IEEE Robotics & Automation Magazine - June 2020 - 35

target point on the manifold within a desired precision matrix,
represented here as a covariance matrix (the inverse of precision matrix) in the tangent space of the target point. In the
model predictive control (MPC) example of Figure 1(d), a reference path is defined as a set of Gaussians acting as via points
to pass through (i.e., within desired covariances). This GMM
is first learned from a set of demonstrated reference paths
(gray lines). The resulting controller computes a series of commands anticipating the next points to reach, resulting in a path
on the manifold (black lines).
The problems depicted in Figure 1 require data to be
handled in a probabilistic manner. For Euclidean data, multivariate Gaussian distributions are typically considered to
encode either the (co)variations of the data or the uncertainty of the estimates. This article discusses how these
approaches can be extended to other manifolds by exploiting a Riemannian extension of Gaussian distributions. A
practitioner perspective is adopted, with the goal of conveying the main intuitions behind the presented algorithms,
sometimes at the expense of a more rigorous treatment of
each topic. Didactic source codes accompany the article,
available as part of PbDlib [9], a collection of source codes
for robot programming by demonstration (learning from
demonstration), including various functionalities for statistical learning, dynamical systems, optimal control, and
Riemannian geometry. Two distinct versions are maintained, and they can be used independently in MATLAB
(with full GNU Octave compatibility) or C++. Scalars are
denoted by lowercase letters (x), vectors by boldface lowercase letters (x), and matrices by boldface uppercase letters
(X), where X < is the transpose of X. Manifolds and tangent spaces are designated by the calligraphic letters M
and TM, respectively.
Riemannian Manifolds
A smooth d-dimensional manifold M is a topological
space that locally behaves like the Euclidean space R d . A

Riemannian manifold is a smooth and differentiable manifold
equipped with a PD metric tensor. For each point p ! M,
there exists a tangent space T p M that locally linearizes the
manifold. On a Riemannian manifold, the metric tensor
induces a PD inner product on each tangent space T p M,
which enables vector lengths and the angles between vectors
to be measured. The affine connection, computed from the
metric, is a differential operator that provides, among other
functionalities, a way to compute geodesics and transport vectors on tangent spaces along any smooth curves on the manifold [10], [11]. It also fully characterizes the intrinsic
curvature and torsion of the manifold. The Cartesian product
of two Riemannian manifolds is also a Riemannian manifold
(often called manifold bundles or manifold composites), which
enables joint distributions to be constructed on any combination of Riemannian manifolds. Two basic notions of Riemannian geometry are crucial for robot learning and adaptive
control applications. They are illustrated in Figure 2 and
described as follows:
● Geodesics: The minimum-length curves between two
points on a Riemannian manifold are called geodesics. Similar to straight lines in the Euclidean space, the second
derivative is zero everywhere along a geodesic. The exponential map Exp x0 : Tx0 M " M charts a point u in the
tangent space of x 0 to a point x on the manifold so that x
lies on the geodesic starting at x 0 in the direction u.
The norm of u is equal to the geodesic distance between x 0 and x. The inverse map is called the logarithmic map, Log x0 : M " Tx0 M. Figure 2(a) depicts these
mapping functions.
● Parallel transport: Parallel transport C g " h : T g M " Th M
moves vectors between tangent spaces such that the inner
product between two vectors in a tangent space is conserved. It employs the notion of connection, defining how
to associate vectors between infinitesimally close tangent
spaces. This connection enables the smooth transport of a
vector from one tangent space to another by sliding it (with

u ∈ Tx0M
x0 ∈ M
u = Logx0(x )

x = Expx0(u )
x∈M

(a)

u = Γg →h (u)
u ∈ Tg M
h ∈M
g ∈M

(b)

Figure 2. Applications in robotics using Riemannian manifolds rely on two well-known principles of Riemannian geometry:
exponential/logarithmic mapping and parallel transport, which are depicted on an S 2 manifold embedded in R 3 . (a) The
bidirectional mappings between the tangent space and manifold. (b) The parallel transport of a vector along a geodesic.

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IEEE ROBOTICS & AUTOMATION MAGAZINE

IEEE Robotics & Automation Magazine - June 2020

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