Partitioning de Bruijn graphs into fixed-length cycles for robot identification and tracking

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, ℓ. 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 dB(q,ℓ).We provide several existence results that give the maximum number of cycles in dB(q,ℓ) in various cases. For example, we give an optimal solution when k=qℓ−1. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if dB(q,ℓ) can be partitioned into k-cycles, then dB(q,tℓ) 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.