Rational topological complexity

Motion planning is a rapidly growing area of research in robotics where topological methods have been applied [14]. From the topological point of view, a motion planning algorithm has as input two states of the motion, i.e., two arbitrary points in the configuration space X . As output it has to provide a continuous path between the two chosen states. In other words, such an algorithm consists of a section (not necessarily continuous) s : X × X −→ XI of the unpointed path fibration p : XI −→ X × X, p(σ) = ( σ(0),σ(1) ) , where XI denotes the space of free paths on X .