Journal of Intelligent and Robotic Systems 39: 43–55, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 43 Stability and Control of Collective Systems R. D. ROBINETT III Energy & Transportation Security Center, Sandia National Laboratories, ⋆ P.O. Box 5800, MS 0741, Albuquerque, NM 87185-0741, U.S.A.; e-mail: rdrobin@sandia.gov J. E. HURTADO Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, U.S.A.; e-mail: jehurtado@tamu.edu Abstract. In this paper, decentralized, distributed feedback control laws are presented for coop- erative robotic systems whose task is to localize unknown sources. The control laws follow from a second order representation of the source field. The stability of the proposed feedback control laws for the individual robots, and for the entire robot collective, is demonstrated using Lyapunov’s direct method and a vector Lyapunov approach. Additional feedback control laws are proposed to achieve an additional level of coordination. In particular, control laws that achieve desired formations surrounding a localized source are developed. Key words: cooperative robots, Lyapunov analysis, decentralized control, distributed systems. 1. Introduction In this paper decentralized, distributed feedback control laws are developed for cooperative robotic systems whose task is to localize unknown sources. Some practical applications and scenarios include a team of robots cooperating to lo- cate buried land mines or chemical plume sources. It is assumed that the robots collect information about the source through a set of sensors and the information is shared among the robots through a communication network. This represents the “distributed” portion of the feedback control laws. Each robot then uses the information it gathers and receives to autonomously determine its position update. This represents the “decentralized” portion of the feedback control laws. The goal is to design stable control laws for the individual members of a robot collective such that the entire collective behaves in a stable manner while accomplishing the task of localizing unknown sources. There have been many studies recently involving cooperative control of robot collectives, or multi-agent systems. One particular area of interest is in formation control. One approach to formation control utilizes the concept of virtual leaders (or virtual vehicles or virtual structures) [4, 11, 12]. Virtual leaders are nonsta- ⋆ Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04- 94AL85000.