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Port-Hamiltonian modeling of rigid multibody systems

Thomas Berger, René Hochdahl, Timo Reis, Robert Seifried

TL;DR

This work develops a port-Hamiltonian framework for nonlinear rigid multibody systems subject to position and velocity constraints, unifying Cartesian and redundant coordinate representations. By leveraging modulated Dirac structures, Lagrangian submanifolds, and modulated resistive relations, it yields differential-algebraic equations that intrinsically conserve energy and admit modular interconnection. Key contributions include translational and redundant-coordinate formulations, explicit handling of gyroscopic effects, a clear interconnection scheme that preserves the port-Hamiltonian structure under mild conditions, and concrete examples (differential-drive robot, gyroscope, slider-crank) that demonstrate the approach. The framework offers a principled, energy-aware basis for simulation, control, and modular composition of complex mechanical systems.

Abstract

We employ a port-Hamiltonian approach to model nonlinear rigid multibody systems subject to both position and velocity constraints. Our formulation accommodates Cartesian and redundant coordinates, respectively, and captures kinematic as well as gyroscopic effects. The resulting equations take the form of nonlinear differential-algebraic equations that inherently preserve an energy balance. We show that the proposed class is closed under interconnection, and we provide several examples to illustrate the theory.

Port-Hamiltonian modeling of rigid multibody systems

TL;DR

This work develops a port-Hamiltonian framework for nonlinear rigid multibody systems subject to position and velocity constraints, unifying Cartesian and redundant coordinate representations. By leveraging modulated Dirac structures, Lagrangian submanifolds, and modulated resistive relations, it yields differential-algebraic equations that intrinsically conserve energy and admit modular interconnection. Key contributions include translational and redundant-coordinate formulations, explicit handling of gyroscopic effects, a clear interconnection scheme that preserves the port-Hamiltonian structure under mild conditions, and concrete examples (differential-drive robot, gyroscope, slider-crank) that demonstrate the approach. The framework offers a principled, energy-aware basis for simulation, control, and modular composition of complex mechanical systems.

Abstract

We employ a port-Hamiltonian approach to model nonlinear rigid multibody systems subject to both position and velocity constraints. Our formulation accommodates Cartesian and redundant coordinates, respectively, and captures kinematic as well as gyroscopic effects. The resulting equations take the form of nonlinear differential-algebraic equations that inherently preserve an energy balance. We show that the proposed class is closed under interconnection, and we provide several examples to illustrate the theory.

Paper Structure

This paper contains 12 sections, 5 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Port-Hamiltonian systems
  4. Newtownian mechanical systems: Point masses in Cartesian coordinates
  5. Multibody systems: Redundant coordinates
  6. Interconnection
  7. Examples
  8. A differential drive robot in the plane
  9. Gyroscope
  10. Slider Crank
  11. Conclusion
  12. Auxiliary results on Dirac structures and Lagrangian submanifolds

Key Result

Proposition 1

Let $U_{\mathrm{pos}}\subset{\mathbb{R}}^n$ be open, let $A:U_{\mathrm{pos}}\to{\mathbb{R}}^{\ell\times n}$, $B:U_{\mathrm{pos}}\to{\mathbb{R}}^{n\times m}$ be continuous, and assume that $A$ has constant rank on $U_\mathrm{pos}$. Then the family $(\mathcal{D}_{\bm{{r}}})_{\bm{{r}}\in U_{\mathrm{pos

Figures (4)

  • Figure 1: Visual representation of a port-Hamiltonian system.
  • Figure 2: Differential drive robot
  • Figure 3: Gyroscope
  • Figure 4: Slider Crank

Theorems & Definitions (20)

  • Definition 1: Submanifold of ${\mathbb{R}}^n$
  • Definition 2: Tangent space of a submanifold
  • Definition 3: Dirac structure
  • Definition 4: Modulated Dirac structure
  • Definition 5: Lagrangian submanifold
  • Definition 6: (Modulated) resistive relation
  • Definition 7: Port-Hamiltonian system
  • Remark 1
  • Proposition 1
  • Proposition 2
  • ...and 10 more