Spatiotemporal Tubes for Probabilistic Temporal Reach-Avoid-Stay Task in Uncertain Dynamic Environment

TL;DR

This paper introduces an extension of the Spatiotemporal Tube framework to probabilistically ensure safe navigation in uncertain, dynamic environments, providing real-time, model-free control with formal safety guarantees.

Contribution

It develops a novel real-time tube synthesis method that accounts for time-varying uncertainties and offers formal probabilistic safety guarantees without requiring approximation or optimization.

Findings

Effective in simulation and hardware experiments

Ensures probabilistic safety and task completion

Scalable to various robotic platforms

Abstract

In this work, we extend the Spatiotemporal Tube (STT) framework to address Probabilistic Temporal Reach-Avoid-Stay (PrT-RAS) tasks in dynamic environments with uncertain obstacles. We develop a real-time tube synthesis procedure that explicitly accounts for time-varying uncertain obstacles and provides formal probabilistic safety guarantees. The STT is formulated as a time-varying ball in the state space whose center and radius evolve online based on uncertain sensory information. We derive a closed-form, approximation-free control law that confines the system trajectory within the tube, ensuring both probabilistic safety and task satisfaction. Our method offers a formal guarantee for probabilistic avoidance and finite-time task completion. The resulting controller is model-free, approximation-free, and optimization-free, enabling efficient real-time execution while guaranteeing convergence to the target. The effectiveness and scalability of the framework are demonstrated through simulation studies and hardware experiments on mobile robots, a UAV, and a 7-DOF manipulator navigating in cluttered and uncertain environments.