  Motion planning under differential constraints is a classic problem in
robotics. To date, the state of the art is represented by sampling-based
techniques, with the Rapidly-exploring Random Tree algorithm as a leading
example. Yet, the problem is still open in many aspects, including guarantees
on the quality of the obtained solution. In this paper we provide a thorough
theoretical framework to assess optimality guarantees of sampling-based
algorithms for planning under differential constraints. We exploit this
framework to design and analyze two novel sampling-based algorithms that are
guaranteed to converge, as the number of samples increases, to an optimal
solution (namely, the Differential Probabilistic RoadMap algorithm and the
Differential Fast Marching Tree algorithm). Our focus is on driftless
control-affine dynamical models, which accurately model a large class of
robotic systems. In this paper we use the notion of convergence in probability
(as opposed to convergence almost surely): the extra mathematical flexibility
of this approach yields convergence rate bounds - a first in the field of
optimal sampling-based motion planning under differential constraints.
Numerical experiments corroborating our theoretical results are presented and
discussed.
