Publications by authors named "Hunter Johnston"

5 Publications

  • Page 1 of 1

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

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-xDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7553110PMC
September 2020

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

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/math7060537DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7263466PMC
June 2019

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

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/make1040060DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259481PMC
December 2019

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

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/math8030397DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7259483PMC
March 2020

High accuracy least-squares solutions of nonlinear differential equations.

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.2018.12.007DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7243685PMC
May 2019
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