AbstractThe problem of robot motion planning in an environment with obstacles can often be reduced to the study of connectivity of the robot's free configuration space. In turn, space connectivity can be analysed via the correspondingconnectivity graph. For two‐degree‐of‐freedom robots, the free configuration space presents atwo‐dimensional(2D)surface—a compact subspace of a 2D orientable compact manifold. This paper addresses the following abstract problem: given a compact 2D surface bounded by simple closed curves and lying in an orientable 2D manifold (a sphere, a torus, etc.) and given two points in the subspace, suggest a systematic way of defining the connectivity graph in the subspace, based on its topological properties. The use of space topology results in powerful, from the robotics standpoint, provable algorithms capable of on‐line motion planning in an environment with unknown obstacles of arbitrary shapes. This makes the method distinct from other techniques, which require complete information, algebraic representation of space geometry, and off‐lin