Optimality, reduction and collective motion

Stigmergy: from mathematical modelling to control

Stigmergy, the indirect communication between agents of a swarm through dynamic environmental modifications, is a fundamental self-organization mechanism of animal swarms. Engineers have drawn inspiration from stigmergy to establish strategies for the coordination of swarms of robots and of mixed societies of robots and animals. Currently, all models of stigmergy are algorithmic, in the form of behavioural rules implemented at an individual level. A critical challenge for the understanding of stigmergic behaviour and translation of stigmergy to engineering is the lack of a holistic approach to determine which modifications of the environment are necessary to achieve desired behaviours for the swarm. Here, we propose a mathematical framework that rigorously describes the relationship between environmental modifications and swarm behaviour. Building on recent strides in continuification techniques, we model the swarm and environmental modifications as continua. This approach allows us to design the environmental modifications required for the swarm to behave as desired. Through analytical derivations and numerical simulations of one- and two-dimensional examples, we show that our framework yields the distribution of traces required to achieve a desired formation. Such an approach provides an adaptable framework for different implementation platforms, from robotic swarms to mixed societies of robots and animals.