The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially Lipschitz and Hurwitz order. The fourth chapter is dedicated to the relationship between unit quaternions and rotations in R3 , thanks to which we can characterize finite subgroups of H1 , or equivalently H× . This result will be used in the last chapter, where we are mainly focused on the problem of characterization of the group of units in orders in Hamiltonian quaternions. 1

Táto diplomová práca sa zaoberá matematickými aparátmi popisujúcimi doprednú a inverznú kinematiku robotického ramena. Pre popis polohy koncového efektoru, teda doprednej kinematiky, je potrebné zaviesť špeciálnu Euklidovskú grupu zobrazení. Táto grupa môže byť reprezentovaná pomocou matíc alebo pomocou duálnych kvaterniónov. Problém inverznej kinematiky, kedy je potrebné z určenej polohy koncového efektoru dopočítať kĺbové parametre robotického ramena, je v tejto práci riešený pomocou exponenciálnych zobrazení a Grobnerovej bázy. Všetky spomenuté popisy doprednej a inverznej kinematiky sú aplikované na robotické rameno s troma rotačnými kĺbami. Odvodené postupy sú následne implementované a vizualizované v prostredí programu Mathematica.

This paper deals with study of inertial navigation, global navigation satellite system, and their fusion into the one navigation solution. The first part of the work is to calculate the trajectory from accelerometers and gyroscopes measurements. Navigation equations calculate rotation with quaternions and remove gravity sensed by accelerometers. The equation’s output is in earth centred fixed navigation frame. Then, inertial navigation errors are discussed and focused to the bias correction. Theory about INS/GNSS inte- gration compares different integration architecture. The Kalman filter is used to obtain navigation solution for attitude, velocity and position with advantages of both systems.