What Symmetry Buys a Learned Motion Planner

RESUMO

Learning-based motion planners pay at training what classical planners pay per query. Trained in world coordinates, they relearn the same motion at every position and orientation. Existing work restores the missing rigid-body equivariance in the training data, in the inference operator, or in the weights, and each carries a cost. We ask how much of that equivariance the planning query supplies for free. A start s and a goal g determine a frame in closed form, with origin at their midpoint and first axis along g-s. Expressing trajectory and obstacles in that frame removes three translations and two rotations of SE(3), at initialisation, for one cross product per query and with no constraint on the architecture. A single rotation about the start-goal axis remains, and no continuous rule removes it. On a cluttered 3D benchmark, holding architecture, data and budget fixed, the frame raises the held-out collision-free rate from 14.60% to 51.10%, where a straight segment from start to goal scores 15.6% and the world-frame model does not beat it. We build all three mechanisms for the residual rotation and each is worth under a point, though the equivariant backbone reaches any given level two to three times sooner. What the representation supplies therefore dominates what any mechanism enforces, and the standard diagnostic does not see the difference: two models with indistinguishable non-equivariance residuals differ by 28 points. Calibrated against a non-symmetry intervention, the frame is not even the largest effect available, since local geometry is worth +40.0 where the frame is worth +36.5.