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

TL;DR

This paper develops a port-Hamiltonian framework for constrained multibody systems, enabling structure-preserving simulation, consistent interconnection of rigid bodies, and robust long-term energy-conserving numerical integration.

Contribution

It introduces a novel port-Hamiltonian formulation with a singularity-free rotation representation, compatible with classical joints, and incorporates an index-reduction strategy for stable simulation.

Findings

Structure-preserving implicit midpoint integration ensures energy and momentum conservation.

The framework is robust for long-term simulations of complex multibody systems.

Numerical examples demonstrate the method's effectiveness and energy-based control suitability.

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.