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The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems

Andreas Mueller, Zdravko Terze

TL;DR

The paper shows that for holonomically constrained multibody systems, the configuration space of rigid-body motions should be modeled as the Lie group $SE(3)$ to ensure exact constraint satisfaction in numerical time stepping, particularly for lower-pair joints that form $SE(3)$ subgroups. It develops both coordinate-based and coordinate-free ODE formulations on $SE(3)$ and compares them to the traditional $SO(3)\times\mathbb{R}^{3}$ approach, analyzing constraint fidelity via kinematic reconstruction and Lie-group integration (e.g., Munthe-Kaas). The main finding is that SE(3) updates yield perfect respect of kinematic constraints when motions are confined to SE(3) subgroups, while for general motions both representations have the same order of accuracy; thus SE(3) offers superior consistency and can be tailored per body for improved simulation fidelity. The results are supported by multiple numerical examples, demonstrating improved constraint satisfaction and energy behavior when using SE(3), with practical guidance on choosing configuration updates and leveraging dependent quaternions or coordinate-free Lie-group integration to handle singularities and large motions.

Abstract

The dynamics simulation of multibody systems (MBS) using spatial velocities (non-holonomic velocities) requires time integration of the dynamics equations together with the kinematic reconstruction equations (relating time derivatives of configuration variables to rigid body velocities). The latter are specific to the geometry of the rigid body motion underlying a particular formulation, and thus to the used configuration space (c-space). The proper c-space of a rigid body is the Lie group SE(3), and the geometry is that of the screw motions. The rigid bodies within a MBS are further subjected to geometric constraints, often due to lower kinematic pairs that define SE(3) subgroups. Traditionally, however, in MBS dynamics the translations and rotations are parameterized independently, which implies the use of the direct product group $SO\left( 3\right) \times {\Bbb R}^{3}$ as rigid body c-space, although this does not account for rigid body motions. Hence, its appropriateness was recently put into perspective. In this paper the significance of the c-space for the constraint satisfaction in numerical time stepping schemes is analyzed for holonomicaly constrained MBS modeled with the 'absolute coordinate' approach, i.e. using the Newton-Euler equations for the individual bodies subjected to geometric constraints. It is shown that the geometric constraints a body is subjected to are exactly satisfied if they constrain the motion to a subgroup of its c-space. Since only the $SE\left( 3\right) $ subgroups have a practical significance it is regarded as the appropriate c-space for the constrained rigid body. Consequently the constraints imposed by lower pair joints are exactly satisfied if the joint connects a body to the ground. For a general MBS, where the motions are not constrained to a subgroup, the SE(3) and $SO\left( 3\right) \times {\Bbb R}^{3}$ yield the same order of accuracy.

The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems

TL;DR

The paper shows that for holonomically constrained multibody systems, the configuration space of rigid-body motions should be modeled as the Lie group to ensure exact constraint satisfaction in numerical time stepping, particularly for lower-pair joints that form subgroups. It develops both coordinate-based and coordinate-free ODE formulations on and compares them to the traditional approach, analyzing constraint fidelity via kinematic reconstruction and Lie-group integration (e.g., Munthe-Kaas). The main finding is that SE(3) updates yield perfect respect of kinematic constraints when motions are confined to SE(3) subgroups, while for general motions both representations have the same order of accuracy; thus SE(3) offers superior consistency and can be tailored per body for improved simulation fidelity. The results are supported by multiple numerical examples, demonstrating improved constraint satisfaction and energy behavior when using SE(3), with practical guidance on choosing configuration updates and leveraging dependent quaternions or coordinate-free Lie-group integration to handle singularities and large motions.

Abstract

The dynamics simulation of multibody systems (MBS) using spatial velocities (non-holonomic velocities) requires time integration of the dynamics equations together with the kinematic reconstruction equations (relating time derivatives of configuration variables to rigid body velocities). The latter are specific to the geometry of the rigid body motion underlying a particular formulation, and thus to the used configuration space (c-space). The proper c-space of a rigid body is the Lie group SE(3), and the geometry is that of the screw motions. The rigid bodies within a MBS are further subjected to geometric constraints, often due to lower kinematic pairs that define SE(3) subgroups. Traditionally, however, in MBS dynamics the translations and rotations are parameterized independently, which implies the use of the direct product group as rigid body c-space, although this does not account for rigid body motions. Hence, its appropriateness was recently put into perspective. In this paper the significance of the c-space for the constraint satisfaction in numerical time stepping schemes is analyzed for holonomicaly constrained MBS modeled with the 'absolute coordinate' approach, i.e. using the Newton-Euler equations for the individual bodies subjected to geometric constraints. It is shown that the geometric constraints a body is subjected to are exactly satisfied if they constrain the motion to a subgroup of its c-space. Since only the subgroups have a practical significance it is regarded as the appropriate c-space for the constrained rigid body. Consequently the constraints imposed by lower pair joints are exactly satisfied if the joint connects a body to the ground. For a general MBS, where the motions are not constrained to a subgroup, the SE(3) and yield the same order of accuracy.
Paper Structure (42 sections, 3 theorems, 80 equations, 22 figures)

This paper contains 42 sections, 3 theorems, 80 equations, 22 figures.

Table of Contents

  1. Introduction
  2. The Configuration Space Lie Group of a Constrained MBS
  3. MBS C-Space in Left-Invariant $SE\left( 3\right)$ Representation
  4. MBS C-Space in Left-Invariant $SO\left( 3\right) \times {\mathbb R}^{3}$ Representation
  5. Motion Equations of Constrained MBS
  6. Motion equations on the parameter space in canonical coordinates (axis-angle, position vector)
  7. Motion equations in dependent coordinates (quaternions)
  8. Motion equations on the c-space Lie group
  9. The significance of the configuration space for numerical reconstruction of rigid body motions constrained to subgroups
  10. Proof.
  11. Examples
  12. Unconstrained Rigid Body -- Reference Frame at COM
  13. Body-Fixed Newton-Euler Equations on $SE(3)$
  14. Newton-Euler Equations on $SO(3)\times {\mathbb R}^{3}$
  15. Numerical Results
  16. ...and 27 more sections

Key Result

Corollary 1

The kinematic motion constraints of a body in a MBS are satisfied by a configuration update step in terms of linear combinations of velocity samples ${\bf V}^{\left( j\right) }$ if its motion is constrained to a subgroup of its c-space Lie group.

Figures (22)

  • Figure 1: a) Frame rotating about a fixed axis with constant angular velocity. b) Position error when using the $SO\left( 3\right) \times {\mathbb R}^{3}$ update.
  • Figure 2: Drift of kinetic energy when integrating a) the $SE(3)$, and b) the $SO(3)\times {\mathbb R}^{3}$ formulation.
  • Figure 3: Drift of COM from analytic solution when integrating a) the $SE(3)$, and b) the $SO(3)\times {\mathbb R}^{3}$ formulation.
  • Figure 4: Drift $\varepsilon _{r}$ of the orientation from analytic solution when integrating a) the $SE(3)$, and b) the $SO(3)\times {\mathbb R}^{3}$ formulation.
  • Figure 5: Drift $\varepsilon$ of the COM when integrating a) the $SE(3)$, and b) the $SO(3)\times {\mathbb R}^{3}$ formulation.
  • ...and 17 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Corollary 1
  • Lemma 2
  • Corollary 3
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 2 more