# On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis

arXiv: https://arxiv.org/abs/2302.10433 · HTML: https://mlsanity.com/paper/2302.10433/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 21 Feb 2023.

## Builds on (most cited first, 14 of 14)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Learning Symmetric and Low-energy Locomotion](https://mlsanity.com/paper/1801.08093/) (2018)
- [Chirality Nets for Human Pose Regression](https://mlsanity.com/paper/1911.00029/) (2019)
- [Benchmarking Structured Policies and Policy Optimization for Real-World Dexterous Object Manipulation](https://mlsanity.com/paper/2105.02087/) (2021)
- [Legged Robot State Estimation using Invariant Kalman Filtering and Learned Contact Events](https://mlsanity.com/paper/2106.15713/) (2021)
- [An Adaptable Approach to Learn Realistic Legged Locomotion without Examples](https://mlsanity.com/paper/2110.14998/) (2021)

## Built on by (most cited first, 8 of 8)

- [Symmetry Considerations for Learning Task Symmetric Robot Policies](https://mlsanity.com/paper/2403.04359/) (2024)
- [Leveraging Symmetry in RL-based Legged Locomotion Control](https://mlsanity.com/paper/2403.17320/) (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)
- [Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling](https://mlsanity.com/paper/2312.07457/) (2023)
- [Leveraging Symmetry to Accelerate Learning of Trajectory Tracking Controllers for Free-Flying Robotic Systems](https://mlsanity.com/paper/2409.11238/) (2024)
- [Symmetries-enhanced Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2501.01136/) (2025)
- [Robust RL Control for Bipedal Locomotion with Closed Kinematic Chains](https://mlsanity.com/paper/2507.10164/) (2025)

## Abstract (arXiv)

We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group G and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and G-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.