rigid-body kinematics

This work addresses the critical influence of configuration space selection on constraint satisfaction accuracy in numerical simulations of fully constrained rigid body dynamics. From a geometric perspective, the study proposes a differential-algebraic equation (DAE) formulation and geometric integration scheme based on the Lie group SE(3). It demonstrates that when kinematic constraints correspond to subgroups of SE(3), these constraints can be preserved exactly over time. The approach elucidates the intrinsic relationship between SE(3) subgroup structures and lower-pair joints, establishing that employing SE(3) as the configuration space enables strict enforcement of constraints. This result provides both a theoretical foundation and numerical guarantees for high-fidelity simulation of rigid multibody systems.

In contact-rich environments, existing zeroth-order optimization methods are robust but computationally inefficient, hindering gradient-based perception, planning, and control. Method: We propose the first fully analytical, twice continuously differentiable (C²) contact dynamics modeling framework that unifies contact detection and rigid-body dynamics. Our formulation ensures smooth differentiability throughout forward and inverse dynamics—without iterative solvers or convex decomposition—and achieves contact-count-independent computational complexity. The method integrates analytical geometry, implicit distance fields, Log-Sum-Exp smoothing, Lagrangian mechanics derivation, and automatic differentiation–friendly design, supporting collisions between arbitrary geometries. Results: Simulation experiments demonstrate that this “reasoning-friendly physics engine” significantly improves planning and control efficiency for both first- and second-order gradient-based methods, establishing a new paradigm for generating intelligent behaviors in highly contact-rich scenarios.

This work addresses the problem of learning equations of motion for mechanical systems solely from discrete positional observations—such as motion-capture trajectories, pixel coordinates, or low-resolution tracking data—without requiring explicit velocity measurements. We propose a neural dynamics modeling framework grounded in the discrete forced Lagrange–d’Alembert principle, the first to integrate this variational principle with feedforward neural networks and autoencoder latent spaces: conservative and non-conservative forces are learned separately, while symplectic geometric structure is inherently preserved. Physical consistency is enforced by constraining the model to satisfy the discrete Euler–Lagrange equations. The method is validated on synthetic mechanical systems, real human motion capture data, and latent embeddings of image sequences, achieving high-fidelity trajectory reconstruction and interpretable decomposition into conservative and external forcing terms. Our core contribution is the first purely position-driven dynamical learning paradigm that simultaneously ensures physical fidelity, data efficiency, and structural interpretability.

Long-term prediction of complex multi-body collisions in rigid-body dynamics remains challenging, with existing graph neural networks (GNNs) exhibiting significant limitations in modeling long-range interactions and generalizing across diverse geometric configurations and initial conditions. Method: We propose a novel physics-informed neural dynamics model that integrates physical priors with high-order topological structures. Specifically, we introduce simplicial complex embeddings into neural networks to enable topology-aware, physically constrained message passing, and jointly regularize the model using rigid-body dynamical equations and physics-informed neural networks (PINNs). Contribution/Results: Our approach achieves substantial improvements in long-horizon rollout accuracy and demonstrates strong generalization to unseen geometries and initial states. By embedding domain knowledge—both physical laws and topological relationships—into the architecture, the model yields interpretable, robust, and broadly applicable simulations for multi-body systems, establishing a generalizable and explainable paradigm for neural dynamical modeling.

This work addresses gait optimization and motion planning for mobile systems with hybrid kinematic–dynamic characteristics—such as nonholonomic wheeled robots and biomimetic swimming robots. We propose a geometric modeling and optimization framework that unifies second-order dynamics with nonholonomic constraints. For the first time, we integrate Lie group numerical integrators with Lagrangian reduction, leveraging manifold symmetries to enable variational gait optimization. The framework supports anisotropic added inertia and fluid drag modeling, overcoming limitations of Euclidean-space-based optimization. Evaluated on roller racer, snakeboard, and Purcell swimmer platforms, our method efficiently generates diverse locomotion behaviors—including acceleration, steady-state cruising, steering, and smooth multi-gait transitions. Both simulation and physical experiments demonstrate high accuracy and strong generalizability across distinct underactuated, nonholonomic systems.

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 study addresses the challenge of efficiently computing high-order dynamics for floating-base tree-structured robots. By leveraging the Lie group SE(3) and spatial vector algebra, the authors formulate, for the first time, unified high-order Newton–Euler, articulated-body inertia, and hybrid recursive algorithms within a consistent Lie group framework, and derive closed-form equations of motion. Key theoretical contributions include proving that the articulated inertia tensor remains invariant under arbitrary-order time derivatives and identifying a Coriolis matrix structure that satisfies passivity. The method is applied to a 12-degree-of-freedom aerial manipulator, enabling analytical computation of forward and inverse dynamics along with their first derivatives, and efficient numerical evaluation up to fifth-order derivatives. The computational complexity grows only quadratically with the derivative order, offering a significant advantage over the exponential complexity of automatic differentiation.

This study addresses the challenge of accurately modeling the dynamics of flexible two-degree-of-freedom robotic arms, where rigid-body assumptions fail to capture link flexibility and unmodeled residual dynamics. To overcome this limitation, the authors propose a semi-parametric hybrid dynamical modeling framework that augments rigid-body dynamics with a Gaussian Mixture Model (GMM) to learn residual terms, while employing a purely data-driven kinematic regression as a baseline. This approach synergistically integrates physical priors with data-driven mechanisms, transcending the constraints of conventional fully parametric models in flexible systems. Experimental results on an open-source dataset demonstrate that the data-driven component—combined with regularization and least-squares estimation—significantly enhances torque prediction accuracy, thereby validating the efficacy and superiority of the proposed hybrid model.

This work addresses the challenge that trajectory representations often fail to maintain consistent identification and generalization performance across different coordinate systems, as existing coordinate-invariant formulations are susceptible to measurement noise and suffer from singularities. To overcome these limitations, the paper proposes a Dual Upper-Triangular Invariant Representation (DUTIR), which leverages differential geometry and invariant theory to construct computable local coordinate-invariant features. This approach enables unified modeling of rigid-body motion and interaction-force trajectories under arbitrary coordinate frames. The method demonstrates significantly enhanced robustness against both singularities and sensor noise, achieving stable and consistent performance in trajectory segmentation, recognition, and prediction across multiple coordinate systems. Its effectiveness and broad applicability are validated through experiments in robotics and biomechanics.

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.