Screw Attention: Rigid-Body Algebra Inside a Transformer

Contact-rich manipulation tasks such as tight-clearance insertion, connector mating, polishing, and surface-conforming wiping remain difficult for data-driven controllers because they couple discontinuous contact dynamics, partial observability, and strict safety constraints. No single sensing modality suffices: vision supplies global context before contact, force/torque (F/T) feedback governs interaction after contact, and proprioceptive pose provides a consistent kinematic backbone. Most prior imitation-learning policies for contact-rich tasks operate on uni- or bi-modal signals, and the few that fuse three modalities typically adopt off-the-shelf attention modules with no explicit prior on how attention mass should be distributed across task-relevant regions. We present Spacetime Optimal-Transport Attention (SO-TA), a tri-modal fusion backbone that replaces softmax-normalized patch attention by an entropy-regularized Optimal Transport (OT) alignment between force-pose-derived sub-queries and visual patches. Explicit marginal constraints act as a structured inductive bias for contact-rich tasks, encouraging conditioning-aware spatial selection that is stable across illumination, distractors, and partial occlusion. SO-TA is paired with a diffusion-based sequence policy mapping observation windows to pose-action chunks. We evaluate SO-TA on three real-robot tasks: tight peg-in-hole assembly, BCM wiring-connector insertion, and curved-surface mark erasing. With ~200 rollouts per condition, SO-TA reaches 100% success on tight peg-in-hole versus 93% for cross-attention at matched capacity, and retains 82.5% success under illumination, distractor, and partial-occlusion perturbations where a concatenation baseline drops to 43.5%. OT-derived patch heatmaps and leave-one-out modality-influence ratios provide interpretable, phase-dependent diagnostics.

Learning motion latents for robotic manipulation heavily relies on extracting motion patterns from visual sequences, yet effective action abstractions require understanding three-dimensional geometric transformations. Here, we introduce GeoMoLa (Geometry-Aware Motion Latents), which learns discrete motion latent codes by predicting how point clouds evolve during manipulation rather than reconstructing visual observations. This four-dimensional objective -- spatial geometry changing through time -- forces latent representations to encode actual physical motion rather than appearance patterns. GeoMoLa achieves state-of-the-art performance using only single-view RGB-D input, while existing methods require multi-view reconstruction, succeeding across diverse manipulation benchmarks. Our ablations reveal that geometric prediction is the key to driving performance, quantitatively validating that manipulation depends on spatial understanding. Furthermore, the learned codes exhibit effective motion abstraction: applying them to novel scenes produces physically consistent transformations regardless of visual context. Our real-world experiments also confirm this robustness capability, achieving robust manipulation with minimal demonstrations in cluttered environments where geometric reasoning determines success. Thus, we demonstrate that effective motion latents for robot control can better emerge from understanding motion through its three-dimensional effects rather than pixel-level patterns.

Visuomotor policies trained from a few demonstrations may reproduce demonstrated trajectories without reliably following changes in object position. Existing approaches with explicit attention typically obtain spatial priors from human annotation or visual models. We introduce TACIT (tactile contact informs attention), which uses measured tactile contacts from teleoperated demonstrations to supervise spatial attention without additional point annotation. Gaussian targets over preceding camera point clouds supervise an attention head whose pooled output conditions a visuotactile diffusion policy. Targets are used only during training; tactile observations remain inputs at inference. In the primary real-robot benchmark, with ten demonstrations per task and five demonstrated placement regions, TACIT achieves 66.7% success on ball placement and 73.3% on peg insertion, compared with 10.0% and 20.0% for input-matched 3D visuotactile fusion and 20.0% and 43.3% for vision-only DP3. TACIT enters the 150 mm palm-to-object approach region within 12 seconds in all 30 trials per task; all remaining failures occur after arrival. Across three training seeds on real ball and simulated peg, TACIT outperforms input-matched fusion and an architecture-matched control without explicit attention supervision, supporting the contribution of supervision beyond branch capacity. Pre-contact and contact-time supervision show no consistent ordering. These results demonstrate that measured tactile contact provides effective spatial supervision for approach behavior from few demonstrations within the evaluated workspace.