# Efficient Hybrid SE(3)-Equivariant Visuomotor Flow Policy via Spherical Harmonics for Robot Manipulation

arXiv: https://arxiv.org/abs/2603.23227 · HTML: https://mlsanity.com/paper/2603.23227/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 35 of its 44 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [OpenVLA: An Open-Source Vision-Language-Action Model](https://mlsanity.com/paper/2406.09246/) (2024)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [MimicGen: A Data Generation System for Scalable Robot Learning using Human Demonstrations](https://mlsanity.com/paper/2310.17596/) (2023)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Consistency Policy: Accelerated Visuomotor Policies via Consistency Distillation](https://mlsanity.com/paper/2405.07503/) (2024)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [FlowPolicy: Enabling Fast and Robust 3D Flow-based Policy via Consistency Flow Matching for Robot Manipulation](https://mlsanity.com/paper/2412.04987/) (2024)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning](https://mlsanity.com/paper/2407.01479/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [AdaFlow: Imitation Learning with Variance-Adaptive Flow-Based Policies](https://mlsanity.com/paper/2402.04292/) (2024)
- [RISE: 3D Perception Makes Real-World Robot Imitation Simple and Effective](https://mlsanity.com/paper/2404.12281/) (2024)
- [AlphaFlow: Understanding and Improving MeanFlow Models](https://mlsanity.com/paper/2510.20771/) (2025)
- [EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulation](https://mlsanity.com/paper/2310.16050/) (2023)
- [ET-SEED: Efficient Trajectory-Level SE(3) Equivariant Diffusion Policy](https://mlsanity.com/paper/2411.03990/) (2024)
- [Consistency Diffusion Bridge Models](https://mlsanity.com/paper/2410.22637/) (2024)
- [Hierarchical Neural Semantic Representation for 3D Semantic Correspondence](https://mlsanity.com/paper/2509.17431/) (2025)
- [On Robot Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2306.06489/) (2023)
- [FreqPolicy: Frequency Autoregressive Visuomotor Policy with Continuous Tokens](https://mlsanity.com/paper/2506.01583/) (2025)
- [SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space](https://mlsanity.com/paper/2507.01723/) (2025)

## Built on by (most cited first, 1 of 1)

- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)

## Abstract (arXiv)

While existing equivariant methods enhance data efficiency, they suffer from high computational intensity, reliance on single-modality inputs, and instability when combined with fast-sampling methods. In this work, we propose E3Flow, a novel framework that addresses the critical limitations of equivariant diffusion policies. E3Flow overcomes these challenges, successfully unifying efficient rectified flow with stable, multi-modal equivariant learning for the first time. Our framework is built upon spherical harmonic representations to ensure rigorous SO(3) equivariance. We introduce a novel invariant Feature Enhancement Module (FEM) that dynamically fuses hybrid visual modalities (point clouds and images), injecting rich visual cues into the spherical harmonic features. We evaluate E3Flow on 8 manipulation tasks from the MimicGen and further conduct 4 real-world experiments to validate its effectiveness in physical environments. Simulation results show that E3Flow achieves a 3.12% improvement in average success rate over the state-of-the-art Spherical Diffusion Policy (SDP) while simultaneously delivering a 7x inference speedup. E3Flow thus demonstrates a new and highly effective trade-off between performance, efficiency, and data efficiency for robotic policy learning. Code: https://github.com/zql-kk/E3Flow.