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

Samuel Tobin; Caleb Rucker

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arXiv · 2608.23606

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

Samuel Tobin·Caleb Rucker

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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.

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BibTeX RIS

Samuel Tobin, Caleb Rucker. 2026-08-21. Orientation in Extended Position-Based Dynamics: Application to Rigid Bodies and Cosserat Rods. https://arxiv.org/abs/2608.23606

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