We present a general framework for motion planning of robots in the presence of obstacles and other robots. We use variational calculus and optimization to find optimal open loop and closed loop plans in the presence of uncertainty. The plans are based on world models with set-valued uncertainty associated with the positions and shape of the obstacles. The open loop plans are generated by an efficient method that allows successive refinements of a nominal motion plan and accommodates finer levels of granularity as additional information becomes available. The closed loop plans are control policies that are based on given sensor models. The optimal plan is a control policy that performs the best in the worst-case situation. We discuss how the open loop and closed loop plans can be viewed as optimal strategies in the framework of two-person, zero-sum, non-cooperative games.