
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.


Rigid body motion from the geometric viewpoint
Karas, Jakub ; Hrdina, Jaroslav (referee) ; Návrat, Aleš (advisor)
Cílem této práce je odvodit rovnice levoinvariantních 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. levoinvariantních úloh. Jako další zjednodušení volíme také levoinvariantní 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 levoinvariantní 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.

 

Robotic mechanisms control
Mareček, Tomáš ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
The aim of this thesis is to discuss kinematic models, their properties and control. For this task, we chose a geometric control theory approach. For a predescribed trajectory of the gripper, a straight path in particular, motion planning algorithm for the UR5e robotic arm from Universal Robots is implemented. For the description of motion, the concept of conformal geometric algebra is used. Properties of the algebra objects were thoroughly described and consequently used to propose a model of forward and reverse kinematics of UR5e. Gains and losses of this approach were discussed. The algorithms are implemented in CLUViz 7.0.

 

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.


Identification of 3D objects for Robotic Applications
Hujňák, Jaroslav ; Návrat, Aleš (referee) ; Matoušek, Radomil (advisor)
This thesis focuses on robotic 3D vision for application in Bin Picking. The new method based on Conformal Geometric Algebra (CGA) is proposed and tested for identification of spheres in Pointclouds created with 3D scanner. The speed, precision and scalability of this method is compared to traditional descriptors based method. It is proved that CGA maintains the same precision as the traditional method in much shorter time. The CGA based approach seems promising for the use in the future of robotic 3D vision for identification and localization of spheres.


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.


Geometric algebra computations
Tomešová, Tereza ; Vechetová, Jana (referee) ; Vašík, Petr (advisor)
This thesis deals with computing in geometric algebra and its illustration in software CLUCalc. Firstly, it introduces fundamental terms and properties of vector space, scalar product and Clifford algebra. Consequently, the term geometric algebra, its products and operations are defined. These terms are illustrated on a specific exampel, i.e. translation and rotation of a sphere along fixed curve in software CLUCalc.


Image corrections by CGA
Machálek, Lukáš ; Hrdina, Jaroslav (referee) ; Vašík, Petr (advisor)
This thesis deals with conformal geometric algebra (CGA) in image processing. We focus on correct definitions of notions in geometric algebra, which we use for correcting image defects. First, the concepts of vector space are mentioned, then, the properties of geometric algebra are observed. Consequently, the 3D point is conformaly embedded into CGA, thereafter, another geometric objects are described with their representations in null spaces. In the end, the thesis deals with object transformation and with image defects correction.
