Solving the Pattern Formation by Mobile Robots With Chirality

Among fundamental problems in the context of distributed computing by mobile robots, the <italic>Pattern Formation</italic> (PF) is certainly the most representative. Given a multi-set <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> of points in the Euclidean plane and a set <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> of robots such that <inline-formula> <tex-math notation="LaTeX">$|R|=|F|$ </tex-math></inline-formula>, PF asks for a distributed algorithm that moves robots so as to reach a configuration similar to <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>. Similarity means that robots must be disposed as <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> regardless of translations, rotations, reflections, uniform scalings. In the literature, PF has been approached by assuming asynchronous robots endowed with chirality, i.e. a common handedness. The proposed algorithm along with its correctness proof turned out to be flawed. In this paper, we propose a new algorithm on the basis of a recent methodology studied for approaching problems in the context of distributed computing by mobile robots. According to this methodology, the correctness proof results to be well-structured and less prone to faulty arguments. We then ultimately characterize PF when chirality is assumed.