Swarm Intelligence

, Volume 2, Issue 2–4, pp 209–239 | Cite as

A framework of space–time continuous models for algorithm design in swarm robotics

  • Heiko HamannEmail author
  • Heinz Wörn


Designing and analyzing self-organizing systems such as robotic swarms is a challenging task even though we have complete knowledge about the robot’s interior. It is difficult to determine the individual robot’s behavior based on the swarm behavior and vice versa due to the high number of agent–agent interactions. A step towards a solution of this problem is the development of appropriate models which accurately predict the swarm behavior based on a specified control algorithm. Such models would reduce the necessary number of time-consuming simulations and experiments during the design process of an algorithm. In this paper we propose a model with focus on an explicit representation of space because the effectiveness of many swarm robotic scenarios depends on spatial inhomogeneity. We use methods of statistical physics to address spatiality. Starting from a description of a single robot we derive an abstract model of swarm motion. The model is then extended to a generic model framework of communicating robots. In two examples we validate models against simulation results. Our experience shows that qualitative correctness is easily achieved, while quantitative correctness is disproportionately more difficult but still possible.


Swarm robotics Design of self-organization Microscopic modeling Macroscopic modeling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bettstetter, C. (2004). On the connectivity of Ad Hoc networks. The Computer Journal, 47(4), 432–447. CrossRefGoogle Scholar
  2. Bjerknes, J. D., Winfield, A., & Melhuish, C. (2007). An analysis of emergent taxis in a wireless connected swarm of mobile robots. In IEEE swarm intelligence symposium (pp. 45–52). Los Alamitos, CA: IEEE Press. CrossRefGoogle Scholar
  3. Brown, R. (1828). A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philosophical Magazine, 4, 161–173. Google Scholar
  4. Correll, N. (2007). Coordination schemes for distributed boundary coverage with a swarm of miniature robots: synthesis, analysis and experimental validation. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne. Google Scholar
  5. Correll, N., & Martinoli, A. (2006). System identification of self-organizing robotic swarms. In M. Gini & R. Voyles (Eds.), Proceedings of the 8th int. symp. on distributed autonomous robotic systems (DARS’06) (pp. 31–40). Berlin: Springer. CrossRefGoogle Scholar
  6. Crank, J., & Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. In Proceedings of the Cambridge philosophical society (Vol. 43, pp. 50–64). Google Scholar
  7. Deguet, J., Demazeau, Y., & Magnin, L. (2006). Elements about the emergence issue: A survey of emergence definitions. Complexus, 3(1–3), 24–31. CrossRefGoogle Scholar
  8. Doob, J. L. (1953). Stochastic processes. New York: Wiley. zbMATHGoogle Scholar
  9. Edelstein-Keshet, L. (2006). Mathematical models of swarming and social aggregation. Robotica, 24(3), 315–324. CrossRefGoogle Scholar
  10. Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 17, 549–560. CrossRefGoogle Scholar
  11. Feddema, J. T., Lewis, C., & Schoenwald, D. A. (2002). Decentralized control of cooperative robotic vehicles: theory and application. IEEE Transactions on Robotics and Automation, 18(5), 852–864. CrossRefGoogle Scholar
  12. Fokker, A. D. (1914). Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik, 348(5), 810–820. CrossRefGoogle Scholar
  13. Galstyan, A., Hogg, T., & Lerman, K. (2005). Modeling and mathematical analysis of swarms of microscopic robots. In Proceedings of IEEE swarm intelligence symposium (SIS’05) (pp. 201–208). Los Alamitos, CA: IEEE Press. CrossRefGoogle Scholar
  14. Gazi, V., & Passino, K. M. (2003). Stability analysis of swarms. IEEE Transactions on Automatic Control, 48(4), 692–697. CrossRefMathSciNetGoogle Scholar
  15. Grünbaum, D., & Okubo, A. (1994). Modeling social animal aggregations. Frontiers in Theoretical Biology, 100, 296–325. Google Scholar
  16. Haken, H. (1977). Synergetics—an introduction. Berlin: Springer. zbMATHGoogle Scholar
  17. Hamann, H., & Wörn, H. (2007a). An analytical and spatial model of foraging in a swarm of robots. In E. Şahin, W. Spears, & A. F. Winfield (Eds.), Lecture notes in computer science : Vol. 4433. Swarm robotics—second SAB 2006 international workshop (pp. 43–55). Berlin: Springer. Google Scholar
  18. Hamann, H., & Wörn, H. (2007b). A space- and time-continuous model of self-organizing robot swarms for design support. In First IEEE international conference on self-adaptive and self-organizing systems (SASO’07) (pp. 23–31). Los Alamitos, CA: IEEE Press. CrossRefGoogle Scholar
  19. Helbing, D., Schweitzer, F., Keltsch, J., & Molnar, P. (1997). Active walker model for the formation of human and animal trail systems. Physical Review E, 56(3), 2527–2539. CrossRefGoogle Scholar
  20. Higham, N. J. (2002). Accuracy and stability of numerical algorithms. Society for industrial and applied mathematics. Google Scholar
  21. Hogg, T. (2006). Coordinating microscopic robots in viscous fluids. Autonomous Agents and Multi-Agent Systems, 14(3), 271–305. CrossRefMathSciNetGoogle Scholar
  22. Holland, J. H. (1998). Emergence—from chaos to order. New York: Oxford University Press. zbMATHGoogle Scholar
  23. Kolmogorov, A. N. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen, 104(1), 415–458. zbMATHCrossRefMathSciNetGoogle Scholar
  24. Langevin, P. (1908). Sur la théorie du mouvement brownien. Comptes-rendus de l’Académie des Sciences, 146, 530–532. zbMATHGoogle Scholar
  25. Langton, C. G. (Eds.). (1989). Artificial life: proceedings of an interdisciplinary workshop on the synthesis and simulation of living systems. Reading, MA: Addison-Wesley. Google Scholar
  26. Lemons, D. S., & Gythiel, A. (1997). Paul Langevin’s 1908 paper “On the theory of Brownian motion” [“Sur la théorie du mouvement brownien,” Comptes-rendus de l’Académie des Sciences (Paris) 146, 530–533 (1908)]. American Journal of Physics, 65(11), 1079–1081. CrossRefGoogle Scholar
  27. Lerman, K., Martinoli, A., & Galstyan, A. (2005). A review of probabilistic macroscopic models for swarm robotic systems. In E. Şahin & W. M. Spears (Eds.), Swarm robotics—SAB 2004 international workshop (pp. 143–152). Berlin: Springer. Google Scholar
  28. Martinoli, A. (1999). Swarm intelligence in autonomous collective robotics: from tools to the analysis and synthesis of distributed control strategies. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne. Google Scholar
  29. Martinoli, A., Easton, K., & Agassounon, W. (2004). Modeling swarm robotic systems: A case study in collaborative distributed manipulation. International Journal of Robotics Research, 23(4), 415–436. CrossRefGoogle Scholar
  30. McCanne, S., Floyd, S., Fall, K., & Varadhan, K. et al. (1997). Network simulator—ns-2.
  31. Mogilner, A., & Edelstein-Keshet, L. (1999). A non-local model for a swarm. Journal of Mathematical Biology, 38(6), 534–570. zbMATHCrossRefMathSciNetGoogle Scholar
  32. Nembrini, J., Winfield, A. F., & Melhuish, C. (2002). Minimalist coherent swarming of wireless networked autonomous mobile robots. In B. Hallam, D. Floreano, J. Hallam, G. Hayes, & J.-A. Meyer (Eds.), Proceedings of the seventh international conference on simulation of adaptive behavior (From animals to animats) (pp. 373–382). Cambridge, MA: MIT Press. Google Scholar
  33. Von Neumann, J. (1966). The theory of self-reproducing automata. Champaign, IL: University of Illinois Press. Arthur Burks (ed.). Google Scholar
  34. Okubo, A. (1986). Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds. Advances in Biophysics, 22, 1–94. CrossRefGoogle Scholar
  35. Okubo, A., & Levin, S. A. (2001). Diffusion and ecological problems: modern perspectives. Berlin: Springer. Google Scholar
  36. Planck, M. (1917). Über einen Satz der statistischen Dynamik and seine Erweiterung in der Quantentheorie. Sitzungsberichte der Preußischen Akademie der Wissenschaften, 24, 324–341. Google Scholar
  37. Risken, H. (1984). The Fokker–Planck equation. Berlin: Springer. zbMATHGoogle Scholar
  38. Schillo, M., Fischer, K., & Klein, C. T. (2000). The micro-macro link in DAI and sociology. In S. Moss & P. Davidsson (Eds.), Lecture notes in computer science : Vol. 1979. Multi-agent-based simulation: second international workshop, (MABS’00) (pp. 303–317). Berlin: Springer. Google Scholar
  39. Schmickl, T., & Crailsheim, K. (2006). Trophallaxis among swarm-robots: A biologically inspired strategy for swarm robotics. In The first IEEE/RAS-EMBS international conference on biomedical robotics and biomechatronics (BioRob’06) (pp. 377–382). Los Alamitos, CA: IEEE Press. CrossRefGoogle Scholar
  40. Schmickl, T., & Crailsheim, K. (2008). Trophallaxis within a robotic swarm: bio-inspired communication among robots in a swarm. Autonomous Robots 25(1–2):171–188. CrossRefGoogle Scholar
  41. Schmickl, T., Möslinger, C., & Crailsheim, K. (2007a). Collective perception in a robot swarm. In E. Şahin, W. M. Spears, & A. F. Winfield (Eds.), Lecture notes in computer science : Vol. 4433. Swarm robotics—second SAB 2006 international workshop (pp. 144–157). Berlin: Springer. Google Scholar
  42. Schmickl, T., Möslinger, C., Thenius, R., & Crailsheim, K. (2007b). Bio-inspired navigation of autonomous robots in heterogenous environments. International Journal of Factory Automation, Robotics and Soft Computing, 3, 164–170. Google Scholar
  43. Schmickl, T., Möslinger, C., Thenius, R., & Crailsheim, K. (2007c). Individual adaptation allows collective path-finding in a robotic swarm. International Journal of Factory Automation, Robotics and Soft Computing, 4, 102–108. Google Scholar
  44. Schweitzer, F. (2003). Brownian agents and active particles. On the emergence of complex behavior in the natural and social sciences. Berlin: Springer. Google Scholar
  45. Von Smoluchowski, M. (1906). Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Annalen der Physik, 21, 756–780. CrossRefGoogle Scholar
  46. Soysal, O., & Şahin, E. (2007). A macroscopic model for self-organized aggregation in swarm robotic systems. In E. Şahin, W. M. Spears, & A. F. Winfield (Eds.), Lecture notes in computer science : Vol. 4433. Swarm robotics—second SAB 2006 international workshop (pp. 27–42). Berlin: Springer. Google Scholar
  47. van Kampen, N. G. (1981). Stochastic processes in physics and chemistry. Amsterdam: North-Holland. zbMATHGoogle Scholar
  48. Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 6(75), 1226–1229. CrossRefGoogle Scholar
  49. Winfield, A. F. T., Sav, J., Fernández-Gago, M.-C., Dixon, C., & Fisher, M. (2005). On formal specification of emergent behaviours in swarm robotic systems. International Journal of Advanced Robotic Systems, 2(4), 363–370. Google Scholar
  50. Yamins, D. (2005). Towards a theory of “local to global” in distributed multi-agent systems. In Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems (AAMS’05) (pp. 183–190). New York: ACM. CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute for Process Control and RoboticsUniversität Karlsruhe (TH)KarlsruheGermany

Personalised recommendations