# Coordinated Humanoid Robot Locomotion with Symmetry Equivariant Reinforcement Learning Policy

arXiv: https://arxiv.org/abs/2508.01247 · HTML: https://mlsanity.com/paper/2508.01247/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL for robot locomotion and manipulation, one of 15 directions in Deep reinforcement learning methods.
- Builds on 26 of its 49 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## 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, 26 of 26)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Improving Sample Efficiency in Model-Free Reinforcement Learning from Images](https://mlsanity.com/paper/1910.01741/) (2019)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Rapid Locomotion via Reinforcement Learning](https://mlsanity.com/paper/2205.02824/) (2022)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [End-to-End Robotic Reinforcement Learning without Reward Engineering](https://mlsanity.com/paper/1904.07854/) (2019)
- [Reinforcement Learning for Versatile, Dynamic, and Robust Bipedal Locomotion Control](https://mlsanity.com/paper/2401.16889/) (2024)
- [HOMIE: Humanoid Loco-Manipulation with Isomorphic Exoskeleton Cockpit](https://mlsanity.com/paper/2502.13013/) (2025)
- [Humanoid-Gym: Reinforcement Learning for Humanoid Robot with Zero-Shot Sim2Real Transfer](https://mlsanity.com/paper/2404.05695/) (2024)
- [Learning Humanoid Locomotion with Perceptive Internal Model](https://mlsanity.com/paper/2411.14386/) (2024)
- [On-Robot Learning With Equivariant Models](https://mlsanity.com/paper/2203.04923/) (2022)
- [CPG-RL: Learning Central Pattern Generators for Quadruped Locomotion](https://mlsanity.com/paper/2211.00458/) (2022)
- [Symmetry Considerations for Learning Task Symmetric Robot Policies](https://mlsanity.com/paper/2403.04359/) (2024)
- [Continuous MDP Homomorphisms and Homomorphic Policy Gradient](https://mlsanity.com/paper/2209.07364/) (2022)
- [Group Equivariant Deep Reinforcement Learning](https://mlsanity.com/paper/2007.03437/) (2020)
- [OrbitGrasp: SE(3)-Equivariant Grasp Learning](https://mlsanity.com/paper/2407.03531/) (2024)
- [Policy Gradient Methods in the Presence of Symmetries and State Abstractions](https://mlsanity.com/paper/2305.05666/) (2023)
- [Understanding when Dynamics-Invariant Data Augmentations Benefit Model-Free Reinforcement Learning Updates](https://mlsanity.com/paper/2310.17786/) (2023)
- [Natural Humanoid Robot Locomotion with Generative Motion Prior](https://mlsanity.com/paper/2503.09015/) (2025)
- [Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension](https://mlsanity.com/paper/2309.02711/) (2023)
- [Equivariant Ensembles and Regularization for Reinforcement Learning in Map-based Path Planning](https://mlsanity.com/paper/2403.12856/) (2024)
- [Breaking Symmetries Leads to Diverse Quadrupedal Gaits](https://mlsanity.com/paper/2303.04857/) (2023)
- [Reinforcement Learning with Euclidean Data Augmentation for State-Based Continuous Control](https://mlsanity.com/paper/2410.12983/) (2024)

## Built on by (most cited first, 2 of 2)

- [Beyond Topology: A Morphological Symmetry Graph Representation for Locomotion Policy Learning](https://mlsanity.com/paper/2512.00727/) (2025)
- [Constraint-Aware Diffusion Priors for High-Fidelity and Versatile Quadruped Locomotion](https://mlsanity.com/paper/2605.08804/) (2026)

## Abstract (arXiv)

The human nervous system exhibits bilateral symmetry, enabling coordinated and balanced movements. However, existing Deep Reinforcement Learning (DRL) methods for humanoid robots neglect morphological symmetry of the robot, leading to uncoordinated and suboptimal behaviors. Inspired by human motor control, we propose Symmetry Equivariant Policy (SE-Policy), a new DRL framework that embeds strict symmetry equivariance in the actor and symmetry invariance in the critic without additional hyperparameters. SE-Policy enforces consistent behaviors across symmetric observations, producing temporally and spatially coordinated motions with higher task performance. Extensive experiments on velocity tracking tasks, conducted in both simulation and real-world deployment with the Unitree G1 humanoid robot, demonstrate that SE-Policy improves tracking accuracy by up to 40% compared to state-of-the-art baselines, while achieving superior spatial-temporal coordination. These results demonstrate the effectiveness of SE-Policy and its broad applicability to humanoid robots.