Patrolling and collision avoidance beyond classical Navigation Functions

In this paper, we consider the problem of navigating a single unicycle-like robot, while avoiding obstacles in a known environment, and, at the same time, of steering the agent itself to monitor and patrol an assigned path. To this end, we propose a novel framework that combines tools and algorithms borrowed from algebraic geometry with techniques inspired by those associated with classical navigation functions. The former aspect permits the systematic construction of Lyapunov functions that certify the convergence with an assignable decaying rate to the desired patrolling path in the absence of obstacles. This control action is then combined with an additional term and a supervisory logic obtained by relying on the collision avoiding abilities of the underlying navigation function. Such a mixed strategy may potentially lead beyond the current understanding and implementation of classical navigation functions. The paper is then concluded by several numerical simulations that corroborate the theoretical results.