Handling obstacles

Now suppose that ${X_{obs}}\not = \emptyset$. In Section 5.5.1, the RDT was extended until a stopping configuration ${q}_s$ was reached, just in front of an obstacle. There are two new complications under differential constraints. The first is that motion primitives are used. If $\Delta t$ is small, then in many cases the time will expire before the boundary is reached. This can be alleviated by using a large $\Delta t$ and then taking only the violation-free portion of the trajectory. In this case, the trajectory may even be clipped early to avoid overshooting ${\alpha }(i)$. The second complication is due to ${X_{ric}}$. If momentum is substantial, then pulling the tree as close as possible to obstacles will increase the likelihood that the RDT becomes trapped. Vertices close to obstacles will be selected often because they have large Voronoi regions, but expansion is not possible. In the case of the Piano Mover's Problem, this was much less significant because the tree could easily follow along the boundary. In most experimental work, it therefore seems best to travel only part of the way (perhaps half) to the boundary.