# Beyond Topology: A Morphological Symmetry Graph Representation for Locomotion Policy Learning

arXiv: https://arxiv.org/abs/2512.00727 · HTML: https://mlsanity.com/paper/2512.00727/

## Key facts

- Cited by 4 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 23 of its 37 references that are in the map.
- First posted on arXiv 30 Nov 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, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Learning robust perceptive locomotion for quadrupedal robots in the wild](https://mlsanity.com/paper/2201.08117/) (2022)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Walk These Ways: Tuning Robot Control for Generalization with Multiplicity of Behavior](https://mlsanity.com/paper/2212.03238/) (2022)
- [One Policy to Control Them All: Shared Modular Policies for Agent-Agnostic Control](https://mlsanity.com/paper/2007.04976/) (2020)
- [In-Hand Object Rotation via Rapid Motor Adaptation](https://mlsanity.com/paper/2210.04887/) (2022)
- [TWIST: Teleoperated Whole-Body Imitation System](https://mlsanity.com/paper/2505.02833/) (2025)
- [MetaMorph: Learning Universal Controllers with Transformers](https://mlsanity.com/paper/2203.11931/) (2022)
- [Body Transformer: Leveraging Robot Embodiment for Policy Learning](https://mlsanity.com/paper/2408.06316/) (2024)
- [Symmetry Considerations for Learning Task Symmetric Robot Policies](https://mlsanity.com/paper/2403.04359/) (2024)
- [GET-Zero: Graph Embodiment Transformer for Zero-shot Embodiment Generalization](https://mlsanity.com/paper/2407.15002/) (2024)
- [Leveraging Symmetry in RL-based Legged Locomotion Control](https://mlsanity.com/paper/2403.17320/) (2024)
- [Learning Modular Robot Control Policies](https://mlsanity.com/paper/2105.10049/) (2021)
- [Morphological Symmetries in Robotics](https://mlsanity.com/paper/2402.15552/) (2024)
- [Multi-embodiment Legged Robot Control as a Sequence Modeling Problem](https://mlsanity.com/paper/2212.09078/) (2022)
- [MI-HGNN: Morphology-Informed Heterogeneous Graph Neural Network for Legged Robot Contact Perception](https://mlsanity.com/paper/2409.11146/) (2024)
- [Morphological-Symmetry-Equivariant Heterogeneous Graph Neural Network for Robotic Dynamics Learning](https://mlsanity.com/paper/2412.01297/) (2024)
- [Morphologically Symmetric Reinforcement Learning for Ambidextrous Bimanual Manipulation](https://mlsanity.com/paper/2505.05287/) (2025)
- [Coordinated Humanoid Robot Locomotion with Symmetry Equivariant Reinforcement Learning Policy](https://mlsanity.com/paper/2508.01247/) (2025)

## Built on by (most cited first, 4 of 4)

- [WestWorld: A Knowledge-Encoded Scalable Trajectory World Model for Diverse Robotic Systems](https://mlsanity.com/paper/2603.14392/) (2026)
- [Articulated-Body Dynamics Network: Dynamics-Grounded Prior for Robot Learning](https://mlsanity.com/paper/2603.19078/) (2026)
- [Morphologically Equivariant Flow Matching for Bimanual Mobile Manipulation](https://mlsanity.com/paper/2605.12228/) (2026)
- [The Robot Is Not Its Description: GaugeBench for Representation Robustness in Morphology-Aware Policies](https://mlsanity.com/paper/2610.07597/) (2026)

## Abstract (arXiv)

Reinforcement learning has enabled impressive locomotion skills on articulated robots, but common policy representations remain only weakly aligned with robot physics. Generic networks ignore kinematic structure, while graph-based policies encode connectivity without specifying how physical quantities transform across symmetric body parts. We introduce a morphological symmetry graph representation for locomotion policy learning and instantiate it in MS-PPO. Starting from the robot's topological graph, our representation augments each observation and action space with the permutation and sign transformations induced by morphological symmetry. This yields a symmetry-equivariant graph actor and a symmetry-invariant graph critic, enforcing the desired policy and value constraints by construction rather than through reward shaping or data augmentation. We evaluate MS-PPO on a variety of locomotion tasks using both Unitree Go2 quadruped and Unitree G1 humanoid, including command tracking, asymmetric joint failures, out-of-distribution command generalization, and zero-shot sim-to-real deployment. Experiments show improved symmetry generalization, robustness, sample efficiency, and model efficiency over topology- and symmetry-aware baselines.