⚡ electrical engineering

Floating-Base Deep Lagrangian Networks

This paper introduces Floating-Base Deep Lagrangian Networks (FeLaN), a grey-box system identification method that enforces specific physical constraints on inertia matrices to improve the performance and interpretability of deep learning models for floating-base robots like humanoids and quadrupeds.

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

Published 2026-03-04

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot dog or a robot human how to move. To do this, you need a "brain" that understands physics: how heavy the robot is, how its joints bend, and how gravity pulls it down.

For a long time, scientists had two main ways to build this brain:

The "Hard-Code" Way (White-Box): You write down every single law of physics by hand. It's very accurate if you know the robot perfectly, but if the robot changes slightly (like adding a backpack), you have to rewrite all the math.
The "Black-Box" Way (Deep Learning): You throw a massive amount of data at a computer and let it guess the rules. It's flexible, but it often makes silly mistakes because it doesn't actually understand physics. It might think a robot can fly or that a heavy object weighs nothing.

The Problem with Floating Robots
Most robots in these studies are "floating-base" systems. Think of a robot dog or a human. They aren't bolted to the floor; they stand on legs and can move anywhere. This makes the physics incredibly tricky.

When a robot dog runs, its legs move independently, but they are all connected to the main body (the torso). The paper calls this branch-induced sparsity.

Analogy: Imagine a tree. The trunk is the robot's body. The branches are the legs. If you shake one branch (a leg), it affects the whole tree, but it doesn't change the shape of a different branch on the other side.
The Issue: Old AI models treated the robot like a solid block of clay. They didn't realize that moving the left leg doesn't change the physics of the right leg. This made their predictions messy and inaccurate.

Also, there's a rule in physics called the Triangle Inequality for how mass is distributed.

Analogy: Imagine a spinning top. If it's too heavy on one side, it wobbles and falls. Physics has strict rules about how heavy different parts can be relative to each other to keep the object stable. Old AI models often ignored these rules, creating "ghost robots" that were physically impossible (like a robot with a head heavier than its entire body).

The Solution: FeLaN (Floating-Base Deep Lagrangian Networks)
The authors created a new method called FeLaN. Think of it as a "Smart Hybrid" approach.

The "Grey-Box" Strategy: Instead of just guessing (Black-Box) or just hard-coding (White-Box), they built a model that knows the rules of the game but learns the specific details from data.
The Special Blueprint: They invented a new way to describe the robot's weight and balance (the "Inertia Matrix").
- Analogy: Imagine you are packing a suitcase.
  - Old AI: Just throws clothes in randomly. Sometimes the suitcase is too heavy on one side and tips over.
  - FeLaN: Uses a special, pre-organized packing system. It knows exactly how to arrange the clothes so the suitcase is balanced, the zipper closes, and the weight is distributed perfectly. It forces the AI to only learn "valid" packing arrangements.
The Result: The AI learns faster, makes fewer mistakes, and works on different robots without needing to be retrained from scratch.

What They Did
They tested this on real robots, including:

Unitree Go2: A robot dog.
Boston Dynamics Spot: Another famous robot dog.
Talos: A humanoid robot (looks like a human).
HyQReal: A heavy-duty robot dog.

They ran simulations and real-world tests. The results showed that FeLaN was the "champion." It predicted how much force the motors needed to move the robots more accurately than any other method, even when the robots were moving on real, uneven ground with noise and friction.

Why This Matters
This is a big deal for the future of robotics.

Safety: If a robot understands physics better, it won't fall over or break its own legs.
Efficiency: It learns faster, meaning we don't need as much data to train new robots.
Versatility: You can take a model trained on a robot dog and adapt it to a robot human much easier than before.

In a Nutshell
The paper is about teaching AI to respect the laws of physics when dealing with robots that walk on legs. By forcing the AI to use a "smart packing list" for weight and balance, they created a robot brain that is smarter, safer, and more adaptable than anything we had before.

1. Problem Statement

Accurate dynamic models are critical for the control and simulation of robotics, particularly for floating-base systems like quadrupeds and humanoids. While "grey-box" methods (combining deep learning with physics) have shown success in fixed-base and single-rigid-body robots, they struggle with floating-base systems due to the neglect of specific physical constraints.

Current approaches often fail to account for:

Branch-induced sparsity: The inertia matrix of a robot with multiple kinematic chains (legs, arms) is sparse, with blocks depending only on specific joint inputs. Standard Cholesky factorizations used in existing methods (like Deep Lagrangian Networks, DeLaN) destroy this sparsity.
Composite Spatial Inertia Constraints: The $6\times6$ inertia matrix of the floating base must satisfy the triangle inequality on its eigenvalues (a property of single-rigid-body inertia) and maintain positive definiteness.
Input Independence: The inertia of a specific kinematic branch should not depend on the configuration of other branches.

Naively applying existing grey-box models to these systems leads to physically inconsistent models, poor out-of-distribution generalization, and a lack of interpretability.

2. Methodology

The authors propose Floating-Base Deep Lagrangian Networks (FeLaN), a novel grey-box architecture that integrates a physically consistent parameterization of the inertia matrix into a deep learning framework.

A. Physically Consistent Parameterization

The core innovation is a new parameterization of the inertia matrix $H(\nu)$ based on reordered Cholesky factorization ( $H = L^T L$ ). Unlike standard Cholesky, the reordered version preserves the sparsity pattern induced by the robot's kinematic tree.

The authors decompose the lower triangular matrix $L$ into specific blocks to enforce physical laws:

Positive Definiteness: Ensured by enforcing positive diagonal elements in the factor $L$ .
Triangle Inequality: The composite rotational inertia $I_B$ must satisfy the triangle inequality ( $I_x + I_y \geq I_z$ , etc.). The authors derive a parameterization where $I_B$ is constructed from a positive semi-definite covariance matrix $\Sigma_R$ , ensuring the inequality holds by definition.
First Mass Moment & Mass: The linear inertia and first mass moment (coupling between linear and angular motion) are parameterized to ensure the total mass $m$ is positive and the system's center of mass is consistent.
Branch-Induced Sparsity: The matrix $L$ is structured such that sub-matrices corresponding to specific kinematic branches depend only on the joints of that branch, preserving the physical decoupling.

B. Deep Learning Architecture

Inertia Prediction: Instead of predicting the full dense matrix, the network predicts the unconstrained parameters of the reordered Cholesky factor $L$ $L$ .
- Separate neural networks ( $NN_k$ ) predict the blocks corresponding to each kinematic branch $k$ , ensuring input independence.
- A global network ( $NN_R$ ) predicts the base-related blocks.
Potential Energy: The potential energy $P$ is not learned by a separate network. Instead, it is computed analytically from the estimated mass and center of mass parameters derived from the inertia matrix, ensuring consistency between kinetic and potential energy terms.
Loss Function: The model is trained to minimize the inverse dynamics error (difference between predicted and actual joint torques). The loss includes auxiliary terms to penalize violations of positive definiteness during training, though the structural parameterization acts as a "hard" constraint for physical consistency.

3. Key Contributions

Extension of Physical Consistency: The paper extends the full physical consistency conditions (specifically the triangle inequality on eigenvalues) from single rigid bodies to the composite spatial inertia of floating-base systems.
Novel Parameterization: Introduction of a parameterization for the inertia matrix that simultaneously guarantees:
- Positive definiteness.
- Branch-induced sparsity.
- Input independence for kinematic branches.
- The triangle inequality for composite rotational inertia.
FeLaN Architecture: A deep learning architecture that combines Lagrangian mechanics with the proposed parameterization, requiring only kinematic chain topology (no prior dynamic parameters).
Comprehensive Dataset & Benchmark: The authors collected and released a dataset covering multiple quadrupeds (Unitree Go2, Boston Dynamics Spot, HyQReal2) and humanoids (Talos) in both simulation and real-world environments.

4. Experimental Results

The method was evaluated on simulated and real data for Unitree Go2, Boston Dynamics Spot, Talos, and HyQReal2.

Simulation Performance:
- FeLaN achieved the lowest Normalized Mean Squared Error (NMSE) on both Go2 and Talos.
- It outperformed standard DeLaN and black-box MLPs, demonstrating that enforcing physical constraints improves generalization.
- FeLaN-BS (a variant with only sparsity and positive definiteness, but without the full triangle inequality) performed well but slightly worse than full FeLaN, highlighting the importance of the triangle inequality constraint.
Real-World Performance:
- On real robots, FeLaN achieved the best relative NMSE (rNMSE) across all robots, indicating superior robustness and generalization compared to baselines.
- White-box methods (DNEA): Struggled on heavier robots (Talos, HyQReal2) due to unmodeled actuator/contact dynamics, showing errors roughly 3x higher than the best grey-box methods.
- Black-box/Standard Grey-box: While MLPs and standard DeLaN sometimes achieved low error on specific robots, they were highly sensitive to initialization and hyperparameters and failed to generalize consistently across different robot morphologies.
- Interpretability: FeLaN provided better predictions for individual dynamic components (gravity, Coriolis, inertial terms) compared to other methods, particularly in estimating the stationary gravity component.

5. Significance

This work bridges a critical gap in robot learning by enabling physically consistent system identification for complex, multi-legged, and humanoid robots.

Generalization: By embedding fundamental physical laws (triangle inequality, sparsity) directly into the network architecture, FeLaN learns models that generalize better to unseen configurations and different robot morphologies.
Data Efficiency: The method requires less data to converge to a physically plausible model compared to black-box approaches.
Interpretability: Unlike black-box models, FeLaN provides explicit estimates of mass, center of mass, and inertia, which are crucial for downstream tasks like model-based control and state estimation.
Future Impact: The authors suggest FeLaN can serve as a learned dynamical model for advanced locomotion and loco-manipulation tasks, potentially replacing or augmenting traditional model-based controllers.