Parametrized Motion Planning and Topological Complexity

Michael Farber; Shmuel Weinberger

doi:10.1007/978-3-031-21090-7_1

Parametrized Motion Planning and Topological Complexity

Michael Farber^*, Shmuel Weinberger

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing the recent paper [2], we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles in R^d. This algorithm is optimal, it has minimal possible topological complexity for any d≥ 3 odd. Besides, we describe a modification of this algorithm which is optimal for d≥ 2 even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes.

Original language	English
Title of host publication	Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics
Editors	Steven M. LaValle, Jason M. O’Kane, Michael Otte, Dorsa Sadigh, Pratap Tokekar
Publisher	Springer Nature
Pages	1-17
Number of pages	17
ISBN (Print)	9783031210891
DOIs	https://doi.org/10.1007/978-3-031-21090-7_1
State	Published - 2023
Externally published	Yes
Event	15th Workshop on the Algorithmic Foundations of Robotics, WAFR 2022 - College Park, United States Duration: 22 Jun 2022 → 24 Jun 2022

Publication series

Name	Springer Proceedings in Advanced Robotics
Volume	25 SPAR
ISSN (Print)	2511-1256
ISSN (Electronic)	2511-1264

Conference

Conference	15th Workshop on the Algorithmic Foundations of Robotics, WAFR 2022
Country/Territory	United States
City	College Park
Period	22/06/22 → 24/06/22

Keywords

Collision free motion of swarms of robots
Motion planning algorithm
Parametrized topological complexity
Stiefel – Whitney characteristic classes
Topological complexity

Access to Document

10.1007/978-3-031-21090-7_1

Cite this

Farber, M., & Weinberger, S. (2023). Parametrized Motion Planning and Topological Complexity. In S. M. LaValle, J. M. O’Kane, M. Otte, D. Sadigh, & P. Tokekar (Eds.), Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics (pp. 1-17). (Springer Proceedings in Advanced Robotics; Vol. 25 SPAR). Springer Nature. https://doi.org/10.1007/978-3-031-21090-7_1

Farber, Michael ; Weinberger, Shmuel. / Parametrized Motion Planning and Topological Complexity. Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics. editor / Steven M. LaValle ; Jason M. O’Kane ; Michael Otte ; Dorsa Sadigh ; Pratap Tokekar. Springer Nature, 2023. pp. 1-17 (Springer Proceedings in Advanced Robotics).

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Farber, M & Weinberger, S 2023, Parametrized Motion Planning and Topological Complexity. in SM LaValle, JM O’Kane, M Otte, D Sadigh & P Tokekar (eds), Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics. Springer Proceedings in Advanced Robotics, vol. 25 SPAR, Springer Nature, pp. 1-17, 15th Workshop on the Algorithmic Foundations of Robotics, WAFR 2022, College Park, United States, 22/06/22. https://doi.org/10.1007/978-3-031-21090-7_1

Parametrized Motion Planning and Topological Complexity. / Farber, Michael; Weinberger, Shmuel.

Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics. ed. / Steven M. LaValle; Jason M. O’Kane; Michael Otte; Dorsa Sadigh; Pratap Tokekar. Springer Nature, 2023. p. 1-17 (Springer Proceedings in Advanced Robotics; Vol. 25 SPAR).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Farber M, Weinberger S. Parametrized Motion Planning and Topological Complexity. In LaValle SM, O’Kane JM, Otte M, Sadigh D, Tokekar P, editors, Algorithmic Foundations of Robotics XV - Proceedings of the Fifteenth Workshop on the Algorithmic Foundations of Robotics. Springer Nature. 2023. p. 1-17. (Springer Proceedings in Advanced Robotics). doi: 10.1007/978-3-031-21090-7_1