Ryan P. Russell

University of Texas at Austin | UT · Department of Aerospace Engineering & Engineering Mechanics

This work is the third in a series on point mascon lunar gravity models. Point mascon models are computationally-efficient replacements for the standard spherical harmonics gravity models used in astrodynamics applications. Weighted cubed-sphere mascon gravity models are introduced as runtime-efficient alternatives to the spherical harmonics repres...

Spacecraft trajectories near the south pole of Enceladus violate the Brillouin sphere associated with the convergence radius of spherical harmonics models. In this study, a shifted coordinate frame is demonstrated to ensure a convergent model is available in regions of operational interest. Hypothetical experiments are performed around a simulated...

Optimal, many-revolution spacecraft trajectories are challenging to solve. A connection is made for a class of models between optimal direct and indirect solutions. For transfers that minimize thrust-acceleration-squared, primer vector theory maps direct, many-impulsive-maneuver trajectories to the indirect, continuous-thrust-acceleration equivalen...

Model-based control for robots has increasingly depended on optimization-based methods like Differential Dynamic Programming (DDP) and iterative LQR (iLQR). These methods can form the basis of Model-Predictive Control (MPC), which is commonly used for controlling legged robots. Computing the partial derivatives of the robot dynamics is often the mo...

Understanding of relative motion dynamics is essential for many spaceflight objectives, from orbit determination to rendezvous. Although these dynamics are well understood in a Keplerian context, non-Keplerian regimes are increasingly relevant. To address relative motion in arbitrary dynamics, a Quadratic Interpolated State Transition (QIST) model...

Differential Dynamic Programming (DDP) is a popular technique used to generate motion for dynamic-legged robots in the recent past. However, in most cases, only the first-order partial derivatives of the underlying dynamics are used, resulting in the iLQR approach. Neglecting the second-order terms often slows down the convergence rate compared to...

Sensitivity analysis with atmospheric chemical transport models may be used to quantify influences of specific emissions on pollutant concentrations. This information facilitates efficient environmental decision‐making regarding emissions control strategies for pollutants that affect human health and public welfare. The multicomplex step method (MC...

A time regularization scheme is introduced that facilitates trajectory optimization in multi-body regimes. The time transformation function allows for fixed-step propagation, while eliminating the need for multiple models in patched conic approaches, and mitigating the risk of stepping over unplanned flybys. The scheme is motivated by Sundman’s two...

Flyby tours are challenging to design due to the extraordinarily large search space. A single algorithm is proposed to answer the question of what low-cost, post-flyby options exist for a spacecraft arriving at a flyby body. The algorithm a) considers the full domain of reachable bodies and transfer types including even- and odd-nπ resonant ballist...

Model-based control for robots has increasingly been dependent on optimization-based methods like Differential Dynamic Programming and iterative LQR (iLQR). These methods can form the basis of Model-Predictive Control (MPC), which is commonly used for controlling legged robots. Computing the partial derivatives of the dynamics is often the most exp...

The renewed interest in Lunar exploration prompts a need to better understand the dynamics of spacecraft in the vicinity of the Moon. Here, a detailed survey is conducted via a broad grid search to find, characterize, and archive symmetric periodic orbits. The resulting database contains over 13 million planar and three-dimensional solutions in the...

Optimization-based robot control strategies often rely on first-order dynamics approximation methods, as in iLQR. Using second-order approximations of the dynamics is expensive due to the costly second-order partial derivatives of the dynamics with respect to the state and control. Current approaches for calculating these derivatives typically use...

This paper describes a method to estimate the visual hull and relative pose of a small celestial body with limb information extracted from images. The visual hull is the shape that can be reconstructed using silhouette, and it contains the actual shape of the object, but it is entirely contained within the convex hull instead. The method is fully a...

The UT Austin team presents their methodology and result for GTOC11, constructing a hypothetical Dyson ring using asteroids encountered by 10 motherships departing from Earth. A pathfinding algorithm for the mothership is designed using a fast lookup table and a robust Lambert solver. Sequencing of unique mothership itineraries is performed by appr...

Direct optimization of many-revolution spacecraft trajectories requires path discretization into segments. The well-known Sundman transformation guides the development of a piecewise function that regularly varies flight time for each segment. This flight-time function approximates the regularization effect on the dynamics for a spatially even disc...

This document provides full details of second-order partial derivatives of rigid-body inverse dynamics. Several properties and identities using an extension of Spatial Vector Algebra for tensorial use are listed, along with their detailed derivations. Using those, the expressions for second-order derivatives are derived step-by-step in detail. The...

Future missions to asteroids and comets will likely encounter bodies which are tumbling (i.e., not in principal axis rotation), and which have poor or non-existent prior shape models. In this work, simulated images of a tumbling comet are processed by a sequential Extended Kalman Filter (EKF) Simultaneous Localization and Mapping (SLAM) method and...

Lunar gravitational models are generated using sets of point mass potentials (mascons) fit to the GRGM1200A representation derived from GRAIL mission data. Point mass ensembles allow for simple modeling of a gravity potential and its partial derivatives. The new point mascon models are intended to be low- to medium-resolution replacements for spher...

An essential need for many model-based robot control algorithms is the ability to quickly and accurately compute partial derivatives of the equations of motion. State of the art approaches often use analytical methods based on the chain rule applied to existing dynamics algorithms. Although these methods are an improvement over finite differences i...

Direct optimization of many-revolution spacecraft trajectories is performed using an unconstrained formulation with many short-arc, embedded Lambert problems. Each Lambert problem shares its terminal positions with neighboring segments to implicitly enforce position continuity. Use of embedded boundary value problems (EBVPs) is not new to spacecraf...

An essential need for many model-based robot control algorithms is the ability to quickly and accurately compute partial derivatives of the equations of motion. State of the art approaches to this problem often use analytical methods based on the chain rule applied to existing dynamics algorithms. Although these methods are an improvement over fini...

Loosely captured orbits with circulating and pulsating eccentricity vectors have a variety of attractive mission design properties, including low insertion costs, near-circular and highly eccentric phases, mid- to high inclinations, long-term stability, and spatially distributed close approaches. Such orbits are the known result of averaging third-...

Equinoctial orbital elements were recently generalized from spherical geometry to the oblate spheroidal geometry of Vinti theory. For the symmetric Vinti potential, which accounts exactly for oblateness, these nonsingular elements are defined for all nondegenerate orbital regimes and resolve the usual problems found in the classical elements associ...

This is the author’s manuscript, before journal typesetting. To find the journal publication, please go to https://doi.org/10.1016/j.actaastro.2019.12.037‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏...

This is the author’s manuscript, before journal typesetting. To find the journal publication, please go to https://doi.org/10.1016/j.actaastro.2019.12.038‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏...

Lambert’s problem is the two-point boundary value problem for Keplerian dynamics. The parameter and solution space is surveyed for both the zero- and multiple-revolution problems, including a detailed look at the stress cases that typically plague Lambert solvers. The problem domain, independent of formulation, is shown to be rectangular for each r...

An optimization algorithm is described and demonstrated that leverages many solutions to embedded boundary value problems along a spacecraft trajectory discretized into many segments. For Keplerian dynamics, the boundary value problems are solved with an iteration-free, interpolated solution to Lambert's problem. The algorithm combines exact first-...

This paper describes a method to estimate shape and pose of a small celestial body with lidar measurements, and using only an extended Kalman filter. The shape of the small body is represented with a Gaussian Process, which naturally provides a continuous representation of the distance of the surface from the origin of the body and corresponding un...

When designing and navigating space missions to asteroids and comets, mascon models can be attractive because they are simple to compute, implement, and parallelize. However, to achieve a reasonable surface accuracy, mascon models typically require too many elements to compete with other models. Here, mascon models are revisited, with the intent to...

The optimization of impulsive, three-dimensional transfer trajectories to capture at Europa is investigated. Two initial conditions are considered: a halo orbit in the vicinity of Europa, and a resonant orbit around Jupiter. Primer vector theory is utilized to determine the number of impulses and the gradient information needed for this highly nonl...

The cannonball assumption of three-degree-of-freedom (3DOF) space object state prediction can lead to large inaccuracies if the sphericity assumption is violated, while numerical propagation of both the translational and rotational (6DOF) equations of motion is computationally expensive. In this paper, a middle ground is proposed, in which the tran...

A closed-form first-order perturbation solution for the attitude evolution of a triaxial space object in an elliptical orbit is presented. The solution, derived using the Lie–Deprit method, takes into account gravity-gradient torque and is facilitated by an assumption of fast rotation of the object. The formulation builds on the earlier implementat...

A myriad of methods, including many optimization and estimation algorithms, require the sensitivities of a dynamic system governed by a set of ordinary differential equations (ODEs). In this paper, the decoupled direct method (DDM) is derived for the calculation of first- and second-order state transition matrices (STMs) using fixed-step-size Runge...

Optimal orbital trajectories are obtained through the solution of highly nonlinear large-scale problems. In the case of low-thrust propulsion applications, the spacecraft benefits from high specific impulses and, hence, greater payload mass. However, these missions require a high count of orbital revolutions and, therefore, display augmented sensit...

A piecewise-constant Sundman transformation is introduced for a spacecraft trajectory dis-cretized into many segments. The piecewise-constant Sundman transformation is a function that automatically produces the time of flight for the boundary value problem between the terminal states of each segment implicitly enforcing position continuity along th...

Future small body missions will benefit from improved gravity field and mass representations that benefit diverse disciplines like autonomous navigation, mission design, geophysical modeling, and guidance, navigation, and control (GNC). Here, a recently introduced hybrid gravity model is applied to several small bodies of interest with the intent o...

Future small body missions will benefit from improved gravity field and mass representations that benefit diverse disciplines like autonomous navigation, mission design, geophysical modeling, and guidance, navigation, and control (GNC). Here, a recently introduced hybrid gravity model is applied to several small bodies of interest with the intent o...

Lissajous orbits and approximation of their invariant manifolds are used to generate landing trajectories to the surface of Europa. Each lissajous is discretized into individual revolutions that each resemble a periodic orbit. The combined approximate unstable manifolds of the individual revolutions of a lissajous orbit generate more surface covera...

A multiple grid search strategy is implemented to generate a broad database of axisymmetric three-body periodic orbits for planets and main planetary satellites in the Solar system. The periodic orbit search is performed over 24 pairs of bodies that are well approximated by the circular restricted three-body problem (CR3BP), resulting in approximat...

A systematic approach is devised to find ballistic captures in the planar elliptic restricted three-body problem. Simple symmetric periodic orbits around the smaller primary in the circular problem are used as generators for ballistic captures. Combining a scaling factor that maps states from the circular to the elliptic model, and restricting the...

Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating J2, J3, and generally a partial J4 in an orbit propagation theory without recourse to perturbat...

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal eleme...

In the context of small bodies, mascon models can be attractive because they are simple to compute, implement, and parallelize. However, to achieve a reasonable surface accuracy, mascon models typically require too many elements to be competitive with other models. Here, mascon models are revisited, with the intent to minimize the number of element...

A broad database of planar, axi-symmetric three-body periodic orbits for planets and main planetary satellites in the Solar System is generated and made available online. The database generation is based on a grid search that incorporates a new discretization scheme centered around fundamental periodic orbit families, and a robust differential corr...

The optimization of impulsive, three-dimensional transfer trajectories to capture at Europa is investigated. Primer vector theory is utilized to determine the number of impulses, and for the gradient information needed to optimize the problem. Two initial boundary conditions are considered: a halo orbit in the vicinity of Europa, and a resonant orb...

The design of low energy transfers is in general a tedious, time consuming task due to the high dynamical complexity of multi-body environments. A new systematic strategy, which seeks to ease the complexity of this task, is presented. In this model, we show how precomputed three-body periodic orbits can be patched together to give rise to complex t...

Equinoctial orbital elements have been generalized from spherical geometry to the oblate spheroidal geometry of Vinti theory, a satellite theory that accounts exactly for oblateness and optionally J3. For the symmetric potential, these nonsingular elements resolve the usual problems found in the classical elements associated with angle ambiguities....

This is an early conference paper. To find the journal publication, please go to https://doi.org/10.1016/j.actaastro.2019.12.038‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎ ‎‏‏‎...

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal eleme...

Process noise is often used in estimation filters to account for unmodeled and mismodeled accelerations in the dynamics. The process noise covariance acts to inflate the state covariance over propagation intervals, increasing the uncertainty in the state. In scenarios where the acceleration errors change significantly over time, the standard proces...

Fast and accurate collision probability computations are essential for protecting space assets. Monte Carlo (MC) simulation is the most accurate but computationally intensive method. A Graphics Processing Unit (GPU) is used to parallelize the computation and reduce the overall runtime. Using MC techniques to compute the collision probability is com...

This is an early conference paper. To find the journal publication, please go to https://doi.org/10.1016/j.actaastro.2019.12.037‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏‎ ‏‏‎ ‎‏‏‎ ‎‏‏...

A systematic approach is devised to find ballistic captures in the planar elliptic restricted three-body problem. Simple symmetric periodic orbits around the secondary body of the circular problem, computed through a global grid search, are used as generators for ballistic captures in the elliptic problem. Combining a scaling factor that maps state...

Ballisticly connecting halo orbits to science orbits in the circular-restricted three-body problem is investigated. Two classes of terminal science orbits are considered: low-altitude, tight orbits that are deep in the gravity well of the secondary body, and high-altitude, loose orbits that are strongly perturbed by the gravity of the primary body....

Preliminary design of low-thrust trajectories generallybenefits from broad searches over the feasible space. Despite convergence issues, indirect methods are generally faster than direct methods and are therefore well-suited for such searches. Nonetheless, indirect solutions typically require expensive numerical integration of at least the state an...

A variable-step Gauss-Legendre implicit Runge-Kutta (GLIRK) propagator is applied to coupled orbit/attitude propagation. Concepts previously shown to improve efficiency in 3DOF propagation are modified and extended to the 6DOF problem, including the use of variable-fidelity dynamics models. The impact of computing the stage dynamics of a single ste...

A comprehensive tour of the complex outer planet systems is a central goal in space science. However, orbiting multiple moons of the same planet would be extremely prohibitive using traditional propulsion and power technologies. In this paper, a new mission concept, named Magnetour, is presented to facilitate the exploration of outer planet systems...

The modified Harris-Priester model is a computationally inexpensive method for approximating atmospheric density in the thermosphere and lower exosphere – a vital step in low Earth orbit trajectory propagation. This work introduces a revision, dubbed cubic Harris-Priester, which ensures continuous first derivatives, eliminates singularities, and ad...

State transition matrices provide sensitivities or partial derivatives between states at different times along a trajectory and are used for a number of applications such as feedback controls, stability analysis, estimation, targeting, parameter optimization, and optimal control. The need for accurate state transition matrices is especially importa...

Polynomial chaos expansion and Gaussian mixture models are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. Polynomial chaos expansion models uncertainty by performing an expansion using orthogonal polynomials. The accuracy of polynomial chaos expansion for a given problem can be improved by i...

Heliotropic orbits provide long-lifetime low-altitude orbits in the presence of large J2 and solar radiation pressure perturbations. Formal inclusion of high-degree zonal gravity harmonics and simple shadowing provides a more realistic model to initiate the search for heliotropic orbits at irregular primitive bodies like Bennu, which is the target...

The number of tracked space objects is trending upward, raising the need for accurate and fast collision probability computations. Gaussian mixture models (GMMs) provide a compromise between accuracy and runtime by better approximating the true non-Gaussian distributions during conjunction. The use of multidirectional GMMs (MGMMs) is proposed to im...

The problem of stabilizing a spacecraft on Circular Restricted Three Body Problem orbits with a nonlinear, feedback controller subject to input constraints is studied in this paper. The proposed solutions are based on Lyapunov methods, which are fused with results from convex optimization. The control inputs are calculated online, through a pointwi...

Vinti's potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti's spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating J2, J3, and approximately two thirds of J4 in an orbit propagation theory without resorting to...

Preliminary design of low-thrust trajectories generally benefits from broad searches over the feasible space. Despite convergence issues, indirect methods are generally faster than direct methods, and therefore well-suited for such searches. Nonetheless , indirect solutions typically require expensive numerical integration of at least the state and...

It is expected that a non-trivial percentage of small bodies that future missions may visit are in non-principal axis rotation (i.e. “tumbling”). The primary contribution of this paper is the application of the Extended Kalman Filter (EKF) Simultaneous Localization and Mapping (SLAM) method to estimate the small body spin state, mass, and moments o...

Monte Carlo simulations are an accurate but computationally expensive procedure for approximating the resultant non-Gaussian probability density function (PDF) after propagation of an initial Gaussian PDF through a nonlinear function. Univariate splitting libraries for Gaussian Mixture Models (GMMs) exist with up to five elements in the literature....

Space missions involving Hall Effect Thrusters are more and more common, and designs of both Hall Effect Thrusters and low-thrust trajectories require more elaborated process to meet the on-going mission demands. The design of a new Hall Effect Thruster can be improved by considering mission goals for high-demand missions rather than by relying on...

With a goal of efficiently trading higher-memory footprints for faster runtimes, a high-fidelity interpolationdoimethod is presented for approximating a scalar quantity and associated gradients in the global three-dimensionaldoidomain external to a sphere. The new "Fetch" interpolation model uses the Junkins weighting function strategy todoiachieve...

Accurate partial derivatives are of the utmost importance for optimization and root-solving algorithms, but can prove challenging and computationally expensive to obtain. Modern space missions often require highly sensitive trajectories, increasing the need for accurate partials. Different techniques for computing state transition matrices for traj...

Two methods for deriving first-order partial derivatives of the outputs with respect to the inputs of the Lambert boundary value problem are presented. The first method assumes the Lambert problem is solved via the universal vercosine formulation. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a...

Seeking improvements in speed and accuracy in multiobject trajectory simulations, a solution methodology is presented that takes advantage of 1) new high-fidelity geopotential and third-body perturbation models that efficiently trade memory for speed, and 2) a graphics processing unit based integrator to achieve parallelism across multiple objects....

Periodic orbits with respect to an object in an eccentric Keplerian reference orbit can be found in a variety of ways, including the use of Tschauner-Hempel equations and orbital element differences, which both admit linearized solutions, as well as through direct analyses of two orbits. An alternative parameterization of the last approach is propo...

Interplanetary and intermoon tour missions have benefited from the implementation of leveraging maneuvers that efficiently change spacecraft energy relative to a flyby body. In the current work, these v∞ leveraging maneuvers are generalized and reformulated into a boundary-value problem suitable for broad trajectory searches using only one addition...

The use of electrodynamic tethers for propulsion and power generation is attractive for missions to the outer planets, which are traditionally handicapped by large propellant requirements, large times of flight, and a scarcity of power available. In this work, the orbital dynamics of a spacecraft using electrodynamic tether propulsion during the mi...

Analytical inclusion of high degree zonal gravity harmonics and solar radiation pressure enables heliotropic orbits to be found at irregular primitive bodies like Bennu, the target of the OSIRIS-REx mission. Heliotropic orbits provide long-lifetime, low-altitude orbits in the presence of these significant perturbations. Using a constrained, doubly-...

Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. In this paper, the effect of tether forces on periodic orbits in the Jupiter-Io system is investigated. A series of simplifications to the Lorentz force-perturbed circular-restricted three-body problem allows the devel...

A method is introduced to transition space trajectories from low fidelity patched conics models to full-ephemeris n-body dynamics. The algorithm incorporates a continuation method that progressively re-converges solution trajectories in systems with incremental changes in the dynamics. Continuation is accomplished through the variation of a control...

A procedure for deriving analytic partial derivatives of the Lambert problem is presented. Using the universal, cosine based Lambert formulation; first order partial derivatives of the velocities with respect to the positions and times are developed. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed i...

A second-order, general dynamics, relative motion framework is formulated to solve for optimal finite-burn transfers in complex gravity fields that are not amenable to analytic solutions. The second-order variational equations are employed in a Cartesian frame that is general in fidelity and simple to implement. For a passive chief orbit we show th...

The classic and Taylor series of Keplerian motion are extended to solve the Stark problem and to use the generalized Sundman transformation. Exact recursion formulas for the series coefficients are derived, and the method is implemented to high order via a symbolic manipulator. The results lead to fast and accurate propagation models with efficient...

The combined effect of significant solar radiation pressure and \(J_2\) perturbations on spacecraft orbits is investigated using both singly and doubly-averaged disturbing potentials with the Lagrange Planetary Equations. The resulting dynamics are applied to a spacecraft around an oblate asteroid. Several Sun-frozen families of orbits are identifi...