1,965 research outputs found
The Study Variety of Conformal Kinematics
We introduce the Study variety of conformal kinematics and investigate some
of its properties. The Study variety is a projective variety of dimension ten
and degree twelve in real projective space of dimension 15, and it generalizes
the well-known Study quadric model of rigid body kinematics. Despite its high
dimension, co-dimension, and degree it is amenable to concrete calculations via
conformal geometric algebra (CGA) associated to three-dimensional Euclidean
space. Calculations are facilitated by a four quaternion representation which
extends the dual quaternion description of rigid body kinematics. In
particular, we study straight lines on the Study variety. It turns out that
they are related to a class of one-parametric conformal motions introduced by
L. Dorst in 2016. Similar to rigid body kinematics, straight lines (that is,
Dorst's motions) are important for the decomposition of rational conformal
motions into lower degree motions via the factorization of certain polynomials
with coefficients in CGA
Swim-like motion of bodies immersed in an ideal fluid
The connection between swimming and control theory is attracting increasing attention in the recent literature. Starting from an idea of Alberto Bressan [A. Bressan, Discrete Contin. Dyn. Syst. 20 (2008) 1\u201335]. we study the system of a planar body whose position and shape are described by a finite number of parameters, and is immersed in a 2-dimensional ideal and incompressible fluid in terms of gauge field on the space of shapes. We focus on a class of deformations measure preserving which are diffeomeorphisms whose existence is ensured by the Riemann Mapping Theorem. After making the first order expansion for small deformations, we face a crucial problem: the presence of possible non vanishing initial impulse. If the body starts with zero initial impulse we recover the results present in literature (Marsden, Munnier and oths). If instead the body starts with an initial impulse different from zero, the swimmer can self-propel in almost any direction if it can undergo shape changes without any bound on their velocity. This interesting observation, together with the analysis of the controllability of this system, seems innovative.
Mathematics Subject Classification. 74F10, 74L15, 76B99, 76Z10. Received June 14, 2016. Accepted March 18, 2017.
1. Introduction
In this work we are interested in studying the self-propulsion of a deformable body in a fluid. This kind of systems is attracting an increasing interest in recent literature. Many authors focus on two different type of fluids. Some of them consider swimming at micro scale in a Stokes fluid [2,4\u20136,27,35,40], because in this regime the inertial terms can be neglected and the hydrodynamic equations are linear. Others are interested in bodies immersed in an ideal incompressible fluid [8,18,23,30,33] and also in this case the hydrodynamic equations turn out to be linear.
We deal with the last case, in particular we study a deformable body -typically a swimmer or a fish- immersed in an ideal and irrotational fluid. This special case has an interesting geometric nature and there is an attractive mathematical framework for it. We exploit this intrinsically geometrical structure of the problem inspired by [32,39,40], in which they interpret the system in terms of gauge field on the space of shapes. The choice of taking into account the inertia can apparently lead to a more complex system, but neglecting the viscosity the hydrodynamic equations are still linear, and this fact makes the system more manageable. The same fluid regime and existence of solutions of these hydrodynamic equations has been studied in [18] regarding the motion of rigid bodies
Conformal symmetry algebra of the quark potential and degeneracies in the hadron spectra
The essence of the potential algebra concept [3] is that quantum mechanical
free motions of scalar particles on curved surfaces of given isometry algebras
can be mapped on 1D Schroedinger equations with particular potentials. As long
as the Laplace-Beltrami operator on a curved surface is proportional to one of
the Casimir invariants of the isometry algebra, free motion on the surface is
described by means of the eigenvalue problem of that very Casimir operator and
the excitation modes are classified according to the irreps of the algebra of
interest. In consequence, also the spectra of the equivalent Schroedinger
operators are classified according to the same irreps. We here use the
potential algebra concept as a guidance in the search for an interaction
describing conformal degeneracies. For this purpose we subject the so(4)
isometry algebra of the S^3 ball to a particular non-unitary similarity
transformation and obtain a deformed isometry copy to S^3 such that free motion
on the copy is equivalent to a cotangent perturbed motion on S^3, and to the 1D
Schroedinger operator with the trigonometric Rosen-Morse potential as well. The
latter presents itself especially well suited for quark-system studies insofar
as its Taylor series decomposition begins with a Cornell-type potential and in
accord with lattice QCD predictions. We fit the strength of the cotangent
potential to the spectra of the unflavored high-lying mesons and obtain a value
compatible with the light dilaton mass. We conclude that while the conformal
group symmetry of QCD following from AdS_5/CFT_4 may be broken by the dilaton
mass, it still may be preserved as a symmetry algebra of the potential, thus
explaining the observed conformal degeneracies in the unflavored hadron
spectra, both baryons and mesons.Comment: Invited talk presented at "Beauty in Physics:Theory and Experiment",
May 14-May 18, Cocoyoc, Mexico, dedicated to the 70th birthday of Francesco
Yachell
OperatorNet: Recovering 3D Shapes From Difference Operators
This paper proposes a learning-based framework for reconstructing 3D shapes
from functional operators, compactly encoded as small-sized matrices. To this
end we introduce a novel neural architecture, called OperatorNet, which takes
as input a set of linear operators representing a shape and produces its 3D
embedding. We demonstrate that this approach significantly outperforms previous
purely geometric methods for the same problem. Furthermore, we introduce a
novel functional operator, which encodes the extrinsic or pose-dependent shape
information, and thus complements purely intrinsic pose-oblivious operators,
such as the classical Laplacian. Coupled with this novel operator, our
reconstruction network achieves very high reconstruction accuracy, even in the
presence of incomplete information about a shape, given a soft or functional
map expressed in a reduced basis. Finally, we demonstrate that the
multiplicative functional algebra enjoyed by these operators can be used to
synthesize entirely new unseen shapes, in the context of shape interpolation
and shape analogy applications.Comment: Accepted to ICCV 201
Geometric Algebra for Optimal Control with Applications in Manipulation Tasks
Many problems in robotics are fundamentally problems of geometry, which lead
to an increased research effort in geometric methods for robotics in recent
years. The results were algorithms using the various frameworks of screw
theory, Lie algebra and dual quaternions. A unification and generalization of
these popular formalisms can be found in geometric algebra. The aim of this
paper is to showcase the capabilities of geometric algebra when applied to
robot manipulation tasks. In particular the modelling of cost functions for
optimal control can be done uniformly across different geometric primitives
leading to a low symbolic complexity of the resulting expressions and a
geometric intuitiveness. We demonstrate the usefulness, simplicity and
computational efficiency of geometric algebra in several experiments using a
Franka Emika robot. The presented algorithms were implemented in c++20 and
resulted in the publicly available library \textit{gafro}. The benchmark shows
faster computation of the kinematics than state-of-the-art robotics libraries.Comment: 16 pages, 13 figures
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