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Lie groups from the point of view of kinematics and applications in robotics
Kalenský, Jan ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.
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Lie groups and their physical applications
Kunz, Daniel ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.
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Control Theory of robotic snakes with more than three links
Tejkal, Martin ; Návrat, Aleš (referee) ; Hrdina, Jaroslav (advisor)
The subject of this Bachelor's thesis is control theory of mechanism that simulates snake's movement. From a viewpoint of control theory the robot is classified as nonholonomic system, controllability of which is determined by vector fields. Based on nonholomic constrain a set of input vector fields is obtained from a system of nonholonomic equations. The other vector fields that are necessary for controllability of the system are derived from the set of input vector fields by application of Lie bracket operation on two input fields. This set of vector fields is further analysed in particular points of the configuration space. Finally we discuss changes that need to be done in order to describe a mechanism created by adding one, or more new links.
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Control Theory of robotic snakes with missing wheels
Reichmanová, Barbora ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
This thesis looks into the mathematical description of a three-sectional robot. The thesis deals with cases of wheels missing either on the middle or the last section or solely on the middle section. At first theoretical basis is mentioned including the terms such as vector and affinne space, Lie algebra, distribution or controllable system. Subsequently, there is presented formulation of equations describing a snake robot with missing wheels, solutions of equations, calculation of Lie brackets and discussion of controllability. The calculations are demonstrated on examples of various configurations of the robot.
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Rigid body motion from the geometric viewpoint
Karas, Jakub ; Hrdina, Jaroslav (referee) ; Návrat, Aleš (advisor)
Cílem této práce je odvodit rovnice levo-invariantních Hamiltonovských systémů na Lieových grupách. Naše motivace je následující. Pohyb tuhého tělesa v 3D prostoru lze formulovat jako úlohu optimálního řízení na $\R^3$. Pro takto formulovanou úlohu lze využít Pontryaginův princip maxima (PMP). Nicméně pohyb tuhého tělesa lze také chápat jako úlohu na Lieově grupě SE(3). Tato úloha patří do skupiny tzv. levo-invariantních úloh. Jako další zjednodušení volíme také levo-invariantní Hamiltoniány. Běžný postup při studiu takových úloh je, že formulujeme Lagrangián této úlohy, odvodíme Hamiltonián a následně formulujeme Hamiltonovy rovnice. Náš postup je opačný. Odvodíme Hamiltonovy rovnice pro obecnou Lieovu grupu a obecný levo-invariantní Hamiltonián a následně zkoumáme, jaké typy úloh můžeme popsat volbou konkrétní Lieovy grupy a konkrétního Hamiltoniánu. Teoretické výsledky poté využijeme k vytvoření simulačního skriptu pohybu tuhého a pružného tělesa, který využije konformní geometrickou algebru (CGA) jako své výpočetní jádro. CGA je totiž nesmírně silný nástroj pro popis této problematiky, jelikož využitím CGA lze vyvinout kód, který je nezávislý na dimenzi uvažovaného prostoru bez větší námahy.
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Lie groups from the point of view of kinematics and applications in robotics
Kalenský, Jan ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.
|
|
|
Lie groups and their physical applications
Kunz, Daniel ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.
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