  Integrated task and motion planning has emerged as a challenging problem in
sequential decision making, where a robot needs to compute high-level strategy
and low-level motion plans for solving complex tasks. While high-level
strategies require decision making over longer time-horizons and scales, their
feasibility depends on low-level constraints based upon the geometries and
continuous dynamics of the environment. The hybrid nature of this problem makes
it difficult to scale; most existing approaches focus on deterministic, fully
observable scenarios. We present a new approach where the high-level decision
problem occurs in a stochastic setting and can be modeled as a Markov decision
process. In contrast to prior efforts, we show that complete MDP policies, or
contingent behaviors, can be computed effectively in an anytime fashion. Our
algorithm continuously improves the quality of the solution and is guaranteed
to be probabilistically complete. We evaluate the performance of our approach
on a challenging, realistic test problem: autonomous aircraft inspection. Our
results show that we can effectively compute consistent task and motion
policies for the most likely execution-time outcomes using only a fraction of
the computation required to develop the complete task and motion policy.
