IEEE Robotics & Automation Magazine - June 2020 - 34

[4], and symmetric positive definite (SPD) matrices, which
can represent sensory data processed as spatial covariances
[5], inertia [6], [7], and stiffness/manipulability ellipsoids [8]
as well as metrics used in the context of similarity measures.
Moreover, many applications require all these heterogeneous data to be handled together. Several robotics techniques employ components from the framework of
Riemannian geometry. But, unfortunately, they are often
implemented without providing an explicit link to this
framework, which can either weaken the links to other techniques or limit the potential extensions. This can, for example, be the case when computing orientation errors with a
logarithmic map in the context of inverse kinematics and
when interpolating between two unit quaternions with
spherical linear interpolation. This article aims to highlight
the links among existing techniques and catalog the missing
links that could be explored in further research. These points
are discussed in the context of varied robot learning and
adaptive control challenges.
Riemannian manifolds are related to a wide range of problems in machine learning and robotics. This article mainly
focuses on robot learning applications based on Gaussian
distributions. This includes techniques requiring that uncertainty and statistical modeling be computed on structured
non-Euclidean data. The article presents an overview of

existing work and further perspectives on jointly exploiting statistics and Riemannian geometry. One appealing use
of Riemannian geometry in robotics is that it provides a
principled and simple way to extend algorithms initially
developed for Euclidean data to other manifolds by efficiently taking into account prior geometric knowledge
about those manifolds.
By using Riemannian manifolds, data of various forms can
be treated in a unified manner with the advantage that existing models and algorithms initially developed for Euclidean
data can be extended to a wider range of data structures. They
can, for example, be used to revisit constrained optimization
problems formulated in the Euclidean space by treating them
as unconstrained problems, inherently taking into account
the geometry of the data.
Figure 1 shows various common problems in robotics that
can directly leverage Riemannian geometry. In Figure 1(a), a
set of data points is clustered as two distributions represented
in red and blue, trained as a Gaussian mixture model (GMM),
where each point is displayed in the color of the most likely
cluster to which it belongs. In Figure 1(b), the intersection of
two distributions (in red and blue) results in a distribution (in
black), which is the equivalent of a product of Gaussians. In
Figure 1(c), a controller is computed by solving a linear quadratic tracking (LQT) problem, with the goal of reaching a

(a)

(b)

(c)

(d)

Figure 1. Several problems in robotics that can leverage the proposed Gaussian-based representation on Riemannian manifolds:
(a) clustering, (b) information fusion, (c) LQT, and (d) MPC. Such an approach can be used to extend clustering, regression, fusion,
control, and planning problems to non-Euclidean data. In these examples, Gaussians are defined with centers on the manifolds and
covariances in the tangent spaces of the centers.

IEEE ROBOTICS & AUTOMATION MAGAZINE

JUNE 2020

IEEE Robotics & Automation Magazine - June 2020

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2020