The Gathering Problem for Two Oblivious Robots with Unreliable Compasses

Abstract

Anonymous mobile robots are often classified into synchronous, semi-synchronous and asynchronous robots when discussing the pattern formation problem. For semi-synchronous robots, all patterns formable with memory are also formable without memory, with the single exception of forming a point (i.e., the gathering) by two robots. (All patterns formable with memory are formable without memory for synchronous robots, and little is known for asynchronous robots.) However, the gathering problem for two semi-synchronous robots without memory (called oblivious robots in this paper) is trivially solvable when their local coordinate systems are consistent, and the impossibility proof essentially uses the inconsistencies in their coordinate systems. Motivated by this, this paper investigates the magnitude of consistency between the local coordinate systems necessary and sufficient to solve the gathering problem for two oblivious robots under semi-synchronous and asynchronous models. To discuss the magnitude of consistency, we assume that each robot is equipped with an unreliable compass, the bearings of which may deviate from an absolute reference direction, and that the local coordinate system of each robot is determined by its compass. We consider two families of unreliable compasses, namely, static compasses with (possibly incorrect) constant bearings, and dynamic compasses the bearings of which can change arbitrarily (immediately before a new look-compute-move cycle starts and after the last cycle ends). For each of the combinations of robot and compass models, we establish the condition on deviation φ that allows an algorithm to solve the gathering problem, where the deviation is measured by the largest angle formed between the x-axis of a compass and the reference direction of the global coordinate system: φ < π/2 for semi-synchronous and asynchronous robots with static compasses, φ < π/4 for semi-synchronous robots with dynamic compasses, and φ < π/6 for asynchronous robots with dynamiccompasses. Except for asynchronous robots with dynamic compasses, these sufficient conditions are also necessary.

When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots

“…Research on this problem was initiated at the MITACS "International Problem Solving Workshop" held on July [16][17][18][19][20] 2012, in Vancouver, BC, Canada. The authors would like to express their deepest appreciation for the generous support of MITACS.…”

“…[8], flocking, e.g., [30] and many other ones. Several papers, e.g., [9,20,29] concerned unreliable or inaccurate robot sensing devices, rather than the robots themselves. Experimental papers related to unreliable robots performing patrolling were considered in the robotics literature [14,15,18,24].…”

Distributed Computing by Mobile Robots: Gathering

“…With respect to the level of global agreement on the local coordinate systems, different assumptions are made, ranging from availability of a global positioning system [31,32,35,40,43], to the agreement on the direction and orientation of both axes but not on the unit of distance nor the origin (e.g., as provided by a compass) [26,27], to partially accurate agreement on the direction and orientation (e.g., inaccurate compass) [34,36,49], to the absence of any relationship among the local coordinate systems of different robots [6,27,52], i.e., when the robots are disoriented.…”

Autonomous Mobile Robots: A Distributed Computing Perspective

“…The Rendezvous has been extensively studied by assuming different level of agreement on the compass systems of the robots. In particular, the problem is solvable in Async when the robots agree on chirality, but the axis are however tilted up to a φ < π 2 degrees [37], and the tilt is fixed. If the robots still agree on chirality, but the tilt of their compasses might be variable, rendezvous can be achieved in Ssync with fully variable compasses if and only if φ < Gathering and Convergence.…”