David Arnas

• Ph.D.
• Professor (Assistant) at Purdue University West Lafayette

This work introduces a methodology for generating linear operators that approximately represent nonlinear systems of perturbed ordinary differential equations. This is done through the application of classical perturbation theory via the Lindstedt–Poincaré expansion, followed by an extension of the space of configuration that guarantees the linear...

As more and more operators propose and deploy large constellations, the finite orbital volume in low Earth orbit (LEO) is becoming increasingly congested. Over the last several years, we have been developing a proposal for LEO orbital coordination that uses carefully designed slots in nested concentric shells to avoid the potential for hazardous cl...

This work proposes two techniques to obtain linear Koopman operators able to represent the non-linear dynamics characteristic of the Circular-Restricted Three- Body Problem. In particular, a pure numerical method, and a semi-analytical ap- proach are proposed to obtain and study such linear operators. These method- ologies are based on the idea of...

Shell-wise orbital coordination in low Earth orbit can improve space safety, simplify space traffic coordination and management, and optimize orbital capacity. This work describes two methods to generate 2D lattice flower constellations (2D-LFCs) that are defined with respect to either an arbitrary zonal, or zonal, sectoral, and tesseral Earth geop...

This work introduces a methodology for generating linear operators that approximately represent nonlinear systems of perturbed ordinary differential equations. This is done through the application of classical perturbation theory via the Lindstedt-Poincar\'e expansion, followed by an extension of the space of configuration that guarantees the linea...

This work introduces a methodology for generating linear operators that approximately represent nonlinear systems of perturbed ordinary differential equations. This is done through the application of classical perturbation theory via the Lindstedt-Poincaré expansion, followed by an extension of the space of configuration that guarantees the linear...

This work provides a set of closed-form analytical expressions to define osculating frozen orbits under the perturbation effects of the oblateness of the main celestial body. To this end, an analytical perturbation method based on osculating elements is proposed to characterize, define, and study the three existing families of frozen orbits in clos...

As more and more operators propose and begin to deploy large constellations, finite orbital volume in Low Earth Orbit (LEO) is becoming increasingly congested. Over the last several years, we have been developing a proposal for LEO orbital coordination that uses carefully designed slots in nested concentric shells to avoid the potential for hazardo...

This work focuses on the identification of reliable and repeatable spatial (three-dimensional) trajectories that link the Earth and the Moon. For this purpose, this paper aims to extend the 2:1 resonant prograde family and 2:1 resonant retrograde family to three dimensions and to introduce spatial orbits that are not currently present in the litera...

This work introduces the Time Symmetric Constellations, an analytical constellation design methodology that provides all the uniform satellite distributions resultant from a set of time rotations. This enables the design and study of uniform constellations from the perspective of any rotating frame of reference, providing all the relations that app...

This work focuses on providing closed form analytical expressions to define frozen orbits under the effects of the Earth's zonal harmonics. Particularly, the perturbation effects from the terms J2, J3, J4, J5, J6, and J7 are considered in this work. This is done using a power series expansion in the small parameter that allows not only to provide a...

This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term [Formula: see text] from the gravitational potential of the main celestial body. In particular, this paper focuses on the generation of the analytical transformations between osculating and mean elements under this perturbation. This...

This work focuses on the identification of reliable and repeatable spatial (three-dimensional) trajectories that link the Earth and the Moon. For this purpose, this paper aims to extend the 2:1 resonant prograde family and 2:1 resonant retrograde family to three dimensions and to introduce spatial orbits that are not currently present in the litera...

This work focuses on providing closed form analytical expressions to define frozen orbits under the effects of the zonal harmonics of an Earth-like planet. Particularly, the perturbation effects from the terms J2, J3, J4, J5, J6, and J7 are considered in this work. This is done using a power series expansion in the small parameter that allow not on...

This work presents an analytical perturbation method to define and study the dynamics of frozen orbits under the perturbation effects produced by the oblatness of the main celestial body. This is done using a perturbation method purely based on osculating elements. This allows to characterize, define, and study the three existing families of frozen...

This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term J2 from the gravitational potential of the main celestial body. This is done using a power series expansion in the perturbation constant J2 on all the variables of the system, and a time regularization based on the argument of latitu...

This work focuses on the generation of non-self-intersecting relative trajectories, and their applications to satellite constellation design, slotting architectures, and space traffic management. To that end, this paper introduces three theorems to determine when two spacecraft share the same relative trajectory, to identify the only conditions tha...

This work introduces a methodology for the generation of an approximate analytical solution to perturbed ordinary differential equations using Schur decomposition. This methodology is based on the use of operator theory to find a linear approximation to the ordinary differential equation in an expanded space of configuration. Once this linearizatio...

Growth in the active Low Earth Orbit (LEO) satellite population will result in conjunctions involving two active satellites making up a larger portion of overall conjunction risk. Greater and more concentrated orbital density also increases the severity of collisions at densely populated altitudes. Prior work on LEO slotting has proposed methods to...

Shell-wise orbital slotting in Low Earth Orbit (LEO) can improve space safety, simplify space traffic coordination and management, and optimize orbital capacity. This paper describes two methods to generate 2D Lattice Flower Constellations (2D-LFCs) that are defined with respect to either an arbitrary degree or an arbitrary degree and order Earth g...

As more and more operators propose and begin to deploy large constellations , finite orbital volume in Low Earth Orbit (LEO) is becoming increasingly congested. Over the last several years, we have been developing a proposal for LEO orbital slotting that uses carefully designed slots in nested concentric shells to avoid the potential for hazardous...

This work focuses on the study of the reconfiguration strategies available for uniformly distributed satellite constellations and slotting architectures. Particularly, this paper deals with the cases of reducing, maintaining, and also increasing the number of available positions for satellites in a space structure, and takes into account the potent...

The Koopman Operator (KO) offers a promising alternative methodology to solve ordinary differential equations analytically. The solution of the dynamical system is analyzed in terms of observables, which are expressed as a linear combination of the eigenfunctions of the system. Coefficients are evaluated via the Galerkin method, using Legendre poly...

This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular system using the Schur decomposition, and second, generating the solution sequentially following the upper triangu...

This work focuses on the study of the reconfiguration strategies available for uniformly distributed satellite constellations and slotting architectures. Particularly, this manuscript deals with the cases of reducing, maintaining, and also increasing the number of available positions for satellites in the space structure, and takes into account the...

This work focuses on the generation of non-self-intersecting relative trajectories, and their applications to satellite constellation design, slotting architectures, and Space Traffic Management. To that end, this paper introduces two theorems to determine when two spacecrafts share the same relative trajectory, and to identify the only conditions...

This paper investigates the application of the Koopman Operator theory to the motion of a satellite about a libration point in the Circular Restricted Three-Body Problem. Recently, the Koopman Operator has emerged as a promising alternative to the geometric perspective for dynamical systems, where the Koopman Operator formulates the analysis and dy...

MPC controllers and Safety Analysis based on Control Barrier Functions (CBFs)

This work focuses on the study of the reconfiguration strategies available for uniformly distributed satellite constellations and slotting architectures. Particularly, this manuscript deals with the cases of reducing, maintaining, and also increasing the number of available positions for satellites in the space structure, and takes into account the...

In this work we study the optimal orbital maneuvers for an on-orbit servicing satellite to rendezvous with a set of client spacecrafts. With the growth of LEO mega-constellations, various types of Space Traffic Management schemes are being proposed. For on-orbit servicer concepts, efficient transfer between different clients will be necessary, and...

This work introduces the use of the Koopman operator theory to generate approximate analytical solutions for the zonal harmonics problem of a satellite orbiting a non-spherical celestial body. Particularly, the solution proposed directly provides the osculating evolution of the system under the effects of any order of the zonal harmonics, and can b...

4D Lattice Flower Constellations is a new constellation design framework, based on the previous 2D and 3D Lattice theories of Flower Constellations, that focus on the generation of constellations whose satellites can have different semi-major axis and still present a constellation structure that is maintained during the dynamic of the system. This...

This work introduces the n-dimensional congruent lattices using necklaces, a general methodology to generate uniform distributions in multidimensional modular spaces. The formulation presented in this manuscript constitutes the mathematical foundation of the most used satellite constellation designs, including Walker Constellations, and Lattice and...

Due to the increase in popularity of satellite constellations, some altitudes in LEO have experienced a large increase in number of satellites. This is a trend expected to continue in the future, which could potentially lead to an increased risk of collision between satellites. Collision avoidance is therefore paramount to maintain normal operation...

This work focuses on the generation and study of approximated analytical solutions to the J2 perturbed problem of a satellite orbiting the Earth. This is done by using a new set of variables based on spherical coordinates to fully represent the J2 dynamics, and the Koopman operator perturbation theory to solve the system of differential equations t...

This manuscript analysis the solution provided by the Koopman operator methodology in its application to the main satellite problem. In this regard, this work performs an spectral study of the solution generated and compares it with the solution provided by the Poincaré-Lindstedt method. In addition, this manuscript introduces a perturbation method...

This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and, in addition, a complete polynomial system when considering the perturbation produced by the zonal harmonics fro...

This work focuses on the nominal design and later maintenance of satellite constellations based on the 2D Necklace Flower Constellations for their application in Earth observation missions. To that end, we introduce a generalization of the 2D Necklace Flower Constellations formulation to adapt the methodology to constellations whose satellites have...

This work introduces the use of the Koopman Operator Theory to generate analytical solutions for the zonal harmonics problem of a satellite orbiting a non spherical celestial body. Particularly, the solution proposed directly provides the osculating evolution of the system and can be automated to generate any approximation order to the solution. Mo...

This work presents a new range searching algorithm for multidimensional databases. The proposed methodology is based on the idea of generating a navigation metadata structure, complementary to the database, that eases the navigation between the elements of the database. This metadata structure can be adapted to different problems and it is generate...

We derive an analytical closed expression to compute the minimum distance (quantified by the angle of separation measured from the center of the Earth) between any two satellites located at the same altitude and in circular orbits. We also exploit several properties of Flower Constellations (FCs) that, combined with our formula for the distance, gi...

This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and, in addition, a complete polynomial system when considering the perturbation produced by the zonal harmonics fro...

This work introduces a linearized analytical model for the study of the dynamic of satellites in near circular orbits under the effects of the atmospheric drag. This includes the evaluation of the station keeping required for each satellite subjected to a control box strategy, and also the study of the dynamic of tandem formations between two or mo...

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-squa...

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) projection mapping, and c) polynomial mapping. In that respect, an accurate least-squares app...

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 introduces a general numerical technique to invert one dimensional analytic or tabulated nonlinear functions in assigned ranges of interest. The proposed approach is based on an optimal version of the k-vector range searching, an ad-hoc modification devised for function inversion. The optimality consists of retrieving always the same numb...

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 focuses on the definition of satellite constellations whose secular relative distributions are invariant under the perturbation produced by the Earth gravitational potential. This is done by defining the satellite distribution directly in the Earth-Centered–Earth-Fixed frame of reference and using the along-track time distances between sa...

This work proposes the use of 2D Lattice Flower Constellations (2D-LFCs) to facilitate the design of a Low Earth Orbit (LEO) slotting system to avoid collisions between compliant satellites and to optimize the available orbital volume. Specifically, this manuscript proposes the use of concentric orbital shells of admissible “slots” with stacked int...

We derive an analytical closed expression to compute the minimum distance (quantified by the angle of separation measured from the center of the Earth) between any two satellites located at the same altitude and in circular orbits. We also exploit several properties of Flower Constellations (FCs) that, combined with our formula for the distance, gi...

This work introduces a linearized analytical model for the study of the dynamic of satellites in near circular orbits under the effects of the atmospheric drag. This includes the evaluation of the station keeping required for each satellite subjected to a control box strategy, and also the study of the dynamic of tandem formations between two or mo...

This paper proposes the use of Flower Constellation (FC) theory to facilitate the design of a Low Earth Orbit (LEO) slotting system to avoid collisions between compliant satellites and optimize the available space. Specifically, it proposes the use of concentric orbital shells of admissible “slots” with stacked intersecting orbits that preserve a m...

This paper proposes the use of Flower Constellation (FC) theory to facilitate the design of a Low Earth Orbit (LEO) slotting system to avoid collisions between compliant satellites and optimize the available space. Specifically, it proposes the use of concentric orbital shells of admissible "slots" with stacked intersecting orbits that preserve a m...

The FLuorescence EXplorer (FLEX) is an Earth observation mission developed by ESA, whose main objective is to perform quantitative measurements of the solar induced vegetation fluorescence with the goal of monitoring the vegetation photosynthetic activity. FLEX will orbit in tandem with one of the Copernicus Sentinel-3 satellites, this will allow t...

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 Fluorescence Explorer (FLEX) is an Earth observation mission currently in development by the European Space Agency to perform quantitative measurements of the solar induce vegetation fluorescence. As a core of the mission concept, FLEX is planned to be launched by the end of 2023 and shall fly in tandem with one of the Copernicus Sentinel-3 sat...

The FLuorescence EXplorer (FLEX) is an Earth observation mission developed by ESA, whose main objective is to perform quantitative measurements of the solar induced vegetation fluorescence with the goal of monitoring the vegetation photosynthetic activity. FLEX will orbit in tandem with one of the Copernicus Sentinel-3 satellites. The swath of the...

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 work introduces a general numerical technique to invert one dimensional analytic or tabulated nonlinear functions in assigned ranges of interest. The proposed approach is based on an “optimal” version of the k-vector range searching, an ad-hoc modification devised for function inversion. The optimality consists of retrieving always the same nu...

The 2D Necklace Flower Constellation theory is a new design framework based on the 2D Lattice Flower Constellations that allows to expand the possibilities of design while maintaining the number of satellites in the configuration. The methodology presented is a generalization of the 2D Lattice design, where the concept of necklace is introduced in...

A new approach in satellite constellation design is presented in this paper, taking as a base the 3D Lattice Flower Constellation Theory and introducing the necklace problem in its formulation. This creates a further generalization of the Flower Constellation Theory, increasing the possibilities of constellation distribution while maintaining the c...

The aim of the time distributionmethodology presented in this paper is to generate constellations whose satellites share a set of relative trajectories in a given time, andmaintain that property over time without orbit corrections. The model takes into account a series of orbital perturbations such as the gravitational potential of the Earth, the a...

This work proposes a numerical technique that can be used to invert one dimensional analytic or tabulated nonlinear functions in assigned ranges of interest. The approach proposed is based on an "optimal" version of the k-vector range searching technique. The optimality consists of retrieving a prescribed number of data (1, 2, · · ·) to initiate th...

Star Identification (Star-ID) is a complex problem, mainly because some of the observations are not generated by actual stars, but by reflecting debris, other satellites, visible planets, or by electronic noise. For this reason, the capability to discriminate stars from non-stars is an important aspect of Star-ID robustness. Usually, the Star-ID ta...

This work focuses on random number generation with any prescribed nonlinear distribution using the k-vector methodology. Two approaches are introduced. The first is based on inverse transform sampling using an optimal k-vector to generate the numbers by the inversion of the cumulative distribution. The second generates samples using random searchin...

Star Identification (Star-ID) is a complex problem, mainly because some of the observations are not generated by actual stars, but by reflecting debris, other satellites, visible planets, or by electronic noise. For this reason, the capability to discriminate stars from non-stars is an important aspect of Star-ID robustness. Usually, the Star-ID ta...

A new approach in satellite constellation design is presented in this paper, taking as a base the 3D Lattice Flower Constellation Theory and introducing the necklace problem in its formulation. This creates a further generalization of the Flower Constellation Theory, increasing the possibilities of constellation distribution and maintaining the cha...

2D-Lattice Flower Constellations present interesting dynamical features that allow to explore a wide range of potential applications. Their particular initial distribution (lattice) and their symmetries disappear when some perturbations are considered, such as the J2 effect. The new lattice-preserving Flower Constellations maintain the initial dist...

The aim of the constellation design model shown in this paper is to generate constellations whose satellites share the same ground-track in a given time, making all the satellites pass over the same points of the Earth surface. The model takes into account a series of orbital perturbations such as the gravitational potential of the Earth, the atmos...