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Progressive Learning for Physics-informed Neural Motion Planning

Ruiqi Ni, Ahmed H. Qureshi

arXiv:2306.00616v1cs.ROcs.LG

TL;DR

Physics-informed neural planners avoid expert demonstrations but have difficulty with complex environments and scaling across multiple scenarios and high-dimensional settings. This paper combines progressive speed scheduling with a viscosity-based Eikonal formulation. It reports faster planning than several baselines and near-100% success rates across the compared methods, while noting that its approach does not generalize to unseen environments.

  • Problem

    Physics-informed NTFields avoid expert trajectories but have low success in complex, cluttered, high-dimensional problems and do not scale to multiple scenarios.

    Source: §I · p. 2

  • Method

    The method combines progressive speed scheduling with a viscosity term in the Eikonal equation to train physics-informed planners without expert demonstrations.

    Source: §I · p. 2

  • Results

    Our method was about 2 times faster than NTFields, IEF3D, and RRT-Connect and over 30 times faster than FMM, RRT*, and LazyPRM*, while all methods had near 100% high success rate.

    Source: B. Comparison Analysis · p. 8

  • Takeaways & Limitations

    One neural model generalizes to eight C3D scenarios without expert data, unlike NTFields, which requires eight separate models.

    Source: B. Comparison Analysis · p. 8

    The method does not generalize to unseen environments and scales only to the given multiple scenarios.

    Source: §VI · p. 9

Abstract

from arXiv · show

Motion planning (MP) is one of the core robotics problems requiring fast methods for finding a collision-free robot motion path connecting the given start and goal states. Neural motion planners (NMPs) demonstrate fast computational speed in finding path solutions but require a huge amount of expert trajectories for learning, thus adding a significant training computational load. In contrast, recent advancements have also led to a physics-informed NMP approach that directly solves the Eikonal equation for motion planning and does not require expert demonstrations for learning. However, experiments show that the physics-informed NMP approach performs poorly in complex environments and lacks scalability in multiple scenarios and high-dimensional real robot settings. To overcome these limitations, this paper presents a novel and tractable Eikonal equation formulation and introduces a new progressive learning strategy to train neural networks without expert data in complex, cluttered, multiple high-dimensional robot motion planning scenarios. The results demonstrate that our method outperforms state-of-the-art traditional MP, data-driven NMP, and physics-informed NMP methods by a significant margin in terms of computational planning speed, path quality, and success rates. We also show that our approach scales to multiple complex, cluttered scenarios and the real robot set up in a narrow passage environment. The proposed method's videos and code implementations are available at https://github.com/ruiqini/P-NTFields.

I. INTRODUCTION

The paper addresses the trade-off between slow classical planning and data-hungry neural planners, as well as the difficulty of scaling physics-informed NTFields to complex, high-dimensional scenarios. It proposes progressive speed scheduling and a viscosity-based Eikonal formulation, reporting stronger performance than prior methods and demonstrations across multiple scenarios and a real robot setup.

I. INTRODUCTION

  • Classical methods can be computationally slow, whereas neural planners are fast at test time but require massive expert trajectory datasets.Source: §I · p. 1
  • NTFields solve the Eikonal equation without expert trajectories, but struggle in complex environments and do not scale well to multiple scenarios or high-dimensional planning.Source: §I · p. 1
  • Progressive speed scheduling moves training from constant high speed toward very low speed near obstacles to prevent incorrect local minima.Source: §I · p. 2
  • A Laplacian-based viscosity term makes the Eikonal formulation semi-linear with a unique, smooth solution around low-speed obstacles.Source: §I · p. 2
  • The proposed method significantly outperforms prior classical and learning-based methods, needs no training trajectories, and scales to multiple scenarios.Source: §I · p. 2
  • The paper reports a 6-DOF UR5e demonstration on real-world narrow-passage motion planning problems.Source: §I · p. 2

II. RELATED WORK

  • Sampling-based planners remain widely used but are slow, while supervised and reinforcement-learning approaches also require data generation or many environment interactions.Source: §II · p. 2
  • FMM is computationally intractable in high-dimensional configuration spaces, and NTFields’ limitations impede scaling to multiple environments and real-world settings.Source: §II · p. 2

III. BACKGROUND

Motion planning seeks a feasible trajectory through collision-free configurations from a start to a goal, with travel time sometimes preferred to incorporate speed constraints. The background presents NTFields as a demonstration-free approach that learns an Eikonal-based time field and derives paths from its gradients.

III. BACKGROUND

  • The section formally introduces robot motion planning and physics-informed neural planners as background for the paper.Source: §III · p. 3

A. Robot Motion Planning

  • Motion planning seeks a trajectory through feasible configurations connecting start and goal, sometimes minimizing travel time to impose speed constraints near obstacles.Source: A. Robot Motion Planning · p. 3

B. Physics-informed Motion Planning Framework

IV. PROPOSED METHOD

The proposed learning algorithm targets NTFields’ low success in complex, cluttered, high-dimensional problems and its limited scalability across scenarios. It combines a viscosity-based Eikonal formulation with progressive speed updates.

IV. PROPOSED METHOD

  • NTFields performs poorly in complex, cluttered, high-dimensional environments and does not scale to multiple scenarios.Source: §IV · p. 3
  • The proposed algorithm combines a viscosity-based Eikonal formulation with a progressive speed update strategy.Source: §IV · p. 3
  • The algorithm is intended to train physics-informed neural planners across multiple complex, high-dimensional scenarios.Source: §IV · p. 3

A. Viscosity-based Eikonal Equation

The exact Eikonal solution has nonsmooth points where its gradient is not unique, making neural fitting difficult. Adding a Laplacian viscosity term yields a smooth, unique solution, while the coefficient controls smoothness and introduces computational overhead that the method mitigates.

A. Viscosity-based Eikonal Equation

B. Progressive speed scheduling

Progressive speed scheduling moves training from an easy constant-speed solution toward low speeds around obstacles, addressing incorrect local minima. The neural framework combines a C-space encoder, symmetric operator, and time-field generator.

B. Progressive speed scheduling

  • As α increases from 0 to 1, speed and time fields interpolate continuously from constant maximum speed to low speeds around obstacles.At α = 0 the Eikonal solution is trivial; at α = 1 the scene has low speeds around obstacles.Source: B. Progressive speed scheduling · p. 4, Figure 3 · p. 4
  • Training begins with a constant-speed solution, then progressively lowers obstacle speeds to guide the network toward fields that capture obstacles.The constant-speed case is a trivial solution; low-speed obstacle conditions can otherwise lead to incorrect local minima.Source: B. Progressive speed scheduling · p. 4
  • Because time fields change more aggressively than speed fields, the method reduces α’s rate of increase as training proceeds.Source: B. Progressive speed scheduling · pp. 4–5
  • When the epoch-to-epoch loss ratio exceeds threshold η, training reshuffles samples across batches; without this strategy, the network fails to converge and recover the time field.The reshuffling distributes difficult samples across batches.Source: B. Progressive speed scheduling · p. 5

C. Neural Architecture

  • The neural framework comprises a Fourier-based C-space encoder, a symmetric operator, and a time-field generator.Source: C. Neural Architecture · p. 5

1) Fourier-based C-space Encoding:

The planner embeds start and goal configurations using random Fourier features conditioned on an environment code, then combines their encodings symmetrically. A generator network uses this representation to produce a factorized time field.

1) Fourier-based C-space Encoding:

  • Random Fourier features embed the start and goal configurations, while a latent code represents the environment and is assigned as a fixed unique random matrix per environment.A ResNet-style C-space encoder further processes the features into a latent embedding.Source: 1) Fourier-based C-space Encoding · p. 5

2) Non-linear Symmetric Operator:

  • The symmetric operator concatenates the elementwise maximum and minimum of the start and goal features; the cited NTFields work reports improved speed- and time-field prediction with this operator [23].The operator is described as inspired by properties of the Eikonal equation [23].Source: 2) Non-linear Symmetric Operator · p. 5

3) Time Field Generator:

  • A ResNet-style generator takes the symmetric start-goal encoding as input and outputs the factorized time field.Figure 4 depicts the encoder, symmetric operator, and generator, alongside progressive speed guidance and a generated start-to-goal time field.Source: 3) Time Field Generator · p. 5, Figure 4 · p. 5

D. Objective function

The paper replaces the prior nonsmooth isotropic loss with a smooth objective and trains through progressively modified speed targets. Training uses start-goal batches, updates model parameters from speed-prediction loss, and reshuffles samples when the loss ratio exceeds a threshold.

D. Objective function

  • Unlike the prior L1-based isotropic loss, the proposed objective is smooth, nonnegative, and reaches its minimum value of 0 easily when α is small.The paper identifies the L1 norm’s lack of smoothness as a challenge to convergence.Source: D. Objective function · p. 5

E. Training pipeline

  • Training increases α over epochs to modify the ground-truth speed model, then forms batches of valid start-goal configurations and predicts factorized time and associated speeds.Source: E. Training pipeline · p. 5
  • The pipeline computes loss against ground-truth speeds, updates model parameters, and reshuffles samples across batches when the epoch loss ratio exceeds threshold η.The algorithm repeats batch training after reshuffling.Source: E. Training pipeline · pp. 5–6

F. Planning pipeline

The planning pipeline predicts a factorized time field and iteratively updates start and goal configurations toward one another. Implementation uses valid start–goal pairs, and evaluation spans multiple environments, with specified training settings and planner metrics.

F. Planning pipeline

  • Start and goal configurations are updated bidirectionally using gradients until their distance falls below d_g, with β as the predefined step size.The pipeline predicts τ(q_s, q_g) and uses it to compute time and speed fields before path extraction.Source: F. Planning pipeline · p. 6

1) Data generation::

  • Training uses valid start–goal configuration pairs across environments; data generation takes less than two minutes for C3D, Gibson, and manipulator scenarios.The datasets use 8×0.5M pairs for C3D, 2×1M for Gibson, and 2×1M for the manipulator.Source: 1) Data generation · p. 6

2) Hyperparameters:

  • The model uses 128 hidden units and five ResNet-style blocks, reducing network size from 50MB to 7MB.Training increases α from 0.5 on a schedule; testing uses β and d_g settings that differ between 3D and 6-DOF environments.Source: 2) Hyperparameters · p. 6

V. EVALUATION

  • Evaluation covers eight cluttered 3D scenarios, two Gibson scenes, and simulated and real narrow-passage manipulator planning, comparing classical and learning-based planners.The reported metrics include planning time, path length, safe margin, and success rate.Source: §V · pp. 6–7

A. Abalation Analysis

The ablation compares the learned fields with FMM and tests the effects of progressive speed scheduling and viscosity. Removing either component worsens field recovery and increases loss.

A. Abalation Analysis

  • 0.015 units is our method’s loss, compared with 0.03 without speed scheduling and 0.15 without viscosity.The method’s fields resemble FMM, while the ablated neural model fails to recover the correct result.Source: A. Abalation Analysis · p. 7
  • NTFields recovers similar fields but uses a 50MB network, compared with our 7MB model, and does not scale to multiple scenarios.The passage attributes the comparison to the ablation analysis in complex environments.Source: A. Abalation Analysis · p. 7
  • Progressive learning aids efficient recovery of near-optimal time and speed fields in complex environments.The comparison tests progressive speed scheduling and the viscosity term as components of the learning framework.Source: A. Abalation Analysis · p. 7

B. Comparison Analysis

The comparison evaluates the method in C3D, Gibson, and narrow-passage manipulator settings. It reports faster planning in the first two and the highest success rate with shortest execution time in the manipulator scenarios.

B. Comparison Analysis

  • Our method is about 2 times faster than NTFields, IEF3D, and RRT-Connect and over 30 times faster than FMM, RRT*, and LazyPRM* in C3D; all methods have near 100% success.One model generalizes across eight C3D scenarios without expert data, whereas NTFields requires eight scenario-specific models and IEF3D requires expert data.Source: B. Comparison Analysis · p. 8
  • In Gibson, our method is about 2 times faster than NTFields and IEF3D and over 30 times faster than FMM, RRT*, LazyPRM*, and RRT-Connect.Our method and FMM achieve about 98% success, while NTFields, IEF3D, RRT*, and LazyPRM* achieve about 92%; RRT-Connect has the highest success rate but is 40 times slower than our method.Source: B. Comparison Analysis · p. 8, Figure 7 · p. 8
  • In simulated and real narrow-passage manipulator scenarios, our method achieves the highest success rate with the shortest execution time.In one simulated case, our method finds a 0.83-length path in 0.07 seconds; RRT-Connect takes 20.13 seconds for a 0.90-length path, and NTField finds no solution.Source: B. Comparison Analysis · p. 8

C. Data Generation and Training Time Analysis

The method reduces training time relative to NTFields by generalizing across environments, although computing the Laplacian makes training slower than IEF3D. Its scope remains limited to given environments, and its 6-DOF manipulator success rate is around 92%.

C. Data Generation and Training Time Analysis

  • Training time is lower than NTFields because one model generalizes across environments, but slower than IEF3D because the method computes the Laplacian during training.Data generation is described as significantly low and similar to NTFields; IEF3D takes several hours because it requires expert trajectories from FMM.Source: C. Data Generation and Training Time… · p. 9

VI. DISCUSSIONS, CONCLUSIONS, AND FUTURE WORK

  • The framework solves the Eikonal equation without expert demonstrations using progressive speed scheduling and a viscosity term, and is reported to scale to complex, multiple environments.The paper also reports real-world narrow-passage planning with a 6-DOF UR5e manipulator.Source: §VI · p. 9
  • The method scales only to given environments and does not generalize to unseen ones.The authors identify environment encoding strategies for generalization to novel environments as future work.Source: §VI · p. 9
  • The 6-DOF manipulator success rate is around 92%, which the authors say needs improvement toward 100%.The authors also identify dynamic and manifold constraints as future directions for motion-planning PDE formulations.Source: §VI · p. 10
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