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

Predictive controllers rely on dynamics models to evaluate candidate action sequences, making accurate multi-step pose prediction critical to reliable planning and control. Robot dynamics present two intertwined structures: their predictions should remain consistent across equivalent reference frames, while their pose states evolve on a non-Euclidean configuration space. Existing $\mathrm{E}(n)$-equivariant graph neural networks encode symmetry for Euclidean point coordinates, but do not respect the manifold geometry of rotational states. We introduce an On-Manifold Equivariant Graph Neural Network (OM-EGNN) that represents robot poses on the product manifold of translation and rotation, computes relative poses in receiver-local tangent coordinates, and retracts learned updates onto the manifold. Moreover, the radial update of standard EGNNs is restricted to scalar-weighted combinations of pairwise displacements. OM-EGNN augments this update with a learned tangent-space residual that escapes this directional span while preserving task-relevant equivariance. On planar Ackermann-steered vehicle dynamics and spatial quadrotor dynamics, we compare OM-EGNN against standard canonicalized predictors and Euclidean equivariance-based graph baselines. OM-EGNN achieves the lowest position RMSE in both settings. Rotation and velocity are not uniformly best: planar velocity is lowest for a Euclidean lift, and 3D rotation is lowest for a canonicalized MLP. Unlike the Euclidean graph baselines, OM-EGNN keeps the frame-attack gap near zero under the task symmetry. The tangent residual substantially improves position prediction. In matched MPPI, OM-EGNN has the best mean progress, success, and tracking RMSE, but those margins lie within run variation of the strongest canonicalized predictors, and open-loop rankings do not determine the control ranking.