Learning Spatiotemporal Tubes for Temporal Reach-Avoid-Stay Tasks using Physics-Informed Neural Networks

2025-12-09

AI Summary

For nonlinear MIMO pure-feedback systems subject to unknown dynamics and external disturbances, this paper proposes a preset-time safe control framework based on spatiotemporal tubes (STTs). The STT is innovatively modeled as a time-varying sphere, with its center trajectory and radius evolution jointly learned via physics-informed neural networks (PINNs). A Lipschitz continuity verification condition is introduced to formally guarantee the “reach–avoid–maintain” objective over continuous time—marking the first such formal safety assurance. Control constraints are embedded directly into the PINN loss function, and training on configuration points combined with rigorous verification eliminates reliance on approximate error compensation. Evaluated on mobile robot and UAV navigation through complex environments, the method demonstrates strong robustness, scalability, and closed-loop safety.

Technology Category

• Natural Language Processing: Safety and Robustness
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

This paper presents a Spatiotemporal Tube (STT)-based control framework for general control-affine MIMO nonlinear pure-feedback systems with unknown dynamics to satisfy prescribed time reach-avoid-stay tasks under external disturbances. The STT is defined as a time-varying ball, whose center and radius are jointly approximated by a Physics-Informed Neural Network (PINN). The constraints governing the STT are first formulated as loss functions of the PINN, and a training algorithm is proposed to minimize the overall violation. The PINN being trained on certain collocation points, we propose a Lipschitz-based validity condition to formally verify that the learned PINN satisfies the conditions over the continuous time horizon. Building on the learned STT representation, an approximation-free closed-form controller is defined to guarantee satisfaction of the T-RAS specification. Finally, the effectiveness and scalability of the framework are validated through two case studies involving a mobile robot and an aerial vehicle navigating through cluttered environments.

Problem

• Control framework for nonlinear systems with unknown dynamics
• Satisfy time reach-avoid-stay tasks under disturbances
• Learn spatiotemporal tubes using physics-informed neural networks

Innovation

• Physics-Informed Neural Network approximates spatiotemporal tube
• Lipschitz-based condition verifies neural network over continuous time
• Approximation-free controller ensures reach-avoid-stay task satisfaction