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

arXiv cs.RO / 4/6/2026

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Key Points

The paper introduces a learning-based all-pairs motion planning method that handles arbitrary initial and goal states within a predefined safe set.
It constructs smooth goal-conditioned neural ODE dynamics using bi-Lipschitz diffeomorphisms, enabling theoretical guarantees tied to the geometry of the safe set.
The authors prove global exponential stability and safety via forward invariance of the safe set, independently of where the goal is located.
They provide explicit bounds for convergence rate, tracking error, and the magnitude of the learned vector field, aiming for predictable closed-loop behavior.
The approach is implemented with bi-Lipschitz neural networks (optionally leveraging demonstration data) and is demonstrated on a 2D corridor navigation task.

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.