3,809 research outputs found
gafro: Geometric Algebra for Robotics
Geometry is a fundamental part of robotics and there have been various
frameworks of representation over the years. Recently, geometric algebra has
gained attention for its property of unifying many of those previous ideas into
one algebra. While there are already efficient open-source implementations of
geometric algebra available, none of them is targeted at robotics applications.
We want to address this shortcoming with our library gafro. This article
presents an overview of the implementation details as well as a tutorial of
gafro, an efficient c++ library targeting robotics applications using geometric
algebra. The library focuses on using conformal geometric algebra. Hence,
various geometric primitives are available for computation as well as rigid
body transformations. The modeling of robotic systems is also an important
aspect of the library. It implements various algorithms for calculating the
kinematics and dynamics of such systems as well as objectives for optimisation
problems. The software stack is completed by python bindings in pygafro and a
ROS interface in gafro_ros
Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems
In this paper we study the problem of Hamiltonization of nonholonomic systems
from a geometric point of view. We use gauge transformations by 2-forms (in the
sense of Severa and Weinstein [29]) to construct different almost Poisson
structures describing the same nonholonomic system. In the presence of
symmetries, we observe that these almost Poisson structures, although gauge
related, may have fundamentally different properties after reduction, and that
brackets that Hamiltonize the problem may be found within this family. We
illustrate this framework with the example of rigid bodies with generalized
rolling constraints, including the Chaplygin sphere rolling problem. We also
see how twisted Poisson brackets appear naturally in nonholonomic mechanics
through these examples
The Dynamics of a Rigid Body in Potential Flow with Circulation
We consider the motion of a two-dimensional body of arbitrary shape in a
planar irrotational, incompressible fluid with a given amount of circulation
around the body. We derive the equations of motion for this system by
performing symplectic reduction with respect to the group of volume-preserving
diffeomorphisms and obtain the relevant Poisson structures after a further
Poisson reduction with respect to the group of translations and rotations. In
this way, we recover the equations of motion given for this system by Chaplygin
and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force
as a curvature-related effect. In addition, we show that the motion of a rigid
body with circulation can be understood as a geodesic flow on a central
extension of the special Euclidian group SE(2), and we relate the cocycle in
the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte
Quantization of Nonstandard Hamiltonian Systems
The quantization of classical theories that admit more than one Hamiltonian
description is considered. This is done from a geometrical viewpoint, both at
the quantization level (geometric quantization) and at the level of the
dynamics of the quantum theory. A spin-1/2 system is taken as an example in
which all the steps can be completed. It is shown that the geometry of the
quantum theory imposes restrictions on the physically allowed nonstandard
quantum theories.Comment: Revtex file, 23 pages, no figure
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
Asymptotic symmetries of three dimensional gravity and the membrane paradigm
The asymptotic symmetry group of three-dimensional (anti) de Sitter space is
the two dimensional conformal group with central charge . Usually
the asymptotic charge algebra is derived using the symplectic structure of the
bulk Einstein equations. Here, we derive the asymptotic charge algebra by a
different route. First, we formulate the dynamics of the boundary as a
1+1-dimensional dynamical system. Then we realize the boundary equations of
motion as a Hamiltonian system on the dual Lie algebra, , of
the two-dimensional conformal group. Finally, we use the Lie-Poisson bracket on
to compute the asymptotic charge algebra. This streamlines the
derivation of the asymptotic charge algebra because the Lie-Poisson bracket on
the boundary is significantly simpler than the symplectic structure derived
from the bulk Einstein equations. It also clarifies the analogy between the
infinite dimensional symmetries of gravity and fluid dynamics.Comment: 15 page
Gauge Theories of Gravitation
During the last five decades, gravity, as one of the fundamental forces of
nature, has been formulated as a gauge theory of the Weyl-Cartan-Yang-Mills
type. The present text offers commentaries on the articles from the most
prominent proponents of the theory. In the early 1960s, the gauge idea was
successfully applied to the Poincar\'e group of spacetime symmetries and to the
related conserved energy-momentum and angular momentum currents. The resulting
theory, the Poincar\'e gauge theory, encompasses Einstein's general relativity
as well as the teleparallel theory of gravity as subcases. The spacetime
structure is enriched by Cartan's torsion, and the new theory can accommodate
fermionic matter and its spin in a perfectly natural way. This guided tour
starts from special relativity and leads, in its first part, to general
relativity and its gauge type extensions \`a la Weyl and Cartan. Subsequent
stopping points are the theories of Yang-Mills and Utiyama and, as a particular
vantage point, the theory of Sciama and Kibble. Later, the Poincar\'e gauge
theory and its generalizations are explored and special topics, such as its
Hamiltonian formulation and exact solutions, are studied. This guide to the
literature on classical gauge theories of gravity is intended to be a
stimulating introduction to the subject.Comment: 169 pages, pdf file, v3: extended to a guide to the literature on
classical gauge theories of gravit
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