lie-group state estimation

This work addresses the problem of identifying and reconstructing orbits of compact Lie group actions from point cloud data. We propose a novel method integrating geometric measure theory, computational geometry, and matrix manifold optimization to precisely recover the isomorphism type of the irreducible representation direct sum underlying an orbit—not merely detecting symmetry—and to invert the associated Lie group structure. The approach provides theoretical robustness guarantees under Hausdorff and Wasserstein distances for canonical compact Lie groups including SO(2), Tᵈ, SU(2), and SO(3). Evaluated on synthetic data up to 16 dimensions and real-world tasks in image analysis, harmonic analysis, and classical mechanics, our method achieves high-accuracy orbit identification and group structure inference. To our knowledge, this is the first end-to-end, provably falsifiable and reconstructible framework that bridges discrete point clouds to continuous group representations.

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 issue of state estimation error accumulation on Lie group manifolds caused by linearization of nonlinear observation models in tangent spaces. To circumvent linearization altogether, the authors propose a natural gradient Gaussian approximation filtering framework. The approach reformulates manifold-based filtering as a parameter optimization problem over Gaussian incremental variables, where increments are mapped onto the prior state via the exponential map and iteratively refined using natural gradients. Under invariant observation models, a closed-form covariance update is derived, achieving a favorable balance between accuracy and computational efficiency. Experimental validation on the Unitree GO2 quadruped robot across diverse terrains demonstrates approximately 40% reduction in estimation error compared to existing filters, with comparable computational overhead.

To address the demand for high-precision and robust trajectory tracking in quadrotor UAVs, this paper proposes a nonlinear Model Predictive Controller (MPC) formulated on the SE(2,3) Lie group manifold. Unlike conventional Euclidean-space modeling, the approach unifies pose and velocity representation on SE(2,3), explicitly preserving the system’s geometric structure while incorporating optimal control objectives and kinematic constraints. Extensive simulation and real-time hardware experiments are conducted on the Quanser QDrone platform. Results demonstrate that the proposed SE(2,3) MPC achieves significantly improved tracking accuracy—reducing average tracking error by 37%—and enhanced disturbance rejection compared to classical LQR and industrial-grade PID controllers, all while satisfying real-time requirements (≤5 ms per optimization step). The key contribution is the first systematic implementation and experimental validation of an SE(2,3)-geometric MPC on a physical quadrotor platform, thereby confirming the feasibility and superiority of Lie group–based control in resource-constrained embedded systems.

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 of generating smooth SE(3) trajectories that satisfy arbitrary boundary conditions while coupling rotation and translation, a task for which conventional numerical optimization methods are computationally prohibitive for real-time applications. The authors propose a learning-based framework that parameterizes body twist trajectories using high-order polynomials, where a neural network predicts a subset of coefficients and the trajectory duration, while the remaining coefficients are analytically determined from the boundary conditions. During training, the model incorporates Euler–Lagrange conditions, a metric-weighted smoothness objective, and feasibility constraints. This approach introduces, for the first time, metric-conditioned learning into SE(3) trajectory generation, supporting arbitrary left-invariant Riemannian metrics and boundary conditions. It achieves geometric consistency and millisecond-level inference speeds while closely approximating numerically optimized solutions, demonstrating success in real-time motion primitive generation and dynamic trajectory optimization for quadrotors.

This work proposes a dual quaternion-based 6-degree-of-freedom visual object tracking framework to address the sensitivity of conventional P$n$P methods to noise and outliers, as well as their difficulty in handling missing observations. By analyzing system observability through Lie algebra, the approach introduces a measurement model based on unit vectors and relative positions, and uniquely integrates Lie group unscented Kalman filtering with dual quaternions to achieve robust state estimation. The method provides a control-theoretic interpretation of the collinearity-induced degeneracy in P3P and is applicable to non-cooperative, non-smooth motion scenarios. Simulations demonstrate significant improvements over existing P$n$P solvers in both pose accuracy and robustness under occlusion, highlighting its suitability for applications such as visual-inertial navigation and SLAM.

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 problem of path-following and convergence control for fully actuated systems evolving on matrix Lie groups, such as SE(3). A novel vector field approach is proposed that extends Euclidean path-following strategies to general connected matrix Lie groups by leveraging their intrinsic geometric structure. The method constructs a control vector field comprising a component orthogonal to the desired path—ensuring convergence—and a tangential component aligned with the path direction, while employing non-redundant control inputs that exactly match the system’s degrees of freedom. In particular, a computationally efficient vector field algorithm tailored for SE(3) is developed. Experimental validation on a robotic manipulator platform demonstrates high-precision path tracking, confirming the method’s effectiveness for systems with coupled translational and rotational dynamics, such as omnidirectional drones.

This work addresses the challenge of achieving safe, interpretable, and real-time trajectory tracking for domestic service robots while preserving the geometric structure of variables such as SE(3) poses and SPD(n) stiffness/damping matrices—a balance that existing methods struggle to maintain between stability and accuracy. To this end, we propose the Curve-Induced Dynamical System on Manifolds (CDSM), which, for the first time, integrates a curve-induced mechanism into dynamical system modeling on Riemannian manifolds and Lie groups. By decomposing motion into tangential progression and normal attraction components, CDSM unifies stable convergence, online adaptability, and high-precision trajectory generation. Experiments demonstrate that CDSM significantly improves trajectory accuracy, reduces path deviation, and accelerates query speed on the S2 benchmark, with successful real-time adaptive control of both SE(3) and SPD(n) variables validated on robotic arms and mobile platforms.