Automatica 48 (2012) 1077–1087 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Self-triggered coordination of robotic networks for optimal deployment ✩ C. Nowzari 1 , J. Cortés Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA, 92093, USA article info Article history: Received 2 January 2011 Received in revised form 11 October 2011 Accepted 28 October 2011 Available online 9 April 2012 Keywords: Robotic networks Self-triggered control Spatial partitioning Outdated information Set-valued analysis abstract This paper studies a deployment problem for a group of robots where individual agents operate with outdated information about each other’s locations. Our objective is to understand to what extent outdated information is still useful and at which point it becomes essential to obtain new, up-to-date information. We propose a self-triggered coordination algorithm based on spatial partitioning techniques with uncertain information. We analyze its correctness in synchronous and asynchronous scenarios, and establish the same convergence guarantees that a synchronous algorithm with perfect information at all times would achieve. The technical approach combines computational geometry, set-valued stability analysis, and event-based systems. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction This paper studies a robotic sensor network performing an optimal static deployment task when individual agents do not have up-to-date information about each other’s locations. Our objective is to design a self-triggered coordination algorithm where agents autonomously decide when they need new, up-to-date location information in order to successfully perform the required task. Our motivation comes from the need for strategies that naturally account for uncertainty in the state of other agents and are able to produce substantial energy savings in the operation of the network. Literature review. There are two main areas related to the contents of this paper. In the context of robotic sensor networks, this work builds on (Cortés, Martínez, Karatas, & Bullo, 2004), where dis- tributed algorithms based on centroidal Voronoi partitions are pre- sented, and (Cortés, Martínez, & Bullo, 2005), where limited-range interactions are considered. Other works on deployment cover- age problems include (Howard, Matarić, & Sukhatme, 2002; Kwok & Martínez, 2010; Pavone, Arsie, Frazzoli, & Bullo, 2011; Schwa- ger, Rus, & Slotine, 2009). We note that the locational optimization problem considered here is a static coverage problem, in contrast ✩ The material in this paper was partially presented at the 2011 American Control Conference, June 29–July 1, 2011, San Francisco, California, USA. This paper was recommended for publication in revised form by Associate Editor C.C. Cheah under the direction of Editor Toshiharu Sugie. E-mail addresses: cnowzari@ucsd.edu (C. Nowzari), cortes@ucsd.edu (J. Cortés). 1 Tel.: +1 858 822 7930; fax: +1 858 822 3107. to dynamic coverage problems, e.g., (Choset, 2001; Hussein & Sti- panovi ` c, 2007), that seek to visit or continuously sense all points in the environment. A feature of the algorithms mentioned above is the common assumption of constant communication among agents and up-to-date information about each other’s locations. The other area of relevance to this work is discrete-event systems (Cassandras & Lafortune, 2007), and the research in trig- gered control (Anta & Tabuada, 2010; Subramanian & Fekri, 2006; Velasco, Marti, & Fuertes, 2003; Wang & Lemmon, 2009), partic- ularly as related to sensor and actuator networks. Of particular relevance are works that study self-triggered or event-triggered decentralized strategies that are based on local interactions with neighbors defined in an appropriate graph. Among them, we highlight (Kang, Yan, & Bitmead, 2008) on collision avoidance while performing point-to-point reconfiguration, (Dimarogonas & Johansson, 2009) on achieving agreement, (Wan & Lemmon, 2009) on distributed optimization, and (Mazo & Tabuada, 2011) on imple- menting nonlinear controllers over sensor and actuator networks. This paper shares with these works the aim of trading computa- tion and decision making at the agent level for less communication, sensing or actuator effort while still guaranteeing a desired level of performance. Statement of contributions. The main contribution of the paper is the design of the self-triggered centroid algorithm to achieve optimal static deployment in a given convex environment. This strategy is based on two building blocks. The first building block is an update policy that helps an agent determine if the information it possesses about the other agents is sufficiently up-to-date. This update policy is based on spatial partitioning techniques with uncertain information, and in particular, on the 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.03.009