Wyniki wyszukiwania - BazTech

1

The group of diffeomorphisms of a non-compact manifold is not regular

Magnot J.-P.

Demonstratio Mathematica

|

2018

|

Vol. 51, nr 1

8--16

EN

We show that a group of diffeomorphisms D on the open unit interval I, equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non-regular: the exponential map is not defined for some path of the Lie algebra. This result extends to the group of diffeomorphisms of finite dimensional, non-compact manifold M.

2

Application of transverse functions to control differentially driven wheeled robots using velocity fields

Pazderski D.

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2016

|

Vol. 64, nr 4

831--851

EN

This paper deals with control of a nonholonomic unicycle-like robot in a cluttered environment with static obstacles. The proposed solution is based on a combination of a universal motion controller taking advantage of transverse functions with a navigation velocity field determining a path in a free task space. The motion controller is used to imitate an omnidirectional planar kinematics such that nonholonomic constraints become hidden for a navigation layer. Then it is possible to generate vector fields which govern motion of the omnidirectional frame. The controller using the transverse function is discussed in depth. In particular, a possible parametrization of this function is considered and analysis of an augmented dynamics is provided for different motion patterns. Next, construction of obstacles and potential design for star-like shapes are presented. The navigation algorithm is verified experimentally and the results are discussed.

3

Geometric interpretation of a non-linear beam finite element on The Lie Group SE(3)

Sonneville V, Cardona A, Brüls O

Archive of Mechanical Engineering

|

2014

|

Vol. LXI, nr 2

305--329

EN

Recently, the authors proposed a geometrically exact beam finite element formulation on the Lie group SE(3). Some important numerical and theoretical aspects leading to a computationally efficient strategy were obtained. For instance, the formulation leads to invariant equilibrium equations under rigid body motions and a locking free element. In this paper we discuss some important aspects of this formulation. The invariance property of the equilibrium equations under rigid body motions is discussed and brought out in simple analytical examples. The discretization method based on the exponential map is recalled and a geometric interpretation is given. Special attention is also dedicated to the consistent interpolation of the velocities.

PL

W ostatnim czasie autorzy zaproponowali geometrycznie dokładne sformułowanie dla belkowego elementu skończonego w oparciu o formalizm grupy Liego SE(3). Otrzymano szereg istotnych wyników numerycznych i teoretycznych prowadzących do efektywnej strategii obliczeniowej. Dla przykładu, formalizm ten pozwala uzyskać niezmiennicze równania równowagi przy ruchach ciała sztywnego i elemencie wolnym od blokowania siłami ścinającymi. W obecnym artykule autorzy zajmują się kilkoma istotnymi aspektami tego formalizmu. Właściwość niezmienniczości równań równowagi w warunkach ruchu ciała sztywnego przedyskutowano i zilustrowano prostymi przykładami analitycznymi. Przypomniano metodę dyskretyzacji opartą na mapowaniu wykładniczym i pokazano jej interpretację geometryczną. Specjalną uwagę poświęcono zgodnej interpolacji prędkości.

4

Some differential equations of elasticity and their lie point symmetry generators

Bocko J., Glodová I., Lengvarský P.

Acta Mechanica et Automatica

|

2014

|

Vol. 8, no. 2

99--102

EN

The formal models of physical systems are typically written in terms of differential equations. A transformation of the variables in a differential equation forms a symmetry group if it leaves the differential equation invariant. Symmetries of differential equations are very important for understanding of their properties. It can be said that the theory of Lie group symmetries of differential equations is general systematic method for finding solutions of differential equations. Despite of this fact, the Lie group theory is relatively unknown in engineering community. The paper is devoted to some important questions concerning this theory and for several equations resulting from the theory of elasticity their Lie group infinitesimal generators are given.

5

Projective coordinates and compactification in elliptic, parabolic and hyperbolic 2-D geometry

Biswas D.

Journal of Applied Analysis

|

2012

|

Vol. 18, nr 1

145-158

EN

The result that the upper half plane is not preserved in the hyperbolic case has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce an appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case.

6

Invariance of integro-differential equations with moving region of integration

Zawistowski Z. J.

Journal of Technical Physics

|

2006

|

Vol. 47, no 2

103-112

EN

General criterion of invariance of integro-differential equations under the Lie symmetry group of point transformations is derived. It is a generalization of the previous form of the criterion to the case of a moving range of integration. This is the situation when a region of integration depends on external, with respect to integration, variables what leads to its explicit dependence on a group parameter, so the region of integration moves under symmetry transformations. General case of dependence on independent and dependent variables and their derivatives is considered.

7

On the extension of Lie group analysis to functional differential equations

Oberlack M., Wacławczyk M.

Archives of Mechanics

|

2006

|

Vol. 58, nr 6

597-618

EN

In the present paper the classical point symmetry analysis is extended from partial differential to functional differential equations. In order to perform the group analysis and deal with the functional derivatives, we extend the quantities such as infinitesimal transformations, prolongations and invariant solutions. For the sake of example, the procedure is applied to the functional formulation of the Burgers equation. The method can further lead to important applications in continuum mechanics.

8

A Stabilizing Control Law for Invariant Systems on Lie Groups

Apostolou N., Kazakos D.

International Journal of Applied Mathematics and Computer Science

|

2000

|

Vol. 10, no 2

273-282

EN

This paper deals with the stabilizability of invariant control systems defined on Lie groups. A stabilization technique is presented which, under certain hypotheses, can lead to a criterion assuring the existence of a feedback controller which steers every initial condition to a specified target point of the state space of these systems.