A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

2026-07-27

AI Summary

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.

Abstract

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. Inmultibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group SO (3)R3, although rigid body motions form the semidirect product Lie group SE (3). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of SE (3) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

Problem

• constrained rigid body
• differential-algebraic equations
• Lie group
• configuration space
• geometric constraints

Innovation

• SE(3)
• Lie group
• holonomic constraints
• differential-algebraic equations
• geometric integration