Exploration and patrolling are central themes in distributed robotics. These deployment scenarios have deep fundamental importance in robotics, beyond the most obvious direct applications, as they can be used to model a wider range of seemingly unrelated deployment objectives. Deploying a group of robots, or any type of agent in general, to explore or patrol in dynamic or unknown environments presents us with some fundamental conceptual steps. Regardless of the problem domain or application, we are required to (a) understand the environment that the agents are being deployed in; (b) encode the task as a set of constraints and guarantees; and (c) derive an effective deployment strategy for the operation of the agents. This thesis presents a coherent treatment of these steps at the theoretical and practical level. First, we address the problem of obtaining a concise description of a physical environment for robotic exploration. Specifically, we aim to determine the number of robots required to be deployed to clear an environment using non-recontaminating exploration. We introduce the medial axis as a configuration space and derive a mathematical representation of a continuous environment that captures its underlying topology and geometry. We show that this representation provides a concise description of arbitrary environments, and that reasoning about points in this representation is equivalent to reasoning about robots in physical space. We leverage this to derive a lower bound on the number of required pursuers. We provide a transformation from this continuous representation into a symbolic representation. We then present a Markov-based model that captures a pickup and delivery (PDP) problem on a general graph. We present a mechanism by which a group of robots can be deployed to patrol the graph in order to fulfill specific service tasks. In particular, we examine the problem in the context of urban transportation, and establish a model that captures the operation of a fleet of taxis in response to incident customer arrivals throughout the city. We consider three different evaluation criteria: minimizing the number of transportation resources for urban planning; minimizing fuel consumption for the drivers; and minimizing customer waiting time to increase the overall quality of service. Finally, we present two deployment algorithms for multi-robot exploration and patrolling. The first is a generalized pursuit-evasion algorithm. Given an environment we can compute how many pursuers we need, and generate an optimal pursuit strategy that will guarantee the evaders are detected with the minimum number of pursuers. We then present a practical patrolling policy for a general graph. We evaluate our policy using real-world data, by comparing against the actual observed redistribution of taxi drivers in Singapore. Through large-scale simulations we show that our proposed deployment strategy is stable and improves substantially upon the default unmanaged redistribution of taxi drivers in Singapore.
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