[Paper] Learning Physically Consistent Lagrangian Control Models Without Acceleration Measurements

Source: arXiv - 2512.03035v1

Overview

The paper presents a new way to learn Lagrangian dynamics for mechanical systems without needing direct acceleration measurements. By designing a loss function that enforces physical consistency, the authors obtain models that are both accurate and suitable for model‑based control (e.g., feedback linearization and energy‑shaping) on real‑world hardware.

Key Contributions

• Physically‑consistent loss: Introduces a novel training objective that penalizes violations of the Euler‑Lagrange equations, ensuring the learned model respects energy conservation and non‑conservative forces.
• Acceleration‑free learning: Shows how to identify Lagrangian models using only positions, velocities, and control inputs—no explicit acceleration data required.
• Hybrid modeling framework: Combines data‑driven neural networks with the analytical structure of Lagrangian mechanics, yielding interpretable parameters (mass matrix, potential, damping).
• Comprehensive benchmarking: Evaluates the method against standard Lagrangian/Hamiltonian neural networks on both simulated pendulum‑cart systems and a real‑world experimental rig.
• Control‑oriented validation: Demonstrates that the learned models can be directly plugged into feedback‑linearization and energy‑based controllers, achieving stable tracking on the hardware benchmark.

Methodology

1. Problem setup – The system obeys the Euler‑Lagrange equation

   [ \frac{d}{dt}!\bigl(\partial_{\dot q}L\bigr)-\partial_{q}L = \tau + f_{\text{nc}}(q,\dot q), ]

   where (L) is the Lagrangian (kinetic – potential energy) and (f_{\text{nc}}) captures friction or other non‑conservative forces.

2. Neural representation – A neural net parameterizes the kinetic energy matrix (M(q)), the potential (V(q)), and the non‑conservative term (f_{\text{nc}}(q,\dot q)). The network is fully differentiable, so analytical expressions for (\partial_{\dot q}L) and (\partial_{q}L) are obtained automatically.

3. Loss design – Instead of fitting accelerations, the loss enforces the residual of the Euler‑Lagrange equation to be small:

   [ \mathcal{L}{\text{phys}} = \bigl| \frac{d}{dt}!\bigl(\partial{\dot q}L\bigr)-\partial_{q}L - \tau - f_{\text{nc}} \bigr|^{2}. ]

   A secondary data‑fit term matches predicted velocities to measured ones, balancing physics and measurement fidelity.

4. Training pipeline – The model is trained on trajectories collected from the robot/experiment (position, velocity, control torque). Automatic differentiation provides the required time‑derivatives of the learned quantities, eliminating the need for noisy acceleration sensors.

5. Control synthesis – Once the model is learned, the authors derive the inverse dynamics analytically and plug it into standard nonlinear controllers (feedback linearization, passivity‑based control).

Results & Findings

• Physical consistency improves dramatically: the learned mass matrix remains positive‑definite, and the total energy behaves as expected even on noisy data.
• Control performance: Controllers built on the new models achieve faster convergence and higher robustness to disturbances compared with controllers using generic black‑box neural dynamics.

Practical Implications

• Accelerometer‑free system identification – Robotics platforms that lack high‑quality inertial sensors (e.g., soft robots, low‑cost manipulators) can still obtain reliable dynamics models.
• Model‑based control pipelines – Engineers can integrate the learned Lagrangian directly into existing control toolchains (MATLAB/Simulink, ROS) without hand‑crafting the equations of motion.
• Data‑efficient learning – The physics‑aware loss reduces the amount of training data, making on‑site calibration feasible for production lines or field robots.
• Safety‑critical applications – By guaranteeing that the model respects fundamental energy laws, the approach mitigates the risk of unstable control actions caused by spurious learned dynamics.

Limitations & Future Work

• Scalability – The current experiments involve low‑dimensional systems (≤ 4 DoF). Extending the method to high‑DoF manipulators or flexible structures may require more sophisticated network architectures or sparsity priors.
• Non‑smooth non‑conservative forces – Friction models with stick‑slip or impacts are not explicitly handled; the authors note that richer (f_{\text{nc}}) representations could improve realism.
• Real‑time inference – While inference is fast enough for the benchmark, the computational load of automatic differentiation could become a bottleneck on embedded hardware.
• Generalization across tasks – The model is trained on a single operating regime; future work could explore continual learning or domain adaptation to maintain consistency when the robot’s payload or environment changes.

Authors

• Ibrahim Laiche
• Mokrane Boudaoud
• Patrick Gallinari
• Pascal Morin

Paper Information

• arXiv ID: 2512.03035v1
• Categories: eess.SY, cs.LG
• Published: December 2, 2025
• PDF: Download PDF