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Mathematics principles of Navigation
Petrovič, Branislav ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
This bachelor's thesis deals with the calculating of the position of the GPS receiver using the method of Cramer's rule. Subsequently, the errors generated during the transmission are described. The geodetic coordinates for the calculated position in space are introduced. The calculations of the receiver position by Cramer's rule and the calculations in geodesy are performed using MATLAB.
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Clifford algebras in colour theory and image analysis
Tichý, Radek ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
This thesis deals with conformal geometric algebra CGA for colour image processing, particularly with colour segmentation. For this reason it is not sufficient to work in RGB colour space. It is more convenient to use a colour space called CIELAB. CIELAB is endowed by Euclidean metric corresponding with human perception of colours. Afterwards an algorithm for an object detection via CGA based on colour differences is included. The final part of the thesis deals with least squares fitting of sphere to points using CGA. The sphere fitting is then used to adjust colour differences in an image to improve the algorithm for an object detection.
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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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Kinematics of a robotic arm by means of geometric algebras
Křápek, Michal ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
In this thesis we are dealing with forward and inverse kinematics of a robotic arm using a model of two dimensional space in conformal geometric algebra. Goal of this thesis is a proposal of algorithms for dealing with inverse kinematics problem and their implementaion. Five algorithms were constructed and implemented in python language. One for computing a position of a robotic arm and four for calculating the trajectory of the gripper. In this thesis, the problem was solved using a theorem about orientation of the line segment normal and with the triangle signature. Because of that, the cumputing load was reduced in implementation of the most complex algorithm which is combining the motion of the gripper along a polygonal chain and motion of the gripper along circular trajectory. The benefit of this thesis is a new approach to solving inverse kinematics problem.
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Geometric optimal control of a snake robot
Vechetová, Jana ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
This thesis deals with the description of robotic snake the trident snake robot. From a viewpoint of control theory the robot is classified as a nonholonomic system whose controllability is determined by vector fields. We use the operation Lie bracket to create other necessary control vector fields to ensure local controllability of this system. Then we propose the motion planning algorithm. Finally some of the motions caused by the control vector fields are verified in a simulation environment called V-rep.
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Applications of Quaternions in Robot Kinematics
Doctor, Diana ; Vašík, Petr (referee) ; Matoušek, Radomil (advisor)
This thesis deals with the usefulness of the application of quaternions in representing robot kinematics. It begins by showing the relationship of quaternions to the more commonly-known complex numbers and how it can represent rotations in three-dimensions. Then, the dual quaternions are introduced to represent both the three-dimensional rotation and translation. It will then be used to derive the forward and inverse kinematics, particularly, for the Universal Robot UR3 which is a 6-DOF robotic arm. Lastly, an actual application of dual quaternions in robot programming will be demonstrated
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Geometric algebras and neural networks
Zapletal, Jakub ; Procházková, Jana (referee) ; Vašík, Petr (advisor)
This thesis deals with the use of geometric algebras in the field of neural networks. First, Conformal Geometric Algebra (CGA) and Geometric Algebra for Conics (GAC) and their Python implementations are introduced. The functioning of neural networks is then described, including an explanatory example. Finally, both topics are connected by using the appropriate library in the Python language, and the possibilities of geometric algebras for different models of neural networks are demonstrated on several examples.
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Geometric control of nonholonomic systems
Ramasubramaniyan, Sri Ram Prasath ; Vašík, Petr (referee) ; Návrat, Aleš (advisor)
This thesis focuses on a mathematical model for a three-body space robot with the objective of reconfiguring its structure using only internal joint torques. The aim is to minimize fuel consumption and achieve efficient reconfiguration without relying on external actuators. The system exhibits one holonomic and non-holonomic constraint, making the analysis and control design challenging. To address the complexity of the non-holonomic system, the local behavior is studied through the nilpotent approximation. The thesis emphasizes understanding the nilpotent approximation and constructing the nilpotent system of the space robot using algebraic coordinates, along with transforming them into exponential coordinates within the Maple environment.
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Spherical geometry
Kokh, Konstantin ; Vašík, Petr (referee) ; Doupovec, Miroslav (advisor)
This bachelor thesis deals with the description of sphere and spherical geometry. The second chapter defines the mathematical apparatus that we will need in the next part of the work. The third part begins with describing the sphere from the point of view of differential geometry of curves and planes. In the middle, we will show the conformal map of the sphere to the plane and the equiareal map of the sphere to the cylinder. Then we will describe the basic properties of spherical geometry. In the end, we will compare the properties of Euclidean geometry and spherical geometry.
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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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