This paper studies fault tolerant algorithms for the problem of gathering N au-tonomous mobile robots. A gathering algorithm, executed independently by each robot, must ensure that all robots are gathered at one point within nite time. In a failure-prone system, a gathering algorithm is required to successfully gather the nonfaulty robots, independently of the behavior of the faulty ones. Both crash and Byzantine faults are considered. It is rst observed that most existing algorithms fail to operate correctly in a setting allowing crash failures. Subsequently, an algorithm tolerant against one crash-faulty robot in a system of three or more robots is presented. It is then observed that all known algorithms fail to operate correctly in a system prone to Byzantine faults, even in the presence of a single fault. Moreover, it is shown that in an asynchronous environment it is impossible to perform a successful gathering in a 3-robot system, even if at most one of them might fail in a Byzantine manner. Thus, the problem is studied in a fully synchronous system. An algorithm is provided in this model for gathering N 3 robots with at most a single faulty robot, and a more general gathering algorithm is given in an N-robot system with up to f faults, where N 3f +1.

Gathering asynchronous oblivious mobile robots in a ring ∗

This paper considers the convergence problem in autonomous mobile robot systems. A natural algorithm for the problem requires the robots to move towards their center of gravity. Previously it was known that the algorithm converges in the synchronous or semi-synchronous model, and that two robots converge in the asynchronous model. The current paper proves the correctness of the gravitational algorithm in the fully asynchronous model for any number of robots. It also analyses its con-vergence rate, and establishes its convergence in the presence of crash faults.

Anumber of recent studies concern algorithms for distributed control and coordination in systems of autonomous mobile robots. The common theoretical model adopted in these studies assumes that the positional input of the robots is obtained by perfectly accurate visual sensors, that robot movements are accurate, and that internal calculations performed by the robots on (real) coordinates are perfectly accurate as well. The current paper concentrates on the e ect of weakening this rather strong set of assumptions, and replacing it with the more realistic assumption that the robot sensors, movement and internal calculations may have slight inaccuracies. Speci cally, the paper concentrates on the ability of robot systems with inaccurate sensors, movements and calculations to carry out the task of convergence. The paper presents several impossibility theorems, limiting the inaccuracy allowing convergence, and prohibiting a general algorithm for gathering, namely, meeting at a point, in a nite number of steps. The main positive result is an algorithm for convergence under bounded measurement, movement and calculation errors.

Abstract. We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimentional plane, but (with one exception) has not been investigated before. We indeed show that, although the lack of these capabilities renders the problems considerably more difficult, ring exploration is still possible. We show that the minimum number ρ(n) of robots that can explore a ring of size n is O(log n) and that ρ(n) = Ω(log n) for arbitrarily large n. On one hand we give an algorithm that explores the ring starting from any initial configuration, provided that n and k are co-prime, and we show that there always exist such k in O(log n). On the other hand we show that Ω(log n) agents are necessary for arbitrarily large n. Notice that, when k and n are not co-prime, the problem is sometimes unsolvable (i.e., there are initial configurations for which the exploration cannot be done). This is the case, e.g., when k divides n.

Abstract. In the effort to understand the algorithmic limitations of computing by a swarm of robots, the research has focused on the minimal capabilities that allow a problem to be solved. The weakest of the commonly used models is Asynch where the autonomous mobile robots, endowed with visibility sensors (but otherwise unable to communicate), operate in Look-Compute-Move cycles performed asynchronously for each robot. The robots are often assumed (or required to be) oblivious: they keep no memory of observations and computations made in previous cycles. We consider the setting when the robots are dispersed in an anonymous and unlabeled graph, and they must perform the very basic task of exploration: within finite time every node must be visited by at least one robot and the robots must enter a quiescent state. The complexity measure of a solution is the number of robots used to perform the task. We study the case when the graph is an arbitrary tree and establish some unexpected results. We first prove that there are n-node trees where Ω(n) robots are necessary; this holds even if the maximum degree is 4. On the other hand, we show that if the maximum degree is 3, it is log n possible to explore with only O ( ) robots. The proof of the result is log log n constructive. Finally, we prove that the size of the team is asymptotically optimal: we show that there are trees of degree 3 whose exploration log n requires Ω ( ) robots. log log n

Abstract. Asetofk mobile agents with distinct identifiers and located in nodes of an unknown anonymous connected network, have to meet at some node. We show that this gathering problem is no harder than its special case for k = 2, called the rendezvous problem, and design deterministic protocols solving the rendezvous problem with arbitrary startups in rings and in general networks. The measure of performance is the number of steps since the startup of the last agent until the rendezvous is achieved. For rings we design an oblivious protocol with cost O(nlog ℓ), wheren is the size of the network and ℓ is the minimum label of participating agents. This result is asymptotically optimal due to the lower bound showed in [18]. For general networks we show a protocol with cost polynomial in n and logℓ, independent of the maximum difference τ of startup times, which answers in affirmative the open question from [22].

Abstract. This paper studies local algorithms for autonomous robot systems, namely, algorithms that use only information of the positions of a bounded number of their nearest neighbors. The paper focuses on the spreading problem. It defines measures for the quality of spreading, presents a local algorithm for the one-dimensional spreading problem, prove its convergence to the equally spaced configuration and discusses its convergence rate in the synchronous and semi-synchronous settings. It then presents a local algorithm achieving the exact equally spaced configuration in finite time in the synchronous setting, and proves it is time optimal for local algorithms. Finally, the paper also proposes an algorithm for the two-dimensional case and presents simulation results of its effectiveness.

Summary. The distributed coordination and control of a team of autonomous mobile robots is a problem widely studied in a variety of fields, such as engineering, artificial intelligence, artificial life, robotics. Generally, in these areas, the problem is studied mostly from an empirical point of view. Recently, a significant research effort has been and continues to be spent on understanding the fundamental algorithmic limitations on what a set of autonomous mobile robots can achieve. In particular, the focus is to identify the minimal robot capabilities (sensorial, motorial, computational) that allow a problem to be solvable and a task to be performed. In this paper we describe the current investigations on the interplay between robots capabilities, computability, and algorithmic solutions of coordination problems by autonomous mobile robots. 1

Reaching agreement among a set of mobile robots is one of the most fundamental issues in distributed robotic systems. This problem is often illustrated by the gathering problem, where the robots must self-organize to eventually meet at some arbitrary location. That problem has the advantage that, while being very simple to express, it retains the inherent difficulty of agreement, namely the problem of breaking symmetry. In their fully asynchronous model with oblivious robots and limited visibility, Flocchini et al. [7] show that gathering is solvable, as long as the robots share the knowledge of some direction, as provided by a compass. It turns out that, in robotic systems, compasses are devices that are often subject to instabilities. In this paper, we thus define a model with unreliable compasses and, focusing on the gathering problem, show that the algorithm of Flocchini et al. is unable to tolerate unstable compasses. We then give a gathering algorithm that solves the problem in a system where compasses are unstable for some arbitrary long periods, provided that they stabilize eventually.