Control Barrier Functions for Disjunctions of Signal Temporal Logic Tasks

In this work we consider the control problem of systems that are subject to disjunctions of Signal Temporal Logic (STL) tasks. Motivated by existing approaches encoding the STL tasks utilizing time-varying control barrier functions (CBFs), we propose a continuously differentiable function for encoding the STL constraints that is defined as the composition of a smooth approximator of the max operator and a set of functions ensuring the satisfaction of the corresponding STL tasks with a desired robustness, and derive conditions for the choice of the class $\mathcal{K}$ function (when the latter is considered to be linear) to ensure that the proposed function is a CBF. Then, a control law ensuring the satisfaction of the STL task is found as a solution to a computationally efficient QP.

I. Introduction

Nowadays, autonomous systems are more and more involved in a variety of applications that require them to perform arbitrarily complex tasks constrained both in space and time. These tasks can be often expressed as boolean compositions of simpler spatio-temporal tasks such as “reach area A between 0 and 5 sec and maintain a distance d from area B between 0 and 10 sec”.