A Graph Theoretic Approach to Optimal Target Tracking for Mobile Robot Teams Jason C. Derenick, John R. Spletzer and M. Ani Hsieh Abstract—In this paper, we present an optimization frame- work for target tracking with mobile robot teams. The target tracking problem is modeled as a generic semidefinite program (SDP). When paired with an appropriate objective function, the solution to the resulting problem instance yields an optimal robot configuration for target tracking at each time-step, while guaranteeing target coverage (each target is tracked by at least one robot) and maintaining network connectivity. Our methodology is based on the graph theoretic result where the second smallest eigenvalue of the interconnection graph Laplacian matrix is a measure for the connectivity of the graph. This formulation enables us to model agent-target coverage and inter-agent communication constraints as linear- matrix inequalities. We also show that when the communication constraints can be relaxed, the resulting problem can be reposed as a second-order cone program (SOCP) which can be solved significantly more efficiently than its SDP counterpart. Simulation results for a team of robots tracking multiple targets are presented. I. INTRODUCTION We are interested in developing robot teams for use in surveillance and monitoring applications. The idea of using teams of small, inexpensive robotic agents to accomplish various tasks is one that has gained increasing currency as embedded processors and sensors become smaller, more capable, and less expensive. To this point, much of the work in multi-robot coordination has focused on control and perception. It has generally been assumed that each team member has the ability to communicate with any other member with little to no consideration for the the quality of the wireless communication network. Such an assumption, although valid in certain situations, does not generally hold - especially when considering the deployment of robot teams within unstructured and unpredictable environments. Our previous work in target tracking made similar simpli- fying assumptions, as no constraints were placed on sensing and communication ranges [1]. This allowed target coverage and network connectivity requirements to be ignored in order to simplify the optimization process. In this paper however, we consider the problem of controlling the configuration of a team of mobile agents for target tracking under both coverage and communication constraints. Our methodology is based on the graph theoretic result where the second smallest eigenvalue of the interconnection graph Laplacian matrix is a measure for the connectivity of the graph. Recent J. C. Derenick and J. R. Spletzer are with the Department of Computer Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA {derenick,josa}@lehigh.edu M. A. Hsieh is with the GRASP Laboratory, University of Pennsylvania, Philadelphia, PA 19104, USA mya@seas.upenn.edu system and control literature has shown that the maximiza- tion of the second smallest eigenvalue for a state dependent graph Laplacian matrix can be formulated as a semidefinite program [2]. We apply these results to the target tracking task and obtain a coordination strategy that maintains target coverage and network connectivity while optimizing a given objective function. Specifically, robot-target assignments and inter-agent communication constraints are embedded in vis- ibility and network graphs, respectively. The target tracking problem is then formulated as a SDP where the coverage and communication constraints are modeled as linear-matrix inequalities (LMI). An important advantage of this formulation is that it is agnostic to the quality metric being optimized. So long as the objective function is convex, and can be expressed in terms of linear, quadratic, or semidefinite constraints, the resulting problem will be a SDP. The convexity of semidefi- nite programs ensures the problem solution will be globally optimal, and solvable in polynomial time in the number of robots and targets. We also show that when communication constraints must be relaxed to ensure target coverage (e.g. to track diverging/evasive targets), the problem can be reposed as a second order cone program (SOCP) that can be solved significantly more efficiently than its SDP counterpart. II. RELATED WORK In the last several years, increased attention has been focused on the effects of communication networks in multi- agent teams. Earlier works generally assumed static com- munication ranges, [3], and/or relied on coordination strate- gies that require direct line-of-sight, [4]. In [5] and [6] decentralized controllers were used for concurrently mov- ing toward goal destinations while maintaining communica- tion constraints by maintaining line-of-sight and assuming static communication/sensor ranges respectively. Coordina- tion strategies based on inter-agent signal strength include [7], [8], and [9]. In [10], low-level reactive controllers capa- ble of responding to changes in signal strength or estimated available bandwidth are used to constrain robots’ movements in surveillance and reconnaissance tasks. Although much of the recent works have focused on the effects of communi- cation maintenance on navigation, few have addressed the issue of communication maintenance in tasks such as col- laborative/collective localization and data fusion where team connectivity is essential to the team’s ability to accomplish the given task. Previous works in collaborative target localization include [11], [12], and [13] where strategies such as maintaining