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Kinematics of a robotic arm
Hekrle, Vojtěch ; Eryganov, Ivan (referee) ; Vašík, Petr (advisor)
This thesis focuses on problematic with the forward and inverse kinematics of a robotic arm with three, four and five arms. The conformal geometric algebra CGA2 is used to solve this issue. The outcome of the thesis consists of four algorithms for computing the given kinematics in the Python language. Specifically, one algorithm for forward kinematics and the others algorithms for the inverse kinematics of the robot with three, four and five arms.
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Robotic manipulator based on CGA
Stodola, Marek ; Salač,, Tomáš (referee) ; Hrdina, Jaroslav (advisor)
Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.
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The kinematics of the three link robotic snake based on CRA algebra
Motyčková, Paulína ; Tichý, Radek (referee) ; Hrdina, Jaroslav (advisor)
This work deals with Clifford's algebra of type CRA and its use for kinematics of a three-link robotic snake. To solve CRA based kinematics, it is necessary to familiarise with elements forming CRA algebra and the relationships between them. Furthermore, it is crucial to master the operations and transformations of the corresponding algebra and their implementation on objects in the CRA algebra. This work deals with the direct kinematics of a robotic snake that we suppose moves on a flat surface. This work concludes with the ideal method of calculating the kinematics of this snake and the corresponding algorithm in GAALOP.
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Three-dimensional kinematics of eye movements
Stodola, Marek ; Velan, Petr (referee) ; Hrdina, Jaroslav (advisor)
The goal of this thesis is to describe eye movements and general eye position using apparatus of geometric algebra. The introduction covers the theory about the appropriate geometric algebra, followed by the classifications of the eye movements and the terms used to describe these movements. Following this, the calculations that describe eye position derived from a single observed point are listed, for distant and close points. In addition, the possible eye movements in respect to the axis in which an eye can rotate is described, for any general position. All the calculations are based on Donders' law and Listing's law.
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Application of Geometric Algebras in Quantum Computing
Michálek, Jan ; Eryganov, Ivan (referee) ; Vašík, Petr (advisor)
Tato práce se zabývá využitím geometrických algeber v oblasti kvantového počítání. Nejprve je definována obecná Cliffordova algebra a následně je odvozena specifická komplexní geometrická algebra, která je vhodná pro reprezentaci kvantových výpočtů. Tento přístup je porovnán s tradiční metodou použití klasické maticové reprezentace. Cílem práce je poskytnout poznatky o potenciálních výhodách použití geometrických algeber pro kvantové výpočty.
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Stein-Weiss gradients
Malý, Marek ; Lávička, Roman (advisor) ; Souček, Vladimír (referee)
In this bachelor thesis, we describe the construction of rotation invariant differential operators of first order on the Euklidean space Rn given by E. Stein and G. Weiss. For this construction we show how to find an irreducible decomposition of a tensor product of re- presentations of group Spin(n) into irreducible subrepresetations. We shall also prove the rotation invariance of the gradient operator. Then we apply the Stein-Weiss construction to produce some of well-known differential operators. Namely, we construct the Dirac operator in Rn and Hodge-de Rham system of differential equations using this method. 1
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Robotic manipulator based on CGA
Stodola, Marek ; Salač,, Tomáš (referee) ; Hrdina, Jaroslav (advisor)
Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.
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The kinematics of the three link robotic snake based on CRA algebra
Motyčková, Paulína ; Tichý, Radek (referee) ; Hrdina, Jaroslav (advisor)
This work deals with Clifford's algebra of type CRA and its use for kinematics of a three-link robotic snake. To solve CRA based kinematics, it is necessary to familiarise with elements forming CRA algebra and the relationships between them. Furthermore, it is crucial to master the operations and transformations of the corresponding algebra and their implementation on objects in the CRA algebra. This work deals with the direct kinematics of a robotic snake that we suppose moves on a flat surface. This work concludes with the ideal method of calculating the kinematics of this snake and the corresponding algorithm in GAALOP.
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Twistors in relativistic field theories
Nárožný, Jiří ; Scholtz, Martin (advisor) ; Souček, Vladimír (referee)
In this thesis, we are concerning about the Twistor theory, field originally motivated purely physically, although these days fully developed into the many fields of mathem- atics and physics. With its complexion Twistor theory influences algebraic geometry, Clifford analysis as well as the String theory or Theory of quantum gravity. In the thesis we describe the origin of twistors projective or not. Mathematical background to the twistor theory is covered in the first chapter, where we study Clifford algebras and their representations. In the first part of the second chapter we are describing non-projective twistors as representation elements of certain Spin-group, and we find the connection with the standard definition of non-projective twistors as a kernel of the twistor operator. In the last part of the second chapter, we create a space of pro- jective twistors and show its certain properties, especially its correspondence with the complexified compactified Minkowski spacetime.
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Three-dimensional kinematics of eye movements
Stodola, Marek ; Velan, Petr (referee) ; Hrdina, Jaroslav (advisor)
The goal of this thesis is to describe eye movements and general eye position using apparatus of geometric algebra. The introduction covers the theory about the appropriate geometric algebra, followed by the classifications of the eye movements and the terms used to describe these movements. Following this, the calculations that describe eye position derived from a single observed point are listed, for distant and close points. In addition, the possible eye movements in respect to the axis in which an eye can rotate is described, for any general position. All the calculations are based on Donders' law and Listing's law.
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