SIGMA: Interaction-Conditioned Geometry Tokenization for Cross-Shape Humanoid Manipulation

Humanoid interaction with diverse objects depends on local surface structures that determine feasible contacts and motion strategies. Yet representing such geometry for policy learning remains challenging: compact global descriptors such as bounding boxes discard interaction-relevant surface details, while raw point-cloud inputs preserve fine-grained geometry but leave local surface structure implicit, increasing the burden on the policy to learn transferable control across shapes. To address this challenge, we propose **SIGMA** (**S**tructured **I**nteraction **G**eometry for humanoid **MA**nipulation), a geometry interface combining Structured Surface Attributes (SSA) with Interaction-Conditioned Geometry Compression (ICGC). SSA organizes object geometry into fixed-size local surface elements that encode position, normal, and represented area. ICGC aggregates these elements according to task compatibility, humanoid body-surface proximity, and surface-area priors, producing a geometry token that adapts to the current interaction state. We integrate SIGMA into a unified multi-task humanoid controller and evaluate frozen-policy transfer without target-object adaptation on objects with identical global extents but different local structures, unseen local geometric perturbations, and diverse furniture geometries. At 50k iterations, SIGMA achieves 58.33% average full-task success across the three evaluation settings, compared with 22.67% for PointNet and 6.76% for AABB-shape-token. On three held-out Sit object groups, SIGMA averages 89.03% target-entry success, compared with 82.52% for PointNet and 61.00% for AABB-shape-token.