Geometry Beyond Symmetry On-Manifold Equivariant Graph Networks for Robot Dynamics

Learned world models for robots must predict action-conditioned motion while respecting both task symmetries and the geometry of pose. Existing equivariant models enforce reference-frame consistency, but commonly treat rotational states through Euclidean coordinate updates or post-hoc normalization. We introduce an On-Manifold Equivariant Graph Neural Network (OM-EGNN)for direct multi-step robot dynamics prediction.OM-EGNN represents pose on the product manifold $\mathrm{R}^n \times \mathrm{SO(n)}$, computes relative poses in receiver-local tangent coordinates, and applies manifold-valued displacement and retraction. A learned tangent residual further removes the directional span restriction of standard EGNN-style pairwise updates while preserving the task-relevant frame symmetry. Controlled experiments on planar MuJoCo vehicle dynamics and spatial NeuroBEM quadrotor dynamics separate the effects of state geometry, frame symmetry, and update expressivity. OM-EGNN achieves the strongest pose prediction in both settings and maintains reference-frame consistency to numerical precision in the planar task, while gains in aggregate state error over strong canonicalized baselines are more modest. Ablations show that manifold-valid updates alone are insufficient without the expressive tangent residual. These results suggest that state geometry, problem symmetry, and update expressivity are distinct inductive biases, and that explicitly modeling pose on its manifold improves the geometric reliability of learned robot dynamics.