In this paper, we present a time-discrete, incremental methodology for modeling, at the microscopic and macroscopic levels, the dynamics of distributed manipulation experiments using swarms of autonomous robots endowed with reactive controllers. The methodology is well suited for non-spatial metrics, as it does not take into account robot trajectories or the spatial distribution of objects in the environment. The strength of the methodology lies in the fact that it has been generated by considering incremental abstraction steps,from real robots to macroscopic models, each with well-defined mappings between successive implementation levels. Precise heuristic criteria based on geometrical considerations and systematic tests with one or two real robots prevent the introduction of free parameters in the calibration procedure of models. As a consequence, we are able to generate highly abstracted macroscopic models that can capture the dynamics of a swarm of robots at the behavioral level while still being closely anchored to the characteristics of the physical setup. Although this methodology has been and can be applied to other experiments in distributed manipulation (e.g. object aggregation and segregation, foraging), in this paper we,focus on a strictly collaborative case study concerned with pulling sticks out of the ground, an action that requires the collaboration of two robots to be successful. Experiments were carried out with teams consisting of two to 600 individuals at different levels of implementation (real robots, embodied simulations, microscopic and macroscopic models). Results show that models can deliver both qualitatively and quantitatively correct predictions in time lapses that are at least four orders of magnitude smaller than those required by embodied simulations and that they represent a useful tool for generalizing the dynamics of these highly stochastic, asynchronous, nonlinear systems, often outperforming intuitive reasoning. Finally, in addition to discussing subtle numerical effects, small prediction discrepancies, and difficulties in generating the mapping between different abstractions levels, we conclude the paper by reviewing the intrinsic limitations of the current modeling methodology and by proposing a few suggestions for future work.
