Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: Perspectives of continuum modeling via higher gradient continua

• New figures were added.
• A more detailed physical justification for our algorithm was provided in a new paragraph.
• Some observations on the computational costs were also added.
• A better description of the continuum models used for comparison was provided.
• The non-linearities of the system were also better highlighted.

In the present paper a discrete robotic system model whose elements interact via a simple geometric law is presented and some numerical simulations are provided and discussed. The main idea of the work is to show the resemblance between the cases of first and second neighbors interaction with (respectively) first and second gradient continuous deformable bodies. Our numerical results showed indeed that the interaction and the evolution process described is suitable to closely reproduce some basic characteristics of the behavior of bodies whose deformation energy depends on first or on higher gradients of the displacement. Moreover, some specific qualitative characteristics of the continuous deformation are also reproduced. The model introduced here will need further investigation and generalization in both theoretical and numerical directions.