14.2.1 Reachable Sets

For the algorithms in Chapter 5, resolution completeness and probabilistic completeness rely on having a sampling sequence that is dense on ${\cal C}$. In the present setting, this would require dense sampling on $X$. Differential constraints, however, substantially complicate the sampling process. It is generally not reasonable to prescribe precise samples in $X$ that must be reached because reaching them may be impossible or require solving a BVP. Since paths in $X$ are obtained indirectly via action trajectories, completeness analysis begins with considering which points can be reached by integrating action trajectories.