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# Low-Thrust Transfers to Southern L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} Near-Rectilinear Halo Orbits Facilitated by Invariant Manifolds

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## Abstract and Figures

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.
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Journal of Optimization Theory and Applications (2021) 191:517–544
https://doi.org/10.1007/s10957-021-01898-9
Low-Thrust Transfers to Southern L2Near-Rectilinear Halo
Orbits Facilitated by Invariant Manifolds
Sandeep K. Singh1·Brian D. Anderson2·Ehsan Taheri3·John L. Junkins1
Received: 12 September 2020 / Accepted: 16 June 2021 / Published online: 1 July 2021
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
Abstract
In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits
(NRHOs) and optimal low-thrust transfer trajectories using a high-ﬁdelity dynami-
cal 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.
Keywords Optimal control ·Minimum-fuel ·Minimum-time ·Invariant manifolds ·
Indirect optimization
1 Introduction
The Global Exploration Roadmap aims at expanding human presence into the Solar
System [10,16]. The next steps toward fulﬁlling these goals focus on designing sus-
tainable crewed missions to enable living and operating around and on the Moon. The
Roadmap also recognizes the importance of synergism between human and robotic
Communicated by Mauro Pontani.
A preliminary version of this paper was presented as AAS 20-565 at the 2020 AAS/AIAA Astrodynamics
Specialist Virtual Lake Tahoe Conference.
BSandeep K. Singh
sandymeche@tamu.edu
1Texas A&M University, 400 Bizzell St, College Station, TX 77843, USA
2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
3Auburn University, Auburn, AL 36849, USA
123
... Recently, Singh et al. [6] demonstrated a methodology for leveraging stable/unstable manifold pairs in a halo orbit around the Earth-Moon L 1 for end-to-end transfers in the cis-lunar space. They also extended their work to study the behavior of the invariant manifolds of the southern L 2 near-rectilinear halo orbits (NRHOs) in a multi-body system for a more accurate representation of manifolds and used them as terminal coast arcs for trajectory design [7]. Similar approaches to low-thrust trajectory design also appear in Refs. ...
... When planning a manifold-aided mission to the Moon, the most important step is to pick target patch points for insertion, which facilitates a ballistic capture transfer to the target through invariant space curves. A number of heuristic methods [6,7,14] for a priori evaluations of patch points have been studied previously. These approaches primarily use osculating element-space phase portraits of the entire set of piercing points, which are defined as the points at which the periodic-orbit manifolds intersect the Earth's plane in the Earth-Moon synodic frame. ...
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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.
... Rectilinear Halo Orbits (NRHOs) in a multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs for trajectory design [7]. Similar approaches to low-thrust trajectory design also appear in [8][9][10]. ...
... While planning a manifold-aided mission to the Moon, the primary step is to pick target patch-points for insertion which allows a ballistic capture transfer to the target via these invariant space curves. A number of heuristic methods [6,7,14] for a priori evaluation of the patch points have been studied previously. These approaches primarily use osculating element-space phase portraits of the whole set of piercing points, defined as the points where the periodic-orbit manifolds intersect the Earth plane in the Earth-Moon synodic frame. ...
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A supervised machine learning approach called the Gaussian Process Regression (GPR) is applied to approximate the optimal bi-impulse rendezvous maneuvers in cis-lunar space. The use of GPR approximation of the optimal bi-impulse transfer to patch-points associated with various invariant manifolds in the cis-lunar space is demonstrated. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solution of the optimal bi-impulsive Lambert transfer because the learned map is efficient to compute. This approach promises to be useful for aiding preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features in reducing propellant consumption while facilitating the solution of the 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 shown to efficiently represent the fuel costs (in terms of the $\Delta$V magnitude) associated with the class of orbital transfers of interest. A multi-input multi-output GPR model is developed and shown to provide efficient approximations. Multi-resolution use of local GPRs over smaller sub domains, and their use to construct a global GPR model is also demonstrated. One of the unique features of GPRs is to provide an estimate on the quality of the approximations in the form of covariance, which is shown to provide statistical consistency to the optimal trajectories generated from the approximation process. Numerical results demonstrate a basis for optimism for the utility of the proposed method.
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