On-Manifold Equivariant Graph Neural Networks for Planar Robot Dynamics

Rigid-body poses combine Euclidean position with rotational states on $\mathrm{SO}(n)$, giving rise to a non-Euclidean configuration space. The radial coordinate term of standard $\mathrm E(n)$-equivariant graph neural networks (EGNNs) uses weighted pairwise displacements, a rule that is not intrinsic to rotations and remains in the span of those displacements. We introduce the On-Manifold Equivariant Graph Neural Network (OM-EGNN), which represents pose displacements in local tangent coordinates using product-manifold maps and augments the displacement update with a learned tangent residual. The resulting product-manifold layer is equivariant under global left $\mathrm{SE}(n)$ actions by construction. On one-shot multi-step prediction of planar vehicle dynamics, OM-EGNN achieves a position RMSE of $0.026\,\mathrm m$ and a direct equivariance residual of $2.26\times10^{-8}$, compared with $0.065\,\mathrm m$ and approximately $8\times10^{-4}$ for Euclidean EGNN variants. Removing the tangent residual preserves equivariance but increases position RMSE to $0.294\,\mathrm m$, showing that manifold-consistent geometry and tangent-space expressivity play complementary roles in equivariant dynamics prediction.