arXiv subjects

Carl Leake

Publications and source records attributed to Carl Leake.

15 recordsLinked to original sources

arXiv2024

Flight Demonstration and Model Validation of a Prototype Variable-Altitude Venus Aerobot

This paper details a significant milestone towards maturing a buoyant aerial robotic platform, or aerobot, for flight in the Venus clouds. We describe two flights of our subscale altitude-controlled aerobot, fabricated from the materials necessary to survive Venus conditions. During these flights over the Nevada Black Rock desert, the prototype flew at the identical atmospheric densities as 54 to 55 km cloud layer altitudes on Venus. We further describe a first-principle aerobot dynamics model which we validate against the Nevada flight data and subsequently employ to predict the performance of future aerobots on Venus. The aerobot discussed in this paper is under JPL and Aerostar development for an in-situ mission flying multiple circumnavigations of Venus, sampling the chemical and physical properties of the planet's atmosphere and also remotely sensing surface properties.

cs.RO↗

arXiv2023

The Theory of Functional Connections on Vector Spaces

The Theory of Functional Connections (TFC) is most often used for constraints over the field of real numbers. However, previous works have shown that it actually extends to arbitrary fields. The evidence for these claims is restricting oneself to the field of real numbers is unnecessary because all of the theorems, proofs, etc. for TFC apply as written to fields in general. Here, that notion is taken a step further, as fields themselves are unnecessarily restrictive. The theorems, proofs, etc. of TFC apply to vector spaces, not just fields. Moreover, the inputs of the functions/constraints do not even need to be restricted to vector spaces; they can quite literally be anything. In this note, the author shows the more general inputs/outputs of the various symbols that comprise TFC. Examples with accompanying, open-access Python code are included to aid the reader's understanding.

math.AP↗

arXiv2021

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.

math.AP↗

arXiv2021

Fast 2-impulse non-Keplerian orbit-transfer using the Theory of Functional Connections

This study applies a new approach, the Theory of Functional Connections (TFC), to solve the two-point boundary-value problem (TPBVP) in non-Keplerian orbit transfer. The perturbations considered are drag, solar radiation pressure, higher-order gravitational potential harmonic terms, and multiple bodies. The proposed approach is applied to Earth-to-Moon transfers, and obtains exact boundary condition satisfaction and with very fast convergence. Thanks to this highly efficient approach, perturbed pork-chop plots of Earth-to-Moon transfers are generated, and individual analyses on the transfers' parameters are easily done at low computational costs. The minimum fuel analysis is provided in terms of the time of flight, thrust application points, and relative geometry of the Moon and Sun. The transfer costs obtained are in agreement with the literature's best solutions, and in some cases are even slightly better.

astro-ph.EP↗

arXiv2020

Motivations and Preliminary Design for Mid-Air Deployment of a Science Rotorcraft on Mars

Mid-Air Deployment (MAD) of a rotorcraft during Entry, Descent and Landing (EDL) on Mars eliminates the need to carry a propulsion or airbag landing system. This reduces the total mass inside the aeroshell by more than 100 kg and simplifies the aeroshell architecture. MAD's lighter and simpler design is likely to bring the risk and cost associated with the mission down. Moreover, the lighter entry mass enables landing in the Martian highlands, at elevations inaccessible to current EDL technologies. This paper proposes a novel MAD concept for a Mars helicopter. We suggest a minimum science payload package to perform relevant science in the highlands. A variant of the Ingenuity helicopter is proposed to provide increased deceleration during MAD, and enough lift to fly the science payload in the highlands. We show in simulation that the lighter aeroshell results in a lower terminal velocity (30 m/s) at the end of the parachute phase of the EDL, and at higher altitudes than other approaches. After discussing the aerodynamics, controls, guidance, and mechanical challenges associated with deploying at such speed, we propose a backshell architecture that addresses them to release the helicopter in the safest conditions. Finally, we implemented the helicopter model and aerodynamic descent perturbations in the JPL Dynamics and Real-Time Simulation (DARTS)framework. Preliminary performance evaluation indicates landing and helicopter operation scan be achieved up to 5 km MOLA (Mars Orbiter Laser Altimeter reference).

cs.RO↗

arXiv2020

The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.

math.NA↗

arXiv2020

Extreme Theory of Functional Connections: A Physics-Informed Neural Network Method for Solving Parametric Differential Equations

In this work we present a novel, accurate, and robust physics-informed method for solving problems involving parametric differential equations (DEs) called the Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving parametric DEs, 1) the Theory of Functional Connections, TFC, and the Physics-Informed Neural Networks, PINN. Although this paper focuses on the solution of exact problems involving parametric DEs (i.e. problems where the modeling error is negligible) with known parameters, X-TFC can also be used for data-driven solutions and data-driven discovery of parametric DEs. In the proposed method, the latent solution of the parametric DEs is approximated by a TFC constrained expression that uses a Neural Network (NN) as the free-function. This approximate solution form always analytically satisfies the constraints of the DE, while maintaining a NN with unconstrained parameters, like the Deep-TFC method. X-TFC differs from PINN and Deep-TFC; whereas PINN and Deep-TFC use a deep-NN, X-TFC uses a single-layer NN, or more precisely, an Extreme Learning Machine, ELM. This choice is based on the properties of the ELM algorithm. In order to numerically validate the method, it was tested over a range of problems including the approximation of solutions to linear and non-linear ordinary DEs (ODEs), systems of ODEs (SODEs), and partial DEs (PDEs). Furthermore, a few of these problems are of interest in physics and engineering such as the Classic Emden-Fowler equation, the Radiative Transfer (RT) equation, and the Heat-Transfer (HT) equation. The results show that X-TFC achieves high accuracy with low computational time and thus it is comparable with the other state-of-the-art methods.

cs.LG↗

arXiv2020

The n-dimensional k-vector and its application to orthogonal range searching

This work focuses on the definition and study of the n-dimensional k-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 k-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 $\mathcal{O}(nd(k/n)^{2/d})$, where $k$ is the number of elements retrieved, $n$ is the number of elements in the database, and $d$ 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.

cs.DS↗

arXiv2020

Random Sampling using k-vector

This work introduces two new techniques for random number generation with any prescribed nonlinear distribution based on the k-vector methodology. The first approach is based on inverse transform sampling using the optimal k-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 k-vector. Both methods are shown suitable for massive generation of random samples. Examples are provided to clarify these methodologies.

cs.DS↗

arXiv2020

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.

astro-ph.IM↗

arXiv2020

Recursive Star-Identification Algorithm using an Adaptive SVD-based Angular Velocity Estimator

This paper describes an algorithm obtained by merging a recursive star identification algorithm with a recently developed adaptive SVD-based estimator of the angular velocity vector (QuateRA). In a recursive algorithm, the more accurate the angular velocity estimate, the quicker and more robust to noise the resultant recursive algorithm is. Hence, combining these two techniques produces an algorithm capable of handling a variety of dynamics scenarios. The speed and robustness of the algorithm are highlighted in a selection of simulated scenarios. First, a speed comparison is made with the state-of-the-art lost-in-space star identification algorithm, Pyramid. This test shows that in the best case the algorithm is on average an order of magnitude faster than Pyramid. Next, the recursive algorithm is validated for a variety of dynamic cases including a ground-based "Stellar Compass" scenario, a satellite in geosynchronous orbit, a satellite during a re-orientation maneuver, and a satellite undergoing non-pure-spin dynamics.

eess.SP↗

arXiv2020

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 $constrained \, expression$) that contains 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.

math.NA↗

arXiv2018

The Tensor 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 "constrained expressions," representing all 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, g(x,y), satisfy all constraints no matter what the g(x,y) 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 "Tensor Theory of Connections," 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 PDEs and stochastic differential equations.

math.GM↗

arXiv2018

Deep Theory of Functional Connections: A New Method for Estimating the Solutions of PDEs

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 TFC. TFC is used to transform PDEs with boundary conditions into unconstrained optimization problems by embedding the boundary conditions into a "constrained expression." In this work, a neural network is chosen as the free function, and used to solve the now unconstrained optimization problem. The loss function is taken as the square of the residual of the PDE. Then, the neural network is trained in an unsupervised manner to solve the unconstrained optimization problem. 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 randomly samples points from the domain during the training phase. Second, after training, this methodology represents a closed form, analytical, differentiable approximation of the solution throughout the entire training domain. In contrast, other popular methods require interpolation if the estimated solution is desired at points that do not lie on the discretized grid. The deep TFC method for estimating the solution of PDEs is demonstrated on four problems with a variety of boundary conditions.

math.NA↗

arXiv2018

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 Theory of Functional Connections (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.

stat.ML↗