Hierarchical Gaussian models for wind field estimation and path planning

Abstract

Improvements in technology, autonomy, and positioning mechanisms have greatly broadened the range of application of unmanned aerial vehicles. These vehicles are now being used in aerial photography, package delivery, infrastructure inspection, and many other areas. Many of these uses demand new techniques for path planning in complex environments-in particular, spatially heterogeneous and time-evolving wind fields [22, 23, 24]. Navigating and planning [26, 25, 28, 12] in wind fields requires reliable and fast predictive models that quantify uncertainty in future wind velocities, and benefits strongly from the ability to incorporate onboard and external wind field measurements in real time. To make real-time inference and prediction possible, we construct simple hierarchical Gaussian models of the wind field as follows. Given realizations of the wind field over a domain of interest, obtained from detailed offline measurements or computational fluid dynamic simulations, we extract empirical estimates of the mean and covariance functions. The associated covariance matrices are anisotropic and non-stationary, and capture interactions among the wind vectors at all points in a discretization of the domain. We make the further assumption that, given a particular prevailing wind heading, the local wind velocities are jointly Gaussian. The result is a hierarchical Gaussian model in which the mean and covariance are functions of the prevailing wind conditions. Since these empirical covariances are known only for a few prevailing wind conditions, we close our model by interpolating covariance matrices on the appropriate manifold of positive semi-definite matrices [44], via a computationally efficient construction that takes advantage of low-rank structure. Finally, assimilation of successive point observations is conducted by embedding a standard Kalman filter within a hierarchical Bayesian inference framework. This representation will then be used for wind field exploitation.