Virtual spring damper method for nonholonomic robotic swarm self-organization and leader following


In this paper, we demonstrate a method for self-organization and leader following of nonholonomic robotic swarm based on spring damper mesh. By self-organization of swarm robots we mean the emergence of order in a swarm as the result of interactions among the single robots. In other words the self-organization of swarm robots mimics some natural behavior of social animals like ants among others. The dynamics of two-wheel robot is derived, and a relation between virtual forces and robot control inputs is defined in order to establish stable swarm formation. Two cases of swarm control are analyzed. In the first case the swarm cohesion is achieved by virtual spring damper mesh connecting nearest neighboring robots without designated leader. In the second case we introduce a swarm leader interacting with nearest and second neighbors allowing the swarm to follow the leader. The paper ends with numeric simulation for performance evaluation of the proposed control method.


  1. 1.

    Trianni, V.: Evolutionary Swarm Robotics: Evolving Self-organising Behaviours in Groups of Autonomous Robots. Studies in Computational Intelligence, Vol. 108. Springer, Berlin (2008)

  2. 2.

    Brambilla, M., Ferrante, E., Birattari, M., Dorigo, M.: Swarm robotics: a review from the swarm engineering perspective. Swarm Intell. 7(1), 1–41 (2013)

    Article  Google Scholar 

  3. 3.

    Sahin, E., Spears, W. M. (Eds).: Swarm Robots. Lecture Notes in Computer Science book series (LNCS, vol. 3342). Springer, Berlin (2005)

  4. 4.

    Moriconi, C. dell’Erb, R.: Social Dependability: a proposed evolution for future Robotics, Sixth IARP-IEEE/RAS–EURON Joint Workshop on Technical Challenges for Dependable Robots in Human Environments May 17–18, (2008), Pasadena, California

  5. 5.

    Bossi, S., Cipollini, A., dell’Erba, R., Moriconi, C.: Robotics in Italy. Education, Research, Innovation and Economics outcomes. Enea, Rome, (2014)

  6. 6.

    dell’Erba, R., Moriconi, C.: HARNESS: a robotic swarm for environmental surveillance. In 6th IARP Workshop on Risky Interventions and Environmental Surveillance (RISE). Warsaw, Poland, (2012)

  7. 7.

    dell’Erba, R.: Determination of spatial configuration of an underwater swarm with minimum data. Int. J. Adv. Robotic Syst. 12(7), 97–114 (2015)

    Article  Google Scholar 

  8. 8.

    Urcola, P., Riazuelo, L., Lazaro, M., Montano, L.: Cooperative navigation using environment compliant robot formations. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2008, pp. 2789–2794, IEEE (2008)

  9. 9.

    Shucker, B., Bennett, J.K.: Virtual spring mesh algorithms for control of distributed robotic macrosensors. University of Colorado at Bulder, Technical Report CU-CS-996-05 (2005)

  10. 10.

    Chen, Q., Veres, S.M., Wang, Y., Meng, Y.: Virtual spring, -damper mesh-based formation control for spacecraft swarms in potential fields. J. Guid. Control Dyn. 38(3), 539–546 (2015)

    ADS  Article  Google Scholar 

  11. 11.

    Balkacem, K., Foudil, C.: A virtual viscoelastic based aggregation model for self-organization of swarm robots system. TAROS 2016: Towards Autonomous Robotic Systems, pp. 202–213, Springer (2016)

  12. 12.

    Della Corte, A., Battista, A., dell’Isola, F.: Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non-Linear Mech. 80, 209–220 (2016)

    ADS  Article  Google Scholar 

  13. 13.

    Battista, A. et al.: Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena. Math. Mech. Solids, (2016)

  14. 14.

    Della Corte, A., Battista, A., dell’Isola, F., Giorgio, I.: Modeling deformable bodies using discrete systems with centroid-based propagating interaction: fracture and crack evolution. In: Mathematical Modelling in Solid Mechanics, pp. 59–88. Springer Singapore, (2017)

  15. 15.

    Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Samuel, F., Sab, K.: Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models. Math. Mech. Solids. (2017).

  17. 17.

    Steigmann, D.J.: Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int. J. Non-Lin. Mech. 47, 742–743 (2012)

    Article  Google Scholar 

  18. 18.

    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. ZAMP 67(4), 1–28 (2016)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Alibert, J.-J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Buttà, P., De Masi, A., Rosatelli, E.: Slow motion and metastability for a nonlocal evolution equation. J. Stat. Phys. 112(3–4), 709–764 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. Part B Eng. 115, 423–448 (2017)

    Article  Google Scholar 

  22. 22.

    dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Giergiel, J., Żylski, W.: Description of motion of a mobile robot by Maggie’s equations. J. Theor. Appl. Mech. 43(3), 511–521 (2005)

    Google Scholar 

  24. 24.

    Gutowski R.: Mechanika Analityczna, 1971, PWN, Warszawa

  25. 25.

    Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014)

    Google Scholar 

  26. 26.

    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford (1988)

    Google Scholar 

  27. 27.

    Slepyan, L.I.: Models and Phenomena in Fracture Mechanics. Springer, Berlin (2002)

    Google Scholar 

  28. 28.

    Mishuris, G.S., Movchan, A.B., Slepyan, L.I.: Waves and fracture in an inhomogeneous lattice structure. Waves Random Complex Media 17, 409–428 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Mishuris, G.S., Movchan, A.B., Slepyan, L.I.: Dynamics of a bridged crack in a discrete lattice. Q. J. Mech. Appl. Math. 61, 151–160 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Slepyan, L.I.: Wave radiation in lattice fracture. Acoust. Phys. 56(6), 962–971 (2010)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Victor A. Eremeyev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Francesco dell’Isola.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wiech, J., Eremeyev, V.A. & Giorgio, I. Virtual spring damper method for nonholonomic robotic swarm self-organization and leader following. Continuum Mech. Thermodyn. 30, 1091–1102 (2018).

Download citation


  • Swarm robots
  • Swarm self-organization
  • Nonholonomic robots
  • Leader following
  • Virtual spring damper mesh