  We introduce and study the proper topological complexity of a given
configuration space, a version of the classical invariant for which we require
that the algorithm controlling the motion is able to avoid any possible choice
of ``unsafe'' area. To make it a homotopy functorial invariant we characterize
it as a particular instance of the exterior sectional category of an exterior
map, an invariant of the exterior homotopy category which is also deeply
analyzed.
