Orientador: Luiz Antonio Barrera San Martin / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T14:02:28Z (GMT). No. of bitstreams: 1 Silva_JulioCesarConegundesda_M.pdf: 1171316 bytes, checksum: 4e6e6eb2a3f1c066ac73e86495c06428 (MD5) Previous issue date: 2012 / Resumo: ...Observações: Por apresentar basicamente fórmulas, o resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: ...Note: The complete abstract is available with the full electronic document / Mestrado / Matematica / Mestre em Matemática

This thesis deals with a design of 9DOF module for the evaluation of orientation in space. The thesis includes a theoretical description of sensors, mathematical apparatus for rotating and the Kalman filter. The practical part deals with the design of hardware, firmware and software, whereby the mathematical formulas set out in the theoretical part are applied. In the last part a testing of the created module is presented.

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

The most common method for securing components to each other during manufacturing of products is by joining these using screws, nuts and bolts. The benefit of using this method is that it is cheap and makes it easy to join and separate components quickly. The clamping force in the threaded joint is critical to the quality and in some respect the life length of the product, which makes it important to have good control of the clamping force. There are two main tightening strategies used when tightening a threaded joint – torque controlled tightening and angle controlled tightening. The first method monitors the applied torque during the entire tightening and halts when the target torque is reached. The second method, angle controlled tightening, measures the rotation of the threaded fastener in the joint. This method generally produces more accurate results with less scatter in the final clamping force. In order to apply angle controlled tightening using a hand-held tool it is required to not only control the output angle of the tool, but also how the tool moves in relation to the joint. This is to ensure that the control signal from the motor actually translates to clamping force in the joint and not to rotation of the tool itself. This thesis project aims to analyze data from an IMU (Inertial Measurement Unit) built into a hand-held tightening tool in order to estimate tool movement and thereby react to undesired tool movement. An analysis has been performed to evaluate how the two sensor fusion methods – Kalman filter and Particle filter – perform in terms of estimating the orientation of the tool by combining measurements from the IMU’s accelerometers and gyroscopes. Data was collected from the tool IMU during a number of angle tightening sequences with varying setups. Test were performed both for when the tool was kept still during the entire tightening and for when the tools was allowed to move freely. Tests were also carried out for a couple of different tool orientations to better understand the behavior of the two sensor fusion models. The results from the tests showed that the Kalman Filter was able to better estimate the tool orientation. Especially in terms of accuracy, repeatability and reliability. / Den vanligaste metoden för att fästa komponenter till varandra vid tillverkning av produkter är genom att sammanfoga komponenterna med hjälp av skruvar, bultar och muttrar. Fördelen med denna metod är att den är billig och gör det enkelt att sammanfoga och lossa komponenter snabbt. Klämkraften i skruvförbandet är avgörande för hur väl en produkt är ihopsatt och påverkar därmed dess kvalitet, samt i viss mån livslängd. Det finns i huvudsak två olika strategier vid åtdragning i ett skruvförband – momentåtdragning och vinkelåtdragning. Den första metoden bygger på att man kontinuerligt mäter momentet under åtdragning och avbryter åtdragningen när rätt moment uppnåtts. Den andra metoden, vinkelåtdragning, mäter hur många grader fästelementet roterat i förbandet. Metoden producerar i regel högre precision med mindre spridning av den slutgiltiga klämraften. För att kunna tillämpa vinkelåtdragning med ett handhållet verktyg räcker det inte att kontroll över rotationen av verktygets utgående axel, utan även hur verktyget rör sig i förhållande till förbandet under åtdragning. Detta för att säkerställa att verktygets motorstyrning resulterar i önskad klämkraft i förbandet och inte rotation av själva verktyget. Detta examensarbete ämnar analysera data från en IMU (Inertial Measurement Unit) integrerad i ett handhållet åtdragningsverktyg för att estimera verktygets rörelse under vinkelåtdragning och därmed kompensera för oönskade rörelser. En analys har gjorts för hur väl de två olika sensorfusions-modellerna - Kalmanfilter och Partikelfilter – presterar när det kommer till att uppskatta orientering för verktyget genom att kombinera data från IMU-enhetens accelerometrar och gyroskop. Data samlades in från verktygets IMU från ett antal dragningar med varierande uppställning. Tester genomfördes dels då verktyget hölls stilla under hela åtdragningen och dels då det tilläts röra sig fritt. Tester genomfördes även för flera olika orienteringar av verktyget för att i större utsträckning kunna säga hur de olika sensorfusions-modellerna presterade. Resultatet av testerna visade att Kalmanfiltret kunde producera bättre estimeringar av verktygets orientering, speciellt i avseende precision, repeterbarhet och tillförlitlighet.

The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we also find the automorphisms of the zeros of the polynomials and the automorphism group.

The focus of this thesis is to provide an introduction to Lie groups—the study of continuous symmetries—and related concepts. Starting with topology and geometry, progressing through the construction of Cayley-Dickson algebras, the thesis ultimately explores a few of the most common Lie groups. Finally, geometric and visual interpretations of SO(3) and SU(2), as well as the relations between them, are investigated.

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.

Des formules en termes élémentaires de la Théorie des Nombres pour les nombres de classes de conjugaison de sous-groupes finis dans le groupe de norme 1 des ordres maximaux d'algèbres de quaternions sont établies.

A demanda crescente por petróleo impulsiona a exploração marítima desta riqueza para águas cada vez mais profundas. O aumento da lâmina dágua exige novas soluções de engenharia principalmente no que se refere à operação de unidades flutuantes de produção. Dentre os desafios impostos pelos novos ambientes de prospecção, destaca-se o processo de ancoragem, neste texto explorado sob a ótica da chamada estaca-torpedo. Embora já tenha sido empregada com sucesso na ancoragem de risers e FPSOs, esta solução encontra-se em constante desenvolvimento, sendo a principal fonte de informação acerca dos lançamentos da estaca-torpedo proveniente de uma unidade de medição inercial (UMI). A presente pesquisa baseou-se no estudo desta UMI e teve como objetivos principais verificar seu desempenho e compreender a empregabilidade deste tipo de monitoração em operações submarinas de maneira mais ampla. Além do estudo detalhado dos sensores, foi dada especial atenção aos algoritmos empregados no tratamento dos sinais provenientes da UMI. Foram estudadas técnicas de correção do sinal, quantificação de ruído, desafios inerentes aos sistemas do tipo strapdown e o processo de integração. Como resultado final foi desenvolvido um algoritmo baseado em quatérnios, alternativo ao atualmente empregado para o processamento dos sinais provenientes da UMI que equipa a estaca-torpedo. / The increasing demand on crude oils constantly pushes the offshore exploitation to deeper waters. As the water depth grows, new engineering challenges arise, especially concerning to the operation of floating production units. Among all the technical issues inherent to the new prospection environment, the mooring system is a significant topic and the development of the torpedo-pile takes place at this scenario. This mooring system has already been successfully applied in anchoring risers and FPSOs; nevertheless, it is in constant study and improvement. The major source of information about the torpedo-pile deployment comes from an inertial measurement unit (IMU). The research presented here is based on this IMU and had as main objective verify its performance and also comprehend the applicability of such kind of unit in other subsea processes. Along with the detailed sensors study, it was given special attention to the algorithms used to process the signals from the IMU. Signal correction techniques and noise quantification were investigated as long as challenges intrinsically related to strapdown navigation systems and the integration process. In the end, an alternative data processing algorithm based on quaternions was produced, to be employed in torpedo-pile launching together with its IMU.

Made available in DSpace on 2016-04-28T14:16:20Z (GMT). No. of bitstreams: 1 Donizetti Fermino Louro.pdf: 5486550 bytes, checksum: 06eabea84200c8c5e9e39a82ee79d8f2 (MD5) Previous issue date: 2014-06-15 / The historiography of triplets of Hamilton aims to address the mathematical reasoning that preceded the development of quaternions and their contribution to the study of three-dimensional objects. The study of triplets as geometrical representation of square roots and negative quantities prior to the formalization of hypercomplex numbers, in particular the algebra of quaternions. The origin and the quaternion's fundaments based on triplets, applied in the rotational motion of threedimensional objects in space, evokes further study of their complex structures, a priori. The search of Hamilton for a hypercomplex number able to represent rotations in space presented geometrical concerns, as well as, a vector approach of plane, then their applications determine isometries that still underlie the mathematical morphology of the digital dynamic image / A historiografia dos tripletos de Hamilton pretende abordar o raciocínio matemático que antecedeu o desenvolvimento dos quaternions e sua contribuição ao estudo de objetos tridimensionais. O estudo dos tripletos como representação geométrica das raízes quadradas e quantidades negativas antecedeu a formalização dos números hipercomplexos, em específico da álgebra de quaternions. A origem e os fundamentos dos quaternions baseados nos tripletos, aplicados no movimento rotacional de objetos tridimensionais no espaço suscitam um estudo mais aprofundado de suas estruturas complexas, a priori. A busca de Hamilton por um número hipercomplexo capaz de representar rotações do espaço apresentava preocupações geométricas, assim como uma abordagem vetorial do plano, logo suas aplicações determinariam isometrias que ainda fundamentam a morfologia da imagem digital dinâmica