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Trajectory Planning without Trajectory Data: A Manifold-Guided Approach

Silong Yong, Anji Liu, Cunxi Dai, Carl Busart, Guanya Shi, Yilun Du, Katia Sycara, Yaqi Xie

arXiv:2610.08863v1cs.RO

TL;DR

Trajectory-planning models trained on trajectories can struggle with incomplete coverage and unseen start–goal pairs. Ariadne learns state-space geometry from state observations and constructs paths on the learned manifold; experiments report feasible paths, generalization to unseen configurations, and competitive dual-arm planning.

  • Problem

    Trajectory-based generative planners may fail on unseen start–goal requests because finite trajectory datasets incompletely cover feasible paths and the learned distribution may not support the required trajectory.

    Source: §1 · p. 1, §2 · p. 3

  • Method

    Ariadne extracts geometry from a pretrained state-space diffusion model and constructs endpoint-constrained trajectories along the learned manifold using state-only supervision.

    Source: §1 · p. 2, §7 · p. 10

  • Results

    Across Maze2D and robotic motion-planning benchmarks, Ariadne constructs feasible trajectories without trajectory training data and generalizes to unseen start–goal configurations; it achieves the highest average success rate across three DualKUKA environments.

    Source: §5.1 · p. 8, §7 · p. 10

  • Takeaways & Limitations

    State-space geometry may provide an alternative inductive bias for trajectory planning beyond explicit trajectory imitation.

    Source: §7 · p. 10

    Ariadne assumes nearby states on the learned manifold admit valid local transitions, an assumption that state validity alone does not ensure for irreversible, discontinuous, or strongly action-dependent dynamics.

    Source: §7 · p. 10

Abstract

from arXiv · show

A common way for trajectory planning is to leverage generative models trained on large collections of expert trajectories. At inference time, the model generates executable trajectories by conditioning on task goal constraints. However, trajectory-based methods rely on costly supervision, scale poorly with sequence length, and often generalize poorly to unseen constraints such as novel start-goal pairs. We propose an alternative to learn the underlying state-space manifold and use the geometry of the manifold for trajectory planning. This approach requires only state observations and enables generalization to unseen constraints by con- structing trajectories on the learned manifold of the state space. Experiments on Maze2D and robotic motion-planning benchmarks show that Ariadne constructs feasible paths from state-only supervision and generalizes to unseen start-goal combinations. On high-dimensional dual-arm planning, it remains competitive with trajectory-supervised and classical planners, while requiring no trajectory data for training.

1 Introduction

Trajectory-supervised generative planners face incomplete trajectory coverage and poor generalization to unseen start–goal pairs. Ariadne instead learns state-space geometry and uses it to construct paths, with experiments spanning mazes and robotic planning.

1 Introduction

  • Trajectory-based generative planners can fail on unseen start–goal pairs even when both endpoint states appeared in training, amid incomplete coverage of feasible trajectories.As sequence length increases, feasible trajectories grow combinatorially, making sufficient data coverage prohibitively costly.Source: §1 · p. 1
  • Ariadne learns valid-state distributions from state observations and frames trajectory construction as searching for paths on the learned state manifold.The manifold hypothesis treats feasible states as lying on a lower-dimensional structure embedded in the ambient state space.Source: §1 · pp. 1–2
  • The method uses generative-model geometry to build a manifold-respecting vector field connecting endpoints, with a score-derived Riemannian metric for stable projection.The score captures local first-order structure, while higher-order information reveals the full normal space.Source: §1 · p. 2
  • Ariadne produces feasible paths from state-only training in Maze2D and robotic tasks, generalizes to unseen start–goal pairs, and achieves results comparable to trajectory-learning methods in dual-arm planning.The introduction also reports evaluation against classical search and geometry-based optimization.Source: §1 · p. 2

2 Background and Motivation

Trajectory models learn distributions over sequences, but finite datasets cover feasible trajectories incompletely and can leave unseen requests outside the learned distribution. The section motivates learning from state data instead and reconstructing paths on its learned manifold.

2 Background and Motivation

  • Trajectory datasets provide incomplete coverage of feasible paths, and the number of feasible trajectories grows combinatorially with sequence length.Missing valid paths can result from limited collection or biased sampling.Source: §2 · p. 3
  • Trajectory-level distribution mismatch can make a planner fail on an unseen request even when a feasible solution exists in the environment.The required trajectory may lie outside the learned trajectory distribution’s support.Source: §2 · p. 3
  • The authors propose learning from state data alone because states in missing trajectories may still be represented in the observed state space.They argue that trajectory-level out-of-distribution failures can reflect trajectory-data bias rather than insufficient state coverage.Source: §2 · p. 3
  • With a learned state-space model, path construction becomes curve generation under conditions such as specified endpoints, using the manifold’s geometry.The section notes that collecting state data can be easier and less restrictive than collecting full trajectory demonstrations.Source: §2 · p. 3

3 Trajectory Construction on the Learned Manifold

Ariadne constructs paths by moving toward a goal while using learned manifold geometry to constrain motion. The section explains why tangent projection alone may drift from the manifold or fail to reach the endpoint.

3 Trajectory Construction on the Learned Manifold

  • The data manifold is the lower-dimensional subset where valid system states concentrate; Ariadne uses a learned generative model to characterize its local geometry.The method does not assume explicit access to the manifold.Source: §3 · p. 3
  • The Correction-Projection ODE adds a correction to projection so the constructed path stays on the example manifold and reaches its specified endpoint.The figure contrasts this with straight-line interpolation, which ignores manifold geometry, and projection alone, which can shrink progress toward the goal.Source: Figure 2 · p. 4
  • The basic construction follows the direction from the start toward the goal, then removes the displacement’s normal component to project motion onto the local tangent space.This move-then-project strategy seeks to keep successive updates locally tangent to the manifold.Source: §3 · pp. 3–4
  • Repeated tangent updates encourage manifold adherence but do not guarantee it; when the reference direction aligns with the normal space, deviations can accumulate.The section reports that this formulation works well in practice in most cases.Source: §3 · p. 4
  • Even approximate manifold adherence does not guarantee reaching the goal, because projection can redirect global motion away from the intended endpoint.The section identifies this as a limitation of local geometric corrections alone.Source: §3 · p. 4

4 Incorporating Endpoint Constraints

Ariadne modifies move-then-project dynamics to construct paths that satisfy endpoint constraints while remaining on the learned manifold. Its geometry uses higher-order score information to estimate normal directions, and its correction-projection dynamics balance manifold adherence with endpoint satisfaction.

4 Incorporating Endpoint Constraints

  • Projected motion can distort the global path, stall, or converge to a manifold-valid point that fails to reach the specified endpoint.The method addresses this by modifying the induced ODE so endpoint satisfaction becomes a structural property of its integral curves.Source: §4 · p. 5

4.1 Operator Splitting

  • Operator splitting separates reference motion from manifold projection, framing move-then-project updates as a discretization of an ODE.This formulation provides a basis for introducing endpoint correction.Source: §4.1 · p. 5, §4 · p. 5

4.2 Interpreting Iterative Projection as ODE

  • The Correction-Projection ODE adds a correction term to restore components removed by projection while designing paths to begin and end at the prescribed states.In continuous form, the projection component preserves local manifold adherence while the correction term enforces the endpoint constraint; numerical discretization and imperfect geometry estimation can leave residual errors.Source: §4.2 · p. 6

4.3 Numerically evolving CP-ODE

  • Numerical evolution uses Euler operator splitting, with initial arc-length, projection scale, and normal-selection threshold as hyperparameters.On UCG, the reported comparison shows Ariadne balancing manifold adherence and endpoint satisfaction.Source: §4.3 · p. 6
  • On UCG, Ariadne reduces manifold deviation relative to the straight-line reference and endpoint error by an order of magnitude relative to Projection-Only.The straight-line reference satisfies the endpoint constraint but deviates substantially from the manifold; Projection-Only stays closest to the manifold but fails to reach the target.Source: §4.3 · p. 6

4.4 Manifold Hypothesis and Geometry from Generative Models

  • For higher-codimension manifolds, the score alone cannot characterize the full normal space; Hessian curvature identifies normal directions, and a score-derived metric penalizes off-manifold deviations.The metric is assumed to remain close to identity on the manifold and grow away from it; the Jacobian of the score is proposed for normal estimation in the figure.Source: Figure 3 · p. 5, §4.4 · p. 7

5 Experiments

Ariadne is evaluated on Maze2D, YAM 6D, and DualKUKA using state-space training data without trajectory or transition supervision. Results compare it with classical, trajectory-supervised, and geometry-based planners, and examine hyperparameter sensitivity.

5 Experiments

  • The practical setup uses a straight-line reference path and arc-length-based step sizes for consistent trajectory progression.The implementation is adapted for diffusion and flow-matching models.Source: §5 · p. 7
  • The benchmarks include Maze2D, YAM 6D tabletop manipulation, and DualKUKA, with state-only training and start-goal sequence evaluation.Evaluation uses standard environment metrics, including task score and collision-free success rate.Source: §5 · pp. 7–8

5.1 Experimental Results

  • Ariadne outperforms Diffuser* when evaluation trajectories are withheld, while fully supervised Diffuser achieves substantially higher D4RL normalized scores.The state-only generalization setting observes constituent states but withholds trajectories reaching evaluation targets.Source: §5.1 · p. 8
  • Follow-the-Energy degrades substantially on Maze2D and two of three DualKUKA settings; trajectory-level interpolation can leave narrow feasible regions.Ariadne instead constructs paths through repeated local Correction-Projection updates.Source: §5.1 · p. 8
  • 60.77% versus 12.31% for the straight-line reference, Ariadne's collision-free planning success on YAM 6D.Table 2 reports YAM 6D motion-planning success rates.Source: §5.1 · p. 8, §5 · p. 8, Table 2 · p. 8
  • Ariadne achieves the highest average success rate across three DualKUKA environments using only state observations during training.It outperforms Diffuser on all three settings despite Diffuser receiving full trajectory supervision, and is competitive with PBDMP, which uses scene information across environments while Ariadne learns one planner per scene.Source: §5.1 · p. 8, Table 3 · p. 9
  • RRT success degrades as search difficulty rises, and it requires many online collision checks.Sensitivity was consistent across five independently trained score models.Source: §5.1 · p. 9

5.2 Ablation Studies

  • Ariadne is not narrowly dependent on hyperparameter tuning, but loose or restrictive normal thresholds, large steps, and excessive projection each have specific drawbacks.Loose thresholds admit weak or noisy directions, restrictive ones produce incomplete normal estimates, large steps can exceed local-approximation validity, and strong projection can hinder goal progress.Source: §5.2 · p. 9

6 Related Work

Related work frames planning as conditional generation over trajectory sequences and connects diffusion models to learned manifold geometry. Ariadne instead extracts geometry directly from a trained diffusion model without a strong assumed manifold structure.

6 Related Work

  • The detailed related-work discussion is deferred to Appendix A.Source: §6 · p. 9

Learning-based Planning

  • Sequence-modeling methods learn trajectory distributions and generate plans by conditioning on task information such as initial states, goals, or constraints [12].Diffuser [20] represents trajectory distributions through denoising and performs conditional sampling at inference; later work addresses long-horizon planning [28] and compositionality [29].Source: §6 · pp. 9–10

The Manifold Hypothesis

  • Prior manifold work assumes structures such as linear subspaces or D −1-dimensional manifolds, whereas Ariadne extracts local geometry from a diffusion-model Jacobian without a strong structural assumption.The manifold hypothesis holds that high-dimensional data lie on a lower-dimensional embedded manifold.Source: §6 · p. 10

7 Conclusion

Ariadne plans without trajectory supervision by extracting geometry from a pretrained state-space diffusion model and constructing paths under manifold and endpoint constraints. Experiments report feasible paths and generalization to unseen start-goal configurations, alongside limitations in transition assumptions, geometry accuracy, coverage, connectivity, horizon, and inference cost.

7 Conclusion

  • Ariadne constructs trajectories from feasible states under manifold and endpoint constraints, without observing trajectory data during training.The conclusion reports feasible trajectories and generalization to unseen start-goal configurations on Maze2D and robotic motion-planning benchmarks.Source: §7 · p. 10
  • Planning assumes nearby learned-manifold states admit valid local transitions, an assumption that may fail for irreversible, discontinuous, or strongly action-dependent dynamics.The conclusion also identifies dependence on learned local-geometry accuracy, state coverage, multimodal connectivity, and reference paths for long horizons; normal-space derivatives increase inference cost with state dimension.Source: §7 · p. 10

A.1 Generative Models for Sequence Modeling

  • Prior work explores diffusion models for sequence modeling and generative-model geometry, while Ariadne directly extracts geometry from a trained diffusion model.Source: §A.1 · p. 14

A.2 Latent Interpolation

This material contrasts latent interpolation with data-space manifold-constrained path construction and supplies the numerical and geometric background for Ariadne's correction-projection procedure. The appendices describe operator splitting, iterative projection, the correction-projection ODE, its implementation, and local geometry extraction.

A.2 Latent Interpolation

  • Latent interpolation methods include linear, geodesic, and spherical interpolation, but operate primarily in noise space rather than explicitly enforcing ambient-space feasibility or manifold connectivity.Ariadne instead generates paths directly in data space by integrating a constrained ODE whose curves remain on the manifold and satisfy endpoint constraints.Source: §A.2 · p. 14

B.1 Operator Splitting Methods

  • Operator splitting separates a summed ODE into two sub-dynamics, applying A and then B over each interval; its suitability depends on avoiding uncontrolled curvature or instability.Ariadne uses the split conceptually for reference motion and manifold projection.Source: §B.1 · p. 14

B.2 Iterative Projection

  • Iterative projection is described as an operator-splitting numerical method, with ε set to the step size and V and n denoting the local normal subspace and its basis.Source: §B.2 · p. 15

B.3 Correction-Projection ODE

  • The correction-projection ODE combines a straight-line interpolation between endpoints with a nonlinear correction term φ(x_t).The derivation relates this formulation to the iterative-projection ODE and its numerical integration.Source: §B.3 · p. 15

C Pseudo Code for the proposed Correction-Projection ODE

  • Algorithm 1 takes a state-distribution flow model, start and end points, and three hyperparameters, then returns a feasible plan.The hyperparameters are initial arc-length, projection scale, and normal-selection threshold; the procedure repeatedly updates the trajectory state.Source: C Pseudo Code for the proposed… · p. 16

D.1 Hessian and local geometry.

  • Under the manifold-plus-noise assumption, log-density curvature is approximately flat along tangent directions and strongly negative along normal directions, so large negative eigenvalues identify the normal space.Positive eigenvalues may indicate more complex local structure, including multimodality or stratified manifolds.Source: §D.1 · pp. 16–17

D.2 Riemannian metric from score function.

Ariadne derives a Riemannian metric from the learned score function to penalize off-manifold deviations, assuming the score magnitude is near zero on the data manifold. Its practical setup uses a straight-line reference path, fixed spatial step size, and a Flow-Matching-specific Hessian approximation.

D.2 Riemannian metric from score function.

  • Under the assumption that the energy gradient is near zero on the manifold, the score norm estimates distance and enables a metric without explicit manifold supervision.The induced metric penalizes off-manifold directions more strongly.Source: §D.2 · p. 17

E Practical Considerations Full Details

  • A straight-line reference path suffices for reasonable Ariadne results, although SLERP, generated trajectories, and parameterized curves are alternatives.The curve’s shape can be adjusted with parameters, including a perpendicular direction to the straight line.Source: E Practical Considerations Full Details · p. 17
  • A fixed spatial step ds preserves state-space connectivity, with the time interval set to dt = ds ||f||.Here, f is the right-hand side of Eq. 6.Source: E Practical Considerations Full Details · p. 17
  • For Flow-Matching models, the implementation approximates the Hessian with (J+J^T)/2, where J is the output Jacobian with respect to the input.Diffusion models directly predict the score and allow a simpler Hessian computation.Source: E Practical Considerations Full Details · p. 17

F Experimental Setup Details

The experiments evaluate state-only training across Maze2D, YAM 6D, and DualKUKA, using benchmark task scores or collision-free planning success. Training and inference setups vary by environment, with DualKUKA evaluated in end-effector space.

F Experimental Setup Details

  • Maze2D training uses maze state-space data without trajectory data or connectivity information, and evaluation reports the environment’s standard score..Source: F Experimental Setup Details · p. 17
  • YAM 6D evaluates collision-free motion-planning success for a 6-DoF arm, training on feasible states without trajectory or transition information.Training states are sampled arm configurations transformed by forward kinematics, then filtered to retain collision-free states.Source: F Experimental Setup Details · p. 18
  • For each test start-goal pair, the plan is checked under the environment’s standard collision and validity checks, with task success or score reported.The benchmark scoring protocol is used despite training without trajectories.Source: F Experimental Setup Details · p. 18
  • DualKUKA tests two 7-DoF arms avoiding obstacles and each other, with one-shot success requiring a collision-free, goal-reaching path.The study uses 7 or 5 obstacles across its reported scenarios and represents states by end-effector position and Euler angles.Source: F Experimental Setup Details · p. 18
  • Baseline evaluation is restricted to one scene, including methods that can accept different scene layouts.The restriction follows the single-scene planner training convention attributed to Diffuser.Source: F Experimental Setup Details · p. 18

G Training Details

  • For flow matching, Maze 2D and YAM 6D use 4 hidden layers and latent dimension 128, whereas DualKUKA uses 8 hidden layers and hidden dimension 1024.DualKUKA inference is conducted in end-effector space, using IK with the start point’s joint-angle reference to derive joint angles for both KUKA arms. Training uses one NVIDIA RTX 6000 Ada Generation GPU and takes 2–12 hours, depending on tasks and dataset size.Source: G Training Details · p. 18

H More Quatitative Results

Figure 6 visualizes a representative Ariadne trajectory in the YAM 6D scene, with waypoint colors encoding progress from start to goal.

H More Quatitative Results

  • The YAM 6D path routes around obstacles while remaining in the collision-free manifold between the specified endpoints.The visualization colors waypoints by task progress, with green at the initial state and red at the final state.Source: H More Quatitative Results · p. 18, Figure 6 · p. 19

I Extented Analysis on Ablation Studies

The extended analysis describes trade-offs in projection scale, step size, and normal-threshold selection, and discusses Ariadne’s state-manifold perspective, efficiency evaluation, and baseline assumptions.

I Extented Analysis on Ablation Studies

  • A moderate projection scale performs best because it balances following the reference direction against enforcing manifold constraints.Too little scale can drift off-manifold; too much can divert motion from the reference path and impede goal progress.Source: I Extented Analysis on Ablation Studies · p. 19
  • Moderate step sizes perform best: smaller steps require more iterations, while larger steps can leave the locally valid manifold approximation before projection.Both extremes degrade performance in the reported ablation.Source: I Extented Analysis on Ablation Studies · p. 19
  • A moderate normal threshold balances noisy extra directions against an incomplete estimate of the normal space.Too-low thresholds can destabilize projections; too-high thresholds weaken the manifold constraint and permit drift.Source: I Extented Analysis on Ablation Studies · p. 19

J Extended Discussion

  • Trajectory incompleteness need not imply state incompleteness: Ariadne constructs trajectories from feasible-state geometry rather than learning trajectory distributions.The discussion also frames its ODE as deterministic transport between feasible states, related conceptually to probability-flow ODEs.Source: J Extended Discussion · pp. 19–20
  • Extension of the manifold hypothesis and its geometry beyond trajectory planning to broader sequential domains is left for future work..Source: J Extended Discussion · p. 20

K Efficiency Comparison

  • The efficiency comparison with RRT reports success rate, collision-check count, and planning time; Ariadne performs online construction without collision-checker calls.The table’s DualKUKA settings are indexed by obstacle count and half-size.Source: K Efficiency Comparison · p. 20, Table 5 · p. 20

L Detailed Explanation of Baselines

  • The baseline set spans classical search, trajectory-supervised generation, and geometry-based planning, but its methods use different supervision and inference assumptions.Diffuser* sees evaluation target states during training but not as trajectory endpoints; Follow-the-Energy learns an energy model from state observations, while PBDMP receives scene layout.Source: L Detailed Explanation of Baselines · p. 21

M Robustness to Score-Model Retraining

The supplied leads address robustness to score-model retraining: one reports relatively small variation across five independently trained models on three DualKUKA settings, while the other states that the passages do not report such robustness.

M Robustness to Score-Model Retraining

NeurIPS Paper Checklist

  • The checklist reports that limitations are discussed in the Conclusion, theoretical results are absent, and no human subjects or crowdsourcing are involved.The paper also answers yes to compliance with the NeurIPS Code of Ethics and marks societal impact as not applicable because it focuses on core deep-learning problems.Source: NeurIPS Paper Checklist · pp. 22, 23, 26, 28
  • The checklist says hyperparameters and experimental details are in the Appendix, while code release is planned upon acceptance.The paper answers yes on experimental reproducibility but no on open access to data and code at submission.Source: NeurIPS Paper Checklist · pp. 23–25
  • The paper marks LLM use as not applicable because LLMs are not an important, original, or non-standard component of the research.Source: NeurIPS Paper Checklist · p. 28
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