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# Contour map of transfer opportunities: (a) the opportunities for the Earth-escaping phase; the blue areas represent the feasible opportunities. (b) The opportunities for the Moon-captured phase; the blue areas represent the feasible opportunities. In this study, (β, τ ) = (150°, 0.5)

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The low-thrust propulsion will be one of the most important propulsion in the future due to its large specific impulse. Different from traditional low-thrust trajectories (LTTs) yielded by some optimization algorithms, the gradient-based design methodology is investigated for LTTs in this paper with the help of invariant manifolds of \(\mathit{LL}_...

## Context in source publication

**Context 1**

... through the LL 1 Halo orbit as pertur- bations to the manifolds cause crashes or escapes over short time scales. Only the pairs that enable the transfer from the Earth to the Halo orbit near the LL 1 point and from the Halo orbit near the LL 1 point to the Moon can be adopted to de- sign the Earth-Moon transfer; these are depicted in Fig. 6. In this study, the transfer opportunities make up 12. 3), SBCM dynamics equa- tions can be integrated. After obtaining a backward inte- gration, the trajectory of the Earth-escaping phase can be obtained. When a forward integration is derived, the trajec- tory of the Moon-captured phase can be obtained. (ii) When reaching the first ...

## Citations

... Similar approaches to low-thrust trajectory design also appear in Refs. [8][9][10]. Additionally, transfers in the lunar domain, particularly those involving periodic orbits such as NRHOs and distant retrograde orbits have also been studied extensively by various researchers [11][12][13]. ...

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.

... Transfers in the low-thrust domain exploiting manifolds of periodic orbits as well as multi-body equilibria in the cislunar space as a research topic have seen an exciting impetus in the recent past [5,29,37]. Although the NRHOs and associated transfers have been studied in some detail, an end-to-end low-thrust transfer from a geocentric orbit to a NRHO, especially one which leverages manifolds is important from the point of view of the Lunar Gateway mission. ...

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.

... 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]. Additionally, transfers in the lunar domain especially involving periodic orbits like NRHOs and Distant Retrograde Orbits (DROs) have also been studied extensively by various researchers [11][12][13]. ...

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.

... They also extended their work to study the behavior of the invariant manifolds of Southern L 2 Near 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 [19]. Similar approaches to low-thrust trajectory design also appear in [20,21]. ...

A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs while incorporating eclipses and perturbations during the optimization process in an ephemeris model; a feature lacking in state of the art software like MYSTIC and Copernicus. The end-to-end trajectories are generated by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of the proposed trajectory optimization framework using advanced indirect methods and by leveraging a Composite Smooth Control (CSC) construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). The results quantify accurate ∆V costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to be used for analogous un-crewed lunar missions.

... They also extended this to study the behavior of the invariant manifolds of Southern 2 Near 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 [21]. Similar approaches to low-thrust trajectory design also appear in [22][23][24]. Additionally, transfers in the lunar domain especially involving periodic orbits like NRHOs and Distant Retrograde Orbits (DROs) have also been studied extensively by various researchers [25][26][27][28]. ...

A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic, near rectilinear Halo orbits that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an indirect formulation of optimal control theory. The end-to-end trajectories are achieved by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of employing such a hybrid optimization algorithm for solving end-to-end analogous fuel-optimal problems using indirect methods and leveraging a composite smooth control construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). A novel process is introduced to incorporate eclipse-induced coast arcs and their impact within optimization. The results quantify accurate Δ costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to find applications for analogous uncrewed missions.

... In both strategies of Earth-Moon transfers, the Earth-Moon libration points L 1 and L 2 always serve as transfer stations for Moon missions due to its accessibility to the Moon. Therefore, some useful mathematical tools such as invariant manifolds, lunar flyby and low thrust have been utilized to design low energy transfers to the Moon and Earth-Moon L 1 or L 2 (Gao et al. 2019;Lei and Xu 2018;Lian et al. 2015;Qi and de Ruiter 2019;Qu et al. 2017). Moreover, invariant manifolds also have been regarded as an effective tool to design low-cost Earth-Moon transfer trajectories (Davis et al. 2011;Howell and Kakoi 2006;Koon et al. 2000Koon et al. , 2011Lei and Xu 2016;Mingotti et al. 2012a;Xu et al. 2012). ...

... More specifically, interior Earth-Moon transfers are always defined in the Earth-Moon CR3BP model without considering any perturbation. Since invariant manifolds associated with the Earth-Moon L 1 point cannot naturally approach the neighborhood of the Earth, additional maneuvers are usually required to connect the invariant manifold and the transfer trajectory starting from low Earth orbit (LEO) (Marson et al. 2010;Mingotti and Topputo 2011;Qu et al. 2017). Then bi-impulsive Earth-Moon transfers have been investigated by using the patched-conic method and differential correction to avoid the mid-course maneuver (da Silva Fernandes and Maranhão Porto Marinho 2012;Lv et al. 2017;Yagasaki 2004a). ...

Low-energy bi-impulsive Earth-Moon transfers are investigated by using periodic orbits. Two Earth-Moon transfer design strategies in the CR3BP are proposed, termed direct and indirect design strategy. In the direct design strategy, periodic orbits which approach both vicinities of the Earth and Moon are selected as candidate periodic orbits, which can provide an initial guess of bi-impulsive Earth-Moon transfers. In the indirect design strategy, new bi-impulsive Earth-Moon transfers can be designed by patching together a bi-impulsive Earth-Moon transfer and a candidate periodic orbit which can approach the vicinity of the Moon. Optimizations in the CR3BP are undertaken based on the Gradient Descent method. Finally, bi-impulsive Earth-Moon transfer design and optimizations in the Sun-Earth-Moon bi-circular model (BCM) are carried out, using bi-impulsive Earth-Moon transfers in the CR3BP as initial guesses. Results show that the bi-impulsive Earth-Moon transfer in the CR3BP can serve as a good approximation for the BCM. Moreover, numerical results indicate that the optimal transfers in the BCM have the potential to be of lower cost in terms of velocity impulse than optimal transfers in the CR3BP.

... The features of the circular restricted three-body problem (CR3BP) [1,2] allow transfer trajectories with small propellant consumption to be designed by exploiting the existence of invariant manifolds, as is discussed in Refs. [3,4,5,6] in the case of the Earth-Moon CR3BP. A potential application of the results of the CR3BP analysis is constituted by space missions orbiting around equilibrium points. ...

A solar sail generates thrust without consuming any propellant, so it constitutes a promising option for mission scenarios requiring a continuous propulsive acceleration, such as the maintenance of a (collinear) L1-type artificial equilibrium point in the Sun-[Earth+Moon] circular restricted three-body problem. The usefulness of a spacecraft placed at such an artificial equilibrium point is in its capabilities of solar observation, as it guarantees a continuous monitoring of solar activity and is able to give an early warning in case of catastrophic solar flares. Because those vantage points are known to be intrinsically unstable, a suitable control system is necessary for station keeping purposes. This work discusses on how to stabilize an L1-type artificial equilibrium point with a solar sail by suitably adjusting its lightness number and thrust vector orientation. A full-state feedback control law is assumed, where the control gains are chosen with a linear-quadratic regulator approach. In particular, the numerical simulation results show that an L1-type artificial equilibrium point can be maintained with small required control torques, by using a set of reflectivity control devices.

... Tang (2013a, 2013b) and Lian et al. (2012) studied the problem of libration point orbit rendezvous using terminal sliding mode control. Qu et al. (2017) investigated a gradient-based design methodology for low-thrust trajectories with the help of invariant manifolds and halo orbit of LL1 point. ...

A methodology is proposed to design optimal time-fixed impulsive transfers in the vicinity of the L2 libration point of the Earth-Moon system, taking the construction of a space station around the collinear libration points as the background. The approximate analytical expression of motions around the L2 point in the CRTBP is given, and the expression in the ERTBP is derived by linearizing the dynamical equations for the purpose of expanding the methodology from the CRTBP to the ERTBP. Thus, the approximate analytical solution of the transfer between two points is obtained by substituting the position vectors of the two points into the expression, which solves the Lambert problem in the three-body system. Furthermore, the transfer between different orbits is constructed by parameterization of the position vectors with the amplitudes and phases of the initial orbit and the final orbit. The transfers are optimized such that the total velocity increment required to implement the transfer exhibits a global minimum. The values of variables involved in the optimal transfers are determined by the unconstrained minimization of a function of one or nine variables using a multivariable search technique. To numerically ensure that the transfers are accurate and to eliminate the linearization bias, the differential correction and SQP method are employed. The optimality of the transfers is determined lastly by the primer vector theory. Simulations of point-to-point transfers, Lissajous-to-Lissajous transfers, halo-to-halo transfers and Lissajous-to-halo transfers are made. The results of this study indicate that the approximate analytical solutions, as well as the differential correction and SQP method, are valid in the design of the optimal transfers around the libration points of the restricted three-body problem.

... Because this PhD research project is dedicated to space science missions in the solar system, it could be argued that the Sun gravitational influence, representing a fourth body perturbation, can not be neglected. This is particularly true when investigating trajectories over long time intervals, as did in the previous section [91,146,147]. ...

In the last decades, the growing interest in investigating natural science in the space environment sets new targets, constraints and challenges in space mission design, defining what is nowadays known as space science. The goal of this PhD research is the development of new techniques of mission analysis, which can lead to further development of space science missions using CubeSat technology. Two main objectives have been pursued, related to both solar system exploration and low Earth orbit missions.
Due to the low power and thrust available on a CubeSat, low energy trajectories are necessary to allow solar system exploration. These are designed here considering a further constraint on the transfer time, which should be minimized to limit the effects of the hostile space environment on the on-board systems, typically based on components off-the-shelf. According to these issues, the topological properties of the linear dynamics in the circular restricted 3-body problem were investigated to develop a method allowing the design of internal transit and captures, including the possibility to select the osculating orbital elements at capture.
Three guidance strategies are proposed, allowing modification on the ultimate behavior of trajectories to match the desired mission requirements, also in the presence of the gravitational perturbations due to a fourth body. These strategies are effective with modest velocity variations (delta-V) and are tailored to be implemented with compact continuous thrusters, compatible with CubeSats. The method was originally developed in the dynamical framework of the spatial circular restricted 3-body problem and later extended to the elliptic restricted 4-body problem.
The final chapters are related to low Earth orbit missions, presenting the development of a purely magnetic attitude determination and control systems, suitable for implementation as a backup solution on CubeSats. Attitude control allows detumbling and pointing towards the magnetic field. At the same time, attitude determination is obtained from the only measurements of a three-axis magnetometer and a model of the geomagnetic field, without implementing any sophisticate filtering solution. To enhance the computational efficiency of the system, complex matrix operations are arranged into a form of the Faddeev algorithm, which can be conveniently implemented on the field programmable gate array core of a CubeSat on-board computer using systolic array architecture. The performance and the robustness of the algorithm are evaluated by means of both numerical analyses in Matlab Simulink and hardware-in-the-loop simulations in a Helmholtz cage facility.

... They also demonstrated the applications in geosynchronous orbit deorbiting and lunar relay satellite system. Qu et al. (2017) have designed an open-loop control law to guide the spacecraft to escape from Earth till it is captured by the Moon. They used a two-dimensional search strategy to find the ON/OFF time of the low-thrust engine during its Earth-escaping and Moon-capture phases. ...

Low-energy transfers (LET) for lunar and interplanetary missions has received immense attention of the scientific community during the last few decades as its importance was understood by the success of JAXA’s Hiten, ESA’s SMART-1, NASA’s GRAIL and ARTHEMIS missions, and several proposals, like the BepiColombo, Multi-Moon Orbiter and Europa Orbiter. In this paper the developments in the area of LET and low-thrust trajectories are reviewed. Starting with the basics of the restricted three-body problem and its use in finding invariant manifolds in phase space, the design of LET trajectories and optimisation methods used to find optimal LET and low-thrust transfers is discussed.