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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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Mathematical security methods in digital data transfer
Bartušek, Petr ; Kureš, Miroslav (referee) ; Vašík, Petr (advisor)
This master’s thesis deals with an analysis of digital security with CRC. In the thesis there is described a principle of coding theory, especially then digital security with CRC, for which there is explained a mathematical principle of their encoding and decoding, software implementation and a description of frequently used generator polynomials. The main aim of the thesis is a testing of undetected errors and a finding of number of these errors. After that it is used for the computation of probability with which undetected errors can occur. The thesis is supplemented with several programs which are programmed in the software Matlab.
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Geometric structures based on quaternions.
Floderová, Hana ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.
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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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Game theory on graphs
Osička, Ondřej ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
The subject of this thesis is to introduce game theory and cooperative game theory in relation to graph theory. Game in characteristic function form is used to model the cooperative game. The optimal division of payoff among the players is determined by means of Shapley value and game kernel. Examples of practical use are presented. To examine more complicated game network or to express relationship between players both directed and undirected graphs are used.
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Lie groups theory in robotics
Horník, Petr ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
In this thesis we focus on the mathematical description of the forward kinematics in three-dimensional space using orthogonal transformations and matrix theory. Applying the acquired knowledge we solve an example using the method of moving frame. Between bases we pass using the passage matrix and we implement the example in the MATLAB environment. Consequently, we focus on deeper relation with exponential functions and extend the theory by the theory of Lie groups and algebras. Especially, we take notice of the special orthogonal group SO(3). Finally, we enrich the theory with homogeneous transformation and special Euclidean group.
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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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Matrices, Determinant and Body Volume
Šomplák, Radovan ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
In his thesis describes the basic properties of matrices, determinants and their use in display issues between vector spaces. In the final chapters devoted to determinants and its applications, which indicates the relation between determinants and calculation of volume. Subsequently determinanti describes how to use in calculating the elements using multiple integrals. Another possible application is the so-called Cramerovo rule which allows us to solving systems of linear equations.
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Geodesic curves and their applications
Orgoník, Svetoslav ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
The aim of the thesis is to give a survey of basic results from the classical theory of curves. A special attention will be paid to geodesics and their properties. In particular, we treat geodesics on some special surfaces. We treat one application with animations. All examples will be illustrated by pictures, which were drawn by means of mathematical software.
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