Dynamic Redistribution of a Swarm of Robots Among Multiple Sites ´ Ad´ am Hal´ asz, M. Ani Hsieh, Spring Berman, and Vijay Kumar Abstract—We present an approach for the dynamic assign- ment and reassignment of a large team of homogeneous robotic agents to multiple locations with applications to search and rescue, reconnaissance and exploration missions. Our work is inspired by experimental studies of ant house hunting and empirical models that predict the behavior of the colony that is faced with a choice between multiple candidate nests. We design stochastic control policies that enable the team of agents to distribute themselves between multiple candidate sites in a specified ratio. Additionally, we present an extension to our model to enable fast convergence via switching behaviors based on quorum sensing. The stability and convergence properties of these control policies are analyzed and simulation results are presented. I. I NTRODUCTION We are interested in deploying swarms of homogeneous robots to distinct locations for simultaneous execution of tasks at each locale. This is relevant in applications such as surveillance of multiple buildings, large scale environmental monitoring or providing aerial coverage for various ground units. In these applications, robots must have the ability to distribute themselves among many locations/sites and have the ability to autonomously redistribute to ensure task com- pletion at each site which may be affected by robot failures or changes in the environment. In addition, there are situations when robots may not be able to easily communicate across sites, e.g. mining at multiple sites, and thus it makes sense to develop a strategy that can achieve (re)distribution of the team with little to no communication. This work draws inspiration from the process through which an ant colony selects a new home from several sites using simple behaviors that arise from local sensing and physical contact [1]. Rather than choosing a “new home”, we propose a strategy for the deployment of a swarm of ho- mogeneous robots such that the team collectively distributes itself to multiple sites in predefined proportions without the use of inter-agent communication. This is similar to the task/resource allocation problem where the objective is to determine the optimal assignment of robots to tasks. As such, the proposed strategy can be readily applied to the problem of dynamic (re)assignment of a large team of homogeneous robotic agents to multiple tasks. Previous works that considered the assignment of multiple robots to one task include [2]–[4]. In particular, market- based approaches have gained much success in various multi- robot applications like robot soccer and treasure hunting [5]– V. Kumar, ´ A. Hal´ asz, M. A. Hsieh, and S. Berman are with the GRASP Laboratory, School of Engineering and Applied Science, University of Pennsylvania, 3330 Walnut Street, Philadelphia PA 19104 {kumar, halasz, mya, spring}@grasp.upenn.edu [7]. However, these existing methods do not address the controller synthesis problem and often suffer in terms of scalability with respect to the number of tasks and robots. As such performance guarantees are often sacrificed to reduce the computation and communication requirements [8], [9]. Other task allocation strategies include [10] where the problem is formulated as a Distributed Constraint Satisfac- tion Problem. This approach, however, requires the explicit modeling of tasks, their requirements, and robot capabilities making implementation to large populations difficult. In [11], large robot populations are modeled using a partial differential equation. Then a centralized optimal control strategy is used to maximize robot occupation at a desired position. In [12], an adaptive multi-foraging task with no explicit communication or global knowledge is modeled as a stochastic process. While the model was verified through simulations, the only way to control robot task reallocation is to modify the task distribution in the environment. Similar to [12], [13], our proposed strategy employs a multi-level representation of swarm activity. Rather than the bottom-up analysis procedure, our methodology is based on a top-down design approach. We build on [14], [15] and extend our synthesis procedure to the problem of deployment to multiple sites and consider the allocation of hundreds and thousands of robots. Our proposed approach is computation- ally inexpensive and scalable, and can be easily modified to simultaneously solve the controller synthesis problem. Furthermore, we are able to control the final allocation among the various sites by translating the global specification into agent closed-loop control laws that produce convergence to the desired allocation. Lastly, our approach enables the system to respond to robot failures in a natural way thus ensuring graceful degradation. This paper is organized as follows: Section II formulates the problem and outlines the system model. Section III shows the stability and convergence properties of our system. Sec- tion IV presents our simulation results. Section V provides an extension of our methodology to include a quorum sensing mechanism to speed up convergence. Finally, we conclude with a discussion of directions for future work in Section VI. II. PROBLEM FORMULATION A. Definitions Consider N agents to be distributed among M sites. We denote the number of agents at site i ∈{1,...,M } at time t by n i (t) and the desired number of agents at site i by ¯ n i . We define the population fraction at each site at time t as x i (t) where x i (t)= n i (t)/ ∑ i ¯ n i . Then the system state vector is given by x =[x 1 ,...,x M ] T . For some initial distribution of