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The Artistic Geometry of Consensus Protocols

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Controls and Art

Abstract

A large class of control problems in multi-agent systems use the so-called consensus protocol to achieve coordinated motion among a team of agents. Inspired by the “standard” consensus protocol ẋ = -Lx, in this paper we propose a decentralized control law for multi-agent formations in two dimensions that allows the participating vehicles to display intricate periodic and quasi-periodic geometric patterns. Similarly to the standard consensus protocol, these controls rely only on the relative position between the networked agents which are neighbors in the underlying communication graph. Several examples are presented, resulting in nontrivial geometric patterns described by trochoidal curves, similar to those generated using a spirograph. These paths can be useful for coordinated, distributed surveillance, and monitoring applications, as well as for the sake of their own esthetic beauty.

Mighty is geometry; joined with art, resistless. Euripides (480–406 BC)

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Notes

  1. 1.

    Owing to the freedom in choosing \({\varvec{q}}_{12}\) and \({\varvec{q}}_{21}\), we define a “position orientation” such that \({\varvec{r}}_{12} \times {\varvec{q}}_{12} ={\varvec{r}}_{21} \times {\varvec{q}}_{21}\).

  2. 2.

    Not everyone is in agreement, of course, with the classical notion of beauty and symmetry. One can easily argue that nonsymmetric patterns may also be esthetically pleasing as long as they do not result in chaos [26]. Recent psychological studies of works of modern art actually claim that it is the cognitive processes themselves that are involved in understanding, classifying, and evaluating a work of art which determine positive, self-rewarding esthetic experiences [20]. Even so, good gestalts tend to give preference to symmetry over nonsymmetry [13].

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Acknowledgments

The authors would like to thank Magnus Egerstedt and Amy LaViers for organizing the session on “Controls and Art” during the 2011 American Control Conference and for inviting the authors of this paper to contribute to the session. The authors would also like to thank Prof. Athanassios Economou from the College of Architecture at Georgia Tech for several informative discussions over the use of group symmetries in art and architecture.

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Authors and Affiliations

  1. Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0150, USA

    Panagiotis Tsiotras

  2. Currently a graduate student at the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Luis Ignacio Reyes Castro

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  1. Panagiotis Tsiotras
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Correspondence to Panagiotis Tsiotras .

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Editors and Affiliations

  1. Systems and Information Engineering Department, University of Virginia, Charlottesville, Virginia, USA

    Amy LaViers

  2. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA

    Magnus Egerstedt

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Tsiotras, P., Reyes Castro, L.I. (2014). The Artistic Geometry of Consensus Protocols. In: LaViers, A., Egerstedt, M. (eds) Controls and Art. Springer, Cham. https://doi.org/10.1007/978-3-319-03904-6_6

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  • DOI: https://doi.org/10.1007/978-3-319-03904-6_6

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