# Real-Time Spatiotemporal Tubes for Dynamic Unsafe Sets

arXiv: https://arxiv.org/abs/2512.06151 · HTML: https://mlsanity.com/paper/2512.06151/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Dec 2025.

## Built on by (most cited first, 1 of 1)

- [Learning from Demonstration via Spatiotemporal Tubes for Unknown Euler-Lagrange Systems](https://mlsanity.com/paper/2607.00534/) (2026)

## Abstract (arXiv)

This paper presents a real-time control framework for nonlinear pure-feedback systems with unknown dynamics to satisfy reach-avoid-stay tasks within a prescribed time in dynamic environments. To achieve this, we introduce a real-time spatiotemporal tube (STT) framework. An STT is defined as a time-varying ball in the state space whose center and radius adapt online using only real-time sensory input. A closed-form, approximation-free control law is then derived to constrain the system output within the STT, ensuring safety and task satisfaction. We provide formal guarantees for obstacle avoidance and on-time task completion. The effectiveness and scalability of the framework are demonstrated through simulations and hardware experiments on a mobile robot and an aerial vehicle, navigating in cluttered dynamic environments.