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Distributed Computing by Mobile Entities pp 109–133Cite as

Symmetry of Anonymous Robots

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11340))

Abstract

Symmetry of anonymous mobile robots imposes many impossibilities. We focus on the formation problem that requires the robots to form a target pattern. We consider the robots moving in the three-dimensional space and the two-dimensional space (3D and 2D space, respectively) and introduce the notion of symmetricity of a set of points that represents the set of rotation groups that the robots cannot resolve. However, the symmetricity does not always match the rotational symmetry of geometric positions of the robots. We demonstrate that the robots are capable of breaking symmetry by their movement in some cases. The goal of this chapter is to present the following characterization of formable patterns; anonymous synchronous mobile robots in 3D space or 2D space can form a target pattern from an initial configuration if and only if the symmetricity of an initial configuration is a subset of the symmetricity of the target pattern.

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Notes

  1. 1.

    The pattern formation problem for the robots with chirality does not allow any reflection of the target pattern by a mirror plane.

  2. 2.

    In Sect. 6, we discuss generalization of these symmetry operations to the robots without chirality, in which model the local coordinate system of a robot is either right-handed or left-handed.

  3. 3.

    We can recognize the robots in 2D space as those that agree on the “top” direction and move on a plane in 3D space.

  4. 4.

    The definition of symmetricity in [16, 19, 23] considers the maximum order of the cyclic groups in the symmetricity.

  5. 5.

    Corollary 1 is a rephrasing of the characterization in [24], where the authors compared the order of 3D rotation groups (i.e., 12, 24, and 60) with the size of each element of the \(\gamma (P)\)-decomposition of an initial configuration P in order to check the symmetricity of P.

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Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP18H03202.

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  1. Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan

    Yukiko Yamauchi

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  1. Yukiko Yamauchi
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Correspondence to Yukiko Yamauchi .

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Editors and Affiliations

  1. University of Ottawa, Ottawa, ON, Canada

    Paola Flocchini

  2. University of Pisa, Pisa, Italy

    Giuseppe Prencipe

  3. Carleton University, Ottawa, ON, Canada

    Nicola Santoro

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Yamauchi, Y. (2019). Symmetry of Anonymous Robots. In: Flocchini, P., Prencipe, G., Santoro, N. (eds) Distributed Computing by Mobile Entities. Lecture Notes in Computer Science(), vol 11340. Springer, Cham. https://doi.org/10.1007/978-3-030-11072-7_6

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  • DOI: https://doi.org/10.1007/978-3-030-11072-7_6

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