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The Port-Hamiltonian Structure of Vehicle Manipulator Systems

Ramy Rashad

Abstract

This paper presents a port-Hamiltonian formulation of vehicle-manipulator systems (VMS), a broad class of robotic systems including aerial manipulators, underwater manipulators, space robots, and omnidirectional mobile manipulators. Unlike existing Lagrangian formulations that obscure the underlying energetic structure, the proposed port-Hamiltonian formulation explicitly reveals the energy flow and conservation properties of these complex mechanical systems. We derive the port-Hamiltonian dynamics from first principles using Hamiltonian reduction theory. Two complementary formulations are presented: a standard form that directly exposes the energy structure, and an inertially-decoupled form that leverages the principal bundle structure of the VMS configuration space and is particularly suitable for control design and numerical simulation. The coordinate-free geometric approach we follow avoids singularities associated with local parameterizations of the base orientation. We rigorously establish the mathematical equivalence between our port-Hamiltonian formulations and existing reduced Euler-Lagrange and Boltzmann-Hamel equations found in the robotics and geometric mechanics literature.

The Port-Hamiltonian Structure of Vehicle Manipulator Systems

Abstract

This paper presents a port-Hamiltonian formulation of vehicle-manipulator systems (VMS), a broad class of robotic systems including aerial manipulators, underwater manipulators, space robots, and omnidirectional mobile manipulators. Unlike existing Lagrangian formulations that obscure the underlying energetic structure, the proposed port-Hamiltonian formulation explicitly reveals the energy flow and conservation properties of these complex mechanical systems. We derive the port-Hamiltonian dynamics from first principles using Hamiltonian reduction theory. Two complementary formulations are presented: a standard form that directly exposes the energy structure, and an inertially-decoupled form that leverages the principal bundle structure of the VMS configuration space and is particularly suitable for control design and numerical simulation. The coordinate-free geometric approach we follow avoids singularities associated with local parameterizations of the base orientation. We rigorously establish the mathematical equivalence between our port-Hamiltonian formulations and existing reduced Euler-Lagrange and Boltzmann-Hamel equations found in the robotics and geometric mechanics literature.
Paper Structure (28 sections, 12 theorems, 124 equations, 5 figures, 4 tables)

This paper contains 28 sections, 12 theorems, 124 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Rigid Body Kinematics and Dynamics on $SE(3)$
  4. Joint Kinematics and Dynamics on Lie Subgroups of $SE(3)$
  5. Standard Formulation of VMS Dynamics
  6. Forward Kinematics
  7. Differential Kinematics
  8. Equations of Motion
  9. Manipulator Port-Hamiltonian Dynamics
  10. Base Port-Hamiltonian Dynamics
  11. Vehicle-Manipulator Port-Hamiltonian Dynamics
  12. Port-Hamiltonian formulation
  13. Inertially-decoupled port-Hamiltonian formulation
  14. Lagrangian Counterparts of VMS Dynamics
  15. Equivalence with Reduced Euler-Lagrange equations of Mishra2023Moghaddam2024
  16. ...and 13 more sections

Key Result

Proposition 3.2

The torque $\tau \in T_q^* Q_m$ in the port-Hamiltonian dynamics eq:manipulator_pH_dynamics that characterizes the power balance eq:manipulator_total_power is given by where $g(q) := \frac{\partial \mathcal{H}_{\text{\normalfont pot}}}{\partial q}(q) \in T_q^* Q_m$ denotes the generalized gravitational torque acting on the manipulator. Furthermore, the port-Hamiltonian dynamics eq:manipulator_pH_

Figures (5)

  • Figure 1: Illustration of vehicle-manipulator systems. Left figure shows a floating-base manipulator $h\in G_b = SE(3)$ while the right figure shows a ground mobile manipulator $h\in G_b = \mathbb{S}^1\times \mathbb{R}^{2}$.
  • Figure 2: Port-Hamiltonian formulation of fixed-base manipulator dynamics.
  • Figure 3: Port-Hamiltonian formulation of moving-base dynamics.
  • Figure 4: Principal bundle structure of the VMS configuration space $Q = G_b \times Q_m$.
  • Figure 5: The proposed port-Hamiltonian formulations of VMS dynamics and their corresponding Dirac structures. Left figure shows the port-Hamiltonian formulation \ref{['eq:floating_base_manipulator_pH']} while the right figure shows the inertially-decoupled port-Hamiltonian formulation \ref{['eq:decoupled_floating_base_manipulator_pH']}. The intermediate figure illustrates the change of coordinates $\phi(q)$.

Theorems & Definitions (29)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Remark 5.1
  • Theorem 5.2
  • proof
  • ...and 19 more