derive invariant eskf

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.

A long-standing debate in robotics concerns whether left- and right-invariant extended Kalman filters (IEKFs) are equivalent and whether their chirality must match the measurement model’s. Method: This paper proves that, on Lie groups, left- and right-IEKFs are mathematically equivalent when a proper state reset step is incorporated—grounded in group-affine system theory and validated via simulations in a GNSS-aided inertial navigation framework. Results: The reset mechanism eliminates chirality-induced discrepancies and substantially improves the asymptotic estimation performance of IEKF. Crucially, the work refutes the common misconception that IEKF chirality must be selected based on the measurement model, establishing the reset step as an essential design element for ensuring consistency and robustness of invariant filtering. This provides both a unifying theoretical perspective and practical guidance for state estimation on Lie groups.

Long-standing inconsistencies in symmetry modeling hinder unified filter design for inertial navigation systems (INS). Method: This work proposes a systematic reconstruction framework grounded in equivariant symmetry. It introduces, for the first time in INS, two novel classes of Lie group equivariant symmetries, establishes a comprehensive taxonomy encompassing all admissible symmetry choices, and integrates equivariant filtering (EqF), matrix Lie group theory, and nonlinear stochastic observer design to achieve a unified modeling framework for both the extended Kalman filter (EKF) and its modern Lie group variants. Contribution/Results: The framework reveals the fundamental mechanisms underlying performance differences among existing filters, precisely delineates their applicability boundaries and intrinsic accuracy limits, and delivers an interpretable, scalable paradigm for designing high-precision, real-time navigation filters—thereby laying a rigorous theoretical foundation for enhancing robustness and consistency in IMU/GNSS integrated navigation.

This paper addresses state estimation under Stiefel manifold geometric constraints. We propose an extended Kalman filter (EKF) framework rigorously defined on the Stiefel manifold, departing from conventional Euclidean-space EKFs. Our method performs linearization in the tangent space, and employs exponential and logarithmic maps for observation updates—thereby preserving orthogonality constraints exactly. Theoretically, we derive the recursive filtering equations and covariance propagation rules intrinsic to the Stiefel manifold. Algorithmically, the framework supports arbitrary St(n,p) manifolds and accommodates common subcases including the unit sphere S² and the 4×2 orthogonal matrix manifold. Simulation results demonstrate that, compared to standard EKF applied naively in the ambient Euclidean embedding space, our approach achieves significantly higher estimation accuracy and superior constraint satisfaction—validating its effectiveness and robustness for non-Euclidean state estimation.

This work addresses the degraded estimation performance in 3D landmark-aided inertial navigation caused by spurious observability in conventional filters. It introduces, for the first time, the Iterated Invariant Extended Kalman Filter (Iterated Invariant EKF) to this setting. By constraining the state to the observation manifold under low-noise conditions and restricting uncertainty to its tangent space, the method effectively preserves the system’s inherent observability structure. The approach integrates Lie group geometry, group-affine system modeling, and a 3D landmark observation model. Numerical simulations demonstrate that it significantly outperforms classical SO(3)-EKF, iterated SO(3)-EKF, and standard IEKF, achieving notable improvements in both estimation accuracy and consistency.

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 challenge of balancing estimation accuracy and computational cost in robotic state estimation by proposing a Smart Scheduling Hybrid (SSH) framework. The approach integrates an Extended Kalman Filter (EKF) for state propagation with periodic invocations of a fixed-structure batch optimization module. Crucially, the scheduling of optimization updates is explicitly modeled as an independent design variable, revealing its pivotal role in governing the trade-off between accuracy and computational expense. Validation on planar SLAM simulations demonstrates that well-designed scheduling strategies can substantially reduce runtime while effectively mitigating pre-optimization drift and transient errors. This enables the system to retain most of the benefits of global optimization at a significantly lower computational cost.