  Over the last 20 years significant effort has been dedicated to the
development of sampling-based motion planning algorithms such as the
Rapidly-exploring Random Trees (RRT) and its asymptotically optimal version
(e.g. RRT*). However, asymptotic optimality for RRT* only holds for linear and
fully actuated systems or for a small number of non-linear systems (e.g.
Dubin's car) for which a steering function is available. The purpose of this
paper is to show that asymptotically optimal motion planning for dynamical
systems with differential constraints can be achieved without the use of a
steering function. We develop a novel analysis on sampling-based planning
algorithms that sample the control space. This analysis demonstrated that
asymptotically optimal path planning for any Lipschitz continuous dynamical
system can be achieved by sampling the control space directly. We also
determine theoretical bounds on the convergence rates for this class of
algorithms. As the number of iterations increases, the trajectory generated by
these algorithms, approaches the optimal control trajectory, with probability
one. Simulation results are promising.
