# Investigating Lagrangian Neural Networks for Infinite Horizon Planning in Quadrupedal Locomotion

arXiv: https://arxiv.org/abs/2506.16079 · HTML: https://mlsanity.com/paper/2506.16079/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 29 references that are in the map.
- First posted on arXiv 19 Jun 2025.

## Builds on (most cited first, 24 of 24)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Learning to Walk in Minutes Using Massively Parallel Deep Reinforcement Learning](https://mlsanity.com/paper/2109.11978/) (2021)
- [TD-MPC2: Scalable, Robust World Models for Continuous Control](https://mlsanity.com/paper/2310.16828/) (2023)
- [Learning agile and dynamic motor skills for legged robots](https://mlsanity.com/paper/1901.08652/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Temporal Difference Learning for Model Predictive Control](https://mlsanity.com/paper/2203.04955/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Asymmetric Actor Critic for Image-Based Robot Learning](https://mlsanity.com/paper/1710.06542/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Model Predictive Path Integral Control using Covariance Variable Importance Sampling](https://mlsanity.com/paper/1509.01149/) (2015)
- [Plan Online, Learn Offline: Efficient Learning and Exploration via Model-Based Control](https://mlsanity.com/paper/1811.01848/) (2018)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [DreamWaQ: Learning Robust Quadrupedal Locomotion With Implicit Terrain Imagination via Deep Reinforcement Learning](https://mlsanity.com/paper/2301.10602/) (2023)
- [Learning Off-Policy with Online Planning](https://mlsanity.com/paper/2008.10066/) (2020)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Hybrid Internal Model: Learning Agile Legged Locomotion with Simulated Robot Response](https://mlsanity.com/paper/2312.11460/) (2023)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [Model Predictive Actor-Critic: Accelerating Robot Skill Acquisition with Deep Reinforcement Learning](https://mlsanity.com/paper/2103.13842/) (2021)
- [Barrier Functions Inspired Reward Shaping for Reinforcement Learning](https://mlsanity.com/paper/2403.01410/) (2024)
- [PIP-Loco: A Proprioceptive Infinite Horizon Planning Framework for Quadrupedal Robot Locomotion](https://mlsanity.com/paper/2409.09441/) (2024)
- [Dynamic Mirror Descent based Model Predictive Control for Accelerating Robot Learning](https://mlsanity.com/paper/2112.02999/) (2021)

## Built on by (most cited first, 3 of 3)

- [STRIDE: Structured Lagrangian and Stochastic Residual Dynamics via Flow Matching](https://mlsanity.com/paper/2603.08478/) (2026)
- [Data-Driven Physics Embedded Dynamics with Predictive Control and Reinforcement Learning for Quadrupeds](https://mlsanity.com/paper/2603.14333/) (2026)
- [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026)

## Abstract (arXiv)

Lagrangian Neural Networks (LNNs) present a principled and interpretable framework for learning the system dynamics by utilizing inductive biases. While traditional dynamics models struggle with compounding errors over long horizons, LNNs intrinsically preserve the physical laws governing any system, enabling accurate and stable predictions essential for sustainable locomotion. This work evaluates LNNs for infinite horizon planning in quadrupedal robots through four dynamics models: (1) full-order forward dynamics (FD) training and inference, (2) diagonalized representation of Mass Matrix in full order FD, (3) full-order inverse dynamics (ID) training with FD inference, (4) reduced-order modeling via torso centre-of-mass (CoM) dynamics. Experiments demonstrate that LNNs bring improvements in sample efficiency (10x) and superior prediction accuracy (up to 2-10x) compared to baseline methods. Notably, the diagonalization approach of LNNs reduces computational complexity while retaining some interpretability, enabling real-time receding horizon control. These findings highlight the advantages of LNNs in capturing the underlying structure of system dynamics in quadrupeds, leading to improved performance and efficiency in locomotion planning and control. Additionally, our approach achieves a higher control frequency than previous LNN methods, demonstrating its potential for real-world deployment on quadrupeds.