Network Connectivity Maintenance via Nonsmooth Control Barrier Functions

Abstract

This paper considers the connectivity maintenance problem for multi-robot systems and proposes a continuous optimization-based controller with Nonsmooth Control Barrier Functions to achieve it. The design is based on the concept of algebraic connectivity of the interaction graph. When viewed as a function of the network state, the algebraic connectivity is not continuously differentiable, a fact neglected in previous controller designs. To illustrate the importance of this observation, we present an example of a simple multi-robot system that fails to maintain connectivity under such controllers. The insights gained allow us to synthesize an optimization-based controller that prescribes that all the nontrivial eigenvalues of the Laplacian remain positive. Using tools from nonsmooth analysis and set-valued theory, we show that the proposed controller is continuous, thereby guaranteeing the existence of the robot trajectories for the closed-loop system and ensuring network connectivity is maintained along them.