Understanding the effect of individual parameters on the collective performance of swarm robotic systems in order to design and optimize individual robot behaviors is a significant challenge. In this paper we present a macroscopic probabilistic model of adaptive collective foraging in a swarm of robots, where each robot in the swarm is capable of adjusting its time threshold parameters following the rules described by Liu et al. 2007. The swarm adapts the ratio of foragers to resters (division of labor) in order to maximize the net swarm energy for a given food density. A probabilistic finite state machine (PFSM) and a number of difference equations are developed to describe collective foraging at a macroscopic level. To model adaptation we introduce the new concepts of the sub-PFSM and private/public time thresholds. The model has been validated extensively with simulation trials, and results show that the model achieves very good accuracy in predicting the group performance of the swarm. Finally, a real-coded genetic algorithm is used to explore the parameter spaces and optimize the parameters of the adaptation algorithm. Although this paper presents a macroscopic probabilistic model for adaptive foraging, we argue that the approach could be applied to any adaptive swarm system in which the heterogeneity of the system is coupled with its time parameters.
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