Time-Varying Soft-Maximum Control Barrier Functions for Safety in an A Priori Unknown Environment

TL;DR

A novel smooth time-varying soft-maximum composite control barrier function (CBF) that can be used to ensure safety in an a priori unknown environment, where local perception information regarding the safe set is periodically obtained is presented.

Abstract

This paper presents a time-varying soft-maximum composite control barrier function (CBF) that can be used to ensure safety in an a priori unknown environment, where local perception information regarding the safe set is periodically obtained. We consider the scenario where the periodically obtained perception feedback can be used to construct a local CBF that models a local subset of the unknown safe set. Then, we use a novel smooth time-varying soft-maximum function to compose the N most recently obtained local CBFs into a single CBF. This composite CBF models an approximate union of the N most recently obtained local subsets of the safe set. Notably, this composite CBF can have arbitrary relative degree r. Next, this composite CBF is used as a rth-order CBF constraint in a real-time optimization to determine a control that minimizes a quadratic cost while guaranteeing that the state stays in a time-varying subset of the unknown safe set. We also present an application of the time-varying soft-maximum composite CBF method to a nonholonomic ground robot with nonnegligible inertia. In this application, we present a simple approach to generate the local CBFs from the periodically obtained perception data.

Figures (10)

• Figure 1: Example of $\eta$.
• Figure 2: Map of a priori unknown environment.
• Figure 3: Three closed-loop trajectories with $360^{\circ}$ LiDAR.
• Figure 4: $h$ and $\psi_1$ for $q_{\rm g} = [\,13\quad5\,]^{\rm T}$ .
• Figure 5: $q_x$, $q_y$, $v$, $\theta$, $u_{\rm d}$, and $u$ for $q_{\rm g} = [\,13\quad5\,]^{\rm T}$.

Theorems & Definitions (1)

• proof