A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body
2026-07-27 • Robotics
Robotics
AI summaryⓘ
The authors explain how to model the movement of a rigid body that has constraints using math equations called differential-algebraic equations (DAEs). They find that the way these constraints hold true depends on which mathematical space, called a Lie group, is used to describe the body's position and orientation. Specifically, they show that using the full group SE(3) better preserves these constraints during numerical simulations than the simpler space SO(3) × R3. The authors conclude that SE(3) is the better choice for accurately modeling constrained rigid body motions, though this may not straightforwardly apply to more complex systems with multiple bodies.
holonomic constraintsrigid body dynamicsNewton-Euler equationsdifferential-algebraic equationsLie groupSE(3)SO(3)configuration spacenumerical integrationmultibody systems
Authors
A. Müller
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.