# Symmetry Considerations for Learning Task Symmetric Robot Policies

arXiv: https://arxiv.org/abs/2403.04359 · HTML: https://mlsanity.com/paper/2403.04359/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 7 Mar 2024.

## Builds on (most cited first, 19 of 19)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Solving Rubik's Cube with a Robot Hand](https://mlsanity.com/paper/1910.07113/) (2019)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [RMA: Rapid Motor Adaptation for Legged Robots](https://mlsanity.com/paper/2107.04034/) (2021)
- [Reinforcement Learning with Augmented Data](https://mlsanity.com/paper/2004.14990/) (2020)
- [Learning robust perceptive locomotion for quadrupedal robots in the wild](https://mlsanity.com/paper/2201.08117/) (2022)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [Learning Symmetric and Low-energy Locomotion](https://mlsanity.com/paper/1801.08093/) (2018)
- [Advanced Skills by Learning Locomotion and Local Navigation End-to-End](https://mlsanity.com/paper/2209.12827/) (2022)
- [Transferring Dexterous Manipulation from GPU Simulation to a Remote Real-World TriFinger](https://mlsanity.com/paper/2108.09779/) (2021)
- [CPG-RL: Learning Central Pattern Generators for Quadruped Locomotion](https://mlsanity.com/paper/2211.00458/) (2022)
- [TriFinger: An Open-Source Robot for Learning Dexterity](https://mlsanity.com/paper/2008.03596/) (2020)
- [Cat-like Jumping and Landing of Legged Robots in Low-gravity Using Deep Reinforcement Learning](https://mlsanity.com/paper/2106.09357/) (2021)
- [On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis](https://mlsanity.com/paper/2302.10433/) (2023)
- [Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension](https://mlsanity.com/paper/2309.02711/) (2023)

## Built on by (most cited first, 28 of 28)

- [Flow Matching Policy Gradients](https://mlsanity.com/paper/2507.21053/) (2025)
- [RSL-RL: A Learning Library for Robotics Research](https://mlsanity.com/paper/2509.10771/) (2025)
- [Leveraging Symmetry in RL-based Legged Locomotion Control](https://mlsanity.com/paper/2403.17320/) (2024)
- [Learning Sim-to-Real Humanoid Locomotion in 15 Minutes](https://mlsanity.com/paper/2512.01996/) (2025)
- [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)
- [Non-conflicting Energy Minimization in Reinforcement Learning based Robot Control](https://mlsanity.com/paper/2509.01765/) (2025)
- [FlashSAC: Fast and Stable Off-Policy Reinforcement Learning for High-Dimensional Robot Control](https://mlsanity.com/paper/2604.04539/) (2026)
- [Beyond Topology: A Morphological Symmetry Graph Representation for Locomotion Policy Learning](https://mlsanity.com/paper/2512.00727/) (2025)
- [AGILE: A Comprehensive Workflow for Humanoid Loco-Manipulation Learning](https://mlsanity.com/paper/2603.20147/) (2026)
- [Symmetry-Guided Memory Augmentation for Efficient Locomotion Learning](https://mlsanity.com/paper/2502.01521/) (2025)
- [Scaling Rough Terrain Locomotion with Automatic Curriculum Reinforcement Learning](https://mlsanity.com/paper/2601.17428/) (2026)
- [Neural Circuit Architectural Priors for Quadruped Locomotion](https://mlsanity.com/paper/2410.07174/) (2024)
- [Overcoming Non-stationary Dynamics with Evidential Proximal Policy Optimization](https://mlsanity.com/paper/2503.01468/) (2025)
- [Learning Bipedal Locomotion on Gear-Driven Humanoid Robot Using Foot-Mounted IMUs](https://mlsanity.com/paper/2504.00614/) (2025)
- [Coordinated Humanoid Robot Locomotion with Symmetry Equivariant Reinforcement Learning Policy](https://mlsanity.com/paper/2508.01247/) (2025)
- [PvP: Data-Efficient Humanoid Robot Learning with Proprioceptive-Privileged Contrastive Representations](https://mlsanity.com/paper/2512.13093/) (2025)
- [Embedding Classical Balance Control Principles in Reinforcement Learning for Humanoid Recovery](https://mlsanity.com/paper/2603.08619/) (2026)
- [APEX: Learning Adaptive High-Platform Traversal for Humanoid Robots](https://mlsanity.com/paper/2602.11143/) (2026)
- [Constraint-Aware Diffusion Priors for High-Fidelity and Versatile Quadruped Locomotion](https://mlsanity.com/paper/2605.08804/) (2026)
- [No More Marching: Learning Humanoid Locomotion for Short-Range SE(2) Targets](https://mlsanity.com/paper/2508.14098/) (2025)
- [GeCCo -- a Generalist Contact-Conditioned Policy for Loco-Manipulation Skills on Legged Robots](https://mlsanity.com/paper/2509.17582/) (2025)
- [Pretraining in Actor-Critic Reinforcement Learning for Locomotion](https://mlsanity.com/paper/2510.12363/) (2025)
- [Morphologically Equivariant Flow Matching for Bimanual Mobile Manipulation](https://mlsanity.com/paper/2605.12228/) (2026)
- [Reflex: Reinforcement Learning with Reflection Symmetry Exploitation in State-Based Continuous Control](https://mlsanity.com/paper/2605.23415/) (2026)
- [SKooP: Symmetric Koopman Predictions for Faster and More Generalizable Legged Robot Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2607.11624/) (2026)
- [Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction](https://mlsanity.com/paper/2608.11318/) (2026)
- [MASkillBlender: Decentralized Whole-Body Coordination for Multi-Humanoid Loco-Manipulation via Skill Blending](https://mlsanity.com/paper/2610.01102/) (2026)

## Abstract (arXiv)

Symmetry is a fundamental aspect of many real-world robotic tasks. However, current deep reinforcement learning (DRL) approaches can seldom harness and exploit symmetry effectively. Often, the learned behaviors fail to achieve the desired transformation invariances and suffer from motion artifacts. For instance, a quadruped may exhibit different gaits when commanded to move forward or backward, even though it is symmetrical about its torso. This issue becomes further pronounced in high-dimensional or complex environments, where DRL methods are prone to local optima and fail to explore regions of the state space equally. Past methods on encouraging symmetry for robotic tasks have studied this topic mainly in a single-task setting, where symmetry usually refers to symmetry in the motion, such as the gait patterns. In this paper, we revisit this topic for goal-conditioned tasks in robotics, where symmetry lies mainly in task execution and not necessarily in the learned motions themselves. In particular, we investigate two approaches to incorporate symmetry invariance into DRL -- data augmentation and mirror loss function. We provide a theoretical foundation for using augmented samples in an on-policy setting. Based on this, we show that the corresponding approach achieves faster convergence and improves the learned behaviors in various challenging robotic tasks, from climbing boxes with a quadruped to dexterous manipulation.