While existing equivariant methods enhance data efficiency, they suffer from high computational intensity, reliance on single-modality inputs, and instability when combined with fast-sampling methods. In this work, we propose E3Flow, a novel framework that addresses the critical limitations of equivariant diffusion policies. E3Flow overcomes these challenges, successfully unifying efficient rectified flow with stable, multi-modal equivariant learning for the first time. Our framework is built upon spherical harmonic representations to ensure rigorous SO(3) equivariance. We introduce a novel invariant Feature Enhancement Module (FEM) that dynamically fuses hybrid visual modalities (point clouds and images), injecting rich visual cues into the spherical harmonic features. We evaluate E3Flow on 8 manipulation tasks from the MimicGen and further conduct 4 real-world experiments to validate its effectiveness in physical environments. Simulation results show that E3Flow achieves a 3.12% improvement in average success rate over the state-of-the-art Spherical Diffusion Policy (SDP) while simultaneously delivering a 7x inference speedup. E3Flow thus demonstrates a new and highly effective trade-off between performance, efficiency, and data efficiency for robotic policy learning. Code: https://github.com/zql-kk/E3Flow.
Figures & tables
Figure 1: (a) Diagram of the E3Flow equivariant policy. E3Flow can learn equivariant trajectories under unseen scene transformations, whereas DP fails due to the lack of symmetry priors. (b) Comparison of average success rates and inference efficiency between state-of-the-art equivariant and non-equivariant policies on MimicGen tasks.
Figure 2: Overall pipeline. E3Flow encodes multimodal inputs through equivariant and non-equivariant visual encoders, aligns invariant visual features across modalities, and constructs a spherical harmonic–equivariant representation to efficiently guide flow matching for generating high-quality equivariant actions.
Figure 3: Illustration of the Feature Enhancement Module (FEM). FEM injects semantic information from images into the equivariant representation of point clouds, achieving efficient fusion of semantic and geometric features.
Figure 4: Visualization of the execution process of E3Flow across eight tasks from MimicGen. Each column depicts the task progression, where the top corresponds to the task initialization and the bottom indicates the final completion state.
Task
EquiBot
DP3
DP
EquiDiff (img)
EquiDiff (voxel)
SDP (DDiM)
SDP (DDPM)
E3Flow (ours)
Coffee_D2
2.06
0.103
0.95
2.44
1.10
0.49
3.79
0.51
Coffee_Preparation_D1
2.13
0.112
0.89
2.07
1.11
0.44
3.46
0.51
Hammer_Cleanup_D1
1.92
0.105
0.94
2.36
1.11
0.48
3.67
0.51
Mug_Cleanup_D1
1.91
0.113
1.01
3.68
1.10
0.46
3.98
0.52
Nut_Assembly_D0
1.80
0.102
0.94
2.34
1.09
0.45
3.71
0.49
Square_D2
2.07
0.112
0.95
2.39
1.11
0.46
3.85
0.54
Table 2: Comparison of inference time (s) between state-of-the-art equivariant and non-equivariant policies and the proposed method. We evaluate eight tasks from MimicGen and compute average inference time across all steps. E3Flow achieves an inference speed 7× faster than the strongest baseline. Inference runs on a single H20 GPU, and we bold the top-3 results.
Figure 5: Ablation on the number of expert demonstrations. As the number of demonstrations increases, all methods show improved performance, while E3Flow consistently achieves higher success rates, highlighting its superior data efficiency.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure A.1 : The overall setup of the real-world robotic experiment platform includes two RealSense D435 RGB-D cameras, a 6-DOF PIPER robotic arm equipped with a two-finger gripper, and a tabletop workspace.
Figure B.2 : Visualization of the execution process for the real-world robotic tasks.
Method
Storing Toys
Bottle Place
Stack Blocks
Assembly
Average SR (%)
E3Flow (ours)
70
80
95
60
76
SDP [ 46 ]
60
65
85
40
62
EquiDiff [ 31 ]
5
25
45
45
30
DP [ 2 ]
0
10
35
20
16
Appendix
Table B.1 : Success rates of different methods on various tasks.
E3Flow
SDP(DDPM)
SDP(DDIM)
EquiDiff
EquiBot
DP
DP3
Batch Size
64
64
64
128
128
128
128
Image Size
84
-
-
-
84
-
-
Point Clouds
1024
1024
1024
1024
1024
1024
1024
Prediction Horizon
16
16
16
16
16
16
16
Action Horizon
8
8
8
8
8
8
8
Learning Rate
1e-4
1e-4
1e-4
1e-4
1e-4
1e-4
1e-4
Appendix
Table C.2 : Hyperparameter settings for different methods.
Visuomotor policy learning trains robots to turn camera observations, often with language and proprioception, into actions for manipulation and navigation.
Developing efficient and accurate visuomotor policies poses a central challenge in robotic imitation learning. While recent rectified flow approaches have advanced visuomotor policy learning, they suffer from a key limitation: After iterative distillation, generated actions may deviate from the ground-truth actions corresponding to the current visual observation, leading to accumulated error as the reflow process repeats and unstable task execution. We present Selective Flow Alignment (SeFA), an efficient and accurate visuomotor policy learning framework. SeFA resolves this challenge by a selective flow alignment strategy, which leverages expert demonstrations to selectively correct generated actions and restore consistency with observations, while preserving multimodality. This design introduces a consistency correction mechanism that ensures generated actions remain observation-aligned without sacrificing the efficiency of one-step flow inference. Extensive experiments across both simulated and real-world manipulation tasks show that SeFA Policy surpasses state-of-the-art diffusion-based and flow-based policies, achieving superior accuracy and robustness while reducing inference latency by over 98%. By unifying rectified flow efficiency with observation-consistent action generation, SeFA provides a scalable and dependable solution for real-time visuomotor policy learning. Code is available on https://github.com/RongXueZoe/SeFA.
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.
S. Talha Bukhari, Austin Garrett, Yi Wei +3
Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA
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.