Variational Hilbert regression for terrain modeling and trajectory optimization

Abstract

The ability to generate accurate terrain models is of key importance in a wide variety of robotics tasks, ranging from path planning and trajectory optimization to environment exploration and mining applications. This paper introduces a novel regression methodology for terrain modeling that can approximate arbitrarily complex functions based on a series of simple kernel calculations, using variational Bayesian inference. A sparse feature vector is used to efficiently project input points into a high-dimensional reproducing kernel Hilbert space, according to a set of inducing points automatically generated from clustering available data. Each inducing point maintains its own regression model in addition to individual kernel parameters, and the entire set is iteratively optimized as more data are collected in order to maximize a global variational lower bound. We also show how kernel and regression model parameters can be jointly learned, to achieve a better approximation of the underlying function. Experimental results show that the proposed methodology consistently outperforms current state-of-the-art techniques, while producing a continuous model with a fully probabilistic treatment of uncertainties, well-defined gradients, and highly scalable to large-scale datasets. As a practical application of the proposed terrain modeling technique, we explore the problem of trajectory optimization, deriving gradients that allow the efficient generation of continuous paths using standard optimization algorithms, minimizing a series of useful properties (i.e. distance traveled, changes in elevation, and terrain variance).