The Gathering Problem for Two Oblivious Robots with Unreliable Compasses

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. However, the gathering problem for two semi-synchronous robots without memory 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 constant bearings, and dynamic compasses the bearings of which can change arbitrarily. For each of the combinations of robot and compass models, we establish the condition on deviation \phi 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: \phi < \pi/2 for semi-synchronous and asynchronous robots with static compasses, \phi < \pi/4 for semi-synchronous robots with dynamic compasses, and \phi < \pi/6 for asynchronous robots with dynamic compasses. Except for asynchronous robots with dynamic compasses, these sufficient conditions are also necessary.

💡 Research Summary

This paper investigates the gathering problem for two oblivious robots under semi‑synchronous (SS and SD) and asynchronous (AS and AD) execution models, focusing on the impact of unreliable compasses. Each robot’s local coordinate system is defined by a compass whose bearing may deviate from the global reference direction by an angle φ. Two compass families are considered: static compasses, whose bearing is fixed for the whole execution, and dynamic compasses, whose bearing may change arbitrarily before each look‑compute‑move cycle.

The authors establish precise bounds on φ that are both necessary and sufficient for gathering in three of the four model‑compass combinations, and they provide a sufficient bound for the remaining case. The main results are:

• Semi‑synchronous robots with static compasses (SS): gathering is possible iff φ < π/2.
• Semi‑synchronous robots with dynamic compasses (SD): gathering is possible iff φ < π/4.
• Asynchronous robots with static compasses (AS): gathering is possible iff φ < π/2.
• Asynchronous robots with dynamic compasses (AD): gathering is possible if φ < π/6; the necessity of this bound remains an open problem.

The core algorithmic idea is a variant of the classic “move to the midpoint” strategy. In the static‑compass cases, even if the two local x‑axes differ by up to 90°, the midpoint computed in each robot’s frame corresponds to the same global point, guaranteeing convergence. When compasses are dynamic, the observed midpoint may be displaced because the bearing can change between cycles. For the semi‑synchronous dynamic case, the authors design a correction step that limits the displacement caused by a bearing change; they prove that as long as the maximum deviation is below π/4, the corrected midpoint remains within a contractive region, leading to eventual gathering.

In the fully asynchronous dynamic scenario, the difficulty is amplified: one robot may be moving while the other observes, so the relative position can be arbitrarily outdated. By carefully bounding the possible error introduced by a bearing change and limiting the step size, the authors show that if φ < π/6, the robots’ trajectories remain confined to a shrinking region and converge to a common point. The proof relies on fairness (both robots are activated infinitely often) and assumes that each move phase can cover at least a small positive distance δ.

The paper also formalizes the robot model: each robot executes an instantaneous look‑compute‑move cycle, during which its scaling factor and compass bearing remain constant. The scaling factor is arbitrary but positive, reflecting that robots may have different unit lengths. The authors assume no direct communication, anonymity, and no memory of past cycles (obliviousness).

Related work is surveyed, highlighting that the gathering problem for two oblivious semi‑synchronous robots is the only known case where memory makes a difference, and that prior results either required perfectly consistent compasses or dealt with eventual consistency. This work fills the gap by quantifying exactly how much inconsistency (measured by φ) can be tolerated.

In conclusion, the paper provides a comprehensive characterization of the minimal compass consistency required for two oblivious robots to gather under various synchrony assumptions. The results are tight for static compasses and for semi‑synchronous dynamic compasses; the asynchronous dynamic case remains partially open, inviting further research into tighter lower bounds. The findings have practical relevance for designing distributed robot algorithms that must operate with imperfect orientation sensors.