  Many of the tools available for robot learning were designed for Euclidean
data. However, many applications in robotics involve manifold-valued data. A
common example is orientation; this can be represented as a 3-by-3 rotation
matrix or a quaternion, the spaces of which are non-Euclidean manifolds. In
robot learning, manifold-valued data are often handled by relating the manifold
to a suitable Euclidean space, either by embedding the manifold or by
projecting the data onto one or several tangent spaces. These approaches can
result in poor predictive accuracy, and convoluted algorithms. In this paper,
we propose an "intrinsic" approach to regression that works directly within the
manifold. It involves taking a suitable probability distribution on the
manifold, letting its parameter be a function of a predictor variable, such as
time, then estimating that function non-parametrically via a "local likelihood"
method that incorporates a kernel. We name the method kernelised likelihood
estimation. The approach is conceptually simple, and generally applicable to
different manifolds. We implement it with three different types of
manifold-valued data that commonly appear in robotics applications. The results
of these experiments show better predictive accuracy than projection-based
algorithms.
