Abstract
We study the problem of determining minimum-length coordinated motions for two axis-aligned square robots translating in an obstacle-free plane: Given feasible start and goal configurations, find a continuous motion for the two squares from start to goal, comprising only robot-robot collision-free configurations, such that the total Euclidean distance traveled by the two squares is minimal among all possible such motions. We present an adaptation of the tools developed for the case of discs by Kirkpatrick and Liu [Characterizing minimum-length coordinated motions for two discs. Proceedings 28th CCCG, 252-259, 2016; CoRR abs/1607.04005, 2016.] to the case of squares. Certain aspects of the case of squares are more complicated, requiring additional and more involved arguments over the case of discs. Our contribution can serve as a basic component in optimizing the coordinated motion of two squares among obstacles, as well as for local planning in sampling-based algorithms, which are often used in practice, in the same setting.
Partially supported by project PID2019-104129GB-I00/MCIN/AEI/10.13039/ 501100011033 of the Spanish Ministry of Science and Innovation. Work by D.H. has been supported in part by the Israel Science Foundation (grant no. 1736/19), by NSF/US-Israel-BSF (grant no. 2019754), by the Israel Ministry of Science and Technology (grant no. 103129), by the Blavatnik Computer Science Research Fund, and by the Yandex Machine Learning Initiative for Machine Learning at Tel Aviv University.
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Notes
- 1.
Formally defined in Sect. 2; roughly, clockwise here refers to the direction of rotation of a vector from the center of one robot to the center of the other robot throughout the motion, from start to goal.
- 2.
One movement is translating a robot from one point in the plane to another location where it does not intersect any other robot by minimizing the distance traveled.
- 3.
If we think of a finite set of lines as graphs of (partially defined) functions, the lower envelope is the graph of the pointwise minimum [12].
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Esteban, G., Halperin, D., Ruíz, V., Sacristán, V., Silveira, R.I. (2023). Shortest Coordinated Motion for Square Robots. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_28
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