# ActionFlow: Equivariant, Accurate, and Efficient Policies with Spatially Symmetric Flow Matching

arXiv: https://arxiv.org/abs/2409.04576 · HTML: https://mlsanity.com/paper/2409.04576/

## Key facts

- Cited by 26 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 47 of its 57 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## 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 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [RT-1: Robotics Transformer for Real-World Control at Scale](https://mlsanity.com/paper/2212.06817/) (2022)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [CLIPort: What and Where Pathways for Robotic Manipulation](https://mlsanity.com/paper/2109.12098/) (2021)
- [Perceiver-Actor: A Multi-Task Transformer for Robotic Manipulation](https://mlsanity.com/paper/2209.05451/) (2022)
- [FoundationPose: Unified 6D Pose Estimation and Tracking of Novel Objects](https://mlsanity.com/paper/2312.08344/) (2023)
- [Transporter Networks: Rearranging the Visual World for Robotic Manipulation](https://mlsanity.com/paper/2010.14406/) (2020)
- [MimicGen: A Data Generation System for Scalable Robot Learning using Human Demonstrations](https://mlsanity.com/paper/2310.17596/) (2023)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [3D Diffuser Actor: Policy Diffusion with 3D Scene Representations](https://mlsanity.com/paper/2402.10885/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [RVT: Robotic View Transformer for 3D Object Manipulation](https://mlsanity.com/paper/2306.14896/) (2023)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [CLIP-Fields: Weakly Supervised Semantic Fields for Robotic Memory](https://mlsanity.com/paper/2210.05663/) (2022)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [VIOLA: Imitation Learning for Vision-Based Manipulation with Object Proposal Priors](https://mlsanity.com/paper/2210.11339/) (2022)
- [Distilled Feature Fields Enable Few-Shot Language-Guided Manipulation](https://mlsanity.com/paper/2308.07931/) (2023)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)

## Built on by (most cited first, 26 of 26)

- [FLOWER: Democratizing Generalist Robot Policies with Efficient Vision-Language-Action Flow Policies](https://mlsanity.com/paper/2509.04996/) (2025)
- [Streaming Flow Policy: Simplifying diffusion/flow-matching policies by treating action trajectories as flow trajectories](https://mlsanity.com/paper/2505.21851/) (2025)
- [3D FlowMatch Actor: Unified 3D Policy for Single- and Dual-Arm Manipulation](https://mlsanity.com/paper/2508.11002/) (2025)
- [FlowMP: Learning Motion Fields for Robot Planning with Conditional Flow Matching](https://mlsanity.com/paper/2503.06135/) (2025)
- [FreqPolicy: Efficient Flow-based Visuomotor Policy via Frequency Consistency](https://mlsanity.com/paper/2506.08822/) (2025)
- [FlowRAM: Grounding Flow Matching Policy with Region-Aware Mamba Framework for Robotic Manipulation](https://mlsanity.com/paper/2506.16201/) (2025)
- [SE(3)-Equivariant Robot Learning and Control: A Tutorial Survey](https://mlsanity.com/paper/2503.09829/) (2025)
- [Fast Flow-based Visuomotor Policies via Conditional Optimal Transport Couplings](https://mlsanity.com/paper/2505.01179/) (2025)
- [FreqPolicy: Frequency Autoregressive Visuomotor Policy with Continuous Tokens](https://mlsanity.com/paper/2506.01583/) (2025)
- [One-Step Flow Policy: Self-Distillation for Fast Visuomotor Policies](https://mlsanity.com/paper/2603.12480/) (2026)
- [ELMUR: External Layer Memory with Update/Rewrite for Long-Horizon RL Problems](https://mlsanity.com/paper/2510.07151/) (2025)
- [3D Equivariant Visuomotor Policy Learning via Spherical Projection](https://mlsanity.com/paper/2505.16969/) (2025)
- [UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning](https://mlsanity.com/paper/2506.02955/) (2025)
- [Time-Unified Diffusion Policy with Action Discrimination for Robotic Manipulation](https://mlsanity.com/paper/2506.09422/) (2025)
- [On the Importance of Tactile Sensing for Imitation Learning: A Case Study on Robotic Match Lighting](https://mlsanity.com/paper/2504.13618/) (2025)
- [SE(3)-PoseFlow: Estimating 6D Pose Distributions for Uncertainty-Aware Robotic Manipulation](https://mlsanity.com/paper/2511.01501/) (2025)
- [Scaling Single Human Demonstrations for Imitation Learning using Generative Foundational Models](https://mlsanity.com/paper/2602.12734/) (2026)
- [PRISM: Performer RS-IMLE for Single-pass Multisensory Imitation Learning](https://mlsanity.com/paper/2602.02396/) (2026)
- [Implicit Drifting Policy: One-Step Action Generation via Conditional Expert Geometry](https://mlsanity.com/paper/2606.01098/) (2026)
- [Towards Safe Imitation Learning via Potential Field-Guided Flow Matching](https://mlsanity.com/paper/2508.08707/) (2025)
- [Morphologically Equivariant Flow Matching for Bimanual Mobile Manipulation](https://mlsanity.com/paper/2605.12228/) (2026)
- [Let the Dynamics Flow: Stable Flow Matching Dynamical Systems](https://mlsanity.com/paper/2606.03834/) (2026)
- [The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups](https://mlsanity.com/paper/2606.20547/) (2026)
- [Scalable Multi-Task Data Generation via Reinforcement Learning for Language-Conditioned Bimanual Dexterous Manipulation](https://mlsanity.com/paper/2606.22471/) (2026)
- [BridgeFlow: Fast and Robust SE(2)-Equivariant Motion Planning with Flow Matching](https://mlsanity.com/paper/2607.14725/) (2026)
- [Temporal Policy: History-Initialized Action Generation for Robotic Learning from Demonstration](https://mlsanity.com/paper/2607.29482/) (2026)

## Abstract (arXiv)

Spatial understanding is a critical aspect of most robotic tasks, particularly when generalization is important. Despite the impressive results of deep generative models in complex manipulation tasks, the absence of a representation that encodes intricate spatial relationships between observations and actions often limits spatial generalization, necessitating large amounts of demonstrations. To tackle this problem, we introduce a novel policy class, ActionFlow. ActionFlow integrates spatial symmetry inductive biases while generating expressive action sequences. On the representation level, ActionFlow introduces an SE(3) Invariant Transformer architecture, which enables informed spatial reasoning based on the relative SE(3) poses between observations and actions. For action generation, ActionFlow leverages Flow Matching, a state-of-the-art deep generative model known for generating high-quality samples with fast inference - an essential property for feedback control. In combination, ActionFlow policies exhibit strong spatial and locality biases and SE(3)-equivariant action generation. Our experiments demonstrate the effectiveness of ActionFlow and its two main components on several simulated and real-world robotic manipulation tasks and confirm that we can obtain equivariant, accurate, and efficient policies with spatially symmetric flow matching. Project website: https://flowbasedpolicies.github.io/