Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat\'ern Kernels

TL;DR

This paper introduces geometry-aware Bayesian optimization for robotics by leveraging Riemannian Matérn kernels, which respect the underlying manifold structures, leading to improved control and planning performance.

Contribution

It develops methods to implement Riemannian Matérn kernels on robotic manifolds and demonstrates their effectiveness in various robotic optimization tasks.

Findings

Enhanced optimization performance on manifold-based problems

Effective kernel implementation on spheres and rotation groups

Improved control and motion planning results

Abstract

Bayesian optimization is a data-efficient technique which can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics. Many of these problems require optimization of functions defined on non-Euclidean domains like spheres, rotation groups, or spaces of positive-definite matrices. To do so, one must place a Gaussian process prior, or equivalently define a kernel, on the space of interest. Effective kernels typically reflect the geometry of the spaces they are defined on, but designing them is generally non-trivial. Recent work on the Riemannian Mat\'ern kernels, based on stochastic partial differential equations and spectral theory of the Laplace-Beltrami operator, offers promising avenues towards constructing such geometry-aware kernels. In this paper, we study techniques for implementing these kernels on manifolds of interest in robotics, demonstrate their performance on a set of artificial benchmark functions, and illustrate geometry-aware Bayesian optimization for a variety of robotic applications, covering orientation control, manipulability optimization, and motion planning, while showing its improved performance.