Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

TL;DR

This paper introduces a novel camera-based system for robot identification and tracking using unique colour sequences, modeled through de Bruijn graph cycles, with mathematical solutions for optimal cycle partitioning.

Contribution

It formulates robot identification as a combinatorial problem involving de Bruijn graph cycles and provides new existence results and constructions for maximum cycle partitions.

Findings

Maximum number of robots identified depends on colours, lights, and visibility.

Provides optimal cycle partition solutions for specific parameters.

Uses finite field algebra and combinatorics to construct cycle partitions.

Abstract

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.