Efficient Hybrid SE(3)-Equivariant Visuomotor Flow Policy via Spherical Harmonics for Robot Manipulation

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 benchmark 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 7$\times$ inference speedup. E3Flow thus demonstrates a new and highly effective trade-off between performance, efficiency, and data efficiency for robotic policy learning. Code and videos will be released.