# Browsing by Subject "Uncertainty propagation"

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Item Improved multidirectional Gaussian Mixture Models applied to probability of collision of resident space objects(2020-08-14) Brown, Chase Patrick; Russell, Ryan Paul, 1976-Show more As the number of Resident Space Objects (RSOs) in Earth Orbit continues to rise by not only increased trackability but also an unprecedented number of commercial launches, conjunction assessment (CA) remains a paramount issue. Maintaining accuracy in probability of collision calculations is specifically of interest because any disparity will cause operators and analysts to lose trust, dismantling the entire CA system. Methods of state uncertainty propagation remain the most tractable way of controlling computational efficiency vs. accuracy. Gaussian Mixture Models (GMMs) have recently been used as an approach to maintain accuracy while decreasing the amount of computation time when compared to a Monte Carlo approach. These GMMs are able to represent the initial probability distribution function (pdf) as a weighted combination of individual Gaussian distributions. When propagated through a nonlinear function, such as the orbital equations of motion, higher order effects are maintained. How that initial pdf is split into a convolution of pdfs is the focus of this and current research. Multidirectional GMMs allow for the pdf to be split along directions of highest nonlinearity in a recursive manner. This study improves on this method by evaluating the directions at every split of every Gaussian mixture and also taking into account the weight of that Gaussian mixture. Equinoctial elements are also explored as a potential element space to perform the splitting due their ability to maintain linearity during propagation. Results show that these improved methods are able to capture the majority of the nonlinear effects very well with relatively few GMs, and therefore can generate accurate Pc calculations, but fail to converge to the exact Monte Carlo value as more mixtures are added in a reasonable time. This is still of use to get within 5% of the Monte Carlo value with very few propagations in highly nonlinear encountersShow more Item New developments in nonlinear filtering using differential algebra(2021-07-24) Servadio, Simone; Zanetti, Renato, 1978-; Akella, Maruthi R.; Jah, Moriba K.; Jones, Brandon A.; D'Souza, ChristopherShow more This dissertation presents five different solutions to the nonlinear filtering problem. Three filtering techniques present a different systematic generalization of the linear update structure associated with the extended Kalman filter for high order polynomial estimation of nonlinear dynamical systems. The minimum mean-square error criterion is used as the cost function to determine the optimal polynomial update during the estimation process. Furthermore, both the propagated and posterior probability density functions (PDFs) can be represented through Taylor expansion polynomials, giving an accurate approximation of the shape of the distribution, and providing the base for a new Maximum A Posteriori estimation technique. The high order series representation is implemented using differential algebra (DA) techniques. Differential algebra has been presented as an efficient tool to map PDFs, propagate central moments, and reduce computational burden on Monte Carlo based filtering techniques. Each proposed algorithm has been tested on strongly nonlinear applications, where the most common filters in the literature fail to provide a correct estimate.Show more Item Uncertainty propagation and conjunction assessment for resident space objects(2015-12) Vittaldev, Vivek; Russell, Ryan Paul, 1976-; Erwin, Richard S; Akella, Maruthi R; Bettadpur, Srinivas V; Humphreys, Todd EShow more Presently, the catalog of Resident Space Objects (RSOs) in Earth orbit tracked by the U.S. Space Surveillance Network (SSN) is greater than 21,000 objects. The size of the catalog continues to grow due to an increasing number of launches, improved tracking capabilities, and in some cases, collisions. Simply propagating the states of these RSOs is a computational burden, while additionally propagating the uncertainty distributions of the RSOs and computing collision probabilities increases the computational burden by at least an order of magnitude. Tools are developed that propagate the uncertainty of RSOs with Gaussian initial uncertainty from epoch until a close approach. The number of possible elements in the form of a precomputed library, in a Gaussian Mixture Model (GMM) has been increased and the strategy for multivariate problems has been formalized. The accuracy of a GMM is increased by propagating each element by a Polynomial Chaos Expansion (PCE). Both techniques reduce the number of function evaluations required for uncertainty propagation and result in a sliding scale where accuracy can be improved at the cost of increased computation time. A parallel implementation of the accurate benchmark Monte Carlo (MC) technique has been developed on the Graphics Processing Unit (GPU) that is capable of using samples from any uncertainty propagation technique to compute the collision probability. The GPU MC tool delivers up to two orders of magnitude speedups compared to a serial CPU implementation. Finally, a CPU implementation of the collision probability computations using Cartesian coordinates requires orders of magnitude fewer function evaluations compared to a MC run. Fast computation of the inherent nonlinear growth of the uncertainty distribution in orbital mechanics and accurately computing the collision probability is essential for maintaining a future space catalog and for preventing an uncontrolled growth in the debris population. The uncertainty propagation and collision probability computation methods and algorithms developed here are capable of running on personal workstations and stand to benefit users ranging from national space surveillance agencies to private satellite operators. The developed techniques are also applicable for many general uncertainty quantification and nonlinear estimation problems.Show more