Lie Group Methods

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Overview

Lie group methods study transformations such as rotations and rigid-body motions using groups and their tangent Lie algebras, preserving composition and geometric constraints. They support robotics, state estimation, trajectory optimization, simulation, and symmetry-aware learning through tools such as exponential and logarithm maps, invariant metrics, geometric integrators, and manifold optimization. Current approaches also incorporate group structure into neural operators, attention, diffusion models, and symmetry discovery, with tests spanning robot control, pose graphs, physical dynamics, and molecular conformations. A central challenge is to preserve geometric guarantees and global validity while keeping derivatives, optimization, and sampling efficient, particularly near singularities or on complex groups.

Latest papers

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42 papers3 papers in the last four weeks, down 50% on the four weeks before. 0.0% of all new papers.

May 7, 2026

Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.

Ahmed Ali, Chiara Gabellieri, Antonio Franchi

Robotics and Mechatronics Department, Electrical Engineering, Mathematics, and Computer Science (EEMCS) Faculty, University of Twente, 7500 AE Enschede, The Netherlands. · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy.

Multibody DynamicsRoboticsLie Group MethodsInverse Dynamics
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May 4, 2026

Exact Higher-Order Derivatives for SE(3) via Analytical/AD Methods

Fast prototyping of new SE(3) estimation objectives remains awkward in practice. Modern Lie-group frameworks -- GTSAM, manif, Sophus, SymForce, Ceres -- target first-order workloads through different code-generation and automatic-differentiation strategies, each optimized for a particular seam between hand-derived geometry and generic differentiation. The remaining gap is a compact, AD-safe path from these first-order primitives to exact Hessians, observed-information matrices, and higher-order derivative tensors: the quantities needed for exact Newton steps, observed-information covariance estimates, and covariance correction. This paper presents a hybrid analytical/AD recipe for SE(3) negative log-likelihoods. The practitioner writes the NLL gradient once, generic over a scalar type, and places the analytical/AD seam at the point-action interface y = Tx. Closed-form Lie-group Jacobians are used up to this interface; AD is applied only beyond it. The same source is then instantiated with ordinary floating-point scalars for gradients, vector-seeded dual numbers for exact Hessians in a single forward-mode pass, and nested dual numbers for higher-order derivative tensors. On a representative 6-DoF, 5-landmark SE(3) NLL, the advocated seeded-Hessian path is approximately 5x faster than finite-differencing the AD gradient on this benchmark while matching a nested-AD oracle to machine precision. The implementation adds roughly 70 lines of analytical-Jacobian code over an AD-only baseline. We also identify and fix a removable singularity in the standard SO(3)/SE(3) scalar basis that would otherwise produce NaNs at the origin under seeded AD, and we audit which Lie-group derivative tensors require this stabilized basis. The result is a practical path from rapidly written SE(3) objectives to exact higher-order derivatives, with predictable runtime and no finite-difference tuning.

Frank O. Kuehnel

Sigma Point Labs, Denver, CO, USA

Automatic DifferentiationLie Group MethodsSecond-Order OptimizationDifferentiable Optimization
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Apr 24, 2026

Closed Form Relations and Higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3)

The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimization schemes necessitates the exponential map, its right-trivialized differential (often referred to as tangent operator), as well as higher derivatives in closed form. The 6×66\times 6 matrix representation of the differential, dexpX:se(3)→se(3)\mathbf{dexp}_{\mathbf{X}}:se\left( 3\right) \rightarrow se\left( 3\right) , and its first derivative were reported using a 3×33\times 3 block partitioning. In this paper, the differential, its first and second derivative, as well as the Jacobian and Hessian of the evaluation maps, dexpXZ\mathbf{dexp}_{\mathbf{X}}\mathbf{Z} and dexpXT\mathbf{dexp}_{\mathbf{X}}^{T}% \mathbf{Z}, are reported avoiding the block partitioning. For all of them, higher-order approximations are derived. Besides the compactness, the advantage of the presented closed form relations is their numerical robustness when combined with the local approximation. The formulations are demonstrated for computation of the deformation field and the strain rates of an elastic Cosserat-Simo-Reissner rod.

Andreas Mueller

Institute of Robotics, Johannes Kepler University Linz, Austria

Representation GeometryLie Group MethodsCosserat Rod Modeling
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Apr 11, 2026

Natural Gradient Gaussian Approximation Filter on Lie Groups for Robot State Estimation

Accurate state estimation for robotic systems evolving on Lie group manifolds, such as legged robots, is a prerequisite for achieving agile control. However, this task is challenged by nonlinear observation models defined on curved manifolds, where existing filters rely on local linearization in the tangent space to handle such nonlinearity, leading to accumulated estimation errors. To address this limitation, we reformulate manifold filtering as a parameter optimization problem over a Gaussian-distributed increment variable, thereby avoiding linearization. Under this formulation, the increment can be mapped to the Lie group through the exponential operator, where it acts multiplicatively on the prior estimate to yield the posterior state. We further propose a natural gradient optimization scheme for solving this problem, whose iteration process leverages the Fisher information matrix of the increment variable to account for the curvature of the tangent space. This results in an iterative algorithm named the Natural Gradient Gaussian Approximation on Lie Groups (NANO-L) filter. Leveraging the perturbation model in Lie derivative, we prove that for the invariant observation model widely adopted in robotic localization tasks, the covariance update in NANO-L admits an exact closed-form solution, eliminating the need for iterative updates thus improving computational efficiency. Hardware experiments on a Unitree GO2 legged robot operating across different terrains demonstrate that NANO-L achieves approximately 40% lower estimation error than commonly used filters at a comparable computational cost.

Tianyi Zhang, Wenhan Cao, Chang Liu +2

School of Vehicle and Mobility, Tsinghua University, Beijing, China · College of Engineering, Peking University, Beijing, China · School of Vehicle and Mobility and College of Artificial Intelligence, Tsinghua University, Beijing, China

Robot State EstimationBayesian FilteringLie Group MethodsNatural Gradient Methods
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Apr 7, 2026

NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization

Group synchronization is a fundamental task involving the recovery of group elements from pairwise measurements. For orthogonal group synchronization, the most common approach reformulates the problem as a constrained nonconvex optimization and solves it using projection-based methods, such as the generalized power method. However, these methods rely on exact SVD or QR decompositions in each iteration, which are computationally expensive and become a bottleneck for large-scale problems. In this paper, we propose a Newton-Schulz-based Riemannian Gradient Scheme (NS-RGS) for orthogonal group synchronization that significantly reduces computational cost by replacing the SVD or QR step with the Newton-Schulz iteration. This approach leverages efficient matrix multiplications and aligns perfectly with modern GPU/TPU architectures. By employing a refined leave-one-out analysis, we overcome the challenge arising from statistical dependencies, and establish that NS-RGS with spectral initialization achieves linear convergence to the target solution up to near-optimal statistical noise levels. Experiments on synthetic data and real-world global alignment tasks demonstrate that NS-RGS attains accuracy comparable to state-of-the-art methods such as the generalized power method, while achieving nearly a 2×\times speedup.

Haiyang Peng, Deren Han, Xin Chen +1

Lie Group MethodsRiemannian Optimization
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Mar 14, 2026

Schrödinger Bridge Over A Compact Connected Lie Group

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on SO(2)\mathsf{SO}(2) and SO(3)\mathsf{SO}(3). The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .

Hamza Mahmood, Abhishek Halder, Adeel Akhtar

Department of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA

Schrödinger BridgesStochastic Optimal ControlLie Group Methods
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Mar 12, 2026

A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

In this paper, we propose a novel mathematical framework for piecewise diffeomorphic image registration that involves discontinuous sliding motion using a diffeomorphism groupoid and algebroid approach. The traditional Large Deformation Diffeomorphic Metric Mapping (LDDMM) registration method builds on Lie groups, which assume continuity and smoothness in velocity fields, limiting its applicability in handling discontinuous sliding motion. To overcome this limitation, we extend the diffeomorphism Lie groups to a framework of discontinuous diffeomorphism Lie groupoids, allowing for discontinuities along sliding boundaries while maintaining diffeomorphism within homogeneous regions. We provide a rigorous analysis of the associated mathematical structures, including Lie algebroids and their duals, and derive specific Euler-Arnold equations to govern optimal flows for discontinuous deformations. Numerical tests are performed to validate the efficiency of the proposed approach.

Lili Bao, Bin Xiao, Shihui Ying +1

Department of Computer Science, University of Copenhagen, Universitetsparken 1, Copenhagen 2100, Denmark · Institute for Medical Imaging Technology, Ruijin Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 201800, China · Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China +1

Deformable Image RegistrationLie Group Methods
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Mar 9, 2026

A General Lie-Group Framework for Continuum Soft Robot Modeling

This paper introduces a general Lie group framework for modeling continuum soft robots, employing Cosserat rod theory combined with cumulative parameterization on the Lie group SE(3). This novel approach addresses limitations present in current strain-based and configuration-based methods by providing geometric local control and eliminating unit quaternion constraints. The paper derives unified analytical expressions for kinematics, statics, and dynamics, including recursive Jacobian computations and an energy-conserving integrator suitable for real-time simulation and control. Additionally, the framework is extended to handle complex robotic structures, including segmented, branched, nested, and rigid-soft composite configurations, facilitating a modular and unified modeling strategy. The effectiveness, generality, and computational efficiency of the proposed methodology are demonstrated through various scenarios, including large-deformation rods, concentric tube robots, parallel robots, cable-driven robots, and articulated fingers. This work enhances modeling flexibility and numerical performance, providing an improved toolset for designing, simulating, and controlling soft robotic systems.

Lingxiao Xun, Benoît Rosa, Jérôme Szewczyk +1

Sorbonne Université, CNRS UMR 7222, Inserm U1150, ISIR, F-75005, Paris, France. · Université de Strasbourg, CNRS, INSERM, ICube, UMR7357, Strasbourg, France.

Soft RoboticsLie Group MethodsContinuum Robot Control
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Mar 7, 2026

Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.

Pavan Karjol, Kumar Shubham, Prathosh AP

Department of Electrical Communication Engineering , Indian Institute of Science , Bengaluru, Karnataka

Symmetry DiscoverySpectral MethodsLie Group Methods
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Feb 25, 2026

Vector Fields for Path Following on Lie Groups with Application in Robot Control

Many robotic systems allow independent control of position and orientation (pose), including omnidirectional aerial vehicles, underwater robots, and manipulator end-effectors. In many applications, these systems must follow a continuous sequence of poses, leading to either trajectory-tracking or path following formulations. Compared to trajectory-tracking, path following offers important practical advantages. In particular, we focus on the problem of path following on Lie groups. Considering the robots as rigid bodies moving in the 3D space, this path-following problem can be posed as a problem of designing guiding vector fields on the matrix Lie group SE(3). In this paper, we develop a general vector-field framework for path following on connected matrix Lie groups, of which SE(3) is a prominent special case. The proposed vector field guarantees convergence to a desired parametric curve from almost all initial conditions while ensuring continuous motion along the path. Furthermore, another interesting feature is that, as opposed to previous works, the control input is "minimal" in terms of representation and closer to the engineering application (e.g., the body twist in the case SE(3)). After establishing the general case, the framework is then specialized to SE(3), of special interest in robotics, yielding an efficient algorithm suitable for real-time robotic control. Experiments with a robotic manipulator tracking complex pose paths demonstrate the effectiveness of the approach. An open-source implementation is also provided.

Felipe Bartelt, Luciano C. A. Pimenta, Weijia Yao +1

Robotic ControlLie Group MethodsRobot Navigation
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Feb 18, 2026

Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient plug-and-play module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50% at the cost of 2.26% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.

Jiaquan Zhang, Fachrina Dewi Puspitasari, Songbo Zhang +7

School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, China · Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China · Institute of Software Chinese Academy of Sciences, Beijing, China +2

Neural PDE SolversPDE Operator LearningNeural OperatorsLie Group Methods
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Sep 22, 2024

Making Space for Time: The Special Galilean Group and Its Application to Some Robotics Problems

The special Galilean group, usually denoted SGal(3), is a 10-dimensional Lie group whose important subgroups include the special orthogonal group, the special Euclidean group, and the group of extended poses. We briefly describe SGal(3) and its Lie algebra and show how the group structure supports a unified representation of uncertainty in space and time. Our aim is to highlight the potential usefulness of this group for several robotics problems.

Jonathan Kelly

Space & Terrestrial Autonomous Robotic Systems (STARS) Laboratory, University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, Canada

Lie Group MethodsSpatiotemporal Reasoning
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