Time-Varying Soft-Maximum Barrier Functions for Safety in Unmapped and Dynamic Environments

TL;DR

This work advances safety for robots operating in unknown or dynamic environments by constructing a time-varying soft-maximum barrier from the most recent perception-derived local barrier functions. A relaxed control barrier function is embedded into a closed-form, safe-and-optimal quadratic program, yielding a control law that guarantees forward invariance of a union-approximate safe set. The approach handles higher relative degree through a high-order CBF framework and uses a smooth temporal homotopy to seamlessly blend old and new perception data. Demonstrations on a ground robot and a quadrotor show safe navigation with online perception, resilience to dynamic obstacles, and performance close to maps-based baselines.

Abstract

We present a closed-form optimal feedback control method that ensures safety in an a prior unknown and potentially dynamic environment. This article considers the scenario where local perception data (e.g., LiDAR) is obtained periodically, and this data can be used to construct a local control barrier function (CBF) that models a local set that is safe for a period of time into the future. Then, we use a smooth time-varying soft-maximum function to compose the N most recently obtained local CBFs into a single barrier function that models an approximate union of the N most recently obtained local sets. This composite barrier function is used in a constrained quadratic optimization, which is solved in closed form to obtain a safe-and-optimal feedback control. We also apply the time-varying soft-maximum barrier function control to 2 robotic systems (nonholonomic ground robot with nonnegligible inertia, and quadrotor robot), where the objective is to navigate an a priori unknown environment safely and reach a target destination. In these applications, we present a simple approach to generate local CBFs from periodically obtained perception data.

Figures (10)

• Figure 1: $\eta$ given by Example \ref{['ex:g']} with $r=2$.
• Figure 2: ${\mathcal{S}}_k$ that approximates the intersection of the zero-superlevel sets of $\xi_k$ and $\sigma_{1,k},\ldots,\sigma_{\ell_k,k}$.
• Figure 3: Three closed-loop trajectories using the control \ref{['eq:softmax_h', 'eq:HOCBF', 'eq:uclose', 'eq:ulambda', 'eq:omegabar', 'eq:dxt']} with the perception feedback $b_k$ generated from $360^{\circ}$ FOV perception in a static environment.
• Figure 4: $q_{\rm x}$, $q_{\rm y}$, $v$, $\theta$, $u_{\rm d}$, and $u$ for $q_{\rm g} = [\,13\quad5\,]^{\rm T}$ m.
• Figure 5: $\psi_0$, $\psi_1$, and $\mu$ for $q_{\rm g} = [\,13\quad5\,]^{\rm T}$ m.

Theorems & Definitions (6)

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