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1
Content available Comparison of long time simulation of Hamilton and Lagrange geometry dynamical models of a multibody system
Bai Long, Ge Xinsheng, Xia Lili
Journal of Theoretical and Applied Mechanics
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2022
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Vol. 60 nr 4
687--704
EN
The geometry dynamical modeling method for a double pendulum is explored with the Lie group and a double spherical space method. Four types of Lagrange equations are built for relative and absolute motion with the above two geometry methods, which are then used to explore the influence of different expressions for motion on the dynamic modeling and computations. With Legendre transformation, the Lagrange equations are transformed to Hamilton ones which are dynamical models greatly reduced. The models are solved by the same numerical method. The simulation results show that they are better for the relative group than for the absolute one in long time simulation with the same numerical computations. The Lie group based result is better than the double spherical space one.
2
Content available Controllabilty and stability analysis on a group associated with Black-Scholes equation
Tiwari Archana, Bhattacharyya Debanjana, Pati K. C.
Archives of Control Sciences
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2020
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Vol. 30, No. 3
553--573
EN
In this paper we have studied the driftless control system on a Lie group which arises due to the invariance of Black-Scholes equation by conformal transformations. These type of studies are possible as Black-Scholes equation can be mapped to one dimensional free Schrödinger equation. In particular we have studied the controllability, optimal control of the resulting dynamics as well as stability aspects of this system. We have also found out the trajectories of the states of the system through two unconventional integrators along with conventional Runge-Kutta integrator.
3
Content available A robust smooth controller for a unicycle-like robot
Pazderski D.
Archives of Control Sciences
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2018
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Vol. 28, No. 1
155-183
EN
In this paper, a stabilizer dedicated for a unicycle-like robot is considered. The proposed smooth control law ensures the global boundedness of position and orientation trajectories to a neighbourhood of the desired point with an arbitrarily selected radius and it is robust to bounded additive measurement noises. The controller consists of a smooth hybrid navigation algorithm and a smooth feedback based on the transverse function approach. The stability proof, simulation and experimental results illustrating properties of the algorithm are discussed.
4
Content available 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
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2014
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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.
5
Content available Some differential equations of elasticity and their lie point symmetry generators
Bocko J., Glodová I., Lengvarský P.
Acta Mechanica et Automatica
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2014
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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.
6
Projective coordinates and compactification in elliptic, parabolic and hyperbolic 2-D geometry
Biswas D.
Journal of Applied Analysis
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2012
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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.
7
Content available Control of a unicycle-like robot with trailers using transverse function approach
Pazderski D., Kozłowski K., Waśkowicz D.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2012
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Vol. 60, nr 3
557-579
EN
The paper presents the application of a smooth kinematic algorithm to control multi-body vehicle which consists of the unicyclelike tractor with three trailers. The controller takes advantage of the transverse functions and properties of the IV-order two input chained system. The derivation of the algorithm is presented in details. In order to improve the performance of the controller in the real application a selected tuning techniques are discussed. The properties of the closed-loop control system are examined based on results of numerical simulations concerning the point stabilization and trajectory tracking tasks.
8
Content available remote Split quaternion and the Lie group S4/3
Kabadayi H., Yayli Y.
Demonstratio Mathematica
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2007
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Vol. 40, nr 3
601-607
EN
In this work we showed that the pseudosphere 3/4 is a Lie group. We obtained a Lie algebra and a Lie product rule of this group. Moreover examining actions of this group we gave some theorems.
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