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Abstract: This paper introduces a high-order nonlinear least-squares method for solving six-degree-of-freedom (6-DOF) navigation of satellite maneuvers. The approach involves developing first through fourth-order Taylor series models, which provide the necessary conditions that are iteratively adjusted to recover the unknown roots for reducing the errors arising from fitting models to a given set of observations. An initial guess is provided for the unknown parameters in the system, developing a correction vector using Taylor expansion, and then manipulating the necessary conditions to provide the least-squares algorithm. Computational differentiation (CD) tools generate the Taylor expansion partial derivative models. This paper presents an experimental work conducted using a 6-DOF platform to demonstrate the performance of the developed high-order nonlinear least-squares navigation method. An initial calibration of the sensing systems is performed in an operationally relevant ground-based environment. The experiments demonstrate that accelerated convergence is achieved for the second-, third-, and fourth-order expansions with various initial guess conditions. PubDate: 2023-05-19
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Abstract: The closed-loop Q-Law guidance algorithm has been shown to be a very capable and efficient method for producing low-thrust trajectories. This paper poses a Q-Law optimization problem for computing locally optimal gain values and for enforcing nonlinear constraints on the initial state using nonlinear programming (NLP). Gradient-based optimization methods have been shown to benefit greatly when analytical partial derivatives are supplied to the optimizer. Therefore, we present derivations of the Q-Law thrust vector partial derivatives with respect to the Q-law gains as well as with respect to the spacecraft’s state. These partials are leveraged to produce a state transition matrix, which contains exact partial derivatives of the terminal state with respect to the NLP problem decision vector. The Q-Law NLP problem can be coupled with the Sims–Flanagan interplanetary model in the same optimization problem. In this approach, the NLP solver uses a Q-Law model to design the planetocentric capture/escape spirals and a Sims–Flanagan model to design the interplanetary legs, resulting in end-to-end trajectory optimization. PubDate: 2023-05-17
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Abstract: This paper presents the study of the perturbed restricted problem of three bodies with an elongated smaller primary and an oblate radiating bigger primary. The albedo effect and small perturbations in the Coriolis and centrifugal forces have also been considered. The equilibrium points of the system and their linear stability are elaborated, and the significant variation in equilibrium points and their stability is observed. It is observed that the critical value of mass parameter \(\mu _c\) increases due to all the considered parameters except oblateness and segment-length. It is found that the critical mass parameter \(\mu _c\) decreases due to the effect of segment-length and oblateness. In addition, periodic orbits are constructed in the neighbourhood of equilibrium points. The effects of perturbing parameters in the periodic orbits are studied. The Poincaré Surface Section is constructed and then used to produce periodic orbits associated with its resonance. Considerable impact on the zero velocity curves and the periodic orbits are observed near the elongated primary. These effects decrease when the infinitesimal particle moves farther away from the elongated primary. PubDate: 2023-05-09
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Abstract: Debris-generating events in orbit pose a significant threat to all space-faring activities. In order to mitigate such events, it is essential to uncover their causes. This paper examines the fragmentations of three Atlas V Centaur upper stages (2009-047B, 2014-55B, 2018-079B) which occured in 2018 and 2019. Three different data sources—data from the Astronomical Institute of the University of Bern (AIUB), the Vimpel catalog, and the Spacetrack catalog—are used for this investigation and the outcomes are compared. Ephemerides and two-line elements (TLEs) from these sources are first propagated to obtain breakup epoch estimates. Subsequently, techniques developed by Tan and Reynolds to obtain Gabbard plots, velocity and angular distributions of fragments, and event intensities, are applied. This work finds that these events stray from existing fragmentation models and patterns exhibited by past events, such as those of the explosive Delta upper stages in the 1970s. Only one of the three events, 2018-079B, may have exploded due to leftover propellant combustion. 2009-047B appears to have endured a structural failure—inferred from clustered fragments with low velocities. A torus-shaped fragment distribution is observed for the 2014-055B event, suggesting it may have been pierced by a small piece of debris. PubDate: 2023-04-26
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Abstract: Trajectories at low energies with respect to the secondary body in the Circular Restricted Three-Body Problem are important for the future of orbital and landing missions. However, at these low energies, we demonstrate that high-latitude regions are unreachable for ballistic trajectories traveling through the \(L_1\) and \(L_2\) necks. This study explores the upper bound of vertical motion for these ballistic trajectories at a particular low energy. An initial grid-search across the \(L_1\) and \(L_2\) necks at Europa reveal that certain patterns appear in the trajectories that maximize inclination and out-of-plane velocity—metrics which are used to quantify vertical motion. These patterns are traced back to two nearby families of planar periodic orbits. These full families are generated and points of bifurcation are located. Certain families that originate as out-of-plane bifurcations from the two planar families are generated, and their unique members that match the energy of the grid-search are isolated. The manifolds of these particular periodic orbits exist along the simulated upper bound of vertical motion from the grid-search results. These results suggest that multiple families of stable and near-stable periodic orbits govern the flow of motion near this boundary. PubDate: 2023-04-11
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Abstract: A generic Bayesian framework is presented to track lost-in-space noncooperative maneuvering satellites. The developed framework predicts the reachability set for a lost-in-space satellite given bounds on maneuver parameters such as maneuver time and maneuver magnitude. Reachability sets are represented as a desired order polynomial series as a function of maneuver parameters. Recent advances in non-product quadrature methods are utilized to compute coefficients of this polynomial series in a computationally efficient manner. A major contribution of this work is to develop quadrature methods to generate samples for spherically uniform distribution for bounded magnitude maneuvers. Samples generated from this polynomial series are used for direct particle propagation in a traditional Bayesian filter rather than solving governing equations of motion for each sample point. An important component of the developed framework is a search strategy which exploits the reachability set calculations to task the sensor to increase the detection probability of the satellite. The samples generated from initial reachability sets are updated to systematically reduce the target search region based on actual detection of the target in a Bayesian framework. Numerical simulations are performed to show the efficacy of the developed ideas for tracking a lost-in-space satellite with the help of space based sensor. Performance of the proposed method varies widely based on factors such as the reachability set polynomial order, maneuver uncertainty bounds, sensor parameters (Field of view, measurement frequency, and detection probability), and initial conditions. For numerical experiments performed, the observer gained the custody of the maneuvering target in \(100\%\) and \(96\%\) of Monte Carlo (MC) simulations for the single maneuver and two maneuver cases, respectively. PubDate: 2023-03-29
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Abstract: This paper analyzes orbit families for the mission of space-based cislunar domain awareness and evaluates a novel set of metrics that can be used to inform the specific orbit parameterization for cislunar SDA constellation design. We present a dynamic simulation of the cislunar environment for use in numerical analysis of various pairings of resident space objects and sensing satellites intended for cislunar space domain awareness. Then we apply numerical observability analysis to calculate the local estimation condition number of observations of satellites on trajectories in the Earth-Moon system. Using our catalogue of simulated cislunar orbits as well as our heuristic and empirical metrics, we describe which orbit families provide the best observability of other cislunar orbit families and suggest a method for architecting constellations of observers in different cislunar orbits to perform the general cislunar space domain awareness mission. PubDate: 2023-03-29
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Abstract: This paper presents an attitude error model of a star tracker, which is induced from the optical system errors, and proposes an attitude Kalman filter considering the star tracker errors. Though it can be calibrated before and after launches, it is impossible to obtain error-free star tracker parameters in practice, which generates non-white noise errors in the star tracker outputs. Moreover, star tracker error models are usually a business secret for the manufacturers, so it is hard to estimate them online on the spacecraft bus. We model the attitude bias caused by the error of the optical parameters as colored noise using the camera model parameters and their covariance. A recursive form of the colored noise is derived based on a vector autoregressive model, and a colored noise Kalman filter is proposed to estimate the attitude error along with the spacecraft attitude and gyro bias. The proposed method only needs three additional states to be estimated and does not contain sensitive information for a star tracker manufacturer, which can ease the burden of its applications. The simulations illustrate the stability and reliability of the proposed algorithm. PubDate: 2023-03-29
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Abstract: 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-body regularization, but accounts for multiple bodies using their spheres of influence and a Heaviside approximation. The transformation enables efficient discretization of the many types of motion that exist in multi-body regimes. The new formulation is analyzed in several restricted three-body dynamical problems, which serve as proxy models that capture the dominant features of N-body ephemeris models within the solar system. The parameters embedded in the transformation are tuned, and its performance is compared against several existing regularizations on a diverse set of examples including periodic orbits in the Earth–Moon and Saturn-Enceladus systems, a low-thrust Earth–Moon spiral, and a low-altitude Jupiter-Europa flyby. The transformation is shown to be robust to the differing conditions, outperforming the benchmarks over wide ranges of the tuning parameters. The results of the numerical experiments are used to justify a set of recommendations for parameter selection in general multi-body applications. The regularization is finally demonstrated on ephemeris-modeled Saturn system trajectories that include unplanned flybys and traverse all three levels of the sun-planet-moon solar system hierarchy. PubDate: 2023-03-21 DOI: 10.1007/s40295-023-00364-0
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Abstract: This work addresses the systematic construction of solar-sail-based stopover cyclers between two bodies. The design of stopover cyclers is based on finding adequate body-to-body transfers with the zero sphere of influence assumption. These transfers strongly depend upon physical parameters that may be different in each trajectory of which the cycler consists. If the solar radiation pressure is the main propulsion system, the problem of finding transfers between the planets has symmetries that allow to predict the transfer time of the travel back from the transfer time of the outgoing journey. In this paper these symmetries are exploited to reduce the computational effort to just computing transfers from the starting to the target body. In case the parameters of the system are the same along the whole cycler, this reduces the computational effort to half. In case these parameters are assumed to change, the results can be used to construct stopover cyclers using only transfers from the starting to the target planet, but with different parameters. Moreover, transfers for different values of the parameters can be rapidly combined to have a global perspective of possible missions. The provided methods are exemplified with prospective cargo transportation missions to the Main Belt with a solar sail. PubDate: 2023-03-09 DOI: 10.1007/s40295-023-00372-0
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Abstract: A Sun–Earth stable manifold-based method for designing planar two-impulse Earth–Moon transfer trajectories is proposed in this paper. In this method, stable manifolds associated with Sun–Earth L1/L2 Lyapunov orbits provide initial guess trajectories for the planar two-impulse Earth–Moon trajectories. The perilune map of the initial guess trajectories is then applied to estimate the Moon’s initial phase angle for the two-impulse Earth–Moon trajectory in the Sun–Earth rotating frame. Then the accurate value of the Moon’s initial phase angle can be calculated to achieve a two-impulse Earth–Moon transfer trajectory. Furthermore, a global set of solutions for two-impulse Earth–Moon transfers are calculated by exploring the Jacobi constant of Lyapunov orbits around the Sun–Earth L1/L2. Numerical results indicate that stable manifolds of Sun–Earth L1/L2 Lyapunov orbits with larger Jacobi constants can be more easily applied to achieve two-impulse Earth–Moon transfers with lower energy. PubDate: 2023-03-09 DOI: 10.1007/s40295-023-00373-z
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Abstract: A nonlinear dynamic modeling method for active vibration suppression of laminated flexible spacecraft with macro fiber composite (MFC) is presented in this paper. High-order shear deformation theory is utilized to accurately describe the flexible deformations caused by the coupling effect of large overall motions and composite laminated material. Meanwhile, strain feedback control law strategy is established by using MFC as control actuators. According to the experiments results, the correctness of the proposed method has been verified. Depend on the proposed method, the system with different ply control strategies of MFCs is studied for obtaining the optimized control results. Integrated considering the kinematic accuracy, profile accuracy and dynamic character of the spacecraft system, the strategy with double MFCs assembling in the lower surface is superior to others. There are theoretical value and practical significance for the compositive mechanical performance of the spacecraft with laminated piezoelectric plates. PubDate: 2023-02-28 DOI: 10.1007/s40295-023-00370-2
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Abstract: This paper presents a study of formation flying design to optimize relative navigation performance when Global Positioning System (GPS) is unavailable. In GPS denied environments, relative navigation can be achieved through relative range measurements obtained from inter-satellite transceivers. Many formation flying missions require strict relative navigation performance. However, navigation performance using range measurements is highly dependent on the formation geometry. This research presents a method for designing formations such that relative navigation is optimized when using relative range measurements. Toward this, relative navigation performance metrics are derived as a function of formation design parameters. The structure and approach for optimizing these metrics are presented. This includes a presentation of the Divided Rectangles numerical optimization method for application in this field. To validate the optimized formations, a relative navigation filter is applied in a Monte Carlo simulation. Filter validation metrics are outlined and the performance of the optimal formations are presented and evaluated against the performance of the Monte Carlo sample formations. The validation results show that the optimization metrics yield formations with optimal relative navigation performance. PubDate: 2023-02-28 DOI: 10.1007/s40295-023-00369-9
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Abstract: In this paper, a highly accurate and efficient Adaptive Local Variational Iteration Method (ALVIM) is presented to fulfil the need of the astrodynamics society for fast and accurate computational methods for guidance and control. The analytical iteration formula of this method is derived by using a general form of the first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian is required. Apart from the simple yet efficient iteration formula, a straightforward adaptive scheme is introduced to refine the step size and the collocation nodes at each time segment. The presented adaptive method guarantees prescribed accuracy without manual tuning of the algorithm. The computational cost of ALVIM, in terms of functional evaluations, is 1–2 orders of magnitude lower than adaptive finite difference methods. Numerical results of a large amplitude pendulum, perturbed two-body problem, and three-body problem validate the high accuracy and efficiency of this easy-to-use adaptive method. PubDate: 2023-02-09 DOI: 10.1007/s40295-023-00366-y
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Abstract: Bertrand theorem’s states that, among central-force potentials with bound orbits, there are only two types of central-force scalar potentials with the property that all bound orbits are also closed orbits: the inverse-square law and Hooke’s law. These solutions are considered basic examples in classical mechanics since they help in understanding the regular and predictable motion of bodies and superintegrable dynamical systems. However, there are strong beliefs that other potentials may arise in dynamical systems which are not predicted by Bertrand’s theorem. Besides, several dynamical systems such as the solar system are characterized by chaotic and unbounded orbits which are not predicted by Bertrand’s theorem. In this work, we prove an extension of Bertrand’s theorem by means of non-standard Lagrangians and show the existence of a family of solutions for chaotic unstable periodic orbits. PubDate: 2023-02-07 DOI: 10.1007/s40295-023-00367-x
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Abstract: There are destabilizing resonances in medium Earth orbit due to third body effects from the Sun and Moon. Objects in the region suffer a growth of eccentricity dependent on their initial conditions. This study focuses on how debris objects from a fragmentation event would interact with the chaotic dynamics of the area. The debris clouds are studied for their interactions with nearby orbits, their eccentricity growth, and how varying initial conditions of an event will change each cloud’s evolution. PubDate: 2022-12-15 DOI: 10.1007/s40295-022-00362-8
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Abstract: Optimization problems involving multiple impulsive maneuvers are, in general, nonlinear and nonconvex. This implies that their resolution is prone to local optimality and convergence issues. This work proposes an optimization method to exploit a specific structure of single-constraint nonlinear programming problems. The proposed algorithm is able to transform an optimization problem with an arbitrary number of variables into a root-finding problem of a univariate algebraic equation. Moreover, it can readily overcome the aforementioned local optimality and convergence issues. This methodology has been applied to three practical application examples. The first application involves the inclination optimization for change of plane maneuvers using drifting orbits with a relative nodal precession. The second application performs the semi-major axis optimization of phasing orbits, using a two-stage approach to solve it; specifically, the dual-based method yields a solution with phasing orbits that perform a fractional number of revolutions, which is then corrected to provide the appropriate integrality condition. The third application carries out the optimization of multi-impulse Hohmann-like transfers with an inclination change, relying on a conservation law that allows to compute a multi-impulse transfer from the solution of a two-impulse transfer. Finally, two highly relevant mission scenarios are described and numerically solved: the first scenario considers a geostationary transfer orbit with a phasing optimization to locate a satellite into a prescribed slot in geostationary orbit; the second scenario considers a multi-target rendezvous of a servicing spacecraft to visit several satellites of a constellation for debris removal or refueling operations. PubDate: 2022-11-28 DOI: 10.1007/s40295-022-00357-5
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Abstract: Star is a mission design tool that globally searches for patched-conic trajectories that satisfy a set of user-defined constraints. It has been used to develop dozens of mission concepts at the Jet Propulsion Laboratory spanning multi-target rendezvous, sample return, multiple gravity assists (ballistic, high- and low-thrust), central-body switch to escape Earth or capture at a planet, planetary moon tours, and small-body tours sequenced from a pool of thousands of candidate targets. Star exhibits polynomial algorithmic complexity by constructing trajectories from independently computed encounter times, transfer legs, and flybys. Example missions of a ballistic transfer to Titan, tour of the Trojan asteroids, and low-thrust rendezvous with Mercury demonstrate the efficacy of the tool. PubDate: 2022-11-11 DOI: 10.1007/s40295-022-00350-y
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Abstract: Asteroids and comets are triggering interest due to the richness of precious materials, their scientific value as well as for their potential hazardousness. Owing to their significant diversity, minor bodies do not exhibit uniform shapes: they can range from spherical to irregularly shaped objects with rocky, uneven, and cratered surface. Nowadays, space probes rely more and more on optical navigation techniques, due to the increasing demand for autonomy. When dealing with minor bodies, the diversified range of shapes can significantly affect the performance of these techniques. In order to enable deep space probes to confidently deal with uncertainties, the need for robust image processing methods arises. Commonly, few image processing methods are designed and tested with limited shapes to meet mission requirements. In this work, we depart from this paradigm by developing a new framework, which includes extensive testing of the image processing algorithms with various shapes. The shapes are not randomly analyzed, yet they are arranged in a hierarchical structure called hyper-cube. The cube allows for a better understanding of the methods performance and to infer the way they shift from one shape to the other. The novelty of this approach lies both in the cube representation, which allows a better understanding of the link between the image processing algorithms and shape of the object, but also in the extensive number of shapes that have been tested, which has never been done before. In this analysis, four methods are considered, namely: center of brightness, intensity weighted centroiding, correlation with Lambertian spheres, and least-squares-based ellipse fitting. Results from this test allow us correlating the methods performances to the bodies shape, to suggest the best performing method for each shape family, and to assess their robustness. PubDate: 2022-11-07 DOI: 10.1007/s40295-022-00348-6
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