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Goal-Conditioned Neural ODEs with Guaranteed Safety and Stability for Learning-Based All-Pairs Motion Planning

Dechuan Liu, Ruigang Wang, Ian R. Manchester

Abstract

This paper presents a learning-based approach for all-pairs motion planning, where the initial and goal states are allowed to be arbitrary points in a safe set. We construct smooth goal-conditioned neural ordinary differential equations (neural ODEs) via bi-Lipschitz diffeomorphisms. Theoretical results show that the proposed model can provide guarantees of global exponential stability and safety (safe set forward invariance) regardless of goal location. Moreover, explicit bounds on convergence rate, tracking error, and vector field magnitude are established. Our approach admits a tractable learning implementation using bi-Lipschitz neural networks and can incorporate demonstration data. We illustrate the effectiveness of the proposed method on a 2D corridor navigation task.

Goal-Conditioned Neural ODEs with Guaranteed Safety and Stability for Learning-Based All-Pairs Motion Planning

Abstract

This paper presents a learning-based approach for all-pairs motion planning, where the initial and goal states are allowed to be arbitrary points in a safe set. We construct smooth goal-conditioned neural ordinary differential equations (neural ODEs) via bi-Lipschitz diffeomorphisms. Theoretical results show that the proposed model can provide guarantees of global exponential stability and safety (safe set forward invariance) regardless of goal location. Moreover, explicit bounds on convergence rate, tracking error, and vector field magnitude are established. Our approach admits a tractable learning implementation using bi-Lipschitz neural networks and can incorporate demonstration data. We illustrate the effectiveness of the proposed method on a 2D corridor navigation task.

Paper Structure

This paper contains 21 sections, 4 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. The Dynamic Approach for Motion Planning
  4. Problem Statement
  5. Preliminaries on Bi-Lipschitz Diffeomorphisms
  6. Main Theoretical Results
  7. All-Pairs Motion Planning via Natural Gradient Flow
  8. Safety Properties
  9. Time-Varying Goal Location
  10. Finite-time Convergence via Euclidean Norm Potential
  11. Natural Gradient Flow v.s. Gradient Flow
  12. Comparison with Navigation Function based Approach
  13. Compared with Existing Diffeomorphism based Dynamical Approaches
  14. Learning an All-Pairs Motion Planner
  15. Training Data
  16. ...and 6 more sections

Key Result

Theorem 1

If there exists a bi-Lipschitz diffeomorphism $g:\mathcal{X}_{\text{safe}}\rightarrow \mathcal{B}^n$, then the following statements hold: $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure A1: Our approach is based on learning a bi-Lipschitz diffeomorphism $g$ that maps a geometrically complex safe set $\mathcal{X}_{\text{safe}}$ in the $\mathcal{X}$-space (left) onto the unit ball in the $\mathcal{Z}$-space (right). Then simple straight-line point-to-point motions in $\mathcal{Z}$-space can be smoothly pulled back to $\mathcal{X}$ space, defining a goal-conditioned neural ODE which guarantees stability and safety and takes the form of a natural gradient flow.
  • Figure C1: Trajectory samples and vector field on the boundary for the natural gradient flow \ref{['eq:natural_gradient']} (blue) and the gradient flow \ref{['eq:grad-flow']} (black) with different goal points, where red curves are the boundaries.
  • Figure E1: RRT data (gray) in the corridor environment. (Left) RRT rooted at $\hat{x}_{\star}$, with a representative trajectory (blue) in $\mathcal{X}$. (Right) Corresponding cost-to-go field $d(\cdot)$ visualized via contour lines over $\mathcal{X}_{\text{safe}}$.
  • Figure E2: Trajectories generated by system \ref{['eq:natural_gradient']} from multiple initial configurations to the goal $\hat{x}_\star$ in the training dataset. (Left) Smooth, safe paths in the $\mathcal{X}$-space. (Right) Those paths are transformed into straight-line trajectories in the $\mathcal{Z}$-space, demonstrating the geometric simplification induced by $g_{\theta}$.
  • Figure E3: Generalization to previously unseen goal $x_\star$ without retraining. (Left) Smooth, safe paths in the $\mathcal{X}$-space converging to a new goal $x_\star$ (red cross) from multiple initial configurations. (Right) In the $\mathcal{Z}$-space, the paths are transformed into straight-line trajectories within the unit ball.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3: hines2011equilibrium
  • Definition 4
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • ...and 6 more