# Morphological Symmetries in Robotics

arXiv: https://arxiv.org/abs/2402.15552 · HTML: https://mlsanity.com/paper/2402.15552/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 80 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Builds on (most cited first, 24 of 24)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [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)
- [On-Robot Learning With Equivariant Models](https://mlsanity.com/paper/2203.04923/) (2022)
- [EDGI: Equivariant Diffusion for Planning with Embodied Agents](https://mlsanity.com/paper/2303.12410/) (2023)
- [Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control](https://mlsanity.com/paper/1909.04947/) (2019)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [Continuous MDP Homomorphisms and Homomorphic Policy Gradient](https://mlsanity.com/paper/2209.07364/) (2022)
- [Symmetry-Based Representations for Artificial and Biological General Intelligence](https://mlsanity.com/paper/2203.09250/) (2022)
- [An Open Torque-Controlled Modular Robot Architecture for Legged Locomotion Research](https://mlsanity.com/paper/1910.00093/) (2019)
- [On the Sample Complexity of Learning under Invariance and Geometric Stability](https://mlsanity.com/paper/2106.07148/) (2021)
- [Legged Robot State Estimation using Invariant Kalman Filtering and Learned Contact Events](https://mlsanity.com/paper/2106.15713/) (2021)
- [Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformers](https://mlsanity.com/paper/2311.04744/) (2023)
- [Motion Planning for Quadrupedal Locomotion: Coupled Planning, Terrain Mapping and Whole-Body Control](https://mlsanity.com/paper/2003.05481/) (2020)
- [Identification of Fully Physical Consistent Inertial Parameters using Optimization on Manifolds](https://mlsanity.com/paper/1610.08703/) (2016)
- [An Adaptable Approach to Learn Realistic Legged Locomotion without Examples](https://mlsanity.com/paper/2110.14998/) (2021)
- [Inverse-Dynamics MPC via Nullspace Resolution](https://mlsanity.com/paper/2209.05375/) (2022)
- [Perceptive Locomotion through Whole-Body MPC and Optimal Region Selection](https://mlsanity.com/paper/2305.08926/) (2023)
- [Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling](https://mlsanity.com/paper/2312.07457/) (2023)

## Built on by (most cited first, 13 of 13)

- [Leveraging Symmetry in RL-based Legged Locomotion Control](https://mlsanity.com/paper/2403.17320/) (2024)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (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)
- [Guided Reinforcement Learning for Omnidirectional 3D Jumping in Quadruped Robots](https://mlsanity.com/paper/2507.16481/) (2025)
- [Beyond Topology: A Morphological Symmetry Graph Representation for Locomotion Policy Learning](https://mlsanity.com/paper/2512.00727/) (2025)
- [Symmetry-Guided Memory Augmentation for Efficient Locomotion Learning](https://mlsanity.com/paper/2502.01521/) (2025)
- [Recurrent Equivariant Constraint Modulation: Learning Per-Layer Symmetry Relaxation from Data](https://mlsanity.com/paper/2602.02853/) (2026)
- [Multi-Task Reinforcement Learning of Drone Aerobatics by Exploiting Geometric Symmetries](https://mlsanity.com/paper/2602.10997/) (2026)
- [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)
- [Extreme dynamic symmetry enables omnidirectional and multifunctional robots](https://mlsanity.com/paper/2605.29254/) (2026)
- [OmniMimic: Dynamics-completed Motion Augmentation for Multi-style Omnidirectional Quadruped Locomotion](https://mlsanity.com/paper/2609.20566/) (2026)

## Abstract (arXiv)

We present a comprehensive framework for studying and leveraging morphological symmetries in robotic systems. These are intrinsic properties of the robot's morphology, frequently observed in animal biology and robotics, which stem from the replication of kinematic structures and the symmetrical distribution of mass. We illustrate how these symmetries extend to the robot's state space and both proprioceptive and exteroceptive sensor measurements, resulting in the equivariance of the robot's equations of motion and optimal control policies. Thus, we recognize morphological symmetries as a relevant and previously unexplored physics-informed geometric prior, with significant implications for both data-driven and analytical methods used in modeling, control, estimation and design in robotics. For data-driven methods, we demonstrate that morphological symmetries can enhance the sample efficiency and generalization of machine learning models through data augmentation, or by applying equivariant/invariant constraints on the model's architecture. In the context of analytical methods, we employ abstract harmonic analysis to decompose the robot's dynamics into a superposition of lower-dimensional, independent dynamics. We substantiate our claims with both synthetic and real-world experiments conducted on bipedal and quadrupedal robots. Lastly, we introduce the repository MorphoSymm to facilitate the practical use of the theory and applications outlined in this work.