Aquesta tesi doctoral està emmarcada en el camp de l'astrodinàmica. Presenta solucions a problemes identificats en el disseny de missions que utilitzen òrbites entorn dels punts de libració, fent servir la teoria de sistemes dinàmics.El problema restringit de tres cossos és un model per estudiar el moviment d'un cos de massa infinitessimal sota l'atracció gravitatòria de dos cossos molt massius. Els cinc punts d'equilibri d'aquest model, en especial L1 i L2, han estat motiu de nombrosos estudis per aplicacions pràctiques en les últimes dècades (SOHO, Genesis...). Genèricament, qualsevol missió en òrbita al voltant del punt L2 del sistema Terra-Sol es veu afectat per ocultacions degudes a l'ombra de la Terra. Si l'òrbita és al voltant de L1, els eclipsis són deguts a la forta influència electromagnètica del Sol. D'entre els diferents tipus d'òrbites de libració, les òrbites de Lissajous resulten de la combinació de dues oscil.lacions perpendiculars. El seu principal avantatge és que les amplituds de les oscil.lacions poden ser escollides independentment i això les fa adapatables als requeriments de cada missió. La necessitat d'estratègies per evitar eclipsis en òrbites de Lissajous entorn dels punts L1 i L2 motivaren la primera part de la tesi. En aquesta part es presenta una eina per la planificació de maniobres en òrbites de Lissajous que no només serveix per solucionar el problema d'evitar els eclipsis, sinó també per trobar trajectòries de transferència entre òrbites d'amplituds diferents i planificar rendez-vous. Per altra banda, existeixen canals de baix cost que uneixen els punts L1 i L2 d'un sistema donat i representen una manera natural de transferir d'una regió de libració a l'altra. Gràcies al seu caràcter hiperbòlic, una òrbita de libració té uns objectes invariants associats: les varietats estable i inestable. Si tenim present que la varietat estable està formada per trajectòries que tendeixen cap a l'òrbita a la qual estan associades quan el temps avança, i que la varietat inestable fa el mateix però enrera en el temps, una intersecció entre una varietat estable i una d'inestable proporciona un camí asimptòtic entre les òrbites corresponents. Un mètode per trobar connexions d'aquest tipus entre òrbites planes entorn de L1 i L2 es presenta a la segona part de la tesi, i s'hi inclouen els resultats d'aplicar aquest mètode als casos dels problemes restringits Sol Terra i Terra-Lluna.La idea d'intersecar varietats hiperbòliques es pot aplicar també en la cerca de camins de baix cost entre les regions de libració del sistema Sol-Terra i Terra-Lluna. Si existissin camins naturals de les òrbites de libració solars cap a les lunars, s'obtindria una manera barata d'anar a la Lluna fent servir varietats invariants, cosa que no es pot fer de manera directa. I a l'inversa, un camí de les regions de libració lunars cap a les solars permetria, per exemple, que una estació fos col.locada en òrbita entorn del punt L2 lunar i servís com a base per donar servei a les missions que operen en òrbites de libració del sistema Sol-Terra. A la tercera part de la tesi es presenten mètodes per trobar trajectòries de baix cost que uneixen la regió L2 del sistema Terra-Lluna amb la regió L2 del sistema Sol-Terra, primer per òrbites planes i més endavant per òrbites de Lissajous, fent servir dos problemes de tres cossos acoblats. Un cop trobades les trajectòries en aquest model simplificat, convé refinar-les per fer-les més realistes. Una metodologia per obtenir trajectòries en efemèrides reals JPL a partir de les trobades entre òrbites de Lissajous en el model acoblat es presenta a la part final de la tesi. Aquestes trajectòries necessiten una maniobra en el punt d'acoblament, que és reduïda en el procés de refinat, arribant a obtenir trajectòries de cost zero quan això és possible. / This PhD. thesis lies within the field of astrodynamics. It provides solutions to problems which have been identified in mission design near libration points, by using dynamical systems theory. The restricted three body problem is a well known model to study the motion of an infinitesimal mass under the gravitational attraction of two massive bodies. Its five equilibrium points, specially L1 and L2, have been the object of several studies aimed at practical applications in the last decades (SOHO, Genesis...). In general, any mission in orbit around L2 of the Sun-Earth system is affected by occultations due to the shadow of the Earth. When the orbit is around L1, the eclipses are caused by the strong electromagnetic influence of the Sun. Among all different types of libration orbits, Lissajous type ones are the combination of two perpendicular oscillations. Its main advantage is that the amplitudes of the oscillations can be chosen independently and this fact makes Lissajous orbits more adaptable to the requirements of each particular mission than other kinds of libration motions. The need for eclipse avoidance strategies in Lissajous orbits around L1 and L2 motivated the first part of the thesis. It is in this part where a tool for planning maneuvers in Lissajous orbits is presented, which not only solves the eclipse avoidance problem, but can also be used for transferring between orbits having different amplitudes and for planning rendez-vous strategies.On the other hand, there exist low cost channels joining the L1 and L2 points of a given sistem, which represent a natural way of transferring from one libration region to the other one. Furthermore, there exist hyperbolic invariant objects, called stable and unstable manifolds, which are associated with libration orbits due to their hyperbolic character. If we bear in mind that the stable manifold of a libration orbit consists of trajectories which tend to the orbit as time goes by, and that the unstable manifold does so but backwards in time, any intersection between a stable and an unstable manifold will provide an asymptotic path between the corresponding libration orbits. A methodology for finding such asymptotic connecting paths between planar orbits around L1 and L2 is presented in the second part of the dissertation, including results for the particular cases of the Sun-Earth and Earth-Moon problems. Moreover, the idea of intersecting hyperbolic manifolds can be applied in the search for low cost paths joining the libration regions of different problems, such as the Sun-Earth and the Earth-Moon ones. If natural paths from the solar libration regions to the lunar ones was found, it would provide a cheap way of transferring to the Moon from the vicinity of the Earth, which is not possible in a direct way using invariant manifolds. And the other way round, paths from the lunar libration regions to the solar ones would allow for the placement of a station in orbit around the lunar L2, providing services to solar libration missions, for instance. In the third part of the thesis, a methodology for finding low cost trajectories joining the lunar L2 region and the solar L2 region is presented. This methodology was developed in a first step for planar orbits and in a further step for Lissajous type orbits, using in both cases two coupled restricted three body problems to model the Sun-Earth-Moon spacecraft four body problem. Once trajectories have been found in this simplified model, it is convenient to refine them to more realistic models. A methodology for obtaining JPL real ephemeris trajectories from the initial ones found in the coupled models is presented in the last part of the dissertation. These trajectories need a maneuver at the coupling point, which can be reduced in the refinement process until low cost connecting trajectories in real ephemeris are obtained (even zero cost, when possible).

Esta tesis doctoral llevó a cabo el estudio, desarrollo e implementación de algoritmos para la navegación con sistemas globales de navegación por satélite (GNSS), enfocándose en la determinación precisa de la posición, velocidad y aceleración usando GPS, en modo post-procesado y lejos de estaciones de referencia. Uno de los objetivos era desarrollar herramientas en esta área y hacerlas disponibles a la comunidad GNSS. Por ello el desarrollo se hizo dentro del marco del proyecto preexistente de software libre llamado GPS Toolkit (GPSTk). Una de las primeras tareas realizadas fue la validación de las capacidades de la GPSTk para el procesado del pseudorango, realizando comparaciones con una herramienta de procesamiento de datos probada (BRUS). La gestión de datos GNSS demostró ser un asunto importante cuando se intentó extender las capacidades de la GPSTk al procesamiento de datos obtenidos de las fases de la señal GPS. Por ello se desarrollaron las Estructuras de Datos GNSS (GDS), que combinadas con su paradigma de procesamiento aceleran el proceso de desarrollo de software y reducen errores. La extensión de la GPSTk a los algoritmos de procesado en fase se hizo mediante la ayuda de las GDS, proporcionándose importantes clases accesorias que facilitan el trabajo. Se implementó el procesado de datos Precise Point Positioning (PPP) con ejemplos relativamente simples basados en las GDS, y al comparar sus resultados con otras aplicaciones de reputación ya establecida, se encontró que destacan entre los mejores. También se estudió cómo obtener la posición precisa, en post-proceso, de un receptor GPS a cientos de kilómetros de la estación de referencia más cercana y usando tasas de datos arbitrarias (una limitación del método PPP). Las ventajas aportadas por las GDS permitieron la implementación de un procesado semejante a un PPP cinemático basado en una red de estaciones de referencia, estrategia bautizada como Precise Orbits Positioning (POP) porque sólo necesita órbitas precisas para trabajar y es independiente de la información de los relojes de los satélites GPS. Los resultados de este enfoque fueron muy similares a los del método PPP cinemático estándar, pero proporcionando soluciones de posición con una tasa mayor y de manera más robusta. La última parte se enfocó en la implementación y mejora de algoritmos para determinar con precisión la velocidad y aceleración de un receptor GPS. Se hizo énfasis en el método de las fases de Kennedy debido a su buen rendimiento, desarrollando una implementación de referencia y demostrando la existencia de una falla en el procedimiento propuesto originalmente para el cálculo de las velocidades de los satélites. Se propuso entonces una modificación relativamente sencilla que redujo en un factor mayor que 35 el RMS de los errores 3D en velocidad. Tomando ideas de los métodos Kennedy y POP se desarrolló e implementó un nuevo procedimiento de determinación de velocidad y aceleración que extiende el alcance. Este método fue llamado Extended Velocity and Acceleration determination (EVA). Un experimento usando una aeronave ligera volando sobre los Pirineos mostró que tanto el método de Kennedy (modificado) como el método EVA son capaces de responder ante la dinámica de este tipo de vuelos. Finalmente, tanto el método de Kennedy modificado como el método EVA fueron aplicados a una red en la zona ecuatorial de Sur América con líneas de base mayores a 1770 km. En este escenario el método EVA mostró una clara ventaja tanto en los promedios como en las desviaciones estándar para todas las componentes de la velocidad y la aceleración. / This Ph.D. Thesis focuses on the development of algorithms and tools for precise GPS-based position, velocity and acceleration determination very far from reference stations in post-process mode. One of the goals of this thesis was to develop a set of state-of-the-art GNSS data processing tools, and make them available for the research community. Therefore, the software development effort was done within the frame of a preexistent open source project called the GPSTk. Therefore, validation of the GPSTk pseudorange-based processing capabilities with a trusted GPS data processing tool was one of the initial task carried out in this work. GNSS data management proved to be an important issue when trying to extend GPSTk capabilities to carrier phasebased data processing algorithms. In order to tackle this problem the GNSS Data Structures (GDS) and their associated processing paradigm were developed. With this approach the GNSS data processing becomes like an assembly line, providing an easy and straightforward way to write clean, simple to read and use software that speeds up development and reduces errors. The extension of GPSTk capabilities to carrier phase-based data processing algorithms was carried out with the help of the GDS, adding important accessory classes necessary for this kind of data processing and providing reference implementations. The performance comparison of these relatively simple GDS-based source code examples with other state-of-the art Precise Point Positioning (PPP) suites demonstrated that their results are among the best. Furthermore, given that the GDS design is based on data abstraction, it allows a very flexible handling of concepts beyond mere data encapsulation, including programmable general solvers, among others. The problem of post-process precise positioning of GPS receivers hundreds of kilometers away from nearest reference station at arbitrary data rates was dealt with, overcoming an important limitation of classical post-processing strategies like PPP. The advantages of GDS data abstraction regarding solvers were used to implement a kinematic PPP-like processing based on a network of stations. This procedure was named Precise Orbits Positioning (POP) because it is independent of precise clock information and it only needs precise orbits to work. The results from this approach were very similar (as expected) to the standard kinematic PPP processing strategy, but yielding a higher positioning rate. Also, the network-based processing of POP seems to provide additional robustness to the results, even for receivers outside the network area. The last part of this thesis focused on implementing, improving and testing algorithms for the precise determination of velocity and acceleration hundreds of kilometers away from nearest reference station. Special emphasis was done on the Kennedy method because of its good performance. A reference implementation of Kennedy method was developed, and several experiments were carried out. Experiments done with very short baselines showed a flaw in the way satellite velocities were computed, introducing biases in the velocity solution. A relatively simple modification was proposed, and it reduced the RMS of 5-min average velocity 3D errors by a factor of over 35. Then, borrowing ideas from Kennedy method and the POP method, a new velocity and acceleration determination procedure named EVA was developed and implemented that greatly extends the effective range. An experiment using a light aircraft flying over the Pyrenees showed that both the modified-Kennedy and EVA methods were able to cope with the dynamics of this type of flight. Finally, both modified-Kennedy and EVA method were applied to a challenging scenario in equatorial South America, with baselines over 1770 km, where EVA method showed a clear advantage in both averages and standard deviations for all components of velocity and acceleration. Lloc i

Satellite constellations are an increasingly attractive option for many commercial and military applications. They provide a robust and distributed method of accomplishing the goals of expensive monolithic satellites. Among the many challenges that satellite constellations engender (challenges in control, coordination, disposal, and other areas), refueling is of particular interest because of the many methods one can use to refuel a constellation and the lifetime implications on the satellites. The present work presents a methodology for carrying out peer-to-peer refueling maneuvers within a constellation. Peer-to-peer (P2P) refueling can be of great value both in cases where a satellite unexpectedly consumes more fuel than it was alloted, and as part of a mixed refueling strategy that will include an outside tanker bringing fuel to the constellation. Without considering mixed-refueling, we formulate the peer-to-peer refueling problem as an assignment problem that seeks to guarantee that all satellites will have the fuel they need to be functional until the next refueling, while concurrently minimizing the cost in fuel that the refueling maneuvers entail. The assignment problem is then solved via auctions, which, by virtue of their distributed nature, can easily and effectively be implemented on a constellation without jeopardizing any robustness properties. Taking as a given that the P2P assignment problem has been solved, and that it has produced some matching among fuel deficient and fuel sufficient satellites, we then seek to sequence those prescribed maneuvers in the most effective manner. The idea is that while a constellation can be expected to have some redundancy, enough satellites leaving their assigned orbital slots will eventually make it impossible for the constellation to function. To tackle this problem, we define a wide class of operability conditions, and present three algorithms that intelligently schedule the maneuvers. We then briefly show how combining the matching and scheduling problems yields a complete methodology for organizing P2P satellite refueling operations.

To prove that a polynomial is nonnegative on Rn, one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semi-definite programming (SDP) computation that is much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables with k = 4 fixed those that are SOS occupy a vanishingly small fraction of those that are nonnegative on Rn, as n -> 1. With an eye toward the case of small n, we refine Blekherman'[s bounds by incorporating the underlying Newton polytope, simultaneously sharpening some of his older bounds along the way. Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs.

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.

When excitable cardiac tissue is electrically paced at a sufficiently high rate, the duration of excitation can alternate from beat to beat despite a constant stimulation period. This rhythm, known as alternans, has been identified as an early stage in a sequence of increasingly complex instabilities leading to the lethal arrhythmia ventricular fibrillation (VF). This connection served as as a motivation for research into the control of alternans as a strategy to prevent VF. Control methods that do not use a model of the dynamics have been used for the suppression of alternans. However, these methods possess limitations. In this thesis we study theoretically model-based control techniques with the goal of developing protocols that would overcome the shortcomings of non model-based approaches. We consider one dimensional tissue in two different geometrical configurations: a ring and a fiber with free ends (open fiber). We apply standard control methods for linear time invariant systems to a stroboscopic map of the linearized dynamics around the normal rhythm. We found that, in the ring geometry, model-based control is able to suppress alternans faster and with lower current, thereby reducing the risk of tissue damage, compared with non-model-based control. In the open fiber, model-based control is able to suppress alternans for longer fibers and higher pacing frequencies in comparison with non-model-based control. The methodology presented here can be extended to two- and three-dimensional tissue, and could eventually lead to the suppression of alternans on the entire ventricles.

M. Tech. Electrical Engineering / Deals with nonlinear methods for magnetic attitude control and reaction wheel attitude control. The work is divided into a number of parts. The first part, deals with the satellite attitude control basic information and development of a mathematical model of a low Earth orbit satellite. The second part introduces the controllers used in this dissertation. The third part deals with the dimension between the output of controller and input of reaction wheel. The fourth part solves the problem of the magnetic torque calculation. The last part carries out the simulation tests of those controllers for small satellite and cube satellite.

M. Tech. Electrical Engineering / The topic of this dissertation is the attitude control of a CubeSat in an elliptic orbit using nonlinear control. The attitude control system (ACS) is a subsystem of a CubeSat. Its principal goal is to stabilise the orientation of the satellite after launch and during the orbital motion of the satellite. Although several methods have been applied to achieve this objective, this still remains a challenging objective and hence plays an integral role in many modern technologies. CubeSat "Cube Satellite" is a miniaturised satellite which, due to its low cost and application potential is often used by academic institutions for research purposes. However, due to its physical size and weight of 1 kilogram, CubeSat have comparatively limited power supply and computational resources; hence the need for an uncomplicated and reliable control system is critical.

We study the classical nonlinear dynamics and the quantum vibrational energy eigenstates of the molecule HOCl. The classical vibrational dynamics, at energies below the HO+Cl dissociation energy, contains several saddle-center and period doubling bifurcations. The saddle-center bifurcations are shown to be due to a 2:1, and at higher energies a 3:1, nonlinear resonance between bend and stretch motions in various periodic orbits. The sequence of bifurcations takes the system from nearly integrable at low energies to almost completely chaotic at energies near the HO+Cl dissociation energy. At energies above dissociation we study the chaotic scattering of the Cl atom off the HO dimer. This scattering is governed by a homoclinic tangle formed by the stable and unstable manifolds of a parabolic periodic orbit at infinity. We construct the first three segments of the homoclinic tangle in phase space and use scattering functions to investigate its higher-order structure. For the quantum system we use a discrete variable representation to efficiently calculate the Hamiltonian matrix. We find 365 even and 357 odd parity eigenstates with energies below the dissociation energy. By plotting the eigenstates in configuration space we show that almost every quantum eigenstate can be associated with one or more of the classical periodic orbits. The classical bifurcations that give rise to new periodic orbits are manifest quantum mechanically through the sudden appearance of new classes of eigenstates. Despite the high degree of chaos in the classical dynamics at energies near the dissociation energy most quantum eigenstates remain highly ordered with recognizable nodal patterns. We use R-matrix theory together with a discrete variable representation to calculate quasibound states with energies above the dissociation energy. We find quasibound states with lifetimes ranging over 5 orders of magnitude. Using configuration space plots and Husimi distributions we show that the long-lived quasibound states are supported by unstable periodic orbits in the classical dynamics and medium-lived quasibound states are spread throughout the chaotic region of the classical phase space. Short-lived quasibound states show some similarity to unstable periodic orbits that stretch along the dissociation channel. / text

This thesis develops methods to identify periodic solutions to the n-body problem by representing gravitational orbits with Fourier series. To find periodic orbits, a minimization function was developed that compares the second derivative of the Fourier series with Newtonian gravitation acceleration and modifies the Fourier coefficients until the orbits match. Software was developed to minimize the function and identify the orbits using gradient descent and quadratic curves. A Newtonian gravitational simulator was developed to read the initial orbit data and numerically simulate the orbits with accurate motion integration, allowing for comparison to the Fourier series orbits and investigation of their stability. The orbits found with the programs correlate with orbits from literature, and a number remain stable when simulated. / February 2016