Influence of Stationary Robots on Continuous Robot Formation Problems

We consider a group of $n$ autonomous mobile robots of which $m$ are stationary thus cannot move. Robots are represented by points in the Euclidean plane. They have no memory, do not communicate or share a common coordinate system and they move solely based on the positioning of other robots within their limited viewing range of 1. The goal is to gather the robots inside of the convex hull of all stationary robots. A variant of this problem, the general gathering problem, has been studied in various different time models. In this work, we consider a continuous time model, where robots continuously observe their neighbors, compute the next target of movement and move with a speed limit of 1 at any time. Regarding the robots' local strategy, we only study contracting algorithms in which every robot that is positioned on the border of the convex hull of all robots moves into this hull. We present a time bound of $\mathcal{O}(nd)$ for any general contracting algorithms in a configuration with only a single stationary robot. For configurations with more stationary robots, we prove that robots converge against the convex hull of all stationary robots and that no upper bound on the runtime exists. For the specific contracting algorithms Go-To-The-Left, Go-On-Bisector and Go-To-The-Middle, we provide linear time bounds.