Faster Visuomotor Policy Learning on Action Manifolds via Riemannian MeanFlow

S. Talha Bukhari; Austin Garrett; Yi Wei; Ruiqi Ni; Zachary Kingston; Aniket Bera

cs.ROSep 24, 2026

Faster Visuomotor Policy Learning on Action Manifolds via Riemannian MeanFlow

Authors: S. Talha Bukhari, Austin Garrett, Yi Wei, Ruiqi Ni, Zachary Kingston, Aniket Bera

Organizations: Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA

Abstract

Visuomotor policies learn a direct map from raw sensory observations to robot action sequences. Policies based on Diffusion and Flow Matching capture the multimodal distribution over action sequences in an end-to-end manner. This expressivity comes at the cost of multi-step numerical integration of the learned vector field for action generation, which can be expensive and time-consuming, impeding fast control rates required in robotics applications. Furthermore, robot action sequences are usually defined on a smooth, differentiable manifold, requiring that the learned policy respects the intrinsic geometry of the robot's action space. Here, we present Riemannian MeanFlow Policy (RMFP), which learns the conditioned flow map of the probability path on the robot action manifold. Our formulation employs a flow map consistency objective grounded in the data by a Riemannian Conditional Flow Matching anchor. The flow map consistency condition is stable to train and constrains the learned model to finite-time transport, which yields on-manifold action sequence generation with as few as one network function evaluation. We present results on the spherical LASA and Push-T benchmarks, on the Tool Hang and Transport tasks of the Robomimic suite, and on the Franka Kitchen task with manifold-constrained action generation, and demonstrate that RMFP attains performance competitive with prior work at a lower sampling cost. We also employ RMFP on a real-world robotic manipulation task to demonstrate fast action generation under imperfect sensor measurements in the physical world.

Figures & tables

Panel of Figure 1 — Figure 1 : We present Riemannian MeanFlow Policy (RMFP) for fast action sequence generation on manifolds. Here we show 1-NFE trajectory generation for the LASA shape W projected onto $\mathcal{S}^{2}$ [ 8 ] . Demonstrations from the dataset are shown in black and the executed trajectory is colorized by time-parameterization. $\star$ denotes the north pole. As shown, RMFP traces a smoother path while abiding by the demonstrations. Median DTW is $0.023$ for RMFP against $0.079$ for RFMP.

Panel of Figure 2 — Figure 2 : The Riemannian flowmap consistency objective ( 9 ) and Riemannian conditional flow matching anchor ( 10 ) are schematically shown. The horizontal axis represents time. A point $a_{s}$ is drawn on the conditional geodesic (dashed) from a prior sample $a_{0}$ to a demonstrated chunk $a_{1}$ . The two flow maps through the intermediate time $r$ are rolled out without gradient (grey arcs), and their endpoint $\tilde{a}_{t}$ defines the chord target $\hat{u}$ of ( 6 ). Tangent vectors at $a_{s}$ are denoted by straight arrows: the frozen chord $(t-s)\hat{u}=\mathrm{Log}_{a_{s}}(\tilde{a}_{t})$ ( dashed grey ), the single jump $(t-s)\,u_{\theta}(a_{s},s,t)$ ( teal ) that the gradient reaches, and the geodesic velocity $\dot{a}_{s}$ ( gold ) against which $u_{\theta}(a_{s},s,s)$ is anchored. $\mathcal{L}_{\mathrm{semi}}$ penalizes the residual between the jump and the chord.

Panel of Figure 3 — Figure 3 : Simulated environments of the evaluation benchmarks. (a) Spherical Push-T: the Push-T canvas under the stereographic map onto $\mathcal{S}^{2}$ , with the agent (blue), the block (grey-blue) and the goal pose (green). (b) Robomimic Tool Hang: a single arm inserts a hook into a base and hangs a wrench on it, from a state observation. (c) Robomimic Transport: two arms transfer a hammer from a covered container to a target bin on the opposite shelf, from camera observations. (d) Franka Kitchen: the policy commands an end-effector pose on $\mathbb{R}^{3}\times\mathcal{S}^{3}\times\mathbb{R}^{2}$ , and an episode counts as a success when at least four of the seven subtasks are completed.

Method	NFE 1		NFE 2		NFE 5		NFE 10
Method	DTW	Jerk	DTW	Jerk	DTW	Jerk	DTW	Jerk
RFM	0.090	857	0.020	153	0.016	83	0.016	59
RMF- $v$	0.033	317	0.017	96	0.019	86	0.020	82
RMF- $x_{1}$	0.022	153	0.022	146	0.026	143	0.024	113

Table I : LASA on

\mathcal{S}^{2}

, ten characters. Median DTW to the demonstration, and the executed path’s jerk (the demonstration equals

1.0

); lower is better. (

\lambda_{\mathrm{semi}}=5

)

Method	Coverage Score $(\%)$ $\uparrow$
Method	$1$ NFE	$2$ NFE	$5$ NFE	$10$ NFE
RDP- $\epsilon$	15.0	51.8	83.9	81.6
RDP- $x$	73.1	74.0	76.0	74.4
RFM	67.9	77.2	84.7	83.9
RMF- $v$	78.5	81.0	85.7	84.1
RMF- $x_{1}$	76.0	78.3	81.7	85.0

Table II : Spherical Push-T coverage score against the sampling budget

Method	Success Rate $(\%)$ $\uparrow$
Method	$1$ NFE	$2$ NFE	$5$ NFE	$10$ NFE
RDP- $\epsilon$	3.7	19.7	36.9	47.1
RDP- $x$	23.7	41.7	43.1	39.1
RFM-uni	14.9	40.3	44.3	41.1
RFM-hemi	20.6	41.7	47.1	48.6
RMF- $v$ -uni	34.9	41.7	44.3	45.1

Table III : Franka Kitchen success rate against the sampling budget

Method	Success Rate $(\%)$ $\uparrow$
Method	$1$ NFE	$2$ NFE	$5$ NFE	$10$ NFE
DP	0.0	46.0	64.0	74.0
RFM	60.0	60.0	60.0	64.0
RMF- $v$	64.0	76.0	72.0	74.0
RMF- $x_{1}$	48.0	52.0	60.0	74.0

Table IV : Robomimic Tool Hang success rate against the sampling budget

Method	Success Rate $(\%)$ $\uparrow$
Method	$1$ NFE	$2$ NFE	$5$ NFE	$10$ NFE
DP	0.0	82.0	96.0	82.0
RFM	94.0	86.0	88.0	86.0
RMF- $v$	90.0	92.0	92.0	92.0
RMF- $x_{1}$	94.0	90.0	92.0	90.0

Table V : Robomimic Transport success rate against the sampling budget

Figure 4 : Riemannian MeanFlow Policy executed in a real-world manipulation task. The policy is trained on $60$ real-world demonstrations and deployed on the YAM robotic platform. Representative stages of the rollout show the robot approaching and grasping the object, transporting it to a goal state, and placing it into the target container.

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May 8, 2026, cs.ROJ/K move · Enter open · S save

Trajectory-Consistent Flow Matching for Robust Visuomotor Policy Learning

Riad Ahmed, Sujosh Nag, Moniruzzaman Akash, Mostafa Hussein+1

University of New Hampshire, Durham, NH, USA

Visuomotor Policy Flow-Matching Vision-Language-Action Flow Policies Conditional Flow Matching Robotic Manipulation Velocity Field

Open paper AskPDF arXiv

Flow matching policies learn continuous velocity fields that transport noise to actions, enabling fast deterministic inference for robot manipulation. However, standard training optimizes a pointwise velocity objective while inference requires numerical integration of that field -- a mismatch that causes compounding trajectory errors. We propose four complementary remedies: (1) auxiliary rectified flow velocity regression that provides uniform temporal supervision across the full time interval; (2) multi-step trajectory consistency training that supervises the integrated displacement of the velocity field over trajectory segments, directly closing the train-inference gap; (3) velocity field regularization that enforces temporal smoothness, preventing oscillations that destabilize integration; and (4) fourth-order Runge-Kutta (RK4) inference that reduces global discretization error by orders of magnitude over Euler methods. Critically, these components are not independently sufficient -- RK4 without a smooth velocity field fails, and smoothness without trajectory-level supervision still drifts, as our ablation study confirms. We further pair these with a dual-view 3D point cloud encoder using two independent PointNet encoders for complementary spatial perception. On four real-robot tasks across a Franka arm and a Boston Dynamics Spot, our method achieves 70% and 60% overall success on two long-horizon multi-phase tasks where both baselines score 0%, and reaches 100% on precision tool placement. Three MetaWorld simulation tasks confirm consistent improvements, validating that trajectory-level supervision is essential for reliable policy execution.

Abstract

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