  Integrated task and motion planning problems describe a multi-modal state
space, which is often abstracted as a set of smooth manifolds that are
connected via sets of transitions states. One approach to solving such problems
is to sample reachable states in each of the manifolds, while simultaneously
sampling transition states. Prior work has shown that in order to achieve
asymptotically optimal (AO) solutions for such piecewise-smooth task planning
problems, it is sufficient to double the connection radius required for AO
sampling-based motion planning. This was shown under the assumption that the
transition sets themselves are smooth. The current work builds upon this result
and demonstrates that it is sufficient to use the same connection radius as for
standard AO motion planning. Furthermore, the current work studies the case
that the transition sets are non-smooth boundary points of the valid state
space, which is frequently the case in practice, such as when a gripper grasps
an object. This paper generalizes the notion of clearance that is typically
assumed in motion and task planning to include such individual, potentially
non-smooth transition states. It is shown that asymptotic optimality is
retained under this generalized regime.
