  In this paper we study a notion of topological complexity for the motion
planning problem. The topological complexity is a number which measures
discontinuity of the process of motion planning in the configuration space X.
More precisely, it is the minimal number k such that there are k different
motion planning rules, each defined on an open subset of XxX, so that each rule
is continuous in the source and target configurations. We use methods of
algebraic topology (the Lusternik - Schnirelman theory) to study the
topological complexity . We give an upper bound (in terms of the dimension of
the configuration space X) and also a lower bound (in the terms of the
structure of the cohomology algebra of X). We explicitly compute the
topological complexity of motion planning for a number of configuration spaces:
for spheres, two-dimensional surfaces, for products of spheres. In particular,
we completely calculate the topological complexity of the problem of motion
planning for a robot arm in the absence of obstacles.
