Article

# Rapid accessibility evaluation for ballistic lunar capture via manifolds: A Gaussian process regression application

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## Abstract

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 space. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solutions of the optimal bi-impulsive Lambert transfer because the learned map is computationally efficient. This approach promises to be useful for aiding in preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features for reducing propellant consumption while facilitating the solution of trajectory optimization problems. Long ballistic capture coasts are also very attractive for mission guidance, navigation, and control robustness. A multi-input single-output GPR model is presented to represent the fuel costs (in terms of the ΔV magnitude) associated with the class of orbital transfers of interest efficiently. The developed model is also proven to provide efficient approximations. The multi-resolution use of local GPRs over smaller sub-domains and their use for constructing a global GPR model are also demonstrated. One of the unique features of GPRs is that they provide an estimate of the quality of approximations in the form of covariance, which is proven to provide statistical consistency with the optimal trajectories generated through the approximation process. The numerical results demonstrate our basis for optimism for the utility of the proposed method.

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... In the field of astrodynamics, Shang and Liu [17] assessed accessibility of Main-Belt asteroids by predicting the optimal bi-impulsive ΔV costs to rendezvous with the asteroids via a GPR model trained on the family of such transfers to rendezvous states in a domain spanning the known main-belt asteroid region. Singh et al. [18] later demonstrated the effectiveness of a multi-input multi-output GPR in predicting the optimal family of Lambert transfer trajectories for cis-lunar transfers from Earth to a periodic 1 Halo orbit; leveraging invariant manifolds. ...
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This paper highlights natural transport pathways between in-plane and out-of-plane states associated with the vertical instability of planar Lyapunov orbits around the Lagrange points L1 and L2 in the Earth-Moon circular restricted three-body problem. Computations of invariant manifolds associated with the vertical instability of planar periodic orbits, vertically" stable and unstable manifolds, enable quantitative analyses of inclination changes. This study finds that multiple lunar flybys gradually change the orbital elements of vertically stable and unstable manifolds, and that the distributions of the affected orbital elements depend on the Jacobi constant and on the associated Lagrange point L1 or L2 of the planar Lyapunov orbits. As an application, this study uses the vertically stable manifolds of the planar Lyapunov orbits as initial guesses for optimizing transfers from near rectilinear halo orbits to planar distant retrograde orbits. Significant delta-v savings as compared with the known solutions demonstrate the usefulness of the vertical instability in spacecraft trajectory designs.
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The main-belt asteroids are of great scientific interest and have become one of the primary targets of planetary exploration. In this paper, the accessibility of more than 600,000 main-belt asteroids is investigated. A computationally efficient approach based on Gaussian process regression is proposed to assess the accessibility. Two transfer models consisting of globally optimal two-impulse and Mars gravity-assist transfers are established, which would serve as a source of training samples for Gaussian process regression. The multistart and deflection technologies are incorporated into the numerical optimization solver to avoid local minima, thereby guaranteeing the quality of the training samples. The covariance function, as well as hyperparameters, which dominate the regression process, are chosen elaborately in terms of the correlation between samples. Numerical simulations demonstrate that the proposed method can achieve the accessibility assessment within tens of seconds, and the average relative error is only 1.33%. Mars gravity assist exhibits significant advantage in the accessibility of main-belt asteroids because it reduces the total velocity increment by an average of 1.23 km/s compared with the two-impulse transfer. Furthermore, it is observed that 3976 candidate targets have potential mission opportunities with a total velocity increment of less than 6 km/s. © 2016 by the American Institute of Aeronautics and Astronautics, Inc.
Chapter
Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented and compute approximations to trajectories for a mission to Venus.
Article
Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.
Article
A procedure is described that provides a universal solution for Lambert's problem. Based on the approach of Lancaster and his colleagues, the procedure uses Halley's cubic iteration process to evaluate the unknown parameter, x, at the heart of the approach, initial estimates for x being selected so that three iterations of the process always suffice to yield an accurate value. The overall procedure has been implemented via three Fortran-77 subroutines, listings of which are appended to the paper, and the way in which the subroutines have been tested is outlined.
Article
Transmission spectroscopy, which consists of measuring the wavelength-dependent absorption of starlight by a planet's atmosphere during a transit, is a powerful probe of atmospheric composition. However, the expected signal is typically orders of magnitude smaller than instrumental systematics, and the results are crucially dependent on the treatment of the latter. In this paper, we propose a new method to infer transit parameters in the presence of systematic noise using Gaussian processes, a technique widely used in the machine learning community for Bayesian regression and classification problems. Our method makes use of auxiliary information about the state of the instrument, but does so in a non-parametric manner, without imposing a specific dependence of the systematics on the instrumental parameters, and naturally allows for the correlated nature of the noise. We give an example application of the method to archival NICMOS transmission spectroscopy of the hot Jupiter HD 189733, which goes some way towards reconciling the controversy surrounding this dataset in the literature. Finally, we provide an appendix giving a general introduction to Gaussian processes for regression, in order to encourage their application to a wider range of problems.
Article
The International Sun-Earth Explorer (ISEE) scientific satellite to be stationed in 1978 in the vicinity of the sun-earth interior libration point to continuously monitor the space between the sun and the earth, including the distant geomagnetic tail is described. Orbit selection considerations for the ISEE-C are discussed along with stationkeeping requirements and fuel-optimal trajectories. Due to the alignment of the interior libration point with the sun as viewed from the earth, it will be necessary to place the satellite into a 'halo orbit' around the libration point, in order to eliminate solar interference with down-link telemetry. Parametric data for transfer trajectories between an earth parking orbit (altitude about 185 km) and a libration-point orbit are presented. It is shown that the insertion magnitude required for placing a satellite into an acceptable halo orbit is rather modest.
The role of cis-lunar space in future global space exploration
• M Bobskill
• M Lupisella
Bobskill, M., Lupisella, M. The role of cis-lunar space in future global space exploration. In: Proceedings of the Global Space Exploration Conference, Washington, DC, USA, 2012: GLEX-2012.05.5.4x12270.
Autonomous time-optimal many-revolution orbit raising for electric propulsion GEO satellites via neural networks
• H Y Li
• F Topputo
• H X Baoyin
Li, H. Y., Topputo, F., Baoyin, H. X. Autonomous time-optimal many-revolution orbit raising for electric propulsion GEO satellites via neural networks. arXiv preprint, 2019, https://doi.org/10.48550/arXiv.1909. 08768.
USA. He recently received the highest honor in his field, the AIAA Robert H
and Astronautics (AIAA), USA. He recently received the highest honor in his field, the AIAA Robert H. Goddard Astronautics Award (2019). E-mail: junkins@tamu.edu.