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Descripción
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| Regularized formulations of orbital motion provide powerful tools for solving various problems in orbital mechanics, both analytically and numerically. They rely on a collection of dynamical and mathematical transformations that yields a more convenient description of the dynamics. The goal of the present thesis is to recover the foundations of regularization, to advance the theory toward practical applications, and to use this mathematical contrivance for solving three key challenges in modern astrodynamics: the dynamics of spacecraft formations, the design of low-thrust trajectories, and the high-performance numerical propagation of orbits. The introduction of a fictitious time is a typical practice when regularizing the equations of motion. This technique leads to a new theory of relative motion, called the theory of asynchronous relative motion. It improves the accuracy of the linear propagation by introducing nonlinear terms through simple dynamical mechanisms, and simplifies significantly the derivation of analytic solutions. In addition, it admits any type of orbital perturbation. The method is compact and seems well suited for its implementation in navigation filters and control laws. Universal and fully regular solutions to the relative dynamics follow naturally from this theory. They are valid for any type of reference orbit (circular, elliptic, parabolic, or hyperbolic) and are not affected by the typical singularities related to the eccentricity or inclination of the orbit. The nonlinear corrections proposed by the method are generic and can be applied to existing solutions to improve their accuracy without the need for a dedicated re-implementation. We present a novel shape-based method for preliminary design of low-thrust trajectories: the family of generalized logarithmic spirals. This new solution arises from the search for sets of orbital elements in the accelerated case. It is fully analytic and involves two conservation laws (related to the equations of the energy and angular momentum) that make the solution surprisingly similar to the Keplerian case and simplify the design process. The properties of the solution to the Keplerian Lambert problem find direct analogues in the continuous-thrust case. An analysis of the dynamical symmetries in the problem shows that the perturbing acceleration can be generalized and provides two additional families of analytic solutions: the generalized cardioids and the generalized sinusoidal spirals. As the complexity of space missions increases, more sophisticated orbit propagators are required. In order to integrate flyby trajectories more efficiently, an improved propagation scheme is presented, exploiting the geometry of Minkowski space-time. The motion of the orbital plane is decoupled from the in-plane dynamics, and the introduction of hyperbolic geometry simplifies the description of the planar motion. General considerations on the accuracy of the propagation of flyby trajectories are presented. In the context of N-body systems, we prove that regularization yields a simplified Lyapunov-like indicator that helps in assessing the validity of the numerical integration. Classical concepts arising from stability theories are extended to higher dimensions to comply with the regularized state-space. In this thesis, we present, for the first time, the gauge-generalized form of some element-based regularized formulations. | |
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Internacional
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ISBN
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Tipo de Tesis
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Doctoral |
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Calificación
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Sobresaliente cum laude |
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Fecha
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30/09/2016 |