# Riemannian Flow Matching Policy for Robot Motion Learning

arXiv: https://arxiv.org/abs/2403.10672 · HTML: https://mlsanity.com/paper/2403.10672/

## Key facts

- Cited by 63 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 25 of its 34 references that are in the map.
- First posted on arXiv 15 Mar 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, 25 of 25)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [Diffusion Policies as an Expressive Policy Class for Offline Reinforcement Learning](https://mlsanity.com/paper/2208.06193/) (2022)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Goal-Conditioned Imitation Learning using Score-based Diffusion Policies](https://mlsanity.com/paper/2304.02532/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Efficient Video Prediction via Sparsely Conditioned Flow Matching](https://mlsanity.com/paper/2211.14575/) (2022)
- [Generative Pre-training for Speech with Flow Matching](https://mlsanity.com/paper/2310.16338/) (2023)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Learning Stable Normalizing-Flow Control for Robotic Manipulation](https://mlsanity.com/paper/2011.00072/) (2020)
- [Wrapped Distributions on homogeneous Riemannian manifolds](https://mlsanity.com/paper/2204.09790/) (2022)
- [Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning](https://mlsanity.com/paper/2310.07902/) (2023)

## Built on by (most cited first, 30 of 63)

- [Real-Time Execution of Action Chunking Flow Policies](https://mlsanity.com/paper/2506.07339/) (2025)
- [ReinFlow: Fine-tuning Flow Matching Policy with Online Reinforcement Learning](https://mlsanity.com/paper/2505.22094/) (2025)
- [Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matching](https://mlsanity.com/paper/2409.07343/) (2024)
- [FLOWER: Democratizing Generalist Robot Policies with Efficient Vision-Language-Action Flow Policies](https://mlsanity.com/paper/2509.04996/) (2025)
- [Affordance-based Robot Manipulation with Flow Matching](https://mlsanity.com/paper/2409.01083/) (2024)
- [Can We Detect Failures Without Failure Data? Uncertainty-Aware Runtime Failure Detection for Imitation Learning Policies](https://mlsanity.com/paper/2503.08558/) (2025)
- [ManiFlow: A General Robot Manipulation Policy via Consistency Flow Training](https://mlsanity.com/paper/2509.01819/) (2025)
- [Failure Prediction at Runtime for Generative Robot Policies](https://mlsanity.com/paper/2510.09459/) (2025)
- [Streaming Flow Policy: Simplifying diffusion/flow-matching policies by treating action trajectories as flow trajectories](https://mlsanity.com/paper/2505.21851/) (2025)
- [Flow Matching Imitation Learning for Multi-Support Manipulation](https://mlsanity.com/paper/2407.12381/) (2024)
- [ActionFlow: Equivariant, Accurate, and Efficient Policies with Spatially Symmetric Flow Matching](https://mlsanity.com/paper/2409.04576/) (2024)
- [VITA: Vision-to-Action Flow Matching Policy](https://mlsanity.com/paper/2507.13231/) (2025)
- [3D FlowMatch Actor: Unified 3D Policy for Single- and Dual-Arm Manipulation](https://mlsanity.com/paper/2508.11002/) (2025)
- [Learning Native Continuation for Action Chunking Flow Policies](https://mlsanity.com/paper/2602.12978/) (2026)
- [Fast and Robust Visuomotor Riemannian Flow Matching Policy](https://mlsanity.com/paper/2412.10855/) (2024)
- [FlowMP: Learning Motion Fields for Robot Planning with Conditional Flow Matching](https://mlsanity.com/paper/2503.06135/) (2025)
- [FlowRAM: Grounding Flow Matching Policy with Region-Aware Mamba Framework for Robotic Manipulation](https://mlsanity.com/paper/2506.16201/) (2025)
- [Learning on the Manifold: Unlocking Standard Diffusion Transformers with Representation Encoders](https://mlsanity.com/paper/2602.10099/) (2026)
- [SafeFlowMatcher: Safe and Fast Planning using Flow Matching with Control Barrier Functions](https://mlsanity.com/paper/2509.24243/) (2025)
- [VFP: Variational Flow-Matching Policy for Multi-Modal Robot Manipulation](https://mlsanity.com/paper/2508.01622/) (2025)
- [Riemannian Consistency Model](https://mlsanity.com/paper/2510.00983/) (2025)
- [One-Step Flow Policy: Self-Distillation for Fast Visuomotor Policies](https://mlsanity.com/paper/2603.12480/) (2026)
- [UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning](https://mlsanity.com/paper/2506.02955/) (2025)
- [Energy-Based Flow Matching for Generating 3D Molecular Structure](https://mlsanity.com/paper/2508.18949/) (2025)
- [SE(3)-PoseFlow: Estimating 6D Pose Distributions for Uncertainty-Aware Robotic Manipulation](https://mlsanity.com/paper/2511.01501/) (2025)
- [WarmPrior: Straightening Flow-Matching Policies with Temporal Priors](https://mlsanity.com/paper/2605.13959/) (2026)
- [The Unreasonable Effectiveness of Discrete-Time Gaussian Process Mixtures for Robot Policy Learning](https://mlsanity.com/paper/2505.03296/) (2025)
- [Extremum Flow Matching for Offline Goal Conditioned Reinforcement Learning](https://mlsanity.com/paper/2505.19717/) (2025)
- [Diffusion-Based Impedance Learning for Contact-Rich Manipulation Tasks](https://mlsanity.com/paper/2509.19696/) (2025)
- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)

## Abstract (arXiv)

We introduce Riemannian Flow Matching Policies (RFMP), a novel model for learning and synthesizing robot visuomotor policies. RFMP leverages the efficient training and inference capabilities of flow matching methods. By design, RFMP inherits the strengths of flow matching: the ability to encode high-dimensional multimodal distributions, commonly encountered in robotic tasks, and a very simple and fast inference process. We demonstrate the applicability of RFMP to both state-based and vision-conditioned robot motion policies. Notably, as the robot state resides on a Riemannian manifold, RFMP inherently incorporates geometric awareness, which is crucial for realistic robotic tasks. To evaluate RFMP, we conduct two proof-of-concept experiments, comparing its performance against Diffusion Policies. Although both approaches successfully learn the considered tasks, our results show that RFMP provides smoother action trajectories with significantly lower inference times.