Compatibility of convergence algorithms for autonomous mobile robots

TL;DR

This paper analyzes the compatibility of convergence algorithms for autonomous mobile robots, classifying various problems based on their ability to always reduce the convex hull under crash failures.

Contribution

It provides a classification of convergence and fault-tolerant problems for mobile robots based on compatibility of target functions under crash failures.

Findings

Convergence, FC(1), FC(1)-PO, and FC(f)-CP form the first compatible group.

Gathering and FC(f)-PO for f ≥ 2 are incompatible with convex hull shrinking.

FC(f) for f ≥ 2 occupies an intermediate position in the compatibility classification.

Abstract

We investigate autonomous mobile robots in the Euclidean plane. A robot has a function called target function to decide the destination from the robots' positions. Robots may have different target functions. If the robots whose target functions are chosen from a set $\Phi$ of target functions always solve a problem $\Pi$, we say that $\Phi$ is compatible with respect to $\Pi$. If $\Phi$ is compatible with respect to $\Pi$, every target function $\phi \in \Phi$ is an algorithm for $\Pi$. Even if both $\phi$ and $\phi'$ are algorithms for $\Pi$, $\{ \phi, \phi' \}$ may not be compatible with respect to $\Pi$. From the view point of compatibility, we investigate the convergence, the fault tolerant ($n,f$)-convergence (FC($f$)), the fault tolerant ($n,f$)-convergence to $f$ points (FC($f$)-PO), the fault tolerant ($n,f$)-convergence to a convex $f$-gon (FC($f$)-CP), and the gathering problems, assuming crash failures. Obtained results classify these problems into three groups: The convergence, FC(1), FC(1)-PO, and FC($f$)-CP compose the first group: Every set of target functions which always shrink the convex hull of a configuration is compatible. The second group is composed of the gathering and FC($f$)-PO for $f \geq 2$: No set of target functions which always shrink the convex hull of a configuration is compatible. The third group, FC($f$) for $f \geq 2$, is placed in between. Thus, FC(1) and FC(2), FC(1)-PO and FC(2)-PO, and FC(2) and FC(2)-PO are respectively in different groups, despite that FC(1) and FC(1)-PO are in the first group.