The subject of this work is the patrolling of an environment with the aid of a team of autonomous agents. We consider both the design of open-loop trajectories with optimal properties, and of distributed control laws converging to optimal trajectories. As performance criteria, the refresh time and the latency are considered, i.e., respectively, time gap between any two visits of the same region, and the time necessary to inform every agent about an event occurred in the environment. We associate a graph with the environment, and we study separately the case of a chain, tree, and cyclic graph. For the case of chain graph, we first describe a minimum refresh time and latency team trajectory, and we propose a polynomial time algorithm for its computation. Then, we describe a distributed procedure that steers the robots toward an optimal trajectory. For the case of tree graph, a polynomial time algorithm is developed for the minimum refresh time problem, under the technical assumption of a constant number of robots involved in the patrolling task. Finally, we show that the design of a minimum refresh time trajectory for a cyclic graph is NP-hard, and we develop a constant factor approximation algorithm. The recent development in the autonomy and the capabilities of mobile robots greatly increases the number of applications suitable for a team of autonomous agents. Particular interest has been received by those tasks requiring continual execution, such as the monitoring of oil spills [1], the detection of forest fires [2], the track of border changes [3] , and the patrol (surveillance) of an environment [4] . The surveillance of an area of interest requires the robots to continuously and repeatedly travel the environment, and the challenging problem consists in scheduling the robots trajectories so as to optimize a certain performance criteria. The reader familiar with network location, multiple traveling salesman, or graph exploration problems may observe a close connection with the patrolling problem we address, e.g., see [5] , [6] , [7] . It is worth noting, however, that these classical optimization problems do not capture the repetitive, and hence dynamic, aspect of the patrolling problem, nor the synchronization issues that arise when a timing among the visits of certain zones is required. A precise formulation of the patrolling problem requires the characterization of the robots capabilities, of the environment to be patrolled, and of the performance criteria. In this work, we assume the robots to be identical and capable of sensing and communicating within a certain spatial range, and of moving according to a first order integrator dynamics with bounded speed. We represent the environment as a graph, in which the vertices correspond to physical and strategically important locations, and in which the edges denote the possibility of moving and communicating between locations. We assume that, when a robot is placed at each of the graph vertices, the union of the sensor footprints provides complete sensor coverage of the environment. Regarding the performance criteria of a patrolling trajectory, we consider (i) the time gap between any two visits of the same region, called refresh time, and (ii) the time needed to inform the team of robots about an event occurred in the environment, called latency. Loosely speaking, refresh time and latency reflect the effectiveness of a patrolling team in detecting events in the environment and in organizing remedial actions. For both the refresh time and latency optimization problem, we focus
In REF , the computational complexity of the patrolling problem is studied as a function of the environment topology, and optimal strategies as well as constant-factor approximations are proposed.
On cooperative patrolling: optimal trajectories, complexity analysis, and approximation algorithms
IEEE Transaction on Robotics, vol. 28, issue 3, pp. 592-606, 06/2012
Mathematics Computer Science