Automatica 43 (2007) 1954 – 1960 www.elsevier.com/locate/automatica Brief paper Generalization of nonlinear cyclic pursuit  A. Sinha ∗ , D. Ghose GCDSL, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India Received 16 July 2006; received in revised form 10 December 2006; accepted 16 March 2007 Available online 23 August 2007 Abstract In this paper, some generalizations of the problem of formation of a group of autonomous mobile agents under nonlinear cyclic pursuit is studied. Cyclic pursuit is a simple distributed control law, in which the agent i pursues agent i + 1 modulo n. Each agent is subjected to a nonholonomic constraint. A necessary condition for equilibrium formation to occur among a group of agents with different speeds and different controller gains is obtained. These results generalize equal speed and equal controller gain results available in the literature.  2007 Elsevier Ltd. All rights reserved. Keywords: Autonomous system; Multi-vehicle formations; Cooperative control; Pursuit problems; Decentralized control; Nonlinear dynamical systems 1. Introduction This paper deals with the problem of cyclic pursuit in a multi-vehicle system. Multi-vehicle systems are groups of au- tonomous mobile agents used for search and surveillance tasks, rescue missions, space and oceanic explorations, and other au- tomated collaborative operations. Cyclic pursuit uses simple lo- cal interaction between these vehicles to obtain desired global behavior. The pursuit strategies are designed to mimic the behavior of biological organisms like dogs, birds, ants or beetles. They are commonly referred to as the ‘bugs’ problem. Bruckstein, Cohen, and Efrat (1991) modeled the behavior of ants, crickets and frogs with continuous and discrete pursuit laws and exam- ined the possible evolution of global behavior such as the con- vergence to a point, collision, limit points or periodic motion. Bruckstein (1993) describes the stability and convergence of a group of ants in linear and cyclic pursuit. Convergence to a point in linear pursuit is the starting point to the analysis of achievable global formation among a group of autonomous mobile agents as discussed in Lin, Broucke, and Francis (2004), which also  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +91 80 2293 3023; fax: +91 80 2360 0134. E-mail addresses: asinha@aero.iisc.ernet.in (A. Sinha), dghose@aero.iisc.ernet.in (D. Ghose). 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.03.024 deals with the evolution of the formation of these agents with respect to the possibility of collision. Kinematics of agents with single holonomic constraint is discussed in Marshall, Broucke, and Francis (2004, 2006). The equilibrium and stability of iden- tical agents with this motion constraint is dealt within these papers. The mathematics governing cyclic pursuit are stud- ied in Richardson (2001a,b), Behroozim and Gagnon (1979), Klamkin and Newman (1971), and Bernhard (1959). A general- ization of the linear cyclic pursuit problem has been presented in Sinha and Ghose (2006). This work has been inspired by the problem addressed in Marshall et al. (2004), which considers n identical autonomous mobile agents in cyclic pursuit with nonholonomic constraint. The agents are assumed to have same speed and controller gains. In our paper, a more general case is discussed where the speeds and controller gains for different agents may vary, thus giving rise to a heterogenous system of agents. In such a case, determination of conditions under which the system converges to an equilibrium becomes more complicated. In this paper, we provide a necessary condition for the existence of equilibrium for the general case. Some preliminary results on this were earlier presented in Sinha and Ghose (2005). We would like to point out that, in this paper, we do not address the issue of stability of the equilibrium points. Stabil- ity issues in the case of equal gain and equal speed of all the agents are discussed in Marshall et al. (2004). The system is linearized about an equilibrium point and the local stability is