lie group modeling

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 paper addresses the problem of high-precision pose and velocity estimation for rigid bodies. We propose a geometric unified observation framework based on the Lie group SE(5). By fusing IMU measurements with generic inertial-frame or body-frame measurements, we construct, for the first time on SE(5), a decoupled geometric error dynamics model—where translational error evolution mimics continuous-time Kalman filtering, enabling Riccati-equation-driven time-varying gain design and guaranteeing almost global asymptotic stability. Our approach overcomes limitations of conventional Euclidean-space modeling, achieving intrinsic decoupling of error dynamics, simplification of observer structure, and enhanced robustness. Extensive simulations—including stereo-camera-aided and GPS-aided inertial navigation systems—demonstrate its effectiveness. The method significantly improves the generality and engineering applicability of nonlinear geometric observers for high-accuracy state estimation.

This paper addresses the joint estimation of orientation, gravity vector, linear velocity, and landmark positions in 3D rigid-body SLAM. We propose a geometric modeling framework based on the novel matrix Lie group $SE_{3+n}(3)$, enabling the first unified, integrated representation of these four state components. Building upon this, we design a nonlinear geometric observer with almost global asymptotic stability, which fuses IMU preintegration with robust landmark measurements—tolerant to outliers—while being inherently insensitive only to ambiguities in heading (rotation about the gravity axis) and global translation. Simulation results demonstrate that the method guarantees consistent convergence of pose and map estimates even under large initial errors, significantly improving geometric consistency and system robustness. It thus meets the stringent requirements of practical SLAM systems regarding both stability and accuracy.

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.

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 work addresses the challenge of achieving accurate, consistent, and convergent pose (orientation, velocity, and position) estimation in multi-IMU articulated rigid-body systems, where incorporating joint kinematic constraints without compromising the convergence and consistency of invariant filtering remains difficult. The authors propose a Lie group representation of relative L-extended poses, formulate a group-affine dynamic model, and embed kinematic constraints as noise-free pseudo-observations within an iterative invariant extended Kalman filter (IterIEKF). This approach introduces, for the first time in an invariant filtering framework, a Lie group–based state representation and constraint fusion mechanism that preserves both convergence and consistency for articulated systems. Evaluated on a UR5e robotic arm and a human leg, the method demonstrates faster convergence, lower inter-run variability, and at least 50% reduction in RMSE compared to various EKF and IterIEKF baselines, consistently achieving state-of-the-art estimation accuracy.

This work addresses the singularities and ill-conditioning arising in rigid-body trajectory optimization when Euclidean-space approaches neglect the underlying manifold structure. To overcome these issues, we propose a structure-aware constrained optimization framework formulated directly on matrix Lie groups. Built upon a second-order rigid-body dynamics model, our method uniquely embeds an interior-point algorithm into the Lie group manifold, integrating a line-search strategy with a Lie group variational integrator to preserve rotational topology while avoiding singularities. By exploiting group symmetries, we derive closed-form intrinsic derivatives and employ intrinsic Newton-type updates for efficient solution computation. Experimental results demonstrate that the proposed framework significantly outperforms both general-purpose solvers and existing structure-aware methods in terms of convergence speed and robustness.

This work addresses the challenge of modeling multibody system dynamics in scenarios where velocity data are missing or corrupted by noise. We propose a learning framework grounded in discrete forced Euler–Lagrange equations on Lie groups, which directly models dynamics in the manifold configuration space. This approach inherently preserves the system’s geometric structure and conservation laws while explicitly incorporating external control inputs. As the first framework to integrate Lie group geometric mechanics with purely position-based data-driven learning, our method synergistically combines discrete variational mechanics, geometric deep learning, and multibody dynamics modeling. Evaluated on both synthetic and real-world datasets, it demonstrates superior accuracy and robustness, effectively retaining physical priors and geometric invariances.

This work addresses the absence of analytical solutions for Bayesian filtering in nonlinear dynamical systems, where existing Gaussian approximation methods often neglect the underlying geometric structure of probability distributions. From an information-geometric perspective, the authors model prediction and measurement update steps as inference processes on the manifold of Gaussian distributions and propose the geometry-aware NANO filter. This approach iteratively refines the posterior mean and covariance via a single-step natural gradient descent, preserving covariance positive definiteness while exactly recovering the Kalman update in the linear Gaussian case. The method demonstrates superior accuracy and robustness across diverse applications, including satellite attitude estimation, SLAM, and state estimation for quadrupedal and humanoid robots.