Nesta dissertação abordamos o conjunto dos números complexos, apresentando sua forma algébrica e geométrica, demonstrando que se trata de um conjunto com estrutura algébrica de corpo. Apresentamos também as características de rotação e homotetia da operação de multiplicação, a contextualização histórica e finalizamos com uma proposta de abordagem para o ensino médio. Vemos também a impossibilidade de realizar rotação em três dimensões culminando com a criação dos quatérnios. / In this masters thesis we discuss the complex numbers set, showing its algebraic and geometric forms, demonstrating which it is a set with algebraic structure of field. We also presente the rotation characteristics and homothety of multiplication operation, the historical contextualization and we finalized with an approach proposal for the high school. We also see the impossibility of performing the rotation in three dimensions resulting the generation of quaternions.

Orientador: Manoel Ferreira Borges Neto / Banca: Gilberto Aparecido Pratavieira / Banca: José Márcio Machado / Resumo: Resumo O objetivo deste trabalho é estabelecer similaridades entre os complexos e os hipercomplexos, motivados em explorar idéias de Murnaghan, que introduziu, pela primeira vez, em uma apresentação elementar, a teoria dos quatérnios baseados no teorema de Moivre. É mostrada em detalhes uma analogia da relação complexa clássica de Moivre para quatérnios, e em brevidade para octônios generalizados, e apresenta-se as conexões com os operadores da teoria de Fueter e as funções transcendentais. A extensão do teorema de Moivre é estudada para quatérnios em definindo-se uma função exponencial quaterniônica. / Abstract: Abstract In this work we establish similarities between the complex and the hipercomplex numbers, motivated in exploring ideas of Murnaghan, that introduced, for the first time, in an elementary presentation, the theory of the quaternions based on the theorem of Moivre. We show an analogy of the classic complex relation of Moivre for quaternions, and briefly discuss generalized octonions, as well as to present connections to operators of the theory of Fueter and transcendental functions. We consider them to study the extension of the theorem of Moivre for quaternions, in defining a exponential function on the quaternions. / Mestre

Cette thèse s'intéresse à la recherche de points rationnels sur les courbes elliptiques. Darmon et Logan ont proposé une construction conjecturale de points rationnels sur des courbes elliptiques modulaires définies sur un corps de nombres totalement réel. Cette construction va au delà de la construction classique des points de Heegner. C'est sur la généralisation de ces travaux que porte cette thèse. Après un premier chapitre de rappels concernant essentiellement les variétés de Shimura, on construit, dans le chapitre deux une forme différentielle dont l'ensemble des périodes est, sous une conjecture due à Yoshida, un réseau. On y définit aussi un ensemble de cycles dont la classe d'homologie est de torsion. A l'aide de ces données, on énonce au chapitre suivant une conjecture généralisant celle de Darmon et Logan. On s'interesse aussi aux propriétés de ces nouveaux points, principalement en lien avec les théorèmes "classiques" de Gross-Zagier et Gross-Kohnen-Zagier. Le chapitre 4 tente de rendre holomorphes les opérations du chapitre 2, et le chapitre 5 de les rendre plus explicites. Cette thèse comporte une annexe concernant les vérifications informatiques de la conjecture de Darmon.

A satellite in orbit about a planet needs some means of attitude control in order to, for instance, get as much sun into its solar-panels as possible. It is easy to understand that, for example, a spy satellite has to point at a certain direction without the slightest trembling to get a photo of a certain point on the earth. This type of mission must not exceed an error in attitude of more then about 1/3600 degrees. But, since high accuracy equals high cost, it is also easy to understand why a research satellite measuring solar particles (or radiation) in space does not need high accuracy at all. A research vessel of this sort can probably do with less accuracy then 1 degree. The first part of this report tries to explain some major aspects of satellite space-flight. It continues to focus on the market for small satellites, i.e. satellites weighing less than 500 kg. The second part of this final thesis work deals with the development of a program that simulates the movement of a satellite about a large celestial body. The program, called AOSP, consists of user-definable packages. Sensors and estimation filters are used to predict the satellites current position, velocity, attitude and angular velocity. The purpose of the program, which is written in MATLAB, is to easily determine the pointing accuracy of a satellite when using different sensors and actuators.

<p>A satellite in orbit about a planet needs some means of attitude control in order to, for instance, get as much sun into its solar-panels as possible. It is easy to understand that, for example, a spy satellite has to point at a certain direction without the slightest trembling to get a photo of a certain point on the earth. This type of mission must not exceed an error in attitude of more then about 1/3600 degrees. But, since high accuracy equals high cost, it is also easy to understand why a research satellite measuring solar particles (or radiation) in space does not need high accuracy at all. A research vessel of this sort can probably do with less accuracy then 1 degree. </p><p>The first part of this report tries to explain some major aspects of satellite space-flight. It continues to focus on the market for small satellites, i.e. satellites weighing less than 500 kg. The second part of this final thesis work deals with the development of a program that simulates the movement of a satellite about a large celestial body. The program, called AOSP, consists of user-definable packages. Sensors and estimation filters are used to predict the satellites current position, velocity, attitude and angular velocity. The purpose of the program, which is written in MATLAB, is to easily determine the pointing accuracy of a satellite when using different sensors and actuators.</p>

Cette thèse concerne l'étude d'une famille de μ-actionneurs magnétiques en lévitation asservie.<br />Le principe est celui d'un aimant mobile ayant deux positions stables et une position instable. Parmi<br />les différentes applications de cette structure, un μ-positionneur est étudié plus en détails, avec la<br />conception, la modélisation et la réalisation d'une maquette centimétrique de ce μ-positionneur<br />asservi. La conception de μ-positionneur comprend l'étude du capteur capacitif de position et une<br />synthèse du correcteur pour l'asservissement. La modélisation introduit deux méthodes de calcul de la<br />force et du couple magnétostatiques exercés sur l'aimant mobile: méthode de charge équivalente et<br />méthode d'intégration volumique. Les deux méthodes sont validées et puis utilisées pour déterminer<br />la trajectoire de l'aimant mobile comportant 6 degrés de liberté. Une méthode mathématique utilisant<br />les quaternions est mise en oeuvre. La modélisation permet l'analyse du comportement de différents<br />μ-actionneurs magnétiques ainsi que la vérification de la stabilité de l'asservissement. La réalisation<br />de la maquette centimétrique décrit la fabrication d'un prototype ainsi que la réalisation de<br />l'électronique pour capter la position de l'aimant mobile. L'aimant mobile sur la maquette est<br />finalement asservi en lévitation.

Reproduction de : Thèse de doctorat : Microondes et Microtechnologies : Lille 1 : 2004. / N° d'ordre (Lille 1) : 3469. Résumé en français. Titre provenant de la page de titre du document numérisé. Bibliogr. p. 173-179.

The term proximity operations has been widely used in recent years to describe a wide range of space missions that require a spacecraft to remain close to another space object. Such missions include, for example, the inspection, health monitoring, surveillance, servicing, and refueling of a space asset by another spacecraft. One of the biggest challenges in autonomous space proximity operations, either cooperative or uncooperative, is the need to autonomously and accurately track time-varying relative position and attitude references, i.e., pose references, with respect to a moving target, in order to avoid on-orbit collisions and achieve the overall mission goals. In addition, if the target spacecraft is uncooperative, the Guidance, Navigation, and Control (GNC) system of the chaser spacecraft must not rely on any help from the target spacecraft. In this case, vision-based sensors, such as cameras, are typically used to measure the relative pose between the spacecraft. Although vision-based sensors have several attractive properties, they introduce new challenges, such as no direct linear and angular velocity measurements, slow update rates, and high measurement noise. This dissertation investigates the problem of autonomously controlling and estimating the pose of a chaser spacecraft with respect to a moving target spacecraft, possibly uncooperative. Since this problem is inherently hard, the standard approach in the literature is to split the attitude-tracking problem from the position-tracking problem. Whereas the attitude-tracking problem is relatively simple, since the rotational motion is independent from the translational motion, the position-tracking problem is more complicated, as the translational motion depends on the rotational motion. Hence, whereas strong theoretical results exist for the attitude problem, the position problem typically requires additional assumptions. An alternative, more general approach to the pose control and estimation problems is to consider the fully coupled 6-DOF motion. However, fewer results exist that directly address this higher dimensional problem. The main contribution of this dissertation is to show that dual quaternions can be used to extend the theoretical results that exist for the attitude motion into analogous results for the combined position and attitude motion. Moreover, this dissertation shows that this can be accomplished by (almost) just replacing quaternions by dual quaternions in the original derivations. This is because dual quaternions are built on and are an extension of classical quaternions. Dual quaternions provide a compact representation of the pose of a frame with respect to another frame. Using this approach, three new results are presented in this dissertation. First, a pose-tracking controller that does not require relative linear and angular velocity measurements is derived with vision-based sensors in mind. Compared to existing literature, the proposed velocity-free pose-tracking controller guarantees that the pose of the chaser spacecraft will converge to the desired pose independently of the initial state, even if the reference motion is not sufficiently exciting. In addition, the convergence region does not depend on the gains of the controller. Second, a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) is developed from the highly successful Quaternion MEKF (Q-MEKF) as an alternative way to achieve pose-tracking without velocity measurements. Existing dual quaternion EKFs are additive, not multiplicative, and have two additional states. The DQ-MEKF is experimentally validated and compared with two conventional EKFs on the 5-DOF platform of the Autonomous Spacecraft Testing of Robotic Operations in Space (ASTROS) facility at the School of Aerospace Engineering at Georgia Tech. Finally, the velocity-free pose-tracking controller is compared qualitatively and quantitatively to a pose-tracking controller that uses the velocity estimates produced by the DQ-MEKF through a realistic proximity operations simulation. Third, a pose-tracking controller that does not require the mass and inertia matrix of the chaser satellite is suggested. This inertia-free controller takes into account the gravitational acceleration, the gravity-gradient torque, the perturbing acceleration due to Earth's oblateness, and constant -- but otherwise unknown -- disturbance forces and torques. Sufficient conditions on the reference pose are also given that guarantee the identification of the mass and inertia matrix of the satellite. Compared to the existing literature, this controller has only as many states as unknown elements and it does not require a priori known upper bounds on any states or parameters. Finally, the inertia-free pose-tracking controller and the DQ-MEKF are tested on a high-fidelity simulation of the 5-DOF platform of the ASTROS facility and also experimentally validated on the actual platform. The equations of motion of the 5-DOF platform, on which the high-fidelity simulation is based, are derived for three distinct cases: a 3-DOF case, a 5-DOF case, and a (2+1)-DOF case. Four real-time experiments were run on the platform. In the first, a sinusoidal reference attitude with respect to the inertial frame is tracked using VSCMGs. In the second, a constant reference attitude is maintained with respect to a target object using VSCMGs and measurements from a camera. In the third, the same sinusoidal reference attitude with respect to the inertial frame tracked in the first experiment is now tracked using cold-gas thrusters. Finally, in the fourth and last experiment, a time-varying 5-DOF reference pose with respect to the inertial frame is tracked using cold-gas thrusters.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02-20Bitstream added on 2014-06-13T18:47:50Z : No. of bitstreams: 1 oliveira_ac_me_sjrp.pdf: 346640 bytes, checksum: dfe236f71e5ef2c050420d64d2c48a70 (MD5) / O objetivo deste trabalho é estabelecer similaridades entre os complexos e os hipercomplexos, motivados em explorar idéias de Murnaghan, que introduziu, pela primeira vez, em uma apresentação elementar, a teoria dos quatérnios baseados no teorema de Moivre. É mostrada em detalhes uma analogia da relação complexa clássica de Moivre para quatérnios, e em brevidade para octônios generalizados, e apresenta-se as conexões com os operadores da teoria de Fueter e as funções transcendentais. A extensão do teorema de Moivre é estudada para quatérnios em definindo-se uma função exponencial quaterniônica. / In this work we establish similarities between the complex and the hipercomplex numbers, motivated in exploring ideas of Murnaghan, that introduced, for the first time, in an elementary presentation, the theory of the quaternions based on the theorem of Moivre. We show an analogy of the classic complex relation of Moivre for quaternions, and briefly discuss generalized octonions, as well as to present connections to operators of the theory of Fueter and transcendental functions. We consider them to study the extension of the theorem of Moivre for quaternions, in defining a exponential function on the quaternions.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-29T16:36:38Z No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-30T11:44:57Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) / Made available in DSpace on 2017-08-30T11:44:57Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) Previous issue date: 2015-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this paper it will be studied the imaginary numbers and how their evolution over time occurred. Such evolution has occurred at a slow pace until it reached at what is known today as the imaginary number i. However, the creation of the complex was not the end of the study of imaginary numbers. These studies have introduced even more comprehensive concepts creating sets as quaternions, extension of four dimensions of the complex. It will be concluded, with the extensions of eight and sixteen dimensions of the complex numbers, known as octonions and sedenions, respectively. Additionally, it will be submitted some applications of these extensions, also known as hypercomplex numbers. / Neste trabalho serão estudados os números imaginários e como se deu a sua evolução ao longo do tempo. Evolução esta que ocorreu de forma bem lenta, até se chegar no que é conhecido hoje como o número imaginário i. Entretanto, a criação dos complexos não foi o ponto nal do estudo dos números imaginários. Estudos seguintes introduziram conceitos ainda mais abrangentes criando conjuntos como os quatérnios, extensão de quatro dimensões dos complexos. Finaliza-se o trabalho, com as extensões de oito e dezesseis dimensões dos complexos, conhecidas como octônios e sedênios, respectivamente. Além de ser apresentado algumas aplicações dessas extensões, também conhecidas como números hipercomplexos.