Continuous time gathering of agents with limited visibility and bearing-only sensing

A group of mobile agents, identical, anonymous, and oblivious (memoryless), able to sense only the direction (bearing) to neighboring 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 act in continuous time, and their rule of motion is as follows: they determine the smallest visibility disk sector in which all their visible neighbors reside. If this disk sector spans an angle smaller than \(\pi \), then they set the velocity vector to be the sum of the two unit vectors in \({\mathbb {R}}^2\) pointing to the extremal neighbors. Otherwise, they do not move. If the initial constellation of agents has a visibility graph that is connected, we prove that the agents gather to a common meeting point in finite time.