Learning Spatiotemporal Tubes for Full Class of Signal Temporal Logic Tasks for Control of Unknown Systems Under Input Constraints

Jul 16, 2026

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This paper develops a Spatiotemporal Tube (STT) based control framework for unknown nonlinear Euler–Lagrange systems with input constraints, targeting satisfaction of Signal Temporal Logic (STL) specifications. Each agent’s STT is modeled as a time-varying ball whose center and radius are jointly learned with a physics-informed neural network. STL robustness metrics drive training as a loss, and for multi-agent tasks an additional global robustness metric prevents inter-tube collisions. A closed-form constrained control law keeps trajectories within the learned tubes, and validation is provided through multiple case studies.

LanguageEnglish

CategoryResearch & Report

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Pages17

Reading Time43 min

Abstract

Introduction

LEARNING SPATIOTEMPORAL TUBES FOR FULL CLASS OF SIGNAL TEMPORAL LOGIC TASKS FOR CONTROL OF UNKNOWN SYSTEMS UNDER INPUT CONSTRAINTS ∗ Ahan Basu Centre for Cyber-Physical Systems Indian Institute of Science, Bengaluru, India [ahanbasu@iisc.ac.in](ahanbasu@iisc.ac.in) Ratnangshu Das Centre for Cyber-Physical Systems Indian Institute of Science, Bengaluru, India [ratnangshud@iisc.ac.in](ratnangshud@iisc.ac.in) arXiv :2607 .07 136v 1 [ cs .RO] 8 Jul 2026 Soumyodipta Nath Centre for Cyber-Physical Systems Indian Institute of Science, Bengaluru, India[soumyodiptan@iisc.ac.in](soumyodiptan@iisc.ac.in) Siyuan Liu Electrical Engineering Department, Eindhoven Institute of Technology, Netherlands [s.liu5@tue.nl](s.liu5@tue.nl) Pushpak Jagtap Centre for Cyber-Physical Systems Indian Institute of Science, Bengaluru, India [pushpak@iisc.ac.in](pushpak@iisc.ac.in) July 9, 2026 ABSTRACT This paper presents a Spatiotemporal Tube (STT)-based control framework for general unknown nonlinear EulerLagrange (EL) systems subject to input constraints, with the objective of satisfying Signal Temporal Logic (STL) speciﬁcations, where conﬁnement of the system trajectory within the STT guarantees the satisfaction of the corresponding STL task. For both single and multi-agent scenarios, the STT corresponding to each agent is modeled as a time-varying ball, whose center and radius are jointly parameterized using a physics-informed neural network (PINN) . The robustness metric associated with the STL speciﬁcation corresponding to the agents is incorporated into the training process as a loss function, enabling the learned tube to encode task-level temporal requirements. For a multi-agent scenario, we introduce an additional robustness metric corresponding to the global task, which, when satisﬁed, ensures the tubes do not collide with each other. To ensure that the system trajectory remains within the learned STT and thereby satisﬁes the local and global STL speciﬁcations, we propose a control strategy that explicitly accounts for input constraints. In particular, a closed-form control law is developed to keep the trajectory inside the tube while regulating the motion of the tube by enforcing bounds on its evolution depending on the input constraints of the system. The proposed approach has been validated over several case studies. 1 Introduction Modern control applications, such as safe and reliable operations of robots and unmanned vehicles, demand satisfying complex time-dependent task speciﬁcations instead of the traditional stabilization and trajectory tracking problem. Signal Temporal Logic (STL) [18] provides a formal language framework to describe these tasks, such as requiring a robot to transport materials between speciﬁed locations within a prescribed time horizon, and avoid time-varying ∗The work of Ratnangshu Das is supported by the Prime Ministers Research Fellowship from the Ministry of Education, Government of India. A PREPRINT-JULY 9, 2026 hazardous regions, like areas undergoing scheduled maintenance. Moreover, STL provides a quantitative measure of task satisfaction through robustness metrics [7] . Essentially, STL has become an increasingly popular tool in areas such as deep learning [15], reinforcement learning [27], learning from demonstration [22], planning and control [8, 30] . Despite its wide applicability, control under STL speciﬁcations is challenging due to algorithmic and computational issues. Mixed-Integer Programming (MIP) [25] and gradient-based methods [9] have been used to enforce STL constraints, but because they rely on optimization, they often do not scale well as the STL speciﬁcations become more complex or the system dynamics become more complicated. In recent years, Model Predictive Control has been deployed as a promising tool to satisfy STL speciﬁcations within a receding-horizon framework [24], but it also often suffers from computational complexity issues for complex tasks

FAQs

What is the core idea of the proposed STT-based control framework?

The method learns a spatiotemporal tube such that keeping the system trajectory inside the tube guarantees satisfaction of the corresponding STL specification.

How are the spatiotemporal tubes represented and learned in single- and multi-agent settings?

Each agent’s tube is a time-varying ball whose center and radius are jointly parameterized and learned using a physics-informed neural network. For multi-agent tasks, a global robustness metric is added to ensure the tubes do not collide.

How does the approach handle input constraints while enforcing STL tasks?

A control strategy explicitly accounts for input constraints by using a closed-form control law that keeps trajectories within the tube and regulates tube evolution with bounds tied to the system’s input constraints.

Jul 16, 2026

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