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Floating-Base Deep Lagrangian Networks

Lucas Schulze, Juliano Decico Negri, Victor Barasuol, Vivian Suzano Medeiros, Marcelo Becker, Jan Peters, Oleg Arenz

TL;DR

FeLaN introduces a physically consistent inertia parameterization for floating-base robots by extending the DeLaN framework with a reordered Cholesky factorization that preserves branch-induced sparsity, positive definiteness, and the inertia triangle inequality. The approach uses per-branch neural nets to learn branch-local inertia terms while a shared network models base-related components, and it enforces mass positivity and inertia constraints through the loss and parameterization. Evaluations on simulated quadrupeds and humanoids, as well as real robots (HyQReal2, Spot, Spot+Arm, Talos), show competitive inverse dynamics accuracy with improved physical interpretability and robustness to real-world effects. The results suggest FeLaN achieves a balance between accuracy and physical fidelity, enabling more reliable model-based control and estimation for floating-base robotics.

Abstract

Grey-box methods for system identification combine deep learning with physics-informed constraints, capturing complex dependencies while improving out-of-distribution generalization. Yet, despite the growing importance of floating-base systems such as humanoids and quadrupeds, current grey-box models ignore their specific physical constraints. For instance, the inertia matrix is not only positive definite but also exhibits branch-induced sparsity and input independence. Moreover, the 6x6 composite spatial inertia of the floating base inherits properties of single-rigid-body inertia matrices. As we show, this includes the triangle inequality on the eigenvalues of the composite rotational inertia. To address the lack of physical consistency in deep learning models of floating-base systems, we introduce a parameterization of inertia matrices that satisfies all these constraints. Inspired by Deep Lagrangian Networks (DeLaN), we train neural networks to predict physically plausible inertia matrices that minimize inverse dynamics error under Lagrangian mechanics. For evaluation, we collected and released a dataset on multiple quadrupeds and humanoids. In these experiments, our Floating-Base Deep Lagrangian Networks (FeLaN) achieve highly competitive performance on both simulated and real robots, while providing greater physical interpretability.

Floating-Base Deep Lagrangian Networks

TL;DR

FeLaN introduces a physically consistent inertia parameterization for floating-base robots by extending the DeLaN framework with a reordered Cholesky factorization that preserves branch-induced sparsity, positive definiteness, and the inertia triangle inequality. The approach uses per-branch neural nets to learn branch-local inertia terms while a shared network models base-related components, and it enforces mass positivity and inertia constraints through the loss and parameterization. Evaluations on simulated quadrupeds and humanoids, as well as real robots (HyQReal2, Spot, Spot+Arm, Talos), show competitive inverse dynamics accuracy with improved physical interpretability and robustness to real-world effects. The results suggest FeLaN achieves a balance between accuracy and physical fidelity, enabling more reliable model-based control and estimation for floating-base robotics.

Abstract

Grey-box methods for system identification combine deep learning with physics-informed constraints, capturing complex dependencies while improving out-of-distribution generalization. Yet, despite the growing importance of floating-base systems such as humanoids and quadrupeds, current grey-box models ignore their specific physical constraints. For instance, the inertia matrix is not only positive definite but also exhibits branch-induced sparsity and input independence. Moreover, the 6x6 composite spatial inertia of the floating base inherits properties of single-rigid-body inertia matrices. As we show, this includes the triangle inequality on the eigenvalues of the composite rotational inertia. To address the lack of physical consistency in deep learning models of floating-base systems, we introduce a parameterization of inertia matrices that satisfies all these constraints. Inspired by Deep Lagrangian Networks (DeLaN), we train neural networks to predict physically plausible inertia matrices that minimize inverse dynamics error under Lagrangian mechanics. For evaluation, we collected and released a dataset on multiple quadrupeds and humanoids. In these experiments, our Floating-Base Deep Lagrangian Networks (FeLaN) achieve highly competitive performance on both simulated and real robots, while providing greater physical interpretability.
Paper Structure (26 sections, 1 theorem, 29 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 26 sections, 1 theorem, 29 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related Work
  3. Preliminaries
  4. Notation and Definitions
  5. Lagrangian Mechanics
  6. Deep Lagrangian Networks
  7. Fully Physically-Consistent Spatial Inertia
  8. Physically Consistent Inertia Matrix Parametrization
  9. Inertia Matrix of a Floating-Base system
  10. Branch-Induced Sparsity
  11. Positive Definiteness
  12. First mass moment
  13. Mass
  14. Branched Kinematic Chains
  15. Parametrization Overview
  16. ...and 11 more sections

Key Result

Lemma 1

A composite inertia matrix is fully physical consistent only if $\mathrm{m} > 0$ and $\mathbf{I}_\mathbf{B}(\mathbf{q})$ satisfies the triangle inequality.

Figures (1)

  • Figure 1: Estimated force $z$ and torques for left arm (LA) second joint, and right leg (RL) fifth joint for simulated Talos.

Theorems & Definitions (2)

  • Lemma 1
  • proof