Several natural systems are characterized by a large number of agents, interacting only locally, whose individual actions produce a macroscopic organized behavior. Examples of this phenomenon are the collective motion of flocks of birds or schools of fishes, the functioning of multicellular biological organisms and genetic networks (Figure 1). All the above mentioned systems exhibit an emergent behavior stemming from local and simple coordination rules among the entities.
The aim of this project is to understand how group of man-made, mobile agents can use these biological principles to collectively perform useful tasks such as flocking, reaching a destination, gathering data or searching for objects in a safe, cooperative, and coordinated manner.
Recent developments in the field of electronics and mechanics allow to construct small mobile entities like robots and unmanned air or underwater vehicles having on-board computing capabilities and communicating through wireless networks (Figure 2). However, the study of decentralized control laws capable to reproduce the self-organizing behaviors is still in its infancy. Apparently this goal relies on the availability of faithful mathematical models for coordination phenomena.
In this project, jointly with INRIA (Rocquencourt unit, France), we have developed a methodological framework for the modeling and the control of mobile agents partially interconnected by communication networks. The network is modeled as an incomplete graph: the nodes represent the dynamical systems characterizing individual agents, and the edges capture the topology of the communication protocol. The overall system is modeled by means of Partial difference Equations (PdE) on graphs. PdEs are mathematical models strongly inspired to partial differential equations and provide a unified and compact description of agents interactions in space and time. Moreover, PdEs allow naturally to interpret self-organization phenomena in terms of well known laws of classical physics. For example, in the problem of flocking, agents alignment corresponds to heat diffusion in a room and the formation velocities profile plays the role of a temperature distribution. Cohesion and collision avoidance can be obtained by analogy to some laws of non-linear elasticity (for an example, see the attached document). Indeed, our research lies at the intersection among different scientific fields: functional analysis, algebraic graph theory, mathematical physics and automatic control.