# Fast and Robust Visuomotor Riemannian Flow Matching Policy

arXiv: https://arxiv.org/abs/2412.10855 · HTML: https://mlsanity.com/paper/2412.10855/

## Key facts

- Cited by 18 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 45 of its 64 references that are in the map.
- First posted on arXiv 14 Dec 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 45)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [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)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Behavior Transformers: Cloning k modes with one stone](https://mlsanity.com/paper/2206.11251/) (2022)
- [Goal-Conditioned Imitation Learning using Score-based Diffusion Policies](https://mlsanity.com/paper/2304.02532/) (2023)
- [Consistency Policy: Accelerated Visuomotor Policies via Consistency Distillation](https://mlsanity.com/paper/2405.07503/) (2024)
- [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)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [One-Step Diffusion Policy: Fast Visuomotor Policies via Diffusion Distillation](https://mlsanity.com/paper/2410.21257/) (2024)
- [EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning](https://mlsanity.com/paper/2407.01479/) (2024)
- [Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matching](https://mlsanity.com/paper/2409.07343/) (2024)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [ManiCM: Real-time 3D Diffusion Policy via Consistency Model for Robotic Manipulation](https://mlsanity.com/paper/2406.01586/) (2024)
- [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)

## Built on by (most cited first, 18 of 18)

- [FASTER: Rethinking Real-Time Flow VLAs](https://mlsanity.com/paper/2603.19199/) (2026)
- [3D FlowMatch Actor: Unified 3D Policy for Single- and Dual-Arm Manipulation](https://mlsanity.com/paper/2508.11002/) (2025)
- [Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry](https://mlsanity.com/paper/2505.18193/) (2025)
- [FreqPolicy: Efficient Flow-based Visuomotor Policy via Frequency Consistency](https://mlsanity.com/paper/2506.08822/) (2025)
- [Reverse Flow Matching: A Unified Framework for Online Reinforcement Learning with Diffusion and Flow Policies](https://mlsanity.com/paper/2601.08136/) (2026)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [Fast Flow-based Visuomotor Policies via Conditional Optimal Transport Couplings](https://mlsanity.com/paper/2505.01179/) (2025)
- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)
- [CoLA-Flow Policy: Temporally Coherent Imitation Learning via Continuous Latent Action Flow Matching for Robotic Manipulation](https://mlsanity.com/paper/2601.23087/) (2026)
- [Towards Safe Imitation Learning via Potential Field-Guided Flow Matching](https://mlsanity.com/paper/2508.08707/) (2025)
- [World-Task Factorization for Robot Learning](https://mlsanity.com/paper/2606.02027/) (2026)
- [Let the Dynamics Flow: Stable Flow Matching Dynamical Systems](https://mlsanity.com/paper/2606.03834/) (2026)
- [Optimal Transport Q-Learning for Flow Policy Steering and Acceleration](https://mlsanity.com/paper/2607.06262/) (2026)
- [TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching](https://mlsanity.com/paper/2607.16894/) (2026)
- [History-Conditioned Flow Matching for Probabilistic Dynamics of Tendon-Driven Continuum Robots](https://mlsanity.com/paper/2609.25658/) (2026)
- [Faster Visuomotor Policy Learning on Action Manifolds via Riemannian MeanFlow](https://mlsanity.com/paper/2609.30127/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)
- [Toward Real-Time VLAs: Stage-Aware Two-Step Flow Denoising and System-Level Evaluation](https://mlsanity.com/paper/2609.39822/) (2026)

## Abstract (arXiv)

Diffusion-based visuomotor policies excel at learning complex robotic tasks by effectively combining visual data with high-dimensional, multi-modal action distributions. However, diffusion models often suffer from slow inference due to costly denoising processes or require complex sequential training arising from recent distilling approaches. This paper introduces Riemannian Flow Matching Policy (RFMP), a model that inherits the easy training and fast inference capabilities of flow matching (FM). Moreover, RFMP inherently incorporates geometric constraints commonly found in realistic robotic applications, as the robot state resides on a Riemannian manifold. To enhance the robustness of RFMP, we propose Stable RFMP (SRFMP), which leverages LaSalle's invariance principle to equip the dynamics of FM with stability to the support of a target Riemannian distribution. Rigorous evaluation on ten simulated and real-world tasks show that RFMP successfully learns and synthesizes complex sensorimotor policies on Euclidean and Riemannian spaces with efficient training and inference phases, outperforming Diffusion Policies and Consistency Policies.