Spatial Interpolation for Robotic Sampling: Uncertainty with Two Models of Variance

Several important forms of robotic environmental monitoring involve estimating a spatial field from comparatively few measurements. A number of researchers use linear least squares estimation techniques, frequently either the geostatistical Kriging framework or a Gaussian Process regression formulation, that provide estimates of quantities of interest at unmeasured locations. These methods enable selection of sample locations (e.g., for adaptive sampling) by quantifying uncertainty across the scalar field. This paper assesses the role of pose uncertainty and measurement error on variance of the estimated spatial field. We do this through a systematic empirical comparison of scalar fields reconstructed from measurements taken with our robot using multiple imperfect sensors and actively estimating its pose. We implement and compare two models of variance: Kriging Variance (KV) and Interpolation Variance (IV), illustrating that the latter —which has not been used in a robotics context before— has several advantages when used for online planning of sampling tasks. Using two separate experimental scenarios, we assess the estimated variance in scalar fields constructed from measurements taken by robots. Physical robots sampling within our office building suggest that using IV to select sampling sites gathers more data for a given time window (45% more than KV), travels a shorter distance to collect the same number of samples (25% less than KV), and has a promising speed-up with multiple robots. Water quality data from an Autonomous Underwater Vehicle survey of Lake Pleasant, AZ. also show that IV produces better qualities for given a distance and time.