensation along with three core guidelines: invariance under orthogonal transformations, sufficient compensation relative to the EKF baseline, and a preference for underconfident compensation magnitudes. Through theoretical analysis and numerical experiments, the study demonstrates that adherence to these principles significantly enhances estimation accuracy and reveals that commonly adopted fixed-parameter strategies in the literature are generally suboptimal.

To address observability mismatch and state estimation inconsistency in visual-inertial navigation systems (VINS) caused by linearization-point dependency, this paper proposes a consistency-aware filtering method based on an error-state linear time-varying transformation. The method employs a tightly coupled error-state Kalman filter (ESKF) framework that fuses visual and inertial measurements. Its core contribution lies in constructing a linearization-point-invariant unobservable subspace, thereby preserving the observability structure of the error-state system across arbitrary operating points; additionally, an efficient covariance propagation algorithm is designed to reduce computational overhead. Extensive evaluations on multiple simulated and real-world datasets demonstrate that the proposed approach achieves positioning accuracy comparable to or superior to state-of-the-art VINS methods. The implementation is publicly available as open-source software.

Traditional Kalman filters for nonlinear systems—such as the Extended, Unscented, and Cubature Kalman Filters—suffer from overconfident state estimates, underestimated covariances, and degraded accuracy due to nonlinear measurement functions. This paper identifies, for the first time, the intrinsic bias in covariance propagation underlying these deficiencies and proposes a general, plug-and-play correction framework. Grounded in Bayesian estimation reformulation and explicit covariance propagation calibration, the framework is compatible with mainstream nonlinear Kalman filter variants. Theoretical analysis and extensive experiments across five canonical tasks demonstrate that the method reduces state estimation error by one to three orders of magnitude—particularly under low measurement noise—while significantly improving covariance fidelity and effectively mitigating overconfidence.

This study addresses the lack of quantitative criteria for selecting between closed-loop and open-loop Kalman filter architectures in airborne aided inertial navigation. To this end, the authors propose a unified simulation framework to systematically evaluate the performance differences of these two error-state Kalman filtering approaches under varying inertial sensor accuracy levels. The methodology employs standard inertial mechanization in the geodetic frame combined with direct position aiding, enabling a comprehensive analysis of the trade-off between fusion smoothness and long-term stability. The work provides the first quantitative evidence that, with high-accuracy IMUs, the open-loop architecture yields smoother state fusion, whereas the closed-loop configuration offers superior long-term navigation stability, thereby offering empirical guidance for filter architecture selection.

This work addresses the limitations of traditional Kalman filtering in accurately modeling uncertainty in pose orientation by proposing a joint estimation framework that integrates FoundationPose with an Ensemble Directional Kalman Filter (EnDKF). The method represents pose using unit quaternions and incorporates directional statistics to overcome the conventional assumptions of Gaussianity and linear covariance propagation inherent in standard filters. Experimental results demonstrate that the proposed approach significantly reduces both positional and orientational tracking errors on synthetic datasets and in digital twin head-tracking tasks, outperforming baseline methods that rely solely on raw observations.

This work addresses the lack of geometric consistency in multi-source information fusion for aided inertial navigation systems by constructing a control-oriented Lie group framework based on the extended special Euclidean group SE₂(3), which explicitly captures the system’s symmetry. By unifying high-order state modeling, synchronous observers, and equivariant filtering, the authors propose a geometrically coherent and invariant fusion mechanism. The resulting approach establishes a systematic and engineering-feasible paradigm for modern navigation design, significantly enhancing both accuracy and robustness while preserving theoretical rigor.

This work addresses the challenge of attitude estimation when only scalar inertial measurements—such as partial vector observations along a single body-fixed axis—are available. The authors propose a novel complementary filter formulated directly on the SO(3) manifold, wherein the innovation term is specifically restructured to accommodate the scalar output structure. They establish almost global asymptotic stability of the attitude estimate under the condition that at least three inertial vectors are measured along the same body axis, and further derive sufficient conditions for convergence in two distinct dual-scalar measurement configurations. Numerical experiments demonstrate that the proposed method maintains robustness and effectiveness even under severe sensor constraints or with emerging scalar sensing modalities.