Fast Generative Grasping via Lie Group-Constrained MeanFlow

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

cs.ROAug 26, 2026

Fast Generative Grasping via Lie Group-Constrained MeanFlow

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

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

Abstract

Grasp synthesis is a core task in robotic manipulation, for which the solution typically forms a multimodal distribution rather than a point estimate. Generative robotic grasping aims to learn this distribution with deep generative models such as diffusion and flow-based approaches. The iterative nature of such generative models makes them flexible and generalizable; however, multi-step sampling impedes the time-critical operation required in robotics. We devise an approach to fast generative grasping based on MeanFlow on the product Lie group $\mathcal{G} = \mathrm{SO}(3) \times \mathbb{R}^3$ . The training objective couples a purely algebraic semigroup consistency condition with Riemannian Conditional Flow Matching on $\mathcal{G}$ that anchors the average velocity to the data distribution. The resulting Lie Group-constrained MeanFlow formulation samples reliable grasps in $\leq 5$ network evaluations, matching the grasp generation performance of state-of-the-art diffusion and flow-based models on the ACRONYM dataset at millisecond-scale inference latency (up to $39\times$ speed-up). We further demonstrate that the approach directly translates to real-world robotic grasping without additional training or domain adaptation, exhibiting robust grasp synthesis under observation noise.

Figures & tables

Panel of Figure 2 — Figure 2: A visual comparison of generative modeling on the $\mathrm{SO}(3)$ matrix Lie group. We design a target distribution (of elements of $\mathrm{SO}(3)$ ) whose support is a closed curve on the unit sphere $S^{2}$ , traced by the $z$ -axis of the frame with the $y$ -axis aligned to the tangent of the curve. Each sampled rotation $R$ is rendered as a cone placed at $Re_{3}$ and oriented along $Re_{2}$ , with $\{e_{1},e_{2},e_{3}\}$ the standard basis of $\mathbb{R}^{3}$ . The cone is colored by a kernel density estimate of the target evaluated at that sample (yellow: high density, purple: low density), and the red cone marks the identity rotation $I$ . Flow Matching ( left ) leaves a large fraction of samples off the seam at low step counts and requires tens of integration steps to concentrate on the support, whereas MeanFlow ( right ) recovers the seam within three steps.

Method	SR $(\%)$ $\uparrow$		EMD $\downarrow$
Method	ID	OOD	ID	OOD
SE3Dif [ 40 ]	$70.40\,\text{\scriptsize$ \pm,18.37 $}$	$57.60\,\text{\scriptsize$ \pm,20.84 $}$	$0.4649\,\text{\scriptsize$ \pm,0.1705 $}$	$0.4880\,\text{\scriptsize$ \pm,0.1009 $}$
BRIDG E R [ 8 ]	$59.38\,\text{\scriptsize$ \pm,34.23 $}$	$63.77\,\text{\scriptsize$ \pm,29.41 $}$	$0.6057\,\text{\scriptsize$ \pm,0.1849 $}$	$0.5589\,\text{\scriptsize$ \pm,0.1390 $}$
EGF [ 22 ]	$\underline{85.48}\,\text{\scriptsize$ \pm,22.77 $}$	$65.09\,\text{\scriptsize$ \pm,23.97 $}$	$0.3866\,\text{\scriptsize$ \pm,0.1008 $}$	$0.4805\,\text{\scriptsize$ \pm,0.1125 $}$
VSIGD [ 5 ]	$68.57\,\text{\scriptsize$ \pm,20.20 $}$	$\underline{69.05}\,\text{\scriptsize$ \pm,20.00 $}$	$0.4966\,\text{\scriptsize$ \pm,0.1692 $}$	$0.4675\,\text{\scriptsize$ \pm,0.0963 $}$
GraspMF ( $T=5$ )	$\mathbf{87.40}\,\text{\scriptsize$ \pm,17.43 $}$	$\mathbf{71.73}\,\text{\scriptsize$ \pm,25.57 $}$	$\underline{0.3702}\,\text{\scriptsize$ \pm,0.0942 $}$	$\mathbf{0.4191}\,\text{\scriptsize$ \pm,0.1108 $}$

Table I: Grasping performance on objects from the ACRONYM dataset: grasp success rate (SR) and coverage (EMD). Each comparison method is reported at the sampling budget of its original formulation, and GraspMF at a low-step (

T=5

) and single-step (

T=1

) setting. SR and EMD are each subdivided into in-domain (ID), evaluated on held-out instances of the training shape categories, and out-of-domain (OOD), evaluated on instances from disjoint shape categories. The inference cost of the same settings is reported in Table II . Best and second-best metrics are highlighted.

Method	NFE $\downarrow$		Latency
Method	Per step	Total	(ms) $\downarrow$
SE3Dif [ 40 ] ( $T=70$ )	$2\times$	$140$	$600.1\,\text{\scriptsize$ \pm,23.8 $}$
BRIDG E R [ 8 ] ( $T=40$ )	$1\times$	$40$	$120.6\,\text{\scriptsize$ \pm,3.5 $}$
EGF [ 22 ] ( $T=20$ )	$4\times$	$80$	$187.9\,\text{\scriptsize$ \pm,1.8 $}$
VSIGD [ 5 ] ( $T=70$ )	$2\times$	$140$	$1124.4\,\text{\scriptsize$ \pm,1.4 $}$
GraspMF ( $T=5$ )	$1\times$	$\underline{5}$	$\underline{15.5}\,\text{\scriptsize$ \pm,0.4 $}$

Table II: Inference cost of grasp generation: network function evaluations (NFEs) and average wall-clock latency for a batch of 100 grasps per object measured on a single NVIDIA RTX 5080 GPU. The methods and sampling budgets

T

follow the rows of Table I . Best and second-best entries are highlighted.

Panel of Figure 3 — Figure 3: Qualitative comparison of generated grasps for Laptop, Pencil, and Car (in order). Dark cyan indicates successful grasps while dark purple indicates failures. The columns follow the rows of Table I : each comparison method is shown at the sampling budget of its original formulation, and GraspMF at its low-step ( $T=5$ ) and single-step ( $T=1$ ) settings. GraspMF generates more successful and stable grasps with broader grasp-mode coverage across diverse object geometries, as evaluated in IsaacGym.

Panel of Figure 4 — Figure 4: Grasping performance on the OOD evaluation protocol across the sampling budget $T\in\{1,2,5,10,20,40,70,100\}$ . SR in % ( 4(a) ), EMD ( 4(b) ), and inference latency in $\log$ scale ( 4(c) ) are plotted against sampling budgets, and SR is plotted against inference latency in $\log$ scale ( 4(d) ) where the top-left corner approaches the Pareto frontier.

Variant	SR $(\%)$ $\uparrow$		EMD $\downarrow$
Variant	ID	OOD	ID	OOD
GraspMF	$87.40\,\text{\scriptsize$ \pm,17.43 $}$	$\mathbf{71.73}\,\text{\scriptsize$ \pm,25.57 $}$	$\mathbf{0.3702}\,\text{\scriptsize$ \pm,0.0942 $}$	$\mathbf{0.4191}\,\text{\scriptsize$ \pm,0.1108 $}$
GraspMF-NoSDF	$83.62\,\text{\scriptsize$ \pm,17.90 $}$	$66.60\,\text{\scriptsize$ \pm,22.78 $}$	$0.3865\,\text{\scriptsize$ \pm,0.1163 $}$	$0.4278\,\text{\scriptsize$ \pm,0.1007 $}$
GraspMF-NoSched	$85.02\,\text{\scriptsize$ \pm,19.12 $}$	$65.95\,\text{\scriptsize$ \pm,24.81 $}$	$0.3777\,\text{\scriptsize$ \pm,0.1025 $}$	$0.4235\,\text{\scriptsize$ \pm,0.1057 $}$
GraspMF-GS	$\mathbf{87.51}\,\text{\scriptsize$ \pm,16.08 $}$	$67.05\,\text{\scriptsize$ \pm,26.62 $}$	$0.3790\,\text{\scriptsize$ \pm,0.1022 $}$	$0.4259\,\text{\scriptsize$ \pm,0.1122 $}$
GraspMF-Decomp	$62.25\,\text{\scriptsize$ \pm,20.43 $}$	$52.29\,\text{\scriptsize$ \pm,19.05 $}$	$0.3848\,\text{\scriptsize$ \pm,0.1084 $}$	$0.4265\,\text{\scriptsize$ \pm,0.0904 $}$

Table III: Ablations of GraspMF at sampling budget

T=5

. GraspMF-NoSDF removes the auxiliary signed-distance regression; GraspMF-NoSched holds the semigroup weight constant instead of annealing it; GraspMF-GS replaces the SVD rotation projection with symmetric Gram–Schmidt orthonormalization; GraspMF-Decomp uses the decomposed objective of Zhong et al. [45] in place of the semigroup loss ( 22 ). Best metrics are highlighted.

Method	SR $(\%)$ $\uparrow$		EMD $\downarrow$
Method	ID	OOD	ID	OOD
SE3Dif [ 40 ]	$65.51\,\text{\scriptsize$ \pm,22.97 $}$	$56.19\,\text{\scriptsize$ \pm,21.08 $}$	$0.4785\,\text{\scriptsize$ \pm,0.1475 $}$	$\underline{0.4926}\,\text{\scriptsize$ \pm,0.1083 $}$
BRIDG E R [ 8 ]	$\mathbf{88.84}\,\text{\scriptsize$ \pm,19.60 $}$	$\underline{65.38}\,\text{\scriptsize$ \pm,25.25 $}$	$0.4621\,\text{\scriptsize$ \pm,0.1121 $}$	$0.5721\,\text{\scriptsize$ \pm,0.1312 $}$
EGF [ 22 ]	$76.87\,\text{\scriptsize$ \pm,23.06 $}$	$47.11\,\text{\scriptsize$ \pm,20.75 $}$	$\mathbf{0.3801}\,\text{\scriptsize$ \pm,0.0958 $}$	$\mathbf{0.4637}\,\text{\scriptsize$ \pm,0.1051 $}$
VSIGD [ 5 ]	$49.54\,\text{\scriptsize$ \pm,18.39 $}$	$52.18\,\text{\scriptsize$ \pm,18.68 $}$	$0.5577\,\text{\scriptsize$ \pm,0.1702 $}$	$0.5074\,\text{\scriptsize$ \pm,0.0965 $}$
GraspMF ( $T=5$ )	$\underline{86.08}\,\text{\scriptsize$ \pm,18.23 $}$	$\mathbf{68.46}\,\text{\scriptsize$ \pm,23.64 $}$	$\underline{0.4030}\,\text{\scriptsize$ \pm,0.1213 $}$	$0.5923\,\text{\scriptsize$ \pm,0.1872 $}$

Table IV: Grasping performance under single-view partial point-cloud observation. The sampling configuration kept the same as reported in Table II . Metrics are computed for performance over 3 random views per object, where for each view, 100 grasps are generated as before. Best and second-best metrics are highlighted.

Figure 5: Real-world grasping demonstration on household objects with a wrist-mounted RGB-D camera. Three diverse grasps generated by our approach are shown.

Method	Trials ( $\cdot$ /10)
Method	Black Mug	Red Mug	Gray Bowl
SE3Dif [ 40 ] ( $T=70$ )	$3$	$6$	$2$
BRIDG E R [ 8 ] ( $T=40$ )	$\mathbf{10}$	$8$	$7$
EGF [ 22 ] ( $T=20$ )	$\mathbf{10}$	$8$	$\mathbf{10}$
VSIGD [ 5 ] ( $T=70$ )	$6$	$9$	$9$
GraspMF ( $T=5$ )	$9$	$9$	$\mathbf{10}$

Table V: Real-world grasping performance across three household objects. Each entry reports the number of successful grasps out of 10 trials. Best results for each object are highlighted.

Robotic Grasping

Robotic grasping studies how robots perceive, plan, and execute stable, task-appropriate contacts with objects across varied shapes and conditions.

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GraspMeanFlow: SE(3)-Equivariant MeanFlow for Few-Step 6-DoF Grasp Generation

Jiyong Kwon, Yikun Bai, Amirhossein Mollaali, Guang Lin

School of Mechanical Engineering, Purdue University · Department of Mathematics, Purdue University

Flow Matching Robotic Grasping Equivariant Neural Networks Mean Flow

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Recent data-driven methods for synthesizing 6-DoF grasp poses use generative models to learn complex grasp pose distributions and generate diverse candidate poses. In particular, SE(3)-equivariant flow-based models generate grasp poses that transform consistently with object rotations and translations. However, these methods sample by iterative numerical integration, requiring tens of function evaluations per grasp and limiting their use in real-time manipulation. We propose GraspMeanFlow, an SE(3)-equivariant MeanFlow framework for few-step 6-DoF grasp generation. Our method learns the average velocity over a finite time interval, defined through the time-ordered exponential so that it reproduces exactly the rigid-body displacement accumulated over that interval. We prove that a point-cloud-conditioned distribution transported by an equivariant average-velocity flow map remains invariant, so equivariance is retained under few-step sampling, and we condition the field on a pair of times by lifting both to equivariant vectors, leaving the backbone otherwise unchanged. For stable training, we pair a flow-matching boundary term with either of two consistency terms: the differential MeanFlow identity, whose target requires a Jacobian-vector product, or an equivalent semigroup loss that avoids it. Experiments on ACRONYM show that a single function evaluation of GraspMeanFlow reaches the EMD that an iterative SE(3) flow model needs five steps to approach, that a second instantiation of the same framework improves grasp success by up to 24.3 points in the few-step regime, and that both generate grasp distributions transforming exactly with the object.

Abstract

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