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Port-Hamiltonian multibody dynamics: Lagrangian formulation, consistent interconnection, structure-preserving simulation and index-reduction

Lisa Latussek, Philipp L. Kinon, Peter Betsch

Abstract

This work introduces a port-Hamiltonian (PH) model for constrained mechanical systems, which is directly derived from the Lagrangian equations of motion. The present PH framework incorporates a singularity-free director representation of rigid body rotations, resulting in constant mass matrices. It is shown that the power-preserving interconnection of PH rigid-body subsystems is mathematically equivalent to the classical description of ideal joints using kinematic pairs. This establishes a PH multibody dynamics framework that is consistent with traditional modeling paradigms. Notably, the PH structure of the governing index-2 differential-algebraic equations enables the application of an implicit, structure preserving midpoint time integration. The proposed scheme is able to satisfy both the balance laws for total energy and angular momentum as well as the position-level constraints. These properties make the proposed method remarkably robust and enable stable long-term simulations. Furthermore, a variationally derived index-reduction strategy is incorporated that enforces velocity-level constraints in addition to position-level constraints while preserving the port-Hamiltonian structure. Numerical examples illustrate the favorable properties of the proposed formulation, which is well-suited for energy-based control design.

Port-Hamiltonian multibody dynamics: Lagrangian formulation, consistent interconnection, structure-preserving simulation and index-reduction

Abstract

This work introduces a port-Hamiltonian (PH) model for constrained mechanical systems, which is directly derived from the Lagrangian equations of motion. The present PH framework incorporates a singularity-free director representation of rigid body rotations, resulting in constant mass matrices. It is shown that the power-preserving interconnection of PH rigid-body subsystems is mathematically equivalent to the classical description of ideal joints using kinematic pairs. This establishes a PH multibody dynamics framework that is consistent with traditional modeling paradigms. Notably, the PH structure of the governing index-2 differential-algebraic equations enables the application of an implicit, structure preserving midpoint time integration. The proposed scheme is able to satisfy both the balance laws for total energy and angular momentum as well as the position-level constraints. These properties make the proposed method remarkably robust and enable stable long-term simulations. Furthermore, a variationally derived index-reduction strategy is incorporated that enforces velocity-level constraints in addition to position-level constraints while preserving the port-Hamiltonian structure. Numerical examples illustrate the favorable properties of the proposed formulation, which is well-suited for energy-based control design.
Paper Structure (25 sections, 78 equations, 17 figures, 4 tables)

This paper contains 25 sections, 78 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Research gap
  3. Contributions
  4. Notation
  5. Outline
  6. Port-Hamiltonian systems
  7. Port-Hamiltonian formulation of constrained mechanical systems in the Lagrangian setting
  8. Constrained mechanical systems in the Lagrangian setting
  9. Port-Hamiltonian representation of multibody systems in the Lagrangian setting
  10. Special case of constant mass matrices
  11. Director formulation of multibody dynamics in the PH framework
  12. Rigid body dynamics based on the director formulation
  13. Multibody systems
  14. Kinematic pairs described as PH interconnection
  15. Structure-preserving time integration
  16. ...and 10 more sections

Figures (17)

  • Figure 1: Procedure of the present approach. Instead of taking a Hamiltonian approach, we retain the Lagrangian description to directly set up the corresponding port-Hamiltonian formulation.
  • Figure 2: Director formulation of the rigid body betsch2006.
  • Figure 3: Cylindrical pair in director formulation betsch2006.
  • Figure 4: Internal (red) and external (blue) port-variables of a cylindrical pair.
  • Figure 5: Commuting interconnction and discretization.
  • ...and 12 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.1