Carl Leake

• Doctor of Philosophy
• Robotics Technologist II at National Aeronautics and Space Administration

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 fle...

EELS-DARTS is a simulator designed for autonomy development and analysis of large degree of freedom snake-like robots for space exploration. A detailed description of the EELS-DARTS simulator design is presented. This includes the versatile underlying multibody dynamics representation used to model a variety of distinct snake robot configurations a...

HeliCAT-DARTS is a high-fidelity rotorcraft dynamics simulator developed for the design and development of rotorcraft for planetary exploration. While initially developed for the life cycle use of the Ingenuity Mars Helicopter mission, the simulator now supports a broad range of rotorcraft configurations and applications. HeliCAT provides a GNC tes...

Ice worlds are at the forefront of astrobiological interest because of the evidence of subsurface oceans. Enceladus in particular is unique among the icy moons because there are known vent systems that are likely connected to a subsurface ocean, through which the ocean water is ejected to space. An existing study has shown that sending small robots...

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 wri...

The first (and only) book available to perform functional interpolation. Functional Interpolation is a generalization of interpolation, deriving functionals representing all functions satisfying a set of constraints. The constraints can be on values, derivatives, and integrals, and on any linear combination of them in univariate or multivariate dim...

The numerical solution of boundary value problems in solid mechanics is dominated by the Finite Element Method (FEM). The present study provides an alternative numerical approach using the Theory of Functional Connections (TFC) for solving problems of this type. Here TFC is used to solve multiple beam problems using the Timoshenko–Ehrenfest beam th...

We present a novel, accurate, fast, and robust physics-informed neural network method for solving problems involving differential equations (DEs), called Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving DEs: the Theory of Functional Connections T...

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 unconstraine...

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 unconstraine...

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-...

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-...

This paper describes an algorithm obtained by merging a recursive star-identification algorithm with a recently developed adaptive singular-value-decomposition-based estimator of the angular-velocity vector (QuateRA). In a recursive algorithm, the more accurate the angular-velocity estimate is, the quicker and more robust to noise is the resultant...

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...

In order to conduct future surface operations on the Moon and Mars, NASA will look to autonomous navigation technologies that can fill the infrastruc- ture limitations of the LORAN-C and Global Positioning System: one such system to meet this need is the Stellar Positioning System (SPS). This paper improves on the original SPS algorithm by leveragi...

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...

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,...

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...

Fuel consumption and time of flight are crucial information for mission design. In this paper, adopting a two-impulse maneuver, we analyze the fuel consumption through evaluations of the equivalent V for several transfers in the Earth-Moon system as function of time of flight and other parameters, like the points of application of the thrusts. Tran...

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...

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 Func...

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 Func...

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 inc...

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 dim...

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...

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 dim...

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,...

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 exp...

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 constra...

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 constra...

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...

This work focuses on the study of orthogonal range searching methodologies for static databases with multiple dimensions. To that end, a new algorithm is introduced , the n-dimensional k-vector. This algorithm represents the evolution of the k-vector, a range searching method originally devised to solve the Star-Identification problem in wide field...

The Theory of Functional Connections is a powerful mathematical framework deriving constrained expressions. These expressions allow to transform constrained optimization problems into unconstrained problems. Until now, the Theory of Functional Connections framework only included equality constraints, that is, constraints defined at specific values...

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...

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, representi...

This paper 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 an inverse transform sampling using the optimal k-vector to generate the samples by inverting the cumulative distribution. The second approach generates samples by perfo...

This article presents a new methodology called deep ToC that estimates the solutions of partial differential equations (PDEs) by combining neural networks with the Theory of Connections (ToC). ToC is used to transform PDEs with boundary conditions into unconstrained optimization problems by embedding the boundary conditions into a "constrained expr...

Differential equations 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. Although there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the...