Loop topological complexity

We introduce here the notion of loop motion planning algorithms and show that it yields to a homotopical invariant: the loop topological complexity, denoted throughout this paper by $rm{TC}^{rm{LP}}(-)$, which measures the algorithmic complexity of the motion of a drone as, for example, an unmanned airplane or a guided TV camera. Our main result states that $rm{TC}(-) = rm{TC}^{rm{LP}}(-)$, where $rm{TC}$ denotes the ordinary topological complexity introduced by M. Farber. Some interesting applications will emerge and will be discussed.