The Artistic Geometry of Consensus Protocols

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.

[1]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[2]  Terry Weissman Knight.  Transformations in Design: A Formal Approach to Stylistic Change and Innovation in the Visual Arts , 1994 .

[3]  Murat Arcak,et al.  Passivity as a Design Tool for Group Coordination , 2007, IEEE Transactions on Automatic Control.

[4]  Naomi Ehrich Leonard,et al.  Virtual leaders, artificial potentials and coordinated control of groups , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[5]  Amy LaViers,et al.  Controls and Art: Inquiries at the Intersection of the Subjective and the Objective , 2014 .

[6]  Jin-Ho Park,et al.  Cycloids in Louis I. Kahn’s kimbell art museum at fort worth, Texas , 2007 .

[7]  I. Mcmanus,et al.  Symmetry and asymmetry in aesthetics and the arts , 2005, European Review.

[8]  Raffaello D'Andrea,et al.  So You Think You Can Dance? Rhythmic Flight Performances with Quadrocopters , 2014 .

[9]  Leon M. Hall,et al.  Trochoids, Roses and Thorns: Beyond the Spirograph , 1992 .

[10]  René Vidal,et al.  Average consensus on Riemannian manifolds with bounded curvature , 2011, IEEE Conference on Decision and Control and European Control Conference.

[11]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[12]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[13]  William J. Mitchell,et al.  The Logic of Architecture: Design, Computation, and Cognition , 1992 .

[14]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[15]  Masafumi Yamashita,et al.  Erratum: Distributed Anonymous Mobile Robots: Formation of Geometric Patterns , 2006, SIAM J. Comput..

[16]  R. Olfati-Saber,et al.  Swarms on Sphere: A Programmable Swarm with Synchronous Behaviors like Oscillator Networks , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[17]  P. Locher,et al.  The perceptual value of symmetry , 1989 .

[18]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[19]  Issam El-Said,et al.  Islamic Art and Architecture: The System of Geometric Design , 1998 .

[20]  H. Leder,et al.  A model of aesthetic appreciation and aesthetic judgments. , 2004, British journal of psychology.

[21]  René Vidal,et al.  Riemannian Consensus for Manifolds With Bounded Curvature , 2012, IEEE Transactions on Automatic Control.

[22]  J. Lawrence,et al.  A Catalog of Special Plane Curves , 2013 .

[23]  Mireille E. Broucke,et al.  Formations of vehicles in cyclic pursuit , 2004, IEEE Transactions on Automatic Control.

[24]  G. Stiny,et al.  Algorithmic Aesthetics: Computer Models for Criticism and Design in the Arts , 1978 .

[25]  E. Paschos,et al.  The schemata of the stars. Byzantine astronomy from A.D. 1300. , 1998 .

[26]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[27]  Charles Bouleau,et al.  The Painter's Secret Geometry: A Study of Composition in Art , 1963 .

[28]  Emilio Frazzoli,et al.  Decentralized Policies for Geometric Pattern Formation and Path Coverage , 2007 .

[29]  A. Economou.  The Symmetry Lessons from Froebel Building Gifts , 1999 .

[30]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[31]  R. Arnheim,et al.  Information Theory and Esthetic Perception , 1968 .

[32]  Carl B. Boyer,et al.  A History of Mathematics. , 1993 .

[33]  P. Krishnaprasad,et al.  Gyroscopic control and stabilization , 1992 .

[34]  Mateu Sbert,et al.  Informational Aesthetics Measures , 2008, IEEE Computer Graphics and Applications.

[35]  Alain Sarlette,et al.  Consensus Optimization on Manifolds , 2008, SIAM J. Control. Optim..

[36]  Branko Grünbaum,et al.  SYMMETRY IN MOORISH AND OTHER ORNAMENTS , 1986 .

[37]  Raymond V. Schoder,et al.  Greek Geometric Art , 1971 .

[38]  J. Coldstream.  Geometric Greece: 900–700 BC , 1977 .

[39]  C. Frith,et al.  What determines aesthetic preferences? , 1974, The Journal of general psychology.

[40]  Symmetry as an aesthetic factor , 1986 .

[41]  Panagiotis Tsiotras,et al.  A Note on the Consensus Protocol with Some Applications to Agent Orbit Pattern Generation , 2010, DARS.

[42]  M. Bense.  Einführung in die informationstheoretische Ästhetik , 1969 .