Orientation in Extended Position-Based Dynamics: Application to Rigid Bodies and Cosserat Rods

2026-08-20

AI Summary

本文使用李群理论解决扩展位置动力学中的旋转自由度问题，提高刚体模拟和Cosserat杆的动态一致性和计算效率。

Abstract

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.

Problem

• Extended Position-Based Dynamics
• rotational degrees of freedom
• nonlinear manifold
• rigid-body constraints
• Cosserat rods

Innovation

• Lie theory
• rotational degrees of freedom
• XPBD
• Cosserat rods
• on-manifold interpolation