Continuous Time Gathering of Agents with Limited Visibility and Bearing-Only Sensing

Abstract: A group of mobile agents, identical, anonymous, and oblivious (memoryless), having the capability to sense only the relative direction (bearing) to neighborhing agents within a finite visibility range, are shown to gather to a meeting point in finite time by applying a very simple rule of motion. The agents' rule of motion is : set your velocity vector to be the sum of the two unit vectors in R^2 pointing to your "extremal" neighbours determining the smallest visibility disc sector in which all your visible neighbors reside, provided it spans an angle smaller than pi, otherwise, since you are "surrounded" by visible neighbors, simply stay put (set your velocity to 0). Of course, the initial constellation of agents must have a visibility graph that is connected, and provided this we prove that the agents gather to a common meeting point in finite time, while the distances between agents that initially see each other monotically decreases. We will also prove a geometrical result, a tight lower bound on the sum of cosines of the interior angles of a convex polygon, that we will use to prove the gathering of our dynamical system.