# Browsing by Subject "State transition matrix"

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Item Automatic algorithm for accurate numerical gradient calculation in general and complex spacecraft trajectories(2010-12) Restrepo Gómez, Ricardo León.; Ocampo, Cesar; Hull, David G.Show more An automatic algorithm for accurate numerical gradient calculations has been developed. The algorithm is based on both finite differences and Chebyshev interpolation approximations. The novelty of the method is an automated tuning of the step size perturbation required for both methods. This automation guaranties the best possible solution using these approaches without the requirement of user inputs. The algorithm treats the functions as a black box, which makes it extremely useful when general and complex problems are considered. This is the case of spacecraft trajectory design problems and complex optimization systems. An efficient procedure for the automatic implementation is presented. Several examples based on an Earth-Moon free return trajectory are presented to validate and demonstrate the accuracy of the method. A state transition matrix (STM) procedure is developed as a reference for the validation of the method.Show more Item Revisiting Vinti theory : generalized equinoctial elements and applications to spacecraft relative motion(2017-08-11) Biria, Ashley Darius; Russell, Ryan Paul, 1976-; Akella, Maruthi R.; Bettadpur, Srinivas V.; Jones, Brandon A.; Sabol, Chris A.Show more Early phases of complex astrodynamics applications often require broad searches of large solution spaces. For these studies, mission complexity generally motivates the use of the coarsest dynamical models with analytical solutions because of the implied lightening of the computational load. In this context, two-body dynamics are typically employed in practice, but higher-fidelity models with analytical solutions exist, an attractive prospect for modern applications that may require or benefit from greater accuracy. Vinti theory, which prescribes one of the many alternative described models known as intermediaries, is revisited because it leads to a direct generalization of two-body dynamics, naturally incorporating the dominant effect of oblateness and optionally the top/bottom-heavy characteristic of a celestial body without recourse to perturbation methods. Prior to the innovations introduced in this dissertation, Vinti theory and associated solutions possessed many singularities in popular orbital regimes. The theory has received limited use. The goals of this dissertation are to assess Vinti theory's effectiveness in a modern application and remove its long-standing disincentives. These objectives inform the two main contributions, respectively: 1) Vinti theory is applied to the relative motion problem through the development of a state transition matrix (STM), enabled by improvements to the existing theory; 2) a new nonsingular element set is introduced. The relative motion application leverages Vinti's approximate analytical solution with J₃. An analytical relative motion model is derived and subsequently reformulated so that Vinti's solution is piecewise differentiable, developed alongside boosts in accuracy and removal of singularities in polar and nearly circular or equatorial orbits. Some of these singularities reside in the solution, others in the partials. Solving the problem in oblate spheroidal elements leads to large linear regions of validity. The new STM is compared with side-by-side simulations of a benchmark STM obtained from perturbation methods and is shown to offer improved accuracy over a broad design space. To defray the costs of software development, robust code is provided online. The second major thrust area is the introduction of a nonsingular element set that is at once novel and familiar. Vinti theory suffers from other well-known singularities, strictly artifacts of classical elements that are detrimental to many applications. To mitigate these singularities, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The new elements are derived without J₃ and concise algorithms presented for common coordinate transformations. The transformations are valid away from the nearly rectilinear orbital regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution. As a result, the singularity on the poles is completely eliminated for the first time. Analytical state propagation of the new elements in time for bounded orbits completes their formal introduction. Benefits of the new elements are identified. The dissertation is organized as follows. To convey Vinti theory's broader context, extensive background on intermediaries and related topics is provided in Chapter 1. General enhancements that grew out of the main efforts, including the removal of some singularities, are consolidated in Chapter 2 along with mathematical preliminaries. Relative motion is explored as the selected application in Chapter 3 and the major deficiencies of Vinti theory are removed in Chapter 4 with the introduction of the new element set. Analytical orbit propagation in the new set is developed in Chapter 5.Show more Item Space object translational and rotational state prediction and sensitivity calculation(2016-12) Hatten, Noble Ariel; Russell, Ryan Paul, 1976-; Akella, Maruthi R; Bettadpur, Srinivas V; Jones, Brandon A; Weisman, Ryan MShow more While computing power has grown monumentally during the space age, the demands of astrodynamics applications have more than kept pace. Resources are taxed by the ever-growing number of Earth-orbiting space objects (SOs) that must be tracked to maintain space situational awareness (SSA) and by increasingly popular but computationally expensive tools like Monte Carlo techniques and stochastic optimization algorithms. In this dissertation, methods are presented to improve the accuracy, efficiency, and utility of SO state prediction and sensitivity calculation algorithms. The dynamical model of the low Earth orbit regime is addressed through the introduction of an upgraded Harris-Priester atmospheric density model, which introduces a smooth polynomial dependency on solar flux. Additional modifications eliminate singularities and provide smooth partial derivatives of the density with respect to SO state, time, and solar conditions. The numerical solution of the equations of motion derived from dynamics models is also addressed, with particular emphasis placed on six-degree-of-freedom (6DOF) state prediction. Implicit Runge-Kutta (IRK) methods are applied to the 6DOF problem, and customizations, including variable-fidelity dynamics models and parallelization, are introduced to maximize efficiency and take advantage of modern computing architectures. Sensitivity calculation -- a necessity for SSA and other applications -- via RK methods is also examined. Linear algebraic systems for first- and second-order state transition matrix calculation are derived by directly differentiating either the first- or second-order form of the RK update equations. This approach significantly reduces the required number of Jacobian and Hessian evaluations compared to the ubiquitous augmented state vector approach for IRK methods, which can result in more efficient calculations. Parallelization is once again leveraged to reduce the runtime of IRK methods. Finally, a hybrid special perturbation/general perturbation (SP/GP) technique is introduced to address the notoriously slow speed of fully coupled 6DOF state prediction. The hybrid method uses a GP rotational state prediction to provide low-fidelity attitude information for a high-fidelity 3DOF SP routine. This strategy allows for the calculation of body forces using arbitrary shape models without adding attitude to the propagated state or taking the small step sizes often required by full 6DOF propagation. The attitude approximation is obtained from a Lie-Deprit perturbation result previously applied to SOs in circular orbits subject to gravity-gradient torque and extended here to SOs in elliptical orbits. The hybrid method is shown to produce a meaningful middle ground between 3DOF SP and 6DOF SP methods in the accuracy vs. efficiency space.Show more