**14** Publications

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Sensors (Basel) 2021 Aug 25;21(17). Epub 2021 Aug 25.

Aerospace Engineering, Texas A&M University, 746C H.R. Bright Building, 3141 TAMU, College Station, TX 77843-3141, USA.

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. The continuous geoid description is needed as the geoid represents the true gravity direction. These tools can be implemented in any problem requiring high-precision estimates of the local gravity direction.

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http://dx.doi.org/10.3390/s21175727 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC8433952 | PMC |

August 2021

Mathematics (Basel) 2020 Sep 16;8(9):1593. Epub 2020 Sep 16.

Aeronautics and Astronautics, Massachussetts Institute of Technology, Cambridge, MA 02139, USA.

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. Advantages and disadvantages of using these mappings are highlighted and a few examples are provided. Additionally, the paper shows how to replace boundary constraints expressed in terms of a piece-wise sequence of functions with a single function, which is compatible and required by the Theory of Functional Connections already developed for rectangular domains.

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http://dx.doi.org/10.3390/math8091593 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7553096 | PMC |

September 2020

J Astronaut Sci 2020 25;67(4). Epub 2020 Sep 25.

Aerospace Engineering, Texas A&M University, College Station, TX, 77843, USA.

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. The problem is formulated using the indirect method which casts the optimal guidance problem as a system of nonlinear two-point boundary value problems. Using the Theory of Functional Connections, the problem's linear constraints are analytically embedded into a functional, which maintains a free-function that is expanded using orthogonal polynomials with unknown coefficients. The constraints are always analytically satisfied regardless of the values of the unknown coefficients (e.g., the coefficients of the free-function) which converts the two-point boundary value problem into an unconstrained optimization problem. This process reduces the whole solution space into the admissible solution subspace satisfying the constraints and, therefore, simpler, more accurate, and faster numerical techniques can be used to solve it. In this paper a nonlinear least-squares method is used. In addition to the derivation of this technique, the method is validated in two scenarios and the results are compared to those obtained by the general purpose optimal control software, GPOPS-II. In general, the proposed technique produces solutions of accuracy. Additionally, for the proposed test cases, it is reported that each individual TFC-based inner-loop iteration converges within 6 iterations, each iteration exhibiting a computational time between 72 and 81 milliseconds, with a total execution time of 2.1 to 2.6 seconds using MATLAB. Consequently, the proposed methodology is potentially suitable for real-time computation of optimal trajectories.

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http://dx.doi.org/10.1007/s40295-020-00228-x | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7553110 | PMC |

September 2020

Mathematics (Basel) 2019 Jun 12;7(6):537. Epub 2019 Jun 12.

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

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http://dx.doi.org/10.3390/math7060537 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7263466 | PMC |

June 2019

Mach Learn Knowl Extr 2020 Mar 12;2(1):37-55. Epub 2020 Mar 12.

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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. This optimization problem consists of minimizing a loss function that is chosen to be the square of the residuals of the PDE. The neural network is trained in an unsupervised manner to minimize this loss function. This methodology has two major differences when compared with popular methods used to estimate the solutions of PDEs. First, this methodology does not need to discretize the domain into a grid, rather, this methodology can randomly sample points from the domain during the training phase. Second, after training, this methodology produces an accurate analytical approximation of the solution throughout the entire training domain. Because the methodology produces an analytical solution, it is straightforward to obtain the solution at any point within the domain and to perform further manipulation if needed, such as differentiation. In contrast, other popular methods require extra numerical techniques if the estimated solution is desired at points that do not lie on the discretized grid, or if further manipulation to the estimated solution must be performed.

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http://dx.doi.org/10.3390/make2010004 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259480 | PMC |

March 2020

Mach Learn Knowl Extr 2019 Dec 9;1(4):1058-1083. Epub 2019 Oct 9.

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches.

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http://dx.doi.org/10.3390/make1040060 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259481 | PMC |

December 2019

Mathematics (Basel) 2020 Mar 11;8(3):397. Epub 2020 Mar 11.

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.

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http://dx.doi.org/10.3390/math8030397 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259483 | PMC |

March 2020

J Comput Appl Math 2019 May 18;352:293-307. Epub 2018 Dec 18.

Mathematics Department, Blinn College, College Station, TX, United States.

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 . In this expression, the differential equation constraints are embedded and are always satisfied. The resulting is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.

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http://dx.doi.org/10.1016/j.cam.2018.12.007 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7243685 | PMC |

May 2019

Appl Math Comput 2020 May 8;372. Epub 2020 Jan 8.

Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA.

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. Then, using this information, the algorithm predicts and selects the best approach to deal with each dimension. The algorithm has a worst case complexity of , where is the number of elements retrieved, is the number of elements in the database, and is the number of dimensions of the database. This work includes a detailed description of the methodology as well as a study of the algorithm performance.

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http://dx.doi.org/10.1016/j.amc.2019.125010 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7243811 | PMC |

May 2020

Sensors (Basel) 2020 May 9;20(9). Epub 2020 May 9.

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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.

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http://dx.doi.org/10.3390/s20092697 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7248689 | PMC |

May 2020

Sensors (Basel) 2019 Dec 5;19(24). Epub 2019 Dec 5.

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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. An estimate of this limit is given for our location in the galaxy.

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http://dx.doi.org/10.3390/s19245355 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6960684 | PMC |

December 2019

Sensors (Basel) 2019 Oct 28;19(21). Epub 2019 Oct 28.

Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA.

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. Two distinct approaches to attitude error kinematics are developed. In the first, the estimated angular velocity is defined in the true attitude axes frame, while in the second, it is defined in the estimated attitude axes frame. The first approach is of interest in simulations where the true attitude is known, while the second approach is for real estimation/control applications. Two nonlinear kinematic models are derived that are valid for arbitrarily large rotations and rotation rates. The results presented are expected to be broadly useful to nonlinear attitude estimation/control filtering formulations. A discussion of the benefits of the derived error kinematic models is included.

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http://dx.doi.org/10.3390/s19214682 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6864719 | PMC |

October 2019

Mathematics (Basel) 2019 Mar 22;7(3):296. Epub 2019 Mar 22.

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

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. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the , validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations.

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http://dx.doi.org/10.3390/math7030296 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259476 | PMC |

March 2019

Comput Sci Eng 2019 Jan-Feb;21(1):94-107. Epub 2019 Mar 6.

Aerospace Engineering, Texas A&M University, College Station, TX.

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. Both methods are shown suitable for massive generation of random samples. Examples are provided to clarify these methodologies.

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http://dx.doi.org/10.1109/MCSE.2018.2882727 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7268911 | PMC |

March 2019

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