# Quantifying Robotic Swarm Coverage

## TL;DR

This paper introduces a new, continuous error metric for evaluating coverage in swarm robotics, along with benchmarks and theoretical analysis, to improve performance assessment of control algorithms.

## Contribution

It presents a novel error metric for coverage evaluation, along with two benchmarks and theoretical insights, enhancing performance analysis in swarm robotics.

## Key findings

- The error metric is continuously sensitive to swarm distribution changes.
- Two benchmarks enable comparison of observed coverage to theoretical and random distributions.
- The error metric obeys a central limit theorem, facilitating statistical analysis.

## Abstract

In the field of swarm robotics, the design and implementation of spatial density control laws has received much attention, with less emphasis being placed on performance evaluation. This work fills that gap by introducing an error metric that provides a quantitative measure of coverage for use with any control scheme. The proposed error metric is continuously sensitive to changes in the swarm distribution, unlike commonly used discretization methods. We analyze the theoretical and computational properties of the error metric and propose two benchmarks to which error metric values can be compared. The first uses the realizable extrema of the error metric to compute the relative error of an observed swarm distribution. We also show that the error metric extrema can be used to help choose the swarm size and effective radius of each robot required to achieve a desired level of coverage. The second benchmark compares the observed distribution of error metric values to the probability density function of the error metric when robot positions are randomly sampled from the target distribution. We demonstrate the utility of this benchmark in assessing the performance of stochastic control algorithms. We prove that the error metric obeys a central limit theorem, develop a streamlined method for performing computations, and place the standard statistical tests used here on a firm theoretical footing. We provide rigorous theoretical development, computational methodologies, numerical examples, and MATLAB code for both benchmarks.