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Port-Hamiltonian Neural ODE Networks on Lie Groups For Robot Dynamics Learning and Control

Thai Duong, Abdullah Altawaitan, Jason Stanley, Nikolay Atanasov

TL;DR

This work tackles the challenge of learning robot dynamics that respect both Lie-group geometry and energy conservation. It introduces a port-Hamiltonian neural ODE on matrix Lie groups, with a structured decomposition of kinetic energy, potential energy, dissipation, and input mappings, enabling data-efficient learning and physically meaningful predictions. A corresponding energy-based control strategy based on IDA-PBC is developed for trajectory tracking on Lie groups, with explicit SE(3) tracking laws and Lyapunov-based guarantees under appropriate matching conditions. The approach is validated across simulated pendulum, SE(2)/SE(3) robots, and real quadrotor experiments, including model updates after hardware changes and payload variations, demonstrating robustness and rapid adaptation in practice.

Abstract

Accurate models of robot dynamics are critical for safe and stable control and generalization to novel operational conditions. Hand-designed models, however, may be insufficiently accurate, even after careful parameter tuning. This motivates the use of machine learning techniques to approximate the robot dynamics over a training set of state-control trajectories. The dynamics of many robots are described in terms of their generalized coordinates on a matrix Lie group, e.g. on $SE(3)$ for ground, aerial, and underwater vehicles, and generalized velocity, and satisfy conservation of energy principles. This paper proposes a port-Hamiltonian formulation over a Lie group of the structure of a neural ordinary differential equation (ODE) network to approximate the robot dynamics. In contrast to a black-box ODE network, our formulation embeds energy conservation principle and Lie group's constraints in the dynamics model and explicitly accounts for energy-dissipation effect such as friction and drag forces in the dynamics model. We develop energy shaping and damping injection control for the learned, potentially under-actuated Hamiltonian dynamics to enable a unified approach for stabilization and trajectory tracking with various robot platforms.

Port-Hamiltonian Neural ODE Networks on Lie Groups For Robot Dynamics Learning and Control

TL;DR

This work tackles the challenge of learning robot dynamics that respect both Lie-group geometry and energy conservation. It introduces a port-Hamiltonian neural ODE on matrix Lie groups, with a structured decomposition of kinetic energy, potential energy, dissipation, and input mappings, enabling data-efficient learning and physically meaningful predictions. A corresponding energy-based control strategy based on IDA-PBC is developed for trajectory tracking on Lie groups, with explicit SE(3) tracking laws and Lyapunov-based guarantees under appropriate matching conditions. The approach is validated across simulated pendulum, SE(2)/SE(3) robots, and real quadrotor experiments, including model updates after hardware changes and payload variations, demonstrating robustness and rapid adaptation in practice.

Abstract

Accurate models of robot dynamics are critical for safe and stable control and generalization to novel operational conditions. Hand-designed models, however, may be insufficiently accurate, even after careful parameter tuning. This motivates the use of machine learning techniques to approximate the robot dynamics over a training set of state-control trajectories. The dynamics of many robots are described in terms of their generalized coordinates on a matrix Lie group, e.g. on for ground, aerial, and underwater vehicles, and generalized velocity, and satisfy conservation of energy principles. This paper proposes a port-Hamiltonian formulation over a Lie group of the structure of a neural ordinary differential equation (ODE) network to approximate the robot dynamics. In contrast to a black-box ODE network, our formulation embeds energy conservation principle and Lie group's constraints in the dynamics model and explicitly accounts for energy-dissipation effect such as friction and drag forces in the dynamics model. We develop energy shaping and damping injection control for the learned, potentially under-actuated Hamiltonian dynamics to enable a unified approach for stabilization and trajectory tracking with various robot platforms.
Paper Structure (33 sections, 1 theorem, 86 equations, 15 figures, 2 tables)

This paper contains 33 sections, 1 theorem, 86 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement
  4. Preliminaries
  5. Matrix Lie Groups
  6. Example: SE(3) Manifold
  7. Hamiltonian Dynamics on Matrix Lie Groups
  8. Reformulation as Port-Hamiltonian Dynamics
  9. Example: Hamiltonian Dynamics on the SE(3) Manifold
  10. Neural ODE Networks
  11. Learning Lie Group Hamiltonian Dynamics
  12. Data Collection
  13. Model Architecture
  14. Training Process
  15. Application to SE(3) Hamiltonian Dynamics Learning
  16. ...and 18 more sections

Key Result

Theorem 1

Consider a port-Hamiltonian system on the $SE(3)$ manifold with dynamics eq:portham_dyn_SE3. Assume that the matching condition eq:matching_condition_se3 is satisfied, the desired momentum's derivative $\dot{{\boldsymbol{\mathfrak{p}}}}^*$ is bounded, and the matrices $\mathbf{K}_\mathbf{p}$, $\math

Figures (15)

  • Figure 1: Quadrotor trajectory tracking using a learned port-Hamiltonian dynamics model.
  • Figure 2: Architecture of port-Hamiltonian neural ODE network on matrix Lie group. The trainable terms are shown in green.
  • Figure 3: Pendulum dynamics estimation using an $SO(3)$ port-Hamiltonian neural ODE network with scale factor $\beta = 1.33$.
  • Figure 4: Evaluation of $SE(2)$ port-Hamiltonian neural ODE network on a simulated omnidirectional ground vehicle with scale factor $\beta = 7.1$.
  • Figure 5: Trajectories (blue) of omnidirectional ground vehicle tracking desired trajectories (orange) with our learned $SE(2)$ port-Hamiltonian dynamics and IDA-PBC control design.
  • ...and 10 more figures

Theorems & Definitions (11)

  • Definition 1: Dot Product
  • Definition 2: General Linear Group hall2013lie
  • Definition 3: Matrix Lie Group hall2013lie
  • Definition 4: Tangent Space and Bundle
  • Definition 5: Lie Algebra and Lie Bracket
  • Definition 6: Left Translation and Invariant Vectors
  • Definition 7: Adjoint Operator
  • Definition 8: Cotangent Space and Bundle
  • Definition 9: Coadjoint Operator
  • Theorem 1
  • ...and 1 more