  We present a theoretical analysis of a recent whole body motion planning
method, the Randomized Possibility Graph, which uses a high-level decomposition
of the feasibility constraint manifold in order to rapidly find routes that may
lead to a solution. These routes are then examined by lower-level planners to
determine feasibility. In this paper, we show that this approach is
probabilistically complete for bipedal robots performing quasi-static walking
in "semi-unstructured" environments. Furthermore, we show that the
decomposition into higher and lower level planners allows for a considerably
higher rate of convergence in the probability of finding a solution when one
exists. We illustrate this improved convergence with a series of simulated
scenarios.
