On how swarm robotics can be used to describe particle system’s deformation

  • Ramiro dell’ErbaEmail author
Original Article


In previous works, we have described time evolution of a two-dimensional particle lattice, subject to deformation, without the use of Newton’s law. According to our experience, in control of robotic swarm, the new position of a particle is determined by the spatial position of its neighbours; therefore, we have used an interaction law based on the spatial position of the particles themselves. The tool that we have realized reproduced some behaviour of deformable bodies both according to the standard Cauchy model and the second gradient theory. In this paper, we try to stress what is still under investigation, like the relationship describing the interaction rule and its physical meaning; moreover, we shall describe as some solutions do not agree with the behaviour of the classical solution coming out from differential equations.


Position-based dynamics Swarm robotics Discrete mechanical systems 



The authors received no financial support for the research, authorship and/or publication of this article.


  1. 1.
    Bender, J., Müller, M., Macklin, M.: Position-based simulation methods in computer graphics. In: Eurographics (Tutorials), 2015 [Online]. [Consultato: 06-set-2017]
  2. 2.
    Umetani, N., Schmidt, R., Stam, J.: Position-based elastic rods. In: Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 21–30 (2014)Google Scholar
  3. 3.
    dell’Erba, R.: Determination of spatial configuration of an underwater swarm with minimum data. Int. J. Adv. Robot. Syst. 12(7), 97 (2015)CrossRefGoogle Scholar
  4. 4.
    Moriconi, C., dell’Erba, R.: The localization problem for harness: a multipurpose robotic swarm. In: SENSORCOMM 2012, The Sixth International Conference on Sensor Technologies and Applications, pp. 327–333 [Online] (2012). [Consultato: 04-apr-2014]
  5. 5.
    Battista, A., Rosa, L., dell’Erba, R., Greco, L.: Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena. Math. Mech. Solids (2016). CrossRefzbMATHGoogle Scholar
  6. 6.
    dell’Erba, R.: Position-based dynamic of a particle system: a configurable algorithm to describe complex behaviour of continuum material starting from swarm robotics. Contin. Mech. Thermodyn. 30(5), 1069–1090 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    dell’Erba, R.: Swarm robotics and complex behaviour of continuum material. Contin. Mech. Thermodyn. 31(4), 989–1014 (2019)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Wiech, J., Eremeyev, V.A., Giorgio, I.: Virtual spring damper method for nonholonomic robotic swarm self-organization and leader following. Contin. Mech. Thermodyn. 30(5), 1091–1102 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Seddik, H., Greve, R., Placidi, L., Hamann, I., Gagliardini, O.: Application of a continuum-mechanical model for the flow of anisotropic polar ice to the EDML core. Antarct. J. Glaciol. 54(187), 631–642 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Placidi, L., Greve, R., Seddik, H., Faria, S.H.: Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor. Contin. Mech. Thermodyn. 22(3), 221–237 (2010)ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct 46(3), 774–787 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech 80(1), 73–92 (2010)ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, New York (2012)zbMATHGoogle Scholar
  14. 14.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P., Rendón, L.A.: Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80(3), 217–227 (2010)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Altenbach, H., Eremeyev, V.A.: Generalized Continua From the Theory to Engineering Applications, CISM Courses and Lectures, vol. 541. Springer, Udine (2013)CrossRefGoogle Scholar
  16. 16.
    Barchiesi, E., dell’Isola, F., Laudato, M., Placidi, L., Seppecher, P.: A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results. In: dell’Isola, F., Eremeyev, V., Porubov, A. (eds.) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, vol. 87, pp. 43–74. Springer, Cham (2018)CrossRefGoogle Scholar
  17. 17.
    Placidi, L., Rosi, G., Barchiesi, E.: Analytical solutions of 2-dimensional second gradient linear elasticity for continua with cubic-D4 microstructure. In: Abali, B., Altenbach, H., dell’Isola, F., Eremeyev, V., Öchsner, A. (eds.) New Achievements in Continuum Mechanics and Thermodynamics. Advanced Structured Materials, vol. 108, pp. 383–401. Springer, Cham (2019)CrossRefGoogle Scholar
  18. 18.
    Rosi, G., Placidi, L., dell’Isola, F.: "Fast" and "slow" pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal. Z. Für Angew. Math. Phys. 68(2), 51 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Turco, E.: How the properties of pantographic elementary lattices determine the properties of pantographic metamaterials. In: Abali, B., Altenbach, H., dell’Isola, F., Eremeyev, V., Öchsner, A. (eds,) New Achievements in Continuum Mechanics and Thermodynamics. Advanced Structured Materials, vol. 108, pp 489–506. Springer, Cham (2019)Google Scholar
  20. 20.
    Turco, E., Golaszewski, M., Giorgio, I., Placidi, L.: Can a Hencky-type model predict the mechanical behaviour of pantographic lattices? In: Mathematical Modelling in Solid Mechanics, pp. 285–311. Springer, Singapore (2017)Google Scholar
  21. 21.
    Franciosi, P., Lebail, H.: Anisotropy features of phase and particle spatial pair distributions in various matrix/inclusions structures. Acta Materialia 52(10), 3161–3172 (2004)CrossRefGoogle Scholar
  22. 22.
    Franciosi, P.: A regularized multi-laminate-like plasticity scheme for polycrystals, applied to the FCC structure. Procedia IUTAM 3, 141–156 (2012)CrossRefGoogle Scholar
  23. 23.
    Abali, B.E., Müller, W.H., Dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Für Angew. Math. Phys. 67(4), 85 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    dell’Isola, F., Madeo, A., Seppecher, P.: Cauchy tetrahedron argument applied to higher contact interactions. Arch. Ration. Mech. Anal. 219(3), 1305–1341 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    dell’Isola, F., Seppecher, P., Corte, A.D.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. Math. Phys. Eng. Sci. (2015). CrossRefzbMATHGoogle Scholar
  28. 28.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Seppecher, P., Alibert, J.-J., Isola, F.D.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys.: Conf. Ser. (2011). CrossRefGoogle Scholar
  30. 30.
    Forest, S., Cordero, N.M., Busso, E.P.: First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput. Mater. Sci. 50(4), 1299–1304 (2011)CrossRefGoogle Scholar
  31. 31.
    Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. 27(4–5), 623–638 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Andreaus, U., dell’Isola, F., Giorgio, I., Placidi, L., Lekszycki, T., Rizzi, N.L.: Numerical simulations of classical problems in two-dimensional (non)linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Abali, B.E., Müller, W.H., Eremeyev, V.A.: Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mech. Adv. Mater. Modern Process. 1(1), 4 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5), 114 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids 22(2), 144–157 (2017)zbMATHCrossRefGoogle Scholar
  36. 36.
    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev 67(6), 060804 (2015)CrossRefGoogle Scholar
  37. 37.
    Bückmann, T., et al.: Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv. Mater. 24(20), 2710–2714 (2012)CrossRefGoogle Scholar
  38. 38.
    dell’Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Gołaszewski, M.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. (2018). CrossRefGoogle Scholar
  39. 39.
    Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212–234 (2019)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218(3), 1239–1262 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Turco, E., Giorgio, I., Misra, A., Dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R. Soc. Open Sci. 4(10), 171153 (2017)ADSCrossRefGoogle Scholar
  42. 42.
    dell’Isola, F., Bucci, S., Battista, A.: Against the fragmentation of knowledge: the power of multidisciplinary research for the design of metamaterials. In: Naumenko, K., Aßmus, M. (eds.) Advanced Methods of Continuum Mechanics for Materials and Structures. Advanced Structured Materials, vol. 60, pp. 523–545. Springer, Singapore (2016)CrossRefGoogle Scholar
  43. 43.
    Milton, G., Seppecher, P.: A metamaterial having a frequency dependent elasticity tensor and a zero effective mass density. Physica Status Solidi (b) 249(7), 1412–1414 (2012)ADSCrossRefGoogle Scholar
  44. 44.
    Lanczos, C.: The Variational Principles of Mechanics. Courier Corporation, North Chelmsford (2012)zbMATHGoogle Scholar
  45. 45.
    Lurie, K.A.: An Introduction to the Mathematical Theory of Dynamic Materials, vol. 15. Springer, New York (2007)zbMATHGoogle Scholar
  46. 46.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Eremeyev, V.A., Sharma, B.L.: Anti-plane surface waves in media with surface structure: discrete vs. continuum model. Int. J. Eng. Sci. 143, 33–38 (2019)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Placidi, L., dell’Isola, F., Ianiro, N., Sciarra, G.: Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A Solids 27(4), 582–606 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. In: dell’Isola, F., Gavrilyuk, S. (eds.) Variational Models and Methods in Solid and Fluid Mechanics. CISM Courses and Lectures, vol. 535, pp. 1–15. Springer, Vienna (2011)zbMATHCrossRefGoogle Scholar
  50. 50.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRefGoogle Scholar
  51. 51.
    dell’Isola, F., Gavrilyuk, S.: Variational Models and Methods in Solid and Fluid Mechanics. Springer, New York (2012)CrossRefGoogle Scholar
  52. 52.
    dell’Isola, F., Auffray, N., Eremeyev, V.A., Madeo, A., Placidi, L., Rosi, G.: Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola’s point of view. In: The Complete Works of Gabrio Piola: Volume I, pp. 606–694. Springer, New York (2014)Google Scholar
  53. 53.
    Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.J.: Buckling modes in pantographic lattices. Comptes Rendus Mécanique 344(7), 487–501 (2016)ADSCrossRefGoogle Scholar
  54. 54.
    Boutin, C., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Giorgio, I., Harrison, P., dell’Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc. R. Soc. A: Math. Phys. Eng. Sci. (2018). CrossRefGoogle Scholar
  56. 56.
    Spagnuolo, M., Barcz, K., Pfaff, A., Dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  57. 57.
    De Angelo, M., Spagnuolo, M., D’Annibale, F., Pfaff, A., Hoschke, K., Misra, A., Pawlikowski, M.: The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens. Contin. Mech. Thermodyn. 31, 1181–1203 (2019)ADSCrossRefGoogle Scholar
  58. 58.
    Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Contin. Mech. Thermodyn. 30, 1103–1123 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Turco, E., Barcz, K., Pawlikowski, M., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Zeitschrift für angewandte Mathematik und Physik 67(5), 122 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Turco, E., Rizzi, N.L.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. 77, 65–69 (2016)CrossRefGoogle Scholar
  61. 61.
    Dong, Y., Zhang, G., Xu, A., Gan, Y.: Cellular automata model for elastic solid material. Commun. Theor. Phys. 59(1), 59–67 (2013)ADSzbMATHCrossRefGoogle Scholar
  62. 62.
    Konovalenko, I.S., Smolin, A.Y., Psakhie, S.G.: Multilevel simulation of deformation and fracture of brittle porous materials in the method of movable cellular automata. Phys. Mesomech. 13(1–2), 47–53 (2010)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.ENEA Technical Unit Technologies for Energy and Industry – Robotics LaboratoryRomeItaly

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