Programming by Demonstration on Riemannian Manifolds
Creators: name: ZEESTRATEN, MARTINUS
Publisher: name: Universit� degli studi di Genova
Year: 2018

description: This thesis presents a Riemannian approach to Programming by Demonstration (PbD).
It generalizes an existing PbD method from Euclidean manifolds to Riemannian manifolds.
In this abstract, we review the objectives, methods and contributions of the presented
approach.
OBJECTIVES
PbD aims at providing a user-friendly method for skill transfer between human and
robot. It enables a user to teach a robot new tasks using few demonstrations. In order
to surpass simple record-and-replay, methods for PbD need to ?understand? what to
imitate; they need to extract the functional goals of a task from the demonstration data.
This is typically achieved through the application of statisticalmethods.
The variety of data encountered in robotics is large. Typical manipulation tasks involve
position, orientation, stiffness, force and torque data. These data are not solely
Euclidean. Instead, they originate from a variety of manifolds, curved spaces that are
only locally Euclidean. Elementary operations, such as summation, are not defined on
manifolds. Consequently, standard statistical methods are not well suited to analyze
demonstration data that originate fromnon-Euclidean manifolds. In order to effectively
extract what-to-imitate, methods for PbD should take into account the underlying geometry
of the demonstration manifold; they should be geometry-aware.
Successful task execution does not solely depend on the control of individual task
variables. By controlling variables individually, a task might fail when one is perturbed
and the others do not respond. Task execution also relies on couplings among task variables.
These couplings describe functional relations which are often called synergies. In
order to understand what-to-imitate, PbDmethods should be able to extract and encode
synergies; they should be synergetic.
In unstructured environments, it is unlikely that tasks are found in the same scenario
twice. The circumstances under which a task is executed?the task context?are more
likely to differ each time it is executed. Task context does not only vary during task execution,
it also varies while learning and recognizing tasks. To be effective, a robot should
be able to learn, recognize and synthesize skills in a variety of familiar and unfamiliar
contexts; this can be achieved when its skill representation is context-adaptive.
THE RIEMANNIAN APPROACH
In this thesis, we present a skill representation that is geometry-aware, synergetic and
context-adaptive. The presented method is probabilistic; it assumes that demonstrations
are samples from an unknown probability distribution. This distribution is approximated
using a Riemannian GaussianMixtureModel (GMM).
Instead of using the ?standard? Euclidean Gaussian, we rely on the Riemannian Gaussian?
a distribution akin the Gaussian, but defined on a Riemannian manifold. A Riev
mannian manifold is a manifold?a curved space which is locally Euclidean?that provides
a notion of distance. This notion is essential for statistical methods as such methods
rely on a distance measure. Examples of Riemannian manifolds in robotics are: the
Euclidean spacewhich is used for spatial data, forces or torques; the spherical manifolds,
which can be used for orientation data defined as unit quaternions; and Symmetric Positive
Definite (SPD) manifolds, which can be used to represent stiffness and manipulability.
The Riemannian Gaussian is intrinsically geometry-aware. Its definition is based on
the geometry of the manifold, and therefore takes into account the manifold curvature.
In robotics, the manifold structure is often known beforehand. In the case of PbD, it follows
from the structure of the demonstration data. Like the Gaussian distribution, the
Riemannian Gaussian is defined by a mean and covariance. The covariance describes
the variance and correlation among the state variables. These can be interpreted as local
functional couplings among state variables: synergies. This makes the Riemannian
Gaussian synergetic. Furthermore, information encoded in multiple Riemannian Gaussians
can be fused using the Riemannian product of Gaussians. This feature allows us to
construct a probabilistic context-adaptive task representation.
CONTRIBUTIONS
In particular, this thesis presents a generalization of existing methods of PbD, namely
GMM-GMR and TP-GMM. This generalization involves the definition ofMaximum Likelihood
Estimate (MLE), Gaussian conditioning and Gaussian product for the Riemannian
Gaussian, and the definition of ExpectationMaximization (EM) and GaussianMixture
Regression (GMR) for the Riemannian GMM. In this generalization, we contributed
by proposing to use parallel transport for Gaussian conditioning. Furthermore, we presented
a unified approach to solve the aforementioned operations using aGauss-Newton
algorithm. We demonstrated how synergies, encoded in a Riemannian Gaussian, can be
transformed into synergetic control policies using standard methods for LinearQuadratic
Regulator (LQR). This is achieved by formulating the LQR problem in a (Euclidean) tangent
space of the Riemannian manifold. Finally, we demonstrated how the contextadaptive
Task-Parameterized Gaussian Mixture Model (TP-GMM) can be used for context
inference?the ability to extract context from demonstration data of known tasks.
Our approach is the first attempt of context inference in the light of TP-GMM. Although
effective, we showed that it requires further improvements in terms of speed and reliability.
The efficacy of the Riemannian approach is demonstrated in a variety of scenarios.
In shared control, the Riemannian Gaussian is used to represent control intentions of a
human operator and an assistive system. Doing so, the properties of the Gaussian can
be employed to mix their control intentions. This yields shared-control systems that
continuously re-evaluate and assign control authority based on input confidence. The
context-adaptive TP-GMMis demonstrated in a Pick &amp; Place task with changing pick and
place locations, a box-taping task with changing box sizes, and a trajectory tracking task
typically found in industry