IEEE Robotics & Automation Magazine - June 2020 - 37

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to represent end-effector (tool-tip) orientations [4], while
S 2 can be employed to signify a unit directional vector
that is perpendicular to surfaces (e.g., for contact planning). Articulatory joints can be represented on the torus
S 1 # S 1 # f # S 1 [12], [13]. The Kendall shape space
used to encode 3D skeletal motion-capture data also relies
on unit spheres [14].
The SO group: SO (d) is the group of rotations around the
origin in a d-dimensional space. SO(2) and SO(3) are
widely used in robotics. For example, in [15], the manifold
structure of the rotation group is exploited for preintegration and uncertainty propagation in SO(3). This is used for
state estimation in visual-inertial odometry with mobile
robots. In [16], Kalman filtering adapted to data in SO(3) is
used for estimating the attitude of robots that can rotate in
space. The optimization problem in [17] employs sequential quadratic programming working directly on the manifold SO (3) # R 3 .
The SE(3): SE(3) is the group of rigid body transformations. A rigid body transformation is composed of a
rotation and a translation. The geometry of SE(3) can
be used to describe the kinematics and Jacobian of
robots [2]. Therefore, it is widely used to describe
robot motion and pose estimation [18]. For example,
in [3], exponential maps are exploited to associate
uncertainty with SE(3) data points in robot pose estimation problems.
The manifold of SPD matrices: S d++ can be employed
in various ways in robotics. For example, human-
robot collaboration applications require the use of
various sensors. These sensory data can be preprocessed with sliding windows to analyze at each time
step the evolution of the signals within a short time
window (e.g., to analyze data flows). Often, such analysis takes the form of spatial covariances, which are
SPD matrices [5]. In robot control, tracking gains can
be defined in the form of SPD matrices. The use of
tracking gains as full SPD matrices instead of scalars
has the advantage of enabling the controller to take
into account the coordination of different control
variables (e.g., motor commands). For articulatory
joints, these coordinations often relate to characteristic synergies in human movements. Manipulability
ellipsoids are representations used to analyze and
control robot dexterity as a function of the articulatory joints configuration. This descriptor can be
designed according to different task requirements,
such as tracking a desired position and applying a
specific force [8], [19]. Manipulator inertia matrices
also belong to S d++ and can, for example, be exploited
in human-like trajectory planning [13]. SPD matrices
are also used in problems related to metric interpolation/extrapolation and metric learning [20]. In covariant Hamiltonian motion planning [21], a precision
matrix (metric tensor) is used to prefer perturbations
resulting in small accelerations in the overall

trajectory. In [22], Riemannian manifold policies are
employed to generate natural obstacle-avoidance
reaching motion through traveling along geodesics of
curved spaces defined by the presence of obstacles.
d
● Hyperbolic manifolds: H are the analogs of spheres with
constant negative curvature instead of constant positive
curvature. They are currently underexploited in robotics
despite their interesting potential in a wide range of representations, including dynamical systems, Toeplitz/Hankel
matrices, and autoregressive models [23]. Notably, hyperbolic geometry could be used to encode and visualize heterogeneous topology data, including graphs and trees
structures, such as rapidly exploring random trees [24],
designed to efficiently search nonconvex, high-dimensional spaces in motion planning by randomly building a
space-filling tree. The interesting property of hyperbolic
manifolds is that the circumference of a circle grows exponentially with its radius, which means that exponentially
more space is available with increasing distance. It provides
a convenient representation for hierarchies, which tend to
expand exponentially with depth.
d, p
● The Grassmannian: G
is the manifold of all p-dimensional subspaces of R d . It can, for example, be used to
extract and cluster planar surfaces in the robot's 3D environment. This manifold is largely underrepresented in
robotics despite the fact that such a structure can be used
in various approaches, such as system identification [25],
spatiotemporal modeling of human gestures [26], and the
encoding of nullspaces and projection operators in a probabilistic way.
● Manifolds with nonconstant curvature: These are also
employed in robotics, such as spaces endowed with the
Fisher information metric [27], [28] or kinetic energy
metric [1], [12], [13], [18]. As described in the "Riemannian Manifolds" section, the curvature of a Riemannian
manifold depends on the selected metric tensor. Consequently, a varying metric will result in a changing curvature. Many problems in robotics can be formulated with a
smoothly varying matrix M (Riemannian metric) that
measures the distance between two points x 1 and x 2 as a
quadratic error term (x 1 - x 2) < M (x 1 - x 2). In this context, the Riemannian formulation has the advantage of
being coordinate independent (i.e., geodesic paths are
invariant to the choice of local coordinates) [1], [12],
[13]. In robot dynamics problems, this is typically useful
for taking into account the inertia in the robot motion
[1]. In policy-learning problems, if the conditional density of the action given the state is Gaussian, the natural
policy gradient is given by the Fisher information matrix
[28], which can, for example, be used in deep reinforcement learning [29].
The use of a Riemannian metric is also relevant for deep
generative models, such as variational autoencoders (VAEs)
and generative adversarial networks, as it provides a geometric interpretation of these models. For example, VAEs learn
nonlinear data distributions through a set of latent variables
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IEEE Robotics & Automation Magazine - June 2020

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