1. INTRODUCTION
In general, motion planning with uncertainty is to design a series of inputs to move the system from the initial region to the goal region with constraints. Motion planning with uncertainty is a critical problem in robotics. Erdmann [Erdmann, 1986], [Erdmann 1988] and LMT [Lozano-Perez, Mason and Taylor, 1983] applied back-projection and pre-image in robotics. The concept of back-chaining is nice and gives the essential condition of strategy formulation. However, due to dependence between action command and back-chaining region, the difficulty of implementation of the concept is clear [Lazanas and Latombe, 1995].