Robotic manipulation

Equivariant Volumetric Grasping

Efficient C4-equivariant tri-plane representations for robust 6-DoF grasp planning in clutter.

Pinhao Song1,3   Yutong Hu1,3   Pengteng Li4   Renaud Detry1,2,3

1 KU Leuven, Mechanical Engineering   2 KU Leuven, Electrical Engineering   3 Flanders Make@KU Leuven   4 HKUST(GZ)

Read the paper Code Real-world code
When a cluttered scene rotates by 90 degrees, the predicted grasp orientations rotate with it.

Why equivariance

Learn once. Grasp at every tabletop orientation.

A grasp planner should transform predictably when the scene rotates. We encode that property in the architecture, rather than asking training data to cover every object orientation.

Full 3D rotation equivariance is costly for real-time volumetric planning. We target the rotation that dominates tabletop manipulation: 90-degree rotations around the vertical axis. This preserves the useful symmetry while keeping computation tractable.

Our representation projects 3D features onto three canonical planes. Under a C4 rotation, the table plane rotates in-plane, while the two vertical planes undergo a structured combination of permutation and reflection. The model is explicitly designed to respect this induced action.

The XY, XZ, and YZ tri-plane features under the four C4 rotations.
A vertical-axis rotation acts differently on the three projections: XY rotates, while XZ and YZ exchange roles with reflections. This predictable transformation is the basis of the equivariant tri-plane representation.

Method

An equivariant tri-plane UNet

A single 3D steerable lifting layer creates volumetric primitive features. We then project them to three planes and use lightweight 2D equivariant processing to build a unified, queryable 3D feature field.

Architecture of the Equivariant Tri-plane UNet.
The XY branch is C4-equivariant; the XZ and YZ branches are reflection-equivariant. The vertical-to-tableplane augmentation module lets the branches exchange structured information across UNet layers.

Structured symmetry

Each plane is processed according to the exact transformation induced by a 90-degree vertical rotation.

2D efficiency

Tri-plane processing replaces expensive stacks of 3D steerable convolutions with three 2D branches.

Planner-agnostic features

The representation supports both direct regression and generative grasp-orientation sampling.

Equivariant planners

One representation, two grasp planners

We instantiate the feature field in GIGA and IGD, yielding EquiGIGA and EquiIGD. The two systems preserve equivariance through either grasp-orientation regression or generative orientation sampling.

EquiGIGA

EquiGIGA grasp prediction workflow.

Equivariant heads predict graspness, occupancy, and grasp orientation directly at queried positions.

EquiIGD

EquiIGD grasp prediction workflow.

Equivariant deformable attention and rotation flow model multi-modal grasp orientations, then score candidate grasps.

Evidence

More sample-efficient, within real-time cost.

Across reduced-data experiments, the equivariant models consistently outperform their non-equivariant counterparts. In fixed-view simulated clutter removal, both planners also improve substantially over GIGA and IGD.

Sample-efficiency curves comparing EquiGIGA and EquiIGD with non-equivariant baselines.
With less training data, equivariant models generally achieve stronger grasp success and declutter rates than their non-equivariant alternatives.
98.0%EquiIGD packed-scene grasp success rate
79.5%EquiIGD pile-scene grasp success rate
25 msEquiGIGA depth-to-grasp latency
MethodPacked GSRPacked DRPile GSRPile DRLatency
GIGA84.8 ± 2.285.1 ± 2.569.5 ± 1.349.0 ± 3.424 ms
IGD (N=1)92.9 ± 1.886.7 ± 1.868.2 ± 1.950.6 ± 1.5217 ms
EquiGIGA94.5 ± 1.391.4 ± 1.279.1 ± 2.277.3 ± 3.925 ms
EquiIGD (N=1)98.0 ± 0.692.0 ± 1.579.5 ± 1.377.0 ± 1.152 ms

GSR: grasp success rate. DR: declutter rate. Results are mean ± standard deviation in the fixed-view simulation benchmark.

Equivariance analysis

Architectural consistency beyond data augmentation.

The following analyses show how the proposed architecture preserves grasp prediction behavior under rotation, learns orientation more efficiently, and transfers more reliably across camera viewpoints.

Figure 11: EquiGIGA, GIGA++, and GIGA++A predictions and Grad-CAM maps under C4 rotations.
Figure 11. Under 0, 90, 180, and 270 degree input rotations, EquiGIGA produces consistently transformed grasp predictions and activation patterns. The non-equivariant baselines do not preserve this behavior across all heads.
Figure 13: training and validation curves for EquiGIGA and baselines.
Figure 13. Training and validation curves for average precision, occupancy, graspness, and rotation losses. The lower rotation loss of EquiGIGA explains much of its performance advantage.
Figure 14: empirical equivariance consistency error across planar rotations.
Figure 14. Empirical equivariance consistency error under planar rotations. EquiGIGA reaches floating-point-level error at non-identity C4 rotations across graspness, occupancy, and rotation heads.
Figure 15: grasp success and declutter rate at changing camera viewpoints.
Figure 15. Camera-viewpoint robustness. With a fixed training view, EquiGIGA achieves the strongest mean grasp success and declutter rate across tested azimuth and elevation changes.

Real-world decluttering

Robust across packed, pile, and adversarial scenes.

We evaluate 15 rounds of five objects per scene using a RealSense D435 camera. The equivariant planners deliver the strongest real-world performance across all three scenarios.

Real-world experimental setup and the packed, pile, and adversarial scenes.
Experimental setup and three clutter-removal scenarios: packed objects, randomly dropped piles, and Berkeley adversarial objects.
90.8%EquiIGD packed-scene grasp success
85.3%EquiGIGA pile-scene grasp success
100.0%EquiIGD adversarial-scene declutter rate
Successful grasp predictions by EquiGIGA and EquiIGD in pile and packed scenes.
Top grasp predictions in challenging scenes. Green grippers indicate successful grasps; red grippers identify failed grasps.

EquiGIGA

Packed, pile, and adversarial scene demonstrations.

EquiIGD

Packed, pile, and adversarial scene demonstrations.

Citation

BibTeX

@article{song2025equivariant,
  title={Equivariant Volumetric Grasping},
  author={Song, Pinhao and Hu, Yutong and Li, Pengteng and Detry, Renaud},
  journal={arXiv preprint arXiv:2507.18847},
  year={2025}
}