# Data-Driven Physics Embedded Dynamics with Predictive Control and Reinforcement Learning for Quadrupeds

arXiv: https://arxiv.org/abs/2603.14333 · HTML: https://mlsanity.com/paper/2603.14333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for robot locomotion and manipulation, one of 15 directions in Deep reinforcement learning methods.
- Builds on 25 of its 31 references that are in the map.
- First posted on arXiv 15 Mar 2026.

## Placed in

- Deep reinforcement learning methods / [RL for robot locomotion and manipulation](https://mlsanity.com/deeprl/rl-for-robot-locomotion-and-manipulation/)

## Builds on (most cited first, 25 of 25)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [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)
- [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)
- [Imagination-Augmented Agents for Deep Reinforcement Learning](https://mlsanity.com/paper/1707.06203/) (2017)
- [Model Predictive Path Integral Control using Covariance Variable Importance Sampling](https://mlsanity.com/paper/1509.01149/) (2015)
- [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)
- [Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems](https://mlsanity.com/paper/1907.04489/) (2019)
- [Physics-Informed Model-Based Reinforcement Learning](https://mlsanity.com/paper/2212.02179/) (2022)
- [Inverse Dynamics Trajectory Optimization for Contact-Implicit Model Predictive Control](https://mlsanity.com/paper/2309.01813/) (2023)
- [PIP-Loco: A Proprioceptive Infinite Horizon Planning Framework for Quadrupedal Robot Locomotion](https://mlsanity.com/paper/2409.09441/) (2024)
- [Versatile Real-Time Motion Synthesis via Kino-Dynamic MPC with Hybrid-Systems DDP](https://mlsanity.com/paper/2209.14138/) (2022)
- [Investigating Lagrangian Neural Networks for Infinite Horizon Planning in Quadrupedal Locomotion](https://mlsanity.com/paper/2506.16079/) (2025)
- [Context-Aware Deep Lagrangian Networks for Model Predictive Control](https://mlsanity.com/paper/2506.15249/) (2025)

## Abstract (arXiv)

State of the art quadrupedal locomotion approaches integrate Model Predictive Control (MPC) with Reinforcement Learning (RL), enabling complex motion capabilities with planning and terrain adaptive behaviors. However, they often face compounding errors over long horizons and have limited interpretability due to the absence of physical inductive biases. We address these issues by integrating Lagrangian Neural Networks (LNNs) into an RL MPC framework, enabling physically consistent dynamics learning. At deployment, our inverse dynamics infinite horizon MPC scheme avoids costly matrix inversions, improving computational efficiency by up to 4x with minimal loss of task performance. We validate our framework through multiple ablations of the proposed LNN and its variants. We show improved sample efficiency, reduced long-horizon error, and faster real time planning compared to unstructured neural dynamics. Lastly, we also test our framework on the Unitree Go1 robot to show real world viability.