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,ℓ) 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.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic