# An opportunistic linear-convex algorithm for localization in mobile robot networks

## TL;DR

This paper introduces a distributed, opportunistic linear algorithm for localizing mobile robots in a network, which converges asymptotically and is robust to noise, especially with known beacon nodes.

## Contribution

It presents a novel linear-convex, distributed localization algorithm for mobile robot networks that operates opportunistically and guarantees convergence under certain conditions.

## Key findings

- Algorithm converges asymptotically to true robot locations.
- Performance is robust to measurement noise with proposed modifications.
- Requires at least one beacon for precise tracking of mobile networks.

## Abstract

In this paper, we develop a \textcolor{black}{\emph{distributed}} algorithm to localize a network of robots moving arbitrarily in a bounded region. In the case of such mobile networks, the main challenge is that the robots may not be able to find nearby robots to implement a distributed algorithm. We address this issue by providing an opportunistic algorithm that only implements a location update when there are nearby robots and does not update otherwise. We assume that each robot measures a noisy version of its motion and the distances to the nearby robots. To localize a network of mobile robots in~$\mathbb{R}^m$, we provide a simple \emph{linear} update, which is based on barycentric coordinates and is linear-convex. We abstract the corresponding localization algorithm as a Linear Time-Varying (LTV) system and show that it asymptotically converges to the true locations~of~the robots. We first focus on the noiseless case, where the distance and motion vectors are known (measured) perfectly, and provide sufficient conditions on the convergence of the algorithm. We then evaluate the performance of the algorithm in the presence of noise and provide modifications to counter the undesirable effects of noise. \textcolor{black}{We further show that our algorithm precisely tracks a mobile network as long as there is at least one known beacon (a node whose location is perfectly known).