扩展位置动力学中的方向：应用于刚体与柯西拉杆
Orientation in Extended Position-Based Dynamics: Application to Rigid Bodies and Cosserat Rods
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中文总结 AI 辅助
该研究将李理论引入扩展位置动力学，推导刚体约束公式提升其动态一致性超10^4倍，还扩展至柯西拉杆模拟，验证了大变形与接触场景下的实用性。
中文摘要 AI 辅助
扩展位置动力学（XPBD）中的旋转自由度需要在三维旋转的非线性流形上进行计算。本文证明李理论为XPBD中的旋转、约束、插值和微分提供了简洁统一的框架，可改进刚体约束并实现高阶有限元柯西拉杆。我们推导了刚体模拟的显式李理论约束公式及其梯度，使XPBD中受约束刚体模拟的动态一致性较现有技术提升超10^4倍。该框架通过实现节点旋转的流形上插值自然扩展至有限元柯西拉杆，线性有限元的表现优于传统的刚体链离散化，高阶基函数可提供更平滑的解并加快收敛速度。其实用性在含大变形和接触的多个示例中得到验证。
英文摘要
Rotational degrees of freedom in Extended Position-Based Dynamics (XPBD) require computations on the nonlinear manifold of 3D rotations. We show that Lie theory provides a clean, unified framework for expressing rotations, constraints, interpolation, and differentiation in XPBD, enabling both improved rigid-body constraints and higher-order finite-element Cosserat rods. We derive explicit Lie-theoretic constraint formulations and their gradients for rigid-body simulation, improving the dynamic consistency of constrained rigid-body simulations in XPBD by a factor of over $10^4$ compared to the state-of-the-art. Our framework naturally extends to finite-element Cosserat rods by enabling on-manifold interpolation of nodal rotations. Linear finite elements outperform the conventional chain-of-rigid-bodies discretization, while higher-order basis functions provide even smoother solutions and faster convergence. Utility is demonstrated in a variety of examples with large deformations and contact.
