The fundamental vision of modern nano-robotics is a swarm of micro-robots, which is capable of performing tasks that are not possible with either a single micro-robot, or even with a small group of micro-robots. Using principles of artificial physics and self-organization for the control algorithms of such a swarm of micro-robots, it is expected, that the swarm controlled in such a way shows self-organized behavior similar to the self-organization phenomena occurring in many biological or ecological systems like ant tribes, bee colonies and other insect aggregations or flocking birds and shoals. There are many potential benefits of such a system including greater flexibility and adaptability to the environment, robustness to failures, compensation of breakdowns from one or several units, etc.

In this work we describe a basic mathematical approach for modeling of the swarm behavior and for developing the control software of individual micro-robots. The approach is based on laws of artificial physics and coupled hybrid automata. Designing the interaction patterns between robots, we are able to achieve the desired self-organized behavior. For the modeling of these interactions we use so-called virtual power functions (also known as social powers). Depending on the complexity of the considered swarm scenarios, specific classes of hybrid automata, namely degenerated, standard or hierarchical ones, are used. In the last case complex swarm scenarios are implemented using the concept of hierarchical self-organization. According to this concept, the whole system consists of a hierarchy of subsystems. The most simple subsystems (atomic entities, robots) build simple formations via a usual self-organization process. In the presented approach these simple formations are chains of robots with a pre-defined or variable length. Then the simple formations operate as entities (non-atomic self-organized entities) and build more complex formations via a self-organization process on the next hierarchy level. Theoretically, the number of the self-organization levels in a hierarchical self-organization process is not restricted, however in our approach we restrict it firstly to two levels. This restriction is sufficient in order to demonstrate the applicability of the presented concept and can be extended in future work.

The application of the presented approach is demonstrated by example scenarios with several complexity levels. Additionally, some techniques for the investigation of the self-organization phenomena, applied from the fields of coupled map lattices and hybrid systems, are discussed.