268 research outputs found
Riemannian Gaussian distributions on the space of positive-definite quaternion matrices
Recently, Riemannian Gaussian distributions were defined on spaces of
positive-definite real and complex matrices. The present paper extends this
definition to the space of positive-definite quaternion matrices. In order to
do so, it develops the Riemannian geometry of the space of positive-definite
quaternion matrices, which is shown to be a Riemannian symmetric space of
non-positive curvature. The paper gives original formulae for the Riemannian
metric of this space, its geodesics, and distance function. Then, it develops
the theory of Riemannian Gaussian distributions, including the exact expression
of their probability density, their sampling algorithm and statistical
inference.Comment: 8 pages, submitted to GSI 201
Geometric Reinforcement Learning For Robotic Manipulation
Reinforcement learning (RL) is a popular technique that allows an agent to
learn by trial and error while interacting with a dynamic environment. The
traditional Reinforcement Learning (RL) approach has been successful in
learning and predicting Euclidean robotic manipulation skills such as
positions, velocities, and forces. However, in robotics, it is common to
encounter non-Euclidean data such as orientation or stiffness, and failing to
account for their geometric nature can negatively impact learning accuracy and
performance. In this paper, to address this challenge, we propose a novel
framework for RL that leverages Riemannian geometry, which we call Geometric
Reinforcement Learning (G-RL), to enable agents to learn robotic manipulation
skills with non-Euclidean data. Specifically, G-RL utilizes the tangent space
in two ways: a tangent space for parameterization and a local tangent space for
mapping to a nonEuclidean manifold. The policy is learned in the
parameterization tangent space, which remains constant throughout the training.
The policy is then transferred to the local tangent space via parallel
transport and projected onto the non-Euclidean manifold. The local tangent
space changes over time to remain within the neighborhood of the current
manifold point, reducing the approximation error. Therefore, by introducing a
geometrically grounded pre- and post-processing step into the traditional RL
pipeline, our G-RL framework enables several model-free algorithms designed for
Euclidean space to learn from non-Euclidean data without modifications.
Experimental results, obtained both in simulation and on a real robot, support
our hypothesis that G-RL is more accurate and converges to a better solution
than approximating non-Euclidean data.Comment: 14 pages, 14 figures, journa
Programming by Demonstration on Riemannian Manifolds
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 \u2018understand\u2019 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\u2014the task context\u2014are 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 \u2018standard\u2019 Euclidean Gaussian, we rely on the Riemannian Gaussian\u2014
a distribution akin the Gaussian, but defined on a Riemannian manifold. A Riev
mannian manifold is a manifold\u2014a curved space which is locally Euclidean\u2014that 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\u2014the 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 & Place task with changing pick and
place locations, a box-taping task with changing box sizes, and a trajectory tracking task
typically found in industr
Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach
This work aims to prove that the classical Gaussian kernel, when defined on a
non-Euclidean symmetric space, is never positive-definite for any choice of
parameter. To achieve this goal, the paper develops new geometric and
analytical arguments. These provide a rigorous characterization of the
positive-definiteness of the Gaussian kernel, which is complete but for a
limited number of scenarios in low dimensions that are treated by numerical
computations. Chief among these results are the L-Godement theorems (where ), which provide
verifiable necessary and sufficient conditions for a kernel defined on a
symmetric space of non-compact type to be positive-definite. A celebrated
theorem, sometimes called the Bochner-Godement theorem, already gives such
conditions and is far more general in its scope, but is especially hard to
apply. Beyond the connection with the Gaussian kernel, the new results in this
work lay out a blueprint for the study of invariant kernels on symmetric
spaces, bringing forth specific harmonic analysis tools that suggest many
future applications
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