  Sampling-based algorithms are viewed as practical solutions for
high-dimensional motion planning. Recent progress has taken advantage of random
geometric graph theory to show how asymptotic optimality can also be achieved
with these methods. Achieving this desirable property for systems with dynamics
requires solving a two-point boundary value problem (BVP) in the state space of
the underlying dynamical system. It is difficult, however, if not impractical,
to generate a BVP solver for a variety of important dynamical models of robots
or physically simulated ones. Thus, an open challenge was whether it was even
possible to achieve optimality guarantees when planning for systems without
access to a BVP solver. This work resolves the above question and describes how
to achieve asymptotic optimality for kinodynamic planning using incremental
sampling-based planners by introducing a new rigorous framework. Two new
methods, Stable Sparse-RRT (SST) and SST*, result from this analysis, which are
asymptotically near-optimal and optimal, respectively. The techniques are shown
to converge fast to high-quality paths, while they maintain only a sparse set
of samples, which makes them computationally efficient. The good performance of
the planners is confirmed by experimental results using dynamical systems
benchmarks, as well as physically simulated robots.
