Recent research in robot exploration and mapping has focused on sampling hotspot fields, which often arise in environmental and ecological sensing applications. Such a hotspot field is characterized by continuous, positively skewed, spatially correlated measurements with the hotspots exhibiting extreme measurements and much higher spatial variability than the rest of the field.;To map a hotspot field of the above characterization, we assume that it is realized from non-parametric probabilistic models such as the Gaussian and log-Gaussian processes (respectively, GP and ℓGP), which can provide formal measures of map uncertainty. To learn a hotspot field map, the exploration strategy of a robot team then has to plan resource-constrained observation paths that minimize the uncertainty of a spatial model of the hotspot field. This exploration problem is formalized in a sequential decision-theoretic planning under uncertainty framework called the multi-robot adaptive sampling problem (MASP). So, MASP can be viewed as a sequential, non-myopic version of active learning.' In contrast to finite-state Markov decision problems, MASP adopts a more complex but realistic continuous-state, non-Markovian problem structure so that its induced exploration policy can be informed by the complete history of continuous, spatially correlated observations for selecting paths. It is unique in unifying formulations of non-myopic exploration problems along the entire adaptivity spectrum, thus subsuming existing non-adaptive formulations and allowing the performance advantage of a more adaptive policy to be theoretically realized. Through MASP, it is demonstrated that a more adaptive strategy can exploit clustering phenomena in a hotspot field to produce lower expected map uncertainty. By measuring map uncertainty using the mean-squared error criterion, a MASP-based exploration strategy consequently plans adaptive observation paths that minimize the expected posterior map error or equivalently, maximize the expected map error reduction.;One advantage stemming from the reward-maximizing dual formulations of MASP and iMASP is that they allow observation selection properties of the induced exploration policies to be realized for sampling the hotspot field. These properties include adaptivity, hotspot sampling, and wide-area coverage. We show that existing GP-based exploration strategies may not explore and map the hotspot field well with the selected observations because they are non-adaptive and perform only wide-area coverage. In contrast, the ℓGP-based exploration policies can learn a high-quality hotspot field map because they are adaptive and perform both wide-area coverage and hotspot sampling.;The other advantage is that even though MASP and iMASP are non-trivial to solve due to their continuous state components, the convexity of their reward-maximizing duals can be exploited to derive, in a computationally tractable manner, discrete-state monotone-bounding approximations and subsequently, approximately optimal exploration policies with theoretical performance guarantees. Anytime algorithms based on approximate MASP and iMASP are then proposed to alleviate the computational difficulty that arises from their non-Markovian structure.;It is of practical interest to be able to quantitatively characterize the "hotspotness" of an environmental field. We propose a novel "hotspotness" index, which is defined in terms of the spatial correlation properties of the hotspot field. As a result, this index can be related to the intensity, size, and diffuseness of the hotspots in the field.;We also investigate how the spatial correlation properties of the hotspot field affect the performance advantage of adaptivity. In particular, we derive sufficient and necessary conditions of the spatial correlation properties for adaptive exploration to yield no performance advantage.;Lastly, we develop computationally efficient approximately optimal exploration strategies for sampling the GP by assuming the Markov property in iMASP planning. We provide theoretical guarantees on the performance of the Markov-based policies, which improve with decreasing spatial correlation. We evaluate empirically the effects of varying spatial correlations on the mapping performance of the Markov-based policies as well as whether these Markov-based path planners are time-efficient for the transect sampling task. (Abstract shortened by UMI.).
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