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A supervised stochastic learning method called the Gaussian Process Regression (GPR) is used to design an autonomous guidance law for low-thrust spacecraft. The problems considered are both of the time- and fuel-optimal regimes and a methodology based on "perturbed back-propagation" approach is presented to generate optimal control along neighboring optimal trajectories which form the extremal bundle constituting the training data-set. The use of this methodology coupled with a GPR approximation of the spacecraft control via prediction of the costate n-tuple or the primer vector respectively for time- and fuel-optimal trajectories at discrete time-steps is demonstrated to be effective in designing an autonomous guidance law using the open-loop bundle of trajectories to-go. The methodology is applied to the Earth-3671 Dionysus time-optimal interplanetary transfer of a low-thrust spacecraft with off-nominal thruster performance and the resulting guidance law is evaluated under different design parameters using case-studies. The results highlight the utility and applicability of the proposed framework with scope for further improvements. [ABSTRACT FROM AUTHOR]

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, the higher the decay rate as a function of increasing altitude, and hence the smaller its contribution to the total gravitational acceleration. To maintain a particular accuracy solution for a satellite in a highly elliptic orbit, a high gravity degree is needed near the perigee, and a low degree is sufficient at the apogee. This paper presents an analytic method for automatically selecting the degree of the spherical harmonic series based on the desired solution accuracy specified by the user and the instantaneous radial distance of the satellite from the central body. We present results for several test case orbits around the Earth, the Moon, and Mars that demonstrate a significant speedup when using our analytical radial adaptive model in orbit propagation. [ABSTRACT FROM AUTHOR]

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 are performed to identify suitable manifold entry/exit conditions based on inclination and minimum distance from the Earth. The relative merits of the stable/unstable manifolds are studied with regard to time- and fuel-optimality criteria, for a set of representative low-thrust family of transfers. [ABSTRACT FROM AUTHOR]

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 elusive for over five decades. We revisit Edelbaum's question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N-impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find that the answer to Edelbaum's question is not generally unique, but is frequently a set of equal-Δv extremals. We further find, when Edelbaum's question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces. [ABSTRACT FROM AUTHOR]

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, such as tracking orbit debris and maintaining a catalogue of over twenty thousand pieces of orbit debris greater than the size of a softball, as well as for prediction and prevention of future satellite collisions. The integrators used in the study are a 5th/4th and 8th/7th order Dormand-Prince, an 8th order Gauss-Jackson, a 12th/10th order Runga-Kutta-Nystrom, Variable-step Gauss Legendre Propagator and the Adaptive-Picard-Chebyshev methods. Four orbit test cases are considered, low Earth orbit, Sun-synchronous orbit, geosynchronous orbit, and a Molniya orbit. A set of tests are done using a high fidelity spherical-harmonic gravity (70 × 70) model with and without an exponential cannonball drag model. We present three metrics for quantifying the solution precision achieved by each integration method. These are conservation of the Hamiltonian for conservative systems, round-trip-closure, and the method of manufactured solutions. The efficiency of each integrator is determined by the number of function evaluations required for convergence to a solution with a prescribed accuracy. The present results show the region of applicability of the selected methods as well as their associated computational cost. Comparison results are concisely presented in several figures and are intended to provide the reader with useful information for selecting the best integrator for their purposes and problem specific requirements in astrodynamics. [ABSTRACT FROM AUTHOR]

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 hyperbolic tangent smoothing law for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and thus promote convergence. The TPBVP is solved using the method of particular solutions (MPS) shooting method and Picard-Chebyshev numerical integration. Application of Picard-Chebyshev integration affords an avenue for increased efficiency that is not available with step-by-step integrators. We demonstrate that computing the particular solutions with only a low-fidelity force model greatly increases the efficiency of the algorithm while ultimately achieving near machine precision accuracy. A salient feature of the MPS is that it is parallelizable, and thus further speedups are available. It is also shown that, for near-Earth orbits and over a small number of en-route revolutions around the Earth, only the zonal perturbation terms are required in the costate equations to obtain a solution that is accurate to machine precision and optimal to engineering precision. The proposed framework can be used for trajectory design around small asteroids and also for orbit debris rendezvous and removal tasks. [ABSTRACT FROM AUTHOR]

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. The first is a Keplerian Lambert solver, which is used to provide a good initial guess (warm start) for solving the perturbed problem. It is also used to determine the appropriate algorithm to call for solving the perturbed problem. The arc length or true anomaly angle spanned by the transfer trajectory is the parameter that governs the automated selection of the appropriate perturbed algorithm, and is based on the respective algorithm convergence characteristics. The second algorithm solves the perturbed Lambert problem using the modified Chebyshev-Picard iteration two-point boundary value solver. This algorithm does not require a Newton-like shooting method and is the most efficient of the perturbed solvers presented herein, however the domain of convergence is limited to about a third of an orbit and is dependent on eccentricity. The third algorithm extends the domain of convergence of the modified Chebyshev-Picard iteration two-point boundary value solver to about 90% of an orbit, through regularization with the Kustaanheimo-Stiefel transformation. This is the second most efficient of the perturbed set of algorithms. The fourth algorithm uses the method of particular solutions and the modified Chebyshev-Picard iteration initial value solver for solving multiple revolution perturbed transfers. This method does require “shooting” but differs from Newton-like shooting methods in that it does not require propagation of a state transition matrix. The unified Lambert tool makes use of the General Mission Analysis Tool and we use it to compute thousands of perturbed Lambert trajectories in parallel on the Space Situational Awareness computer cluster at the LASR Lab, Texas A&M University. We demonstrate the power of our tool by solving a highly parallel example problem, that is the generation of extremal field maps for optimal spacecraft rendezvous (and eventual orbit debris removal). In addition we demonstrate the need for including perturbative effects in simulations for satellite tracking or data association. The unified Lambert tool is ideal for but not limited to space situational awareness applications. [ABSTRACT FROM AUTHOR]

The solution of initial value problems provide the state history for a given dynamic system, for prescribed initial conditions. Existing methods for solving these problems have not been very successful in exploiting parallel computation architectures, mainly because most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. This article proposes parallel-structured modified Chebyshev-Picard iteration (MCPI) methods, which iteratively refine estimates of the solutions until the iteration converges. Using Chebyshev polynomials as the orthogonal approximation basis, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. A vector-matrix form is introduced that makes the methods computationally efficient. The power of the methods is illustrated through satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, the proposed methods generate solutions with improved accuracy as well as several orders of magnitude speedup, even before parallel implementation. Allowing only to integrate position states or perturbation motion achieve further speedup. Parallel realization of the methods is implemented using a graphics processing unit to provide inexpensive parallel computation architecture. Significant further speedup is achieved from the parallel implementation. [ABSTRACT FROM AUTHOR]

Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw--Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin's principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin's method for a classical Lambert's problem and with a Chebyshev pseudospectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited. [ABSTRACT FROM AUTHOR]

Modified Chebyshev-Picard Iteration methods are presented for solving high precision, long-term orbit propagation problems. Fusing Chebyshev polynomials with the classical Picard iteration method, the proposed methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to traditional, step-wise, forward integration methods. Numerical results demonstrate that for orbit propagation problems, the presented methods are comparable to or superior to a state-of-the-art 12th order Runge-Kutta-Nystrom method in a serial processor as measured by both precision and efficiency. We have found revolutionary long solution arcs with more than eleven digit path approximations over one to three lower-case Earth orbit periods, multiple solution arcs can be patched continuously together to achieve very long-term propagation, leading to more than ten digit accuracy with built-in precise interpolation. Of revolutionary practical promise to much more efficiently solving high precision, long-term orbital trajectory propagation problems is the observation that the presented methods are well suited to massive parallelization because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. [ABSTRACT FROM AUTHOR]

In static systems, state values are estimated using traditional least squares techniques based on a redundant set of measurements. Inaccuracies in measurement model parameter estimates can lead to significant errors in the state estimates. This paper describes a technique that considers these parameters in a modified least squares framework. It is also shown that this framework leads to the minimum variance solution. Both batch and sequential (recursive) least squares methods are described. One static system and one dynamic system are used as examples to show the benefits of the consider least squares methodology. [ABSTRACT FROM AUTHOR]

A singularity-free solution is presented for inverting the Cartesian to Geodetic transformation. Two rapidly converging generalized continued fraction algorithms are presented for handling applications spanning the LEO to GEO range of applications. Comparisons with state-of-the-art algorithms are provided for solution accuracy and run-time performance. [ABSTRACT FROM AUTHOR]

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter. [ABSTRACT FROM AUTHOR]

The implicit function theorem due to Lagrange is generalized to enable high order implicit rate calculations of general implicit functions about pre-computed solutions of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about an existing root (for algebraic systems) or trajectory (in case of dynamical systems). The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities of the solutions of boundary value problems with respect to the parameters of the system model and/or functions describing the boundary condition. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was found that useful information relating the neighboring extremal paths can be deduced from these implicit rates characterizing the behavior in the neighborhood of the existing solutions. The accuracy of solutions obtained is subsequently enhanced using an averaging scheme based on the Global Local Orthogonal Polynomial (GLO-MAP) weight functions developed by the first author to blend many local approximations in a continuous fashion. Example problems illustrate the wide applicability of the presented generalizations of Lagrange's classical results to static and dynamic optimization problems. [ABSTRACT FROM AUTHOR]

An analytical approach is presented for developing an estimation framework (called the Jth Moment Extended Kalman Filter (JMEKF)). This forms an important component of a class of architectures under investigation to study the interplay of major issues in nonlinear estimation such as model nonlinearity, measurement sparsity and initial condition uncertainty in the presence of low process noise. Utilizing art automated nonlinear expansion of the model about the current best estimated trajectory, a Jth order approximate solution for the departure motion dynamics about a nominal trajectory is derived in the form of state transition tensors. This solution is utilized in evaluating the evolution of statistics of the departure motion as a function of the statistics of initial conditions. The statistics thus obtained are used in the determination of a state estimate assuming a Kalman update structure. Central to the state transition tensor integration about a nominal trajectory, is the high order sensitivity calculations of the nonlinear models (dynamics and measurement), being automated by OCEA (Object Oriented Coordinate Embedding Method), a computational tool generating the required various order partials of the system differential equations without user intervention. Working in tandem with an OCEA automation of the derivation of the state transition tensor differential equations is a vector matrix representation structure of tensors of arbitrary rank, facilitating faster and more accurate computations. High order moment update equations are derived to incorporate the statistical effects of the innovations process more rigorously, improving the effectiveness of the estimation scheme. Numerical simulations on an orbit estimation example investigate the gain obtained in using the proposed methodology in situations where the classical extended Kalman filter's domain of convergence is smaller. The orbit estimation example presented examines a situation that requires us to determine the position and velocity state of the orbiter from range, azimuth and elevation measurements being made available sparsely. [ABSTRACT FROM AUTHOR]

An efficient Kalman filter based algorithm has been proposed for the spacecraft attitude estimation problem using a novel split-field-of-view star camera and three-axis rate gyros. The conventional spacecraft attitude algorithm has been modified for on-orbit estimation of interlock angles between the two fields of view of star camera, gyro axis. and the spacecraft body frame. Real time estimation of the interlock angles makes the attitude estimates more robust to thermal and environmental effects than in-ground estimation, and makes the overall system more tolerant of off-nominal structural mechanical, and optical assembly anomalies. [ABSTRACT FROM AUTHOR]

The nonlinearity index is a measure of the nonlinearity of dynamical systems based on computing the initial-condition sensitivity of the state-transition matrix. The Cayley form is a representation for dynamical systems that relates their motion to N-dimensional rotations. The generalized coordinates of the system are used to define an N-dimensional orientation, and a set of quasi velocities is defined equal to the corresponding angular velocity. The nonlinearity index of the Cayley-form representation is computed for an elastic spherical pendulum and a planar satellite example. These results are compared to values for alternative dynamical representations. Additionally, the nonlinearity is evaluated by analyzing how well the linearized equations of each representation capture certain properties of the motion. These results show that the Cayley form can have lower nonlinearity than traditional representations, in particular those representations that suffer from kinematic singularities. [ABSTRACT FROM AUTHOR]

Star identification is the most critical and important process for attitude estimation, given data from any star sensor. The main purpose of the Star Identification (Star-ID) process is to identify the observed/measured stars with the corresponding cataloged stars. The precision of the observed star directions highly depend on the calibrated accuracy of the star camera parameters, mainly the focal length f, and the optical axis offsets (x0, y0). When these parameters are not accurate or when the camera is not well calibrated, the proposed Non-dimensional Star-ID method becomes very suitable, because it does not require accurate knowledge of these parameters. The Nondimensional Star-ID method represents a unique tool to identify the stars of uncalibrated or inaccurate parameters cameras. The basic idea derives the identification process from the observed focal plane angles which are, to first order, independent from both the focal length and the optical axis offsets. The adoption of the k-vector range search technique, makes this method very fast. Moreover, it is easy to implement, accurate, and the probability of failing Star-ID is less than 0.1% for typical star tracker design parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new direction for generating and simultaneously solving equations of motion for dynamical systems, using automatic differentiation. We overview the current computational approaches for solving multibody dynamics problems and discuss several choices for equation of motion formulation. We present an operator-overloading method for generating equations of motion automatically via Lagrange's Equations and solve them in a direct fashion. Several numerical examples are presented to demonstrate the accuracy and efficiency of the method for simulating the motion of multibody dynamical systems. [ABSTRACT FROM AUTHOR]

A new star pattern recognition method is developed using singular value decomposition of a measured unit column vector matrix in a measurement frame and the corresponding cataloged vector matrix in a reference frame. It is shown that singular values and right singular vectors are invariant with respect to coordinate transformation and robust under uncertainty. One advantage of singular value comparison is that a pairing process for individual measured and cataloged stars is not necessary, and the attitude estimation and pattern recognition process are not separated. An associated method for mission catalog design is introduced and simulation results are presented. [ABSTRACT FROM AUTHOR]

Some observations are made on coordinate selection for the two most fundamental problems of astrodynamics, namely orbit dynamics and attitude dynamics, and some interesting connections and analogies between the two problems are explored. While an infinity of coordinate choices are feasible for each of these problems, we review four coordinate choices for each problem, including several that lead to governing differential equations that are regular, and in some cases, rigorously linear. Some methodology is introduced that has a universal flavor with implications for dynamical systems broadly. We show how dramatic qualitative and quantitative alteration of the mathematical description of the motion can be accomplished by introduction of redundant coordinates to describe the evolution of the dynamics in a higher dimensioned space. Some observations are made on an analogy between the two central problems of interest. One of the regularizing transformations studied for orbital dynamics is motivated directly by this analogy. Finally broadly applicable analytical and computational developments are presented that provide a measure of nonlinearity over a worst-case region of state space in the vicinity of a reference trajectory; this measure is used to evaluate the several coordinate choices. [ABSTRACT FROM AUTHOR]

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 present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure. [ABSTRACT FROM AUTHOR]

Presents an acknowledgement of those who organized the John L. Junkins Astrodynamics Symposium on May 24, 2003.

The article introduces a series of articles on astronautics published within the issue that were presented during the American Astronautical Society (AAS) F. Landis Markley Astronautics Symposium held in Cambridge, Maryland from June 29 to July 2, 2008.

A preface for the July-December 2006 issue of "The Journal of the Aeronautical Sciences" is presented.