On the targeted complexity of a map

Abstract

We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion planners for work maps. In this regard, we focus on the useful set of works. We check some basic properties of the targeted complexity of maps, such as homotopical invariance, reduction, the product of maps, and so on. Then we compare these targeted complexities, and we find some inequalities in reducing the number of motion planners. We show that the relative topological complexity of pair of spaces defined by Short is a special case of the targeted complexity of work maps.

Keywords

• Topological complexity
• work map
• configuration space
• motion planning.

Supporting Institution

Ferdowsi University of Mashhad

Project Number

3/57099

References

1. [1] I. Basabe, J. González, Y. Rudyak, and D. Tamaki, Higher topological complexity and its symmetrization, Algebraic Geom. Topol. 14 (4), 2103-2124, 2014.
2. [2] Z. Baszczyk and M. Kaluba, Effective topological complexity of spaces with symmetries, Publ. Mat. 62, 55-74, 2018.
3. [3] A. Borat and T. Vergili, Higher homotopic distance, Topol. Methods Nonlinear Anal. 57 (2), 525-534, 2021.
4. [4] Z. Doulgeri and L. Droukas, Robot task space pid type regulation with prescribed performance guaranties, In 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, 1644–1649, 2010.
5. [5] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2), 211-221, 2003.
6. [6] M. Is and I. Karaca, Higher topological complexity for fibrations, arXiv preprint arXiv:2107.04465, 2021.
7. [7] E. Mac’ias-Virg’os and D. Mosquera-Lois, Homotopic distance between maps, Mathematical Proceedings of the Cambridge Philosophical Society, 172, 73-93, 2021.
8. [8] A. Murillo and J. Wu, Topological complexity of the work map, J. Topol. Anal. 13 (1), 219-238, 2021.

9. [9] P. Pavešić, Complexity of the forward kinematic map, Mech. Mach. Theory 117, 230- 243, 2017.
10. [10] P. Pavešić, A topologists view of kinematic maps and manipulation complexity, Contemp. Math. 702, 61-83, 2018.
11. [11] Petar Pavešić, Topological complexity of a map, Homol. Homotopy Appl. 21 (2), 107- 130, 2019.
12. [12] Y. Rami and Y. Derfoufi, A variant of the topological comlexity of a map, arXiv preprint arXiv:1809.10174, 2018.
13. [13] Y. Rudyak and S. Sarkar, Maps of degree one, relative ls category and higher topological complexities, arXiv preprint arXiv:2011.13531, 2020.
14. [14] J. Scott, On the topological complexity of maps, arXiv preprint arXiv:2011.10646, 2020.
15. [15] R. Short, Relative topological complexity of a pair, Topol. Appl. 248, 7-23, 2018.