### abstract 17

17
Loop topological complexity
Younes Derfoufi, My Ismail Mamouni

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.

Keywords: Motion planning algorithm, topological robotics, topological complexity,
loop topological complexity, monoidal topological complexity, Iwase-Sakai conjecture.

Cite this paper:
Derfoufi Y., Mamouni M.I., Loop topological complexity
Bull. Comput. Appl. Math. (Bull CompAMa),
Vol. 3, No. 2, Jul-Dec, pp.31-36, 2015