The Artistic Geometry of Consensus Protocols

Chapter

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 = -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.

Keywords

Consensus Protocols Pattern generation Multi-agent systems Esthetics Trochoids 

References

  1. 1.
    Arcak M (2007) Passivity as a design tool for group coordination. IEEE Trans Autom Control 52(8):1380–1390CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bense M (1969) Einführung in die informationstheoretische Asthetik. Grundlegung und Anwendung in der Texttheorie. Rowohlt Taschenbuch Verlag, HamburgGoogle Scholar
  3. 3.
    Birkhoff GD (1933) Aesthetic measure. Harvard University Press, CambridgeGoogle Scholar
  4. 4.
    Bouleau C (1980) The painter’s secret geometry: a study of composition in art. Hacker Art Books, New YorkGoogle Scholar
  5. 5.
    Boyer CB (1968) A history of mathematics. Wiley, New YorkMATHGoogle Scholar
  6. 6.
    Brewer J (1978) Kronecker products and matrix calculus in system theory. IEEE Trans Circ Syst 25(9):772–781CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Coldstream JN (2003) Geometric Greece: 900–700 BC. Psychology Press, AbingdonGoogle Scholar
  8. 8.
    Dawson I (2013) Spirographs. http://www.iandawson.net/spiro.php. Accessed 20 May 2013
  9. 9.
    Economou A (2011) The symmetry lessons from Froebel building gifts. Environ Plann B: Plann Des 26(1):75–90CrossRefMathSciNetGoogle Scholar
  10. 10.
    El-Said I, El-Bouri T, Critchlow K, Damlūjī SS (1993) Islamic art and architecture: the system of geometric design. Garnet Publishing, LondonGoogle Scholar
  11. 11.
    Ettinghausen R, Grabar O, Jenkins M (2001) Islamic art and architecture: 650–1250, vol 51. Yale University Press, LondonGoogle Scholar
  12. 12.
    Fax JA, Murray RM (2004) Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control 49(9):1465–1476CrossRefMathSciNetGoogle Scholar
  13. 13.
    Frith CD, Nias DKB (1974) What determines aesthetic preferences? J Gen Psychol 91:163–173Google Scholar
  14. 14.
    Godsil CD, Royle G (2001) Algebraic graph theory. Springer, New YorkGoogle Scholar
  15. 15.
    Grünbaum B, Grünbaum Z, Shepard GC (1986) Symmetry in moorish and other ornaments. Comput Math Appl 12(3):641–653CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hall LM (1992) Trochoids, roses, and thorns-beyond the spirograph. Coll Math J 23(1):20–35CrossRefGoogle Scholar
  17. 17.
    Halliwell L (2013) http://www.lesleyhalliwell.co.uk/. Accessed 20 May 2013
  18. 18.
    Knight TW (1995) Transformations in design: a formal approach to stylistic change and innovation in the visual arts. Cambridge University Press, CambridgeGoogle Scholar
  19. 19.
    Lawrence JD (1972) A catalog of special plane curves. Dover Publications, New YorkMATHGoogle Scholar
  20. 20.
    Leder H, Belke B, Oeberst A, Augustin D (2004) A model of aesthetic appreciation and aesthetic judgments. Br J Psychol 95(4):489–508CrossRefGoogle Scholar
  21. 21.
    Leonard NE, Fiorelli E (2001) Virtual leaders, artificial potentials, and coordinated control of groups. 40th IEEE Conference on Decision and Control, pp 2968–2973, 4–7 Dec 2001Google Scholar
  22. 22.
    Leonard NE, Young GF, Hochgraf K, Swain DT, Trippe A, Chen W, Fitch K, Marshall S (2014) Controls and art. In the dance studio: an art and engineering exploration of human flocking. In: Lecture Notes in Computer Sciences. Springer-Verlag, New York, p xxxGoogle Scholar
  23. 23.
    Locher P, Nodine C (1989) The perceptual value of symmetry. Comput Math Appl 17(4):475–484CrossRefMathSciNetGoogle Scholar
  24. 24.
    Locher P, Nodine C (1987) Eye movements: from physiology to cognition. Chapter Symmetry Catches the Eye. Elsevier, Holland, pp 353–361Google Scholar
  25. 25.
    Marshall JA, Broucke ME, Francis BA (2004) Formations of vehicles in cyclic pursuit. IEEE Trans Autom Control 49(11):1963–1974CrossRefMathSciNetGoogle Scholar
  26. 26.
    McManus C (2005) Symmetry and asymmetry in aesthetics and the arts. Eur Rev 13:157–180CrossRefGoogle Scholar
  27. 27.
    Mesbahi M, Egerstedt M (2010) Graph theoretic methods in multiagent networks. Princeton University Press, PrincetonMATHGoogle Scholar
  28. 28.
    Mitchell WJ (1990) The logic of architecture: design, computation, and cognition. MIT Press, CambridgeGoogle Scholar
  29. 29.
    Moles A (1968) Information theory and esthetic perception. University of Illinois Press, ChampaignGoogle Scholar
  30. 30.
    Olfati-Saber R (2006) Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. In: 45th IEEE Conference on Decision and Control, pp 5060–5066Google Scholar
  31. 31.
    Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533CrossRefMathSciNetGoogle Scholar
  32. 32.
    Olfati-Saber R, Fax JA, Murray RM (2006) Consensus and cooperation in multi-agent networked systems. Proc IEEE 97:215–233Google Scholar
  33. 33.
    Osborne H (1986) Symmetry as an aesthetic factor. Comput Math Appl 12(1):77–82CrossRefGoogle Scholar
  34. 34.
    Parchos EA, Sotiroudis P (1999) The schemata of the stars: Byzantine astronomy from A.D. 1300. World Scientific, SingaporeGoogle Scholar
  35. 35.
    Park J-H, Joo Y, Yang J-G (2007) Cycloids in Louis I. Kahns Kimbell Art Museum at Fort Worth, Texas. Math Intell 29:42–48Google Scholar
  36. 36.
    Pavone M, Frazzoli E (2007) Decentralized policies for geometric pattern formation and path coverage. J Dyn Syst Meas Contr 129:633–643CrossRefGoogle Scholar
  37. 37.
    Pohl D (2013) The loop yoga drawing project. http://spirographart.blogspot.com/. Accessed 20 May 2013
  38. 38.
    Public domain image “\({\rm File:Triquetra}\_{\rm on}\_{\rm book}\_{\rm cover.jpg}\)” from user Chameleon on the Wikimedia Commons. http://commons.wikimedia.org/wiki/File:Triquetra_on_book_cover.jpg;. Accessed 10 Sept 2013
  39. 39.
    Public domain image ID \(289442\_{\rm WDC25C21}\) (acquisition date 01 Sept 2005). United States Geological Survey. http://earthexplorer.usgs.gov/;. Accessed 21 Nov 2013
  40. 40.
    Ren W, Beard RW (2005) Consensus seeking in multi-agent systems using dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661CrossRefMathSciNetGoogle Scholar
  41. 41.
    Rigau J, Feixas M, Sbert M (2008) Informational aesthetics measures. IEEE Comput Graph Appl 28(2):24–34CrossRefGoogle Scholar
  42. 42.
    Sarlette A, Sepulchre R (2009) Consensus optimization on manifolds. SIAM J Control Optim 48(1):56–76CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Schattschneider D, Hofstadter D (2004) In: Escher MC (ed) Visions of symmetry. Thames & Hudson, LondonGoogle Scholar
  44. 44.
    Schoelling AP, Siegel H, Augugliaro F, D’Andrea R (2014) Controls and art. So you think you can dance? Rhythmic flight performances with quadrocopters. In: Lecture Notes in Computer Sciences. Springer-Verlag, New York, p xxxGoogle Scholar
  45. 45.
    Schweitzer B (1971) Greek geometric art. Phaidon, LondonGoogle Scholar
  46. 46.
    Simmons Jeffrey. Paintings: 1999–2000: Trochoid. http://jeffreysimmonsstudio.com/project/trochoids-paintings-1999/. Accessed 20 May 2013
  47. 47.
    Simoson AJ (2008) Albrecht Dürer’s trochoidal woodcuts. Probl Resour Issues Math Undergraduate Stud (PRIMUS) 18(6):489–499Google Scholar
  48. 48.
    Stiny G, Gips J (1978) Algorithmic aesthetics: computer models for criticism and design in the arts. University of California Press, BerkeleyGoogle Scholar
  49. 49.
    Suzuki I, Yamashita M (1999) Distributed anonymous mobile robots: formation of geometric patterns. SIAM J Comput 28(4):1347–1363CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Tron R, Afsari B, René Vidal (2012) Riemannian consensus for manifolds with bounded curvature. IEEE Trans Autom ControlGoogle Scholar
  51. 51.
    Tsiotras P, Reyes Castro LI. A note on the consensus protocol with some applications to agent orbit pattern generation. In: 10th Symposium on Distributed Autonomous Robotic Systems (DARS). Lausanne, Switzerland, 1–3 Nov 2010Google Scholar
  52. 52.
    Wang L-S, Krishnaprasad PS (1992) Gyroscopic control and stabilization. J Nonlinear Sci 2:367–415CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Wessén E (1952) Det svenska runverket. http://fornvannen.se/pdf/1950talet/1952_193.pdf (in Swedish)
  54. 54.
    Weyl H (1952) Symmetry. Princeton University Press, LondonGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Panagiotis Tsiotras
    • 1
  • Luis Ignacio Reyes Castro
    • 2
  1. 1.Daniel Guggenheim School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Currently a graduate student at the Department of Aeronautics and AstronauticsMassachusetts Institute of TechnologyCambridgeUSA

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