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What Symmetry Buys a Learned Motion Planner

Andrea Emir Sevincel

arXiv:2609.10033v1cs.RO

TL;DR

Learning-based motion planners lack the rigid-body equivariance that classical geometric planners obtain naturally, motivating the question of how much the planning query supplies for free. The paper uses a start-goal frame to remove five degrees of freedom at initialization and evaluates residual-rotation mechanisms under fixed training conditions. The frame raises held-out collision-free rate from 14.60% to 51.10%, while residual mechanisms add under a point in accuracy but an equivariant backbone reaches target levels two to three times sooner.

  • Problem

    Learning-based planners trained in world coordinates relearn identical motions across positions and orientations instead of inheriting the rigid-body equivariance of classical geometric planners.

  • Method

    A start-goal canonical frame expresses trajectories and obstacles relative to their midpoint and direction, removing five SE(3) degrees of freedom before inference and leaving one residual rotation to evaluate.

  • Results

    51.10% held-out collision-free rate versus 14.60% for the world-frame model; residual-rotation mechanisms add under a point, while the equivariant backbone reaches any given level two to three times sooner.

  • Takeaways & Limitations

    The query-derived representation contributes more benchmark accuracy than mechanisms enforcing the residual symmetry, while the equivariant backbone primarily improves optimization speed.

  • Takeaways & Limitations

    The demonstrated scope is limited to one workspace family, clutter generator, and architecture, and joint-space manipulator planning does not admit a clean rigid-world action on joint angles.

Abstract

from arXiv · show

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.

I. Introduction

The paper asks how much rigid-body equivariance a planning query supplies before symmetry is restored through data, inference, or architecture. A start-goal frame removes five degrees of freedom cheaply, while residual-rotation mechanisms add little accuracy but can improve optimization.

  • Motivation: Learning-based planners relearn structurally identical motions across world positions and orientations because they do not inherit classical planners’ geometric equivariance.
  • Method: The start and goal define a canonical frame that removes three translations and two rotations at initialization, using one cross product per query without changing architecture, data, or budget.The frame places the origin at the start-goal midpoint and aligns its first axis with the goal direction.
  • Results: The frame is worth +36.5 points over the plain world frame, whereas showing the obstacle encoder the query without changing frames is worth only +1.0.
  • Results: 51.10% held-out collision-free rate versus 14.60% for the world-frame model, while a straight start-goal segment scores 15.6%.The world-frame model does not beat the straight-segment baseline on the identical problems.
  • Residual rotation: The three residual-rotation mechanisms are alternatives, each worth under a point; the SO(2)-equivariant backbone reaches any given level two to three times sooner.Roll augmentation was tested only at a fixed 20-epoch budget, where its +0.8 effect was not significant across three seeds.
  • Interpretation: The non-equivariance residual does not predict mechanism value: two models with indistinguishable residuals differ by 28 points, while local geometry is worth +40.0 versus +36.5 for the frame.

II. Related Work

Classical planners obtain rigid-transformation equivariance from direct geometric operation, whereas learning-based planners generate trajectories efficiently but must restore the missing symmetry through data, inference, or weights.

  • Classical planners: Sampling-based and optimization-based planners act directly on geometry, so rigidly transformed problems produce rigidly transformed solutions.They offer probabilistic completeness or local convergence, but incur per-query search or optimization costs.
  • Learning-based planners: Learning-based planners move planning into fast network evaluation or whole-trajectory generation, but do not inherit the symmetry classical geometry-based planners obtain automatically.
  • Restoring symmetry: Existing symmetry-restoration methods operate through augmentation, inference-time frame operations, or equivariant weights, each imposing a larger data distribution, extra forward passes, or a restricted hypothesis class.The paper measures the value of a query-derived frame against these alternatives rather than claiming the construction itself is new.

III. Method

The method uses the start and goal to canonicalize motion-planning tasks, removing five rigid-body degrees of freedom at initialization while leaving an SO(2) residual. It then evaluates three ways to impose the remaining symmetry and bounds what post-training symmetrization can recover.

  • III-B. Decomposing the group: The canonicalization leaves rotation about the start-goal axis, so the planner must still be equivariant to SO(2).The residual persists for every rule selecting the second axis; the chosen rule only determines the residual angle.
  • III-B. Decomposing the group: The task supplies a canonical frame with midpoint origin and start-goal axis, removing five SE(3) degrees of freedom at initialization for one cross product per query.The architecture, data, and training budget remain unconstrained by this normalization.
  • III-B. Decomposing the group: The transformed planner receives the reduced trajectory and obstacles, conditions on the start-goal distance, and maps predicted velocities back using the frame rotation.Obstacle centres transform as points, box edges as free vectors, and radii remain unchanged.
  • III-C. Residual symmetry: No continuous second-axis rule removes the residual for all query directions, and scenes with their own symmetry can make two frames indistinguishable.The topological obstruction rules out a continuous tangent direction on S2, while symmetric reduced scenes create an additional ambiguity.
  • III-D. Three ways to impose the symmetry: A missing symmetry can be imposed in training data, the inference operator, or the weights, respectively trading guarantees, computation, or hypothesis-class flexibility.Field averaging projects onto an equivariant subspace, while the non-equivariance residual limits what post-training symmetrization can remove.

IV. Setup and Calibration

The benchmark evaluates conditional flow-matching planners on held-out cluttered 3D environments using collision-free path metrics and paired comparisons against a straight-line baseline. The world-frame model converges to the baseline floor, while best-of-20 evaluation shows the reduced model’s practical sampling utility.

  • Benchmark: PointMass3D contains forty obstacles in [-1, 1]^3, and trajectories use N = 64 waypoints in R3 with resampled collision checking.The robot is a sphere with radius ρ = 0.03; the signed-distance field is tested at spacing 0.01.
  • Data: The dataset has 300 environments with 600 start-goal pairs each, while environments 250–299 are held out from training.Unless otherwise stated, models train on the first 60 of the remaining 250 environments.
  • Model: The planner is a 2.16M-parameter conditional flow-matching model with temporal convolution, FiLM conditioning, and a PointNet-style obstacle encoder.The encoder pools all forty obstacles into one 128-dimensional vector applied identically to all 64 waypoints.
  • Metrics: Evaluation uses 500 identical held-out problems, distinguishing whole-path collision-free samples from best-of-20 query success.Uncertainty for method claims is measured across seeds, while fixed-baseline comparisons use paired bootstrap resampling over environments.
  • Baseline: 15.6% of held-out problems are solved by a straight segment, while the world-frame model converges to 14.60 ± 0.20%.The direct path provides the cheapest baseline for judging what the learned planner adds.
  • Baseline: The paired bootstrap difference between the world-frame model and the straight-line baseline is −1.02 points, with a 95% interval of [−3.06, +0.90].The interval spans zero; at 20 epochs, the world-frame model reaches 15.37 ± 0.36%.
  • Practical metric: Behind a collision check, best-of-20 evaluation gives the reduced model 87.2% held-out query success versus 15.8% for the best untrained prior.This is the reported practitioner-facing measure of sampler usefulness.

V. Results

The query frame supplies most of the benchmark’s gain by removing five rigid-body degrees of freedom, while residual-rotation mechanisms add little accuracy. Controls show that query access and local geometry explain distinct but overlapping effects, and the residual is not a reliable predictor of mechanism value.

  • Scaling: The frame’s gain grows from +6.37 ± 0.17 to +30.08 ± 0.44 points as training environments increase, while the world-frame rate moves only from 12.9 to 15.4.Across the 12.5× increase, the reduction rises from 19.2 to 45.5 without plateauing.
  • Scaling: Random SE(3) data augmentation moves the world-frame model by at most 2.1 points to 17.5, whereas the reduced model reaches 55.8 when trained out.The fixed 20-epoch budget isolates representation in the scaling comparison.
  • Query dependence: The world-frame obstacle encoder is constant across queries within an environment, with measured within-environment standard deviation 0.000000.Under reduction, the same encoder sees obstacles in the query frame and its code varies with the query.
  • Query dependence: Adding the raw query to every obstacle in the world frame reaches 15.59 ± 0.38%, about one point above the world-frame model and essentially the 15.6% straight-line floor.This isolates query dependence without changing frames.
  • Local geometry: Local geometry is worth +40.0 points versus +36.5 for the frame, while supplying both reaches 79.37 instead of the independent prediction 91.1.With local geometry supplied, the frame still adds +24.8, indicating sub-additivity.
  • Local geometry: At matched 80-epoch budgets, the frame beside local geometry falls from +22.37 ± 0.46 points at 60 environments to +17.24 ± 0.17 at 250.The reported 5.1-point decay exceeds either standard error, while local geometry gains 11 points.
  • Replications: The rigid-body replication gives the reduced model 2.9× the world-frame model at matched budgets, but these are the paper’s weakest measurements.The cells use one seed, and both models remain below their own floor at 20 epochs.
  • Residual: The reduced model’s residual r = 0.017 bounds removable squared error below 3 × 10^-4 of field magnitude, despite 48.9% failed samples.Post-hoc symmetrisation therefore cannot close the observed performance gap.

VI. Limitations

The study’s limitations constrain where the canonical-frame result transfers and how broadly its benchmark effects should be interpreted. These boundaries include task representation, benchmark design, data coverage, missing comparisons, seed counts, scaling, and inference information.

  • Scope of applicability: The construction applies when rigid motions act on the state space, covering mobile bases, free-flying bodies, and end-effector-space planning, but not joint-space planning for arms.For arms, a workspace rigid motion induces no clean action on joint angles.
  • Benchmark scope: The benchmark isolates representation effects rather than difficulty: RRT-Connect solves 99.6% of problems in under 0.3 s on one CPU core.The point-mass benchmark is intentionally controlled for clutter, budget, seed, and representation.
  • Data coverage: The claim that the residual sixth degree of freedom needs no treatment relies on implicit roll diversity from many start-goal directions, so few-direction datasets could reverse it.The evaluation uses one workspace family, one clutter generator, and one architecture.
  • Unrun comparison: The study does not run a world-frame equivariant model, so it establishes only that architectural equivariance adds nothing once the frame is present.A world-frame equivariant model would distinguish whether the architecture can supply what the frame supplies.
  • Statistical coverage: Seed coverage is uneven: most cells use two or three seeds, while the converged 250-environment run and rigid-body cells use one.The authors state that every claim they rely on has at least two seeds.
  • Scaling: The frame’s value beside local geometry falls from +22.4 points at 60 environments to +17.2 at 250, and the stopping point of this decay is unknown.Readers extrapolating to much larger datasets should expect the margin to narrow rather than hold.
  • Inference assumptions: Local geometry requires explicit obstacle primitives at inference, making it unavailable to planners operating directly from perception.The signed distance is a deterministic featurisation of obstacle data already received by the network, not an oracle.

VII. Conclusion

The conclusion argues that the planning query itself removes most rigid-body symmetry at negligible architectural cost, while residual symmetry mechanisms mainly affect optimisation. The result transfers only to state spaces acted on by the group, and its benefit shrinks when local geometry is supplied.

  • Conclusion: Five of six SE(3) degrees of freedom are removable exactly from the query-defined frame, yielding +36.5 points in held-out collision-free rate while the world-frame model does not beat a straight line.The frame costs one cross product per query; the straight-line baseline scores 15.6%.
  • Conclusion: Residual-rotation mechanisms add little accuracy, but the SO(2)-equivariant backbone reaches any given level two to three times sooner.Frame averaging and the backbone each recover less than a point, while roll augmentation contributes +0.8 at a fixed budget.
  • Conclusion: The construction transfers to mobile bases, free-flying bodies, and end-effector-space planning, but excludes joint-space planning for arms.The group must act on the state space, not merely the workspace.
  • Conclusion: Local geometry is worth more than the frame and reduces the frame’s gain from +36.5 to +24.8 points when supplied.The authors interpret this as the payoff narrowing when the conditioning path from scene to trajectory is widened.
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