Motion planning for a quantized control system on SO(3)

We present an analysis of the motion planning problem for a driftless quantized control system on SO(3). The system under investigation depends on two control quanta, beta₁ and beta₂. We are able to provide efficient finite complexity motion plans which steer the system arbitrarily close to any given target configuration as soon as at least one of the two control quanta satisfies a strong approximating property of number theoretic nature. On the other hand we prove that whenever both control quanta do not satisfy this strong approximating property, then in order to reach a configuration arbitrarily close to any given target configuration it is necessary to use motion plans of unbounded complexity.