Non-dimensional Star-Identification

This study introduces a new "Non-Dimensional" star identification algorithm to reliably identify the stars observed by a wide field-of-view star tracker when the focal length and optical axis offset values are known with poor accuracy. This algorithm is particularly suited to complement nominal lost-in-space algorithms, which may identify stars incorrectly when the focal length and/or optical axis offset deviate from their nominal operational ranges. These deviations may be caused, for example, by launch vibrations or thermal variations in orbit. The algorithm performance is compared in terms of accuracy, speed, and robustness to the Pyramid algorithm. These comparisons highlight the clear advantages that a combined approach of these methodologies provides.Comment: 17 pages, 10 figures, 4 table

The Multivariate Theory of Functional Connections: An n-Dimensional Constraint Embedding Technique Applied to Partial Differential Equations

The Theory of Functional Connections (TFC) is a functional interpolation framework founded upon the so-called constrained expression: a functional that expresses the family of all possible functions that satisfy some user-specified, linear constraints. These constrained expressions can be utilized to transform constrained problems into unconstrained ones. The benefits of doing so include faster solution times, more accurate solutions, and more robust convergence. This dissertation contains a comprehensive, self-contained presentation of the TFC theory beginning with simple univariate point constraints and ending with general linear constraints in n-dimensions; relevant mathematical theorems and clarifying examples are included throughout the presentation to expand and solidify the reader's understanding. Furthermore, this dissertation describes how TFC can be applied to estimate differential equations' solutions, its primary application to date. In addition, comparisons with other state-of-the-art algorithms that estimate differential equations' solutions are included to showcase the advantages and disadvantages of the TFC approach. Lastly, the aforementioned concepts are leveraged to estimate solutions of differential equations from the field of flexible body dynamics

The Multivariate Theory of Functional Connections: An n-Dimensional Constraint Embedding Technique Applied to Partial Differential Equations

The Theory of Functional Connections (TFC) is a functional interpolation framework founded upon the so-called constrained expression: a functional that expresses the family of all possible functions that satisfy some user-specified, linear constraints. These constrained expressions can be utilized to transform constrained problems into unconstrained ones. The benefits of doing so include faster solution times, more accurate solutions, and more robust convergence. This dissertation contains a comprehensive, self-contained presentation of the TFC theory beginning with simple univariate point constraints and ending with general linear constraints in n-dimensions; relevant mathematical theorems and clarifying examples are included throughout the presentation to expand and solidify the reader's understanding. Furthermore, this dissertation describes how TFC can be applied to estimate differential equations' solutions, its primary application to date. In addition, comparisons with other state-of-the-art algorithms that estimate differential equations' solutions are included to showcase the advantages and disadvantages of the TFC approach. Lastly, the aforementioned concepts are leveraged to estimate solutions of differential equations from the field of flexible body dynamics