A Return to the Sextant-Maritime Navigation Using Celestial Bodies and the Horizon.

Satellite navigation over recent decades has become the default and, in some cases, sole source of positioning for maritime vessels. The classic sextant has been all but forgotten by a significant number of ship navigators. However, recent risks to RF-derived positioning by jamming and spoofing have resurfaced the need to train sailors again in the art.

High Accurate Mathematical Tools to Estimate the Gravity Direction Using Two Non-Orthogonal Inclinometers.

This study provides two mathematical tools to best estimate the gravity direction when using a pair of non-orthogonal inclinometers whose measurements are affected by zero-mean Gaussian errors. These tools consist of: (1) the analytical derivation of the gravity direction expectation and its covariance matrix, and (2) a continuous description of the geoid model correction as a linear combination of a set of orthogonal surfaces. The accuracy of the statistical quantities is validated by extensive Monte Carlo tests and the application in an Extended Kalman Filter (EKF) has been included.

Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-Rectangular 2-Dimensional Domains.

This work presents an initial analysis of using bijective mappings to extend the Theory of Functional Connections to non-rectangular two-dimensional domains. Specifically, this manuscript proposes three different mappings techniques: (a) complex mapping, (b) the projection mapping, and (c) polynomial mapping. In that respect, an accurate least-squares approximated inverse mapping is also developed for those mappings with no closed-form inverse.

Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via Theory of Functional Connections.

In this paper we present a new approach to solve the fuel-efficient powered descent guidance problem on large planetary bodies with no atmosphere (e.g., Moon or Mars) using the recently developed Theory of Functional Connections.

Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding.

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied.

Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations.

This article presents a new methodology called Deep Theory of Functional Connections (TFC) that estimates the solutions of partial differential equations (PDEs) by combining neural networks with the TFC. The TFC is used to transform PDEs into unconstrained optimization problems by analytically embedding the PDE's constraints into a "constrained expression" containing a free function. In this research, the free function is chosen to be a neural network, which is used to solve the now unconstrained optimization problem.

Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines Using the Theory of Functional Connections.

Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS).

Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections.

This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a constrained expression) containing a function that the user is free to choose.

High accuracy least-squares solutions of nonlinear differential equations.

This study shows how to obtain to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a , derived from .

The -dimensional -vector and its application to orthogonal range searching.

This work focuses on the definition and study of the -dimensional -vector, an algorithm devised to perform orthogonal range searching in static databases with multiple dimensions. The methodology first finds the order in which to search the dimensions, and then, performs the search using a modified projection method. In order to determine the dimension order, the algorithm uses the -vector, a range searching technique for one dimension that identifies the number of elements contained in the searching range.

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.

Theoretical Limits of Star Sensor Accuracy.

To achieve mass, power and cost reduction, there is a trend to reduce the volume of many instruments aboard spacecraft, especially for small spacecraft (cubesats or nanosats) with very limited mass, volume and power budgets. With the current trend of miniaturizing spacecraft instruments one could naturally ask if is there a physical limit to this process for star sensors. This paper shows that there is a fundamental limit on star sensor accuracy, which depends on stellar distribution, star sensor dimensions and exposure time.

Derivation of All Attitude Error Governing Equations for Attitude Filtering and Control.

This article presents the full analytical derivations of the attitude error kinematics equations. This is done for several attitude error representations, obtaining compact closed-forms expressions. Attitude error is defined as the rotation between true and estimated orientations.

The Multivariate Theory of Connections.

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called , representing possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, (, ), satisfy all constraints no matter what the (, ) is.

Random Sampling using -vector.

This work introduces two new techniques for random number generation with any prescribed nonlinear distribution based on the -vector methodology. The first approach is based on inverse transform sampling using the optimal -vector to generate the samples by inverting the cumulative distribution. The second approach generates samples by performing random searches in a pre-generated large database previously built by massive inversion of the prescribed nonlinear distribution using the -vector.