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## Publications

Publications (465)

The origin and geometric interpretation of the equinoctial elements is explained with a connection to orthogonal rotations and attitude dynamics in Euclidean 3-space. An identification is made between the equinoctial elements and classical Rodrigues parameters. A new set of equinoctial elements are then developed using the modified Rodrigues parame...

Accurate orbit propagation for satellites in motion around a massive central body requires the inclusion of a high-fidelity gravity model for that central body. Including such a model significantly increases computational costs as a sufficiently large degree for the spherical harmonic series is required. The higher the degree of a specific series,...

In this study, a supervised machine learning approach called Gaussian process regression (GPR) was applied to approximate optimal bi-impulse rendezvous maneuvers in the cis-lunar space. We demonstrate the use of the GPR approximation of the optimal bi-impulse transfer to patch points associated with various invariant manifolds in the cis-lunar spac...

View Video Presentation: https://doi.org/10.2514/6.2022-2458.vid The Hamiltonian formalism in over-parameterized, regularized, and redundant phase spaces is developed and applied to the two-body problem. A class of canonical and point transformations from minimal to non-minimal coordinates is derived that provides an explicit identification of the...

In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits (NRHOs) and optimal low-thrust transfer trajectories using a high-fidelity dynamical model. Time- and fuel-optimal low-thrust transfers to (and from) these NRHOs are generated leveraging their ‘invariant’ manifolds, which serve as long terminal coast arcs. Analyses ar...

Numerical solutions of optimal control problems are influenced by the appropriate choice of coordinates. The proposed method based on the variational approach to map costates between sets of coordinates and/or elements is suitable for solving optimal control problems using the indirect formalism of optimal control theory. The Jacobian of the nonlin...

A supervised machine learning approach called the Gaussian Process Regression (GPR) is applied to approximate the optimal bi-impulse rendezvous maneuvers in cis-lunar space. The use of GPR approximation of the optimal bi-impulse transfer to patch-points associated with various invariant manifolds in the cis-lunar space is demonstrated. The proposed...

A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs while incorporating eclipses and perturbations during the optimization process in an ephemeris model; a feature lacking in state of the art software like MYSTIC and Cope...

A time-varying output covariance assignment problem in the presence of a stochastic disturbance is solved using finite-horizon optimal control formulation. It is shown that an assignment of time-varying output error covariance is possible in the presence of model error by utilizing a time-varying linear quadratic regulator (LQR) controller with a c...

A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic, near rectilinear Halo orbits that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an...

The complexities in using indirect optimization methods get compounded for practical co-optimization problems in the presence of continuous and discrete design variables. In this paper, realistic multimode electric propulsion systems are incorporated within the formulation of gravity-assist, low-thrust trajectory design problems. Electric thrusters...

In this work, end-to-end low-thrust transfers from a GTO orbit to a low-altitude lunar orbit by exploiting the manifolds of a chosen Earth-Moon L1 halo orbit was studied. The practicality of piece-wise, minimum-time transfers that exploit halo orbit manifolds is demonstrated, which offers more flexibility to meet mission objectives. It is known tha...

A novel acceleration-based formulation is proposed to construct minimum-∆v bang-off-bang thrust profiles and impulsive maneuvers in a rapid manner. The proposed methodology leads to substantial simplifications by removing mass state, thrust magnitude and specific impulse values from the ensuing boundary-value problems. Standard acceleration-based m...

A method is proposed to map costates between two sets of coordinates. The proposed method is suitable for solving optimal control problems using indirect methods and with different sets of coordinates or elements. The Jacobian of the non-linear map between any two sets of coordinates/elements plays a pivotal role in the costate vector transformatio...

Application of idealized constant-specific-impulse, constant-thrust electric thruster performance models or curve-fitted polynomials is quite common for spacecraft trajectory design. However, the incorporation of realistic performance models of multi-mode electric thrusters leads to notable challenges, and at the same time, offers unprecedented sys...

Indirect optimization methods convert optimal control problems (OCPs) into two-or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Comp...

Emerging science and on-orbit service missions require a number of satellites in formation to maintain various shapes with given tolerances appropriate for the mission and onboard instruments; therefore, swarm station keeping plays a signicant role in the success of the mission. In this work, we propose two techniques to maintain the desired initia...

Near-Rectilinear Halo Orbits (NRHOs) are deemed to be favorable candidates for establishing a near-future crewed space station in the cis-lunar space. Although the 9:2 resonant southern $L_2$ NRHO has been earmarked as the working orbit for the Lunar Gateway Mission, a plethora of other neighboring resonant NRHOs are also viable options. The invari...

Lyapunov methods are well established as a versatile approach for generating feasible and robustly converging spiral-type low-thrust trajectories. The present study introduces Lya-punov optimal methods for low-thrust guidance. The approach makes use of the regularized modified equinoctial orbit elements in such a way that a nominal trajectory can b...

A renewed interest in revisiting the Moon has blown wide open the previously ajar door to research avenues in the field of Earth-Moon transfer trajectories. While the advent of low-thrust propulsion systems has opened up possibilities to undertake more complicated missions, designing optimal transfer trajectories in this domain is no easy feat. His...

Indirect optimization methods convert optimal control problems (OCPs) into two-or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. We utilize the features o...

Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/o...

Efficient performance of a number of engineering systems is achieved through different modes of operation - yielding systems described as “hybrid”, containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable-Isp, variable-thrust engine or 2)...

A central problem in orbit transfer optimization is to determine the number, time, direction, and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum’s question for multiple-revolution maneuvers has remained e...

Designing long-duration lunar orbiter missions is challenging due to the Moon’s highly nonlinear gravity ﬁeld and the third-body perturbations induced by the Earth, Sun and other large bodies. The absence of a Lunar atmosphere has oﬀered the possibility for mission designers to search for extremely low-altitude, quasi-stable lunar orbits. In additi...

The presence of extremely low-altitude, lunar quasi-frozen orbits (QFOs) has given rise to interesting mission opportunities. These QFOs are ideal for close-range, high-resolution mapping of the lunar south pole, and their inherent stability translates into minimal station-keeping efforts. Despite the aforementioned desirable characteristics, desig...

Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/o...

Efficient performance of a number of engineering systems is achieved through different modes of operation-yielding systems described as "hybrid", containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable-Isp , variable-thrust engine or 2) m...

Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/o...

Low-thrust propulsion technology and planetary gravity-assist maneuvers make a promising combination for deep space explorations. Hybrid optimal control methods have proven to be an excellent solution framework which exploits the advantages of both direct and indirect optimization methods, while alleviating their drawbacks. We employ a recently int...

Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The pr...

A central problem in orbit transfer optimization is to determine the number, time, direction and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum's question for multiple-revolution maneuvers has remained el...

We present the results of a comprehensive study in which the precision and efficiency of six numerical integration techniques, both implicit and explicit, are compared for solving the gravitationally perturbed two-body problem in astrodynamics. Solution of the perturbed two-body problem is fundamental for applications in space situational awareness...

We have developed a new method for solving low-thrust fuel-optimal orbit transfer problems in the vicinity of a large body (planet or asteroid), considering a high-fidelity spherical harmonic gravity model. The algorithm is formulated via the indirect optimization method, leading to a two-point boundary value problem (TPBVP). We make use of a hyper...

Indirect optimization methods hold a special place among the techniques used for solving optimal control problems since they guarantee local optimality of the resulting solutions. On the other hand, complications occur during numerical calculations when optimal control has bang-bang or bang-off-bang structure. Traditionally, smoothing techniques su...

Design of long-duration lunar orbiter missions is challenging due to the Moon's highly non-linear gravity field and third-body perturbations induced by the Earth, Sun and other large bodies, on the orbiting spacecraft. The absence of a Lunar atmosphere, and hence the lack of orbital atmospheric drag, has encouraged mission designers to search for e...

An adaptive self-tuning Picard–Chebyshev numerical integration method is presented for solving initial and boundary value problems by considering high-fidelity perturbed two-body dynamics. The current adaptation technique is self-tuning and adjusts the size of the time interval segments and the number of nodes per segment automatically to achieve n...

An adaptive Collision Risk Assessment Tool (CRATER) is introduced in order to analyze conjunctions featuring non Gaussian distributions, long encounter times, and considerable model uncertainty. The algorithm makes use of the Fokker Planck equation to quantify the error in various assumptions, refine approximations of the propagated probability den...

Coordinate choices have significant consequences in the analytical and computational approaches to solve celestial mechanics problems. The present study focuses on the impact of various coordinate representations of the dynamics on the solution of the ensuing state/co-state two-point boundary-value problems that arise when solving the indirect opti...

In a large number of dynamical systems and depending on the form of performance index,
their Hamiltonian may turn out to be affine in control. Extremal control of such systems may switch between the bounds of the admissible set or take on values interior to its admissible set, i.e., singular arcs. On the other hand, the existence of singular arcs i...

We present an iterative five-element Lyapunov control for low thrust rendezvous that uses Modified Chebyshev Picard Iteration (MCPI), an iterative solver of linear and nonlinear ordinary differential equations (ODEs). MCPI uses Chebyshev polynomials to approximate the orbital trajectory and then uses Picard iteration to improve the approximation it...

This paper introduces a parallel-compiled tool that combines several of our recently developed methods for solving the perturbed Lambert problem using modified Chebyshev-Picard iteration. This tool (unified Lambert tool) consists of four individual algorithms, each of which is unique and better suited for solving a particular type of orbit transfer...

A new method for propagating uncertainty through a general nonlinear dynami-cal model with a parametric model of perturbations (such as aerodynamic drag) is developed. The model is constructed such that all of the model uncertainty is assumed to be embodied in a random vector of parameters. Initial state errors and the uncertain parameters are assu...

We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not...

We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel w...

We present a method for performing low cost attitude estimation for CubeSat type missions. Our algorithm uses measurements from a custom built sun sensor, a star camera, and inertial measurements. These sensing measurements are supplied in real-time to an Multiplicative Kalman Filter for the purpose of generating continuous attitude estimates. The...

In this paper the two-body problem with atmospheric drag is considered. A cannonball drag model is utilized and the problem is solved with the analytic continuation power series technique. Recent developments of the method have made it possible to sum the series to arbitrary order enabling machine precision power series solutions for the two-body p...

Prior works have shown promising efficiency while propagating perturbed two-body motion using orbital elements combined with a novel integration technique. While previous studies show that Modified Chebyshev Picard Iteration (MCPI) is a powerful tool used to propagate position and velocity, instead using orbital elements to propagate the state vect...

The demand of having fast and efficient numerical propagators to solve engineering problems has become essential. The level of system complexity increases the cost of obtaining the solution. Many real world problems require developing efficient, precise and fast solutions. For example, efficient, high precision orbit propagation has gained renewed...

This paper extends previous work on parallel-structured Modified Chebyshev Picard Iteration (MCPI) Methods. The MCPI approach iteratively refines path approximation of the state trajectory for smooth nonlinear dynamical systems and this paper shows that the approach is especially suitable for initial value problems of astrodynamics. Using Chebyshev...

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo-Stiefel transformation and Modified Chebyshev-Picard iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo-Stiefel regularization. This t...

Recent work has shown that two-body motion can be analytically modeled using analytic continuation models, which utilize kinematic transformation scalar variables that can be differentiated to an arbitrary order using the well-known Leib-niz product rule. This method allows for large integration step sizes while still maintaining high accuracy. Wit...

A symmetric flexible rotating spacecraft can be modeled as a distributed parameter system of a rigid hub attached to two flexible appendages with tip masses. First, Hamilton's extended principle is utilized to establish a general treatment for deriving the dynamics of multi-body dynamical systems to establish a hybrid system of integro-partial diff...

A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations.
The key benefit of using the CGL data sampling is that the nodal points are...

The Modified Chebyshev Picard Iteration (MCPI) method has recently proven to be more efficient for a given accuracy than the most commonly adopted numerical integration methods, as a means to solve for perturbed orbital motion. This method utilizes Picard iteration, which generates a sequence of path approximations, and discrete Chebyshev Polynomia...

Several analytical and numerical methods exist to solve the orbit propagation of the two-body problem. Analytic solutions are mainly implemented for the un-perturbed/classical two-body problem. Numerical methods can handle both the unperturbed and the perturbed two-body problem. The literature is rich with numerical methods addressing orbit propaga...

In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functio...

In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed v...

The role of nonlinearity of the measurement model and its interactions with the uncertainty of measurements and geometry of the problem is studied in this paper. An examination of the transformations of the probability density function in various coordinate systems is presented for several astrodynamics applications. Smooth and analytic nonlinear f...