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In this work, we analyze \textit{stochastic coverage schemes} (SCS) forrobotic swarms in which the robots randomly attach to a one-dimensionalboundary of interest using local communication and sensing, without relying onglobal position information or a map of the environment. Robotic swarms may berequired to perform boundary coverage in a variety of applications, includingenvironmental monitoring, collective transport, disaster response, andnanomedicine. We present a novel analytical approach to computing and designingthe statistical properties of the communication and sensing networks that areformed by random robot configurations on a boundary. We are particularlyinterested in the event that a robot configuration forms a connectedcommunication network or maintains continuous sensor coverage of the boundary.Using tools from order statistics, random geometric graphs, and computationalgeometry, we derive formulas for properties of the random graphs generated byrobots that are independently and identically distributed along a boundary. Wealso develop order-of-magnitude estimates of these properties based on Poissonapproximations and threshold functions. For cases where the SCS generates auniform distribution of robots along the boundary, we apply our analyticalresults to develop a procedure for computing the robot population size,diameter, sensing range, or communication range that yields a randomcommunication network or sensor network with desired properties.

Design of Stochastic Robotic Swarms for Target Performance Metrics in Boundary Coverage Tasks