  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.
