14.2 Reachability and Completeness

This section provides preliminary concepts for sampling-based planning algorithms. In Chapter 5, sampling over ${\cal C}$ was of fundamental importance. The most important consideration was that a sequence of samples should be dense so that samples get arbitrarily close to any point in ${\cal C}_{free}$. Planning under differential constraints is complicated by the specification of solutions by an action trajectory instead of a path through ${X_{free}}$. For sampling-based algorithms to be resolution complete, sampling and searching performed on the space of action trajectories must somehow lead to a dense set in ${X_{free}}$.