Floating-Base Deep Lagrangian Networks

Lucas Schulze1, Juliano Decico Negri2, Victor Barasuol3, Vivian Suzano Medeiros2, Marcelo Becker2, Jan Peters1, Oleg Arenz1
1Intelligent Autonomous Systems Lab, TU Darmstadt, Germany
2Mobile Robotics Group, University of São Paulo (EESC-USP), Brazil
3Dynamic Legged Systems Lab, Istituto Italiano di Tecnologia (IIT), Italy

Accepted to the 2026 IEEE International Conference on Robotics & Automation (ICRA 2026)

Paper arXiv
Code & datasets will be released soon.

Overview

We introduce Floating-Base Deep Lagrangian Networks (FeLaN): a grey-box method for physically consistent system identification of floating-base robots (e.g., humanoids, quadrupeds).

Contributions:

  • Novel inertia matrix parametrization: preserves branch-induced sparsity and decoupling inherent to multiple kinematic chains, while ensuring full physical consistency of the composite spatial inertia.
  • FeLaN: a deep learning model that combines Lagrangian mechanics with the proposed parametrization, achieving greater interpretability.
  • Evaluation: validated on four real robots (Pal Robotics Talos, Boston Dynamics Spot, Spot + arm, IIT HyQReal2) and two simulated robots (Talos and Unitree Go2).
  • Full physical consistency of the Composite Spatial Inertia: a theoretical extension of the single rigid-body case.

Inertia Matrix Structure

Floating-Base systems have specific dynamics properties that are typically not modeled in current grey-box methods:

  • Base is not fixed → base motion affects dynamics via composite spatial inertia
  • Multiple kinematic branches (leg, arms, head) → functional decoupling and branch-induced sparsity

These properties are structural constraints that show up directly in the robot's generalized inertia matrix.

For example, consider the inertia matrix of the Unitree Go2 quadruped robot:

FeLaN Architecture
The top-left 6x6 block corresponds to the composite spatial inertia matrix and has three properties:
  • Stationary total mass
  • A skew-symmetric matrix capturing the coupling between linear and angular coordinates through the first mass moment
  • A positive definite composite rotational inertia, whose eigenvalues satisfy a triangle inequality
Inertia Sparsity
Notice the zeros in each leg's rows, reflecting the partial decoupling between legs.
Inertia Sparsity
Additionally, each leg's row depends only on its own joint configuration.
Inertia Functional

Proposed Parametrization

A common way to learn a physically valid inertia matrix is to predict a Cholesky factor to guarantees positive definiteness:

Default Cholesky

However, standard Cholesky factorization does not preserve sparsity patterns: even if the original matrix is sparse, the resulting factor is generally dense. This not only compromises sparsity for a learning-based model, but also prevents the proper decoupling between kinematic branches.

Featherstone (2005) proposed an efficient factorization of the joint-space inertia matrix using a reordered Cholesky factorization, thereby exploiting branch-induced sparsity:

Reordered Cholesky

R. Featherstone, “Efficient factorization of the joint-space inertia matrix for branched kinematic trees,” The International Journal of Robotics Research, vol. 24, no. 6, pp. 487-500, 2005.

Observing that any sparsity or structural property in L is inherited by H, we parameterize the block components of L to satisfy all the inertia matrix properties, including branch-induced sparsity and the structural constraints of the composite spatial inertia.

Floating-Base Deep Lagrangian Networks (FeLaN)

FeLaN employs one MLP for the terms associated with the composite spatial inertia, and one additional MLP per kinematic branch that takes only the corresponding joint coordinates as input. Using our novel parametrization, we construct the factor L, ensuring fully physically consistent spatial inertia while preserving branch-induced sparsity. We then compute the kinetic and potential energies and derive the equations of motion via the Euler-Lagrange equations.

FeLaN Architecture

Experiments

Simulated Talos

We compare FeLaN with several baselines on system identification for Talos in simulation, including a white-box approach based on a differentiable Recursive Newton-Euler algorithm (DNEA). We show the total signal and its components for the linear force z and the left arm's second joint (LA2). While all grey-box methods fit the total signal well, the gravity component of z is stationary only for DNEA and FeLaN, due to the proposed parametrization.

Simulated Talos results

Real Talos

Due to real-world effects (e.g., actuator and contact dynamics, friction, measurement noise, state/force estimation errors), DNEA struggles to accurately estimate the torques/forces. In contrast, the grey-box methods achieve better estimation with FeLaN providing a more interpretable and physically consistent model.

Real Talos results

For a complete report of the results, including additional robots and experimental details, please refer to the preprint.

BibTeX

If you find this work useful, please consider citing:

@misc{schulze2025felan,
      title={Floating-Base Deep Lagrangian Networks}, 
      author={Lucas Schulze and Juliano Decico Negri and Victor Barasuol and Vivian Suzano Medeiros and Marcelo Becker and Jan Peters and Oleg Arenz},
      year={2025},
      eprint={2510.17270},
      archivePrefix={arXiv},
      primaryClass={cs.RO},
      url={https://arxiv.org/abs/2510.17270}, 
}

Method Overview Video

Acknowledgment

This project has been funded by the German Federal Ministry of Research, Technology and Space (BMFTR) - Project number 01IS23057B.