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The problem is addressed of transferring a spacecraft from a low Earth to a low lunar orbit in a planar circular restricted three-body framework. A closed-form approximate expression for the total velocity variation is developed under the assumption of minimum Delta V biimpulsive maneuvers. This approximation quantifies the link between the transfe...
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... T (O; x, y) be an orthogonal reference frame rotating with the two primaries, with the origin in the center of mass of the system and x axis pointing from Earth to moon. It is assumed (Fig. 1) that the coordinates of M ⊕ are (−µ, 0) and the coordinates of M c ...
Context 2
... ρ ⊕ and ρ are the distances of the spacecraft from the Earth and the moon. Referring to Fig. 1, one ...
Context 3
... is nearly identical to the value obtained by Yamakawa et al. 3,10 using a WSB approach. Also, for J t = −2.67, a reduction of the mission time of 39% (when compared to the result by Belbruno and Miller 2 ) is obtained at the expense of an increase of only 0.0227 DU/TU (23.2 m/s) in the V tot . The corresponding ballistic trajectory is shown in Fig. ...
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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...
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Citations
... m/s and the transfer time t = 255.5 d. Again assuming the same two target orbits, Mengali et al. [82] developed a two-impulse optimization method. First, a closed-form approximate expression for the total velocity variation was deduced under the assumption of minimum ΔV biimpulse maneuver. ...
The Moon is the only celestial body that human beings have visited. The design of the Earth-Moon transfer orbits is a critical issue in lunar exploration missions. In the 21st century, new lunar missions including the construction of the lunar space station, the permanent lunar base, and the Earth-Moon transportation network have been proposed, requiring low-cost, expansive launch windows and a fixed arrival epoch for any launch date within the launch window. The low-energy and low-thrust transfers are promising strategies to satisfy the demands. This review provides a detailed landscape of Earth-Moon transfer trajectory design processes, from the traditional patched conic to the state-of-the-art low-energy and low-thrust methods. Essential mechanisms of the various utilized dynamic models and the characteristics of the different design methods are discussed in hopes of helping readers grasp the basic overview of the current Earth-Moon transfer orbit design methods and a deep academic background is unnecessary for the context understanding.
... Earth/Moon transfers have recently attracted the interest of the scientific community following the work done by [42]. This transfer can be done through the use of low thrust maneuvers [35,12,11] or two-impulse thrust, a Hohmann maneuver, solved via the patched restricted three-body problem [6,34], full ephemeris n-body problem [2,21], restricted three body-problem [36,48,44,23,6], restricted fourbody problem [1], and bi-circular restricted four-body problem [50,49,24,5,26,25,45,33,31,40,32]. ...
... This type of transfer is largely used to compare fuel cost efficiency ∆V and time of flight [36,2,4,37,45,32]. The PCR3BP is used with hybrid genetic algorithms and deterministic simplex methods to solve the boundary-value problem and obtain the data in [23]. The PCR3BP is also used with a computer software called AUTO to obtain the data in [48]. ...
This study applies a new approach, the Theory of Functional Connections (TFC), to solve the two-point boundary-value problem (TPBVP) in non-Keplerian orbit transfer. The perturbations considered are drag, solar radiation pressure, higher-order gravitational potential harmonic terms, and multiple bodies. The proposed approach is applied to Earth-to-Moon transfers, and obtains exact boundary condition satisfaction and with very fast convergence. Thanks to this highly efficient approach, perturbed pork-chop plots of Earth-to-Moon transfers are generated, and individual analyses on the transfers' parameters are easily done at low computational costs. The minimum fuel analysis is provided in terms of the time of flight, thrust application points, and relative geometry of the Moon and Sun. The transfer costs obtained are in agreement with the literature's best solutions, and in some cases are even slightly better.
... Besides, the flight time of the WSB transfer in is about 140 days, which is approximately 66 days longer than the optimal transfer in this paper. Comparing the optimal transfer in this paper and the optimal result in work (Mengali and Quarta 2005), it can be found that the optimal transfer (i.e., transfer 7 in Table 4) can save about 33.6 m/s of total cost and 11 days of flight time. Furthermore, the optimal transfer in the paper (Mingotti et al. 2012a) needs less cost but longer flight time than transfer 7 in Table 4. On the other hand, from Table 4, we can note that the total cost of the optimal transfer (i.e., transfer 7) is only 1.79 m/s smaller than that of the suboptimal transfer (i.e., transfer 6). ...
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.
... Earth/Moon transfers have recently attracted the interest of the scientific community following the work done by [42]. This transfer can be done through the use of low thrust maneuvers [35,12,11] or two-impulse thrust, a Hohmann maneuver, solved via the patched restricted three-body problem [6,34], full ephemeris n-body problem [2,21], restricted three body-problem [36,48,44,23,6], restricted fourbody problem [1], and bi-circular restricted four-body problem [50,49,24,5,26,25,45,33,31,40,32]. ...
... This type of transfer is largely used to compare fuel cost efficiency ∆V and time of flight [36,2,4,37,45,32]. The PCR3BP is used with hybrid genetic algorithms and deterministic simplex methods to solve the boundary-value problem and obtain the data in [23]. The PCR3BP is also used with a computer software called AUTO to obtain the data in [48]. ...
This study applies a new approach, the Theory of Functional Connections (TFC), to solve the two-point boundary-value problem (TPBVP) in non-Keplerian orbit transfer. The perturbations considered are drag, solar radiation pressure, higher-order gravitational potential harmonic terms, and multiple bodies. The proposed approach is applied to Earth-to-Moon transfers and obtains exact boundary condition satisfaction and with very fast convergence. Thanks to this highly efficient approach, perturbed pork-chop plots of Earth-to-Moon transfers are generated, and individual analyses on the transfers’ parameters are easily done at low computational costs. The minimum fuel analysis is provided in terms of the time of flight, thrust application points, and relative geometry of the Moon and Sun. The transfer costs obtained are in agreement with the literature’s best solutions and in some cases are even slightly better.
... Since the work done by Poincaré [1], the three body problem has been largely used to describe astrodynamics problems [2,3]. In particular, the restricted three body problem has been used to evaluate V costs for Earth-Moon transfers, as can be seen in several works available in the literature [4,5,6,7]. Investigating these transfers has been done both through the use of low thrust [8,9,10] and two-impulse type thrust [11,12,13]. Additionally, a more complete planar bi-circular 5 restricted four-body model including the Sun's influence has been a topic of study [14,15,16]. ...
... When compared with the uniform distribution, the collocation point distribution results in a much 55 slower increase of the condition number of the matrix to be inverted in the least-squares as the number of basis functions, m, increases. The collocation points can be realized in the problem domain through the relationship provided in Eq. (6). ...
Fuel consumption and time of flight are crucial information for mission design. In this paper, adopting a two-impulse maneuver, we analyze the fuel consumption through evaluations of the equivalent V for several transfers in the Earth-Moon system as function of time of flight and other parameters, like the points of application of the thrusts. Transfer costs from a LEO to the Lagrangian point L 1 are analyzed as functions of the departure position and the time of flight. Transfers from a near-Earth orbit to a near-Moon orbit are also analyzed. The influence of perturbations due to the gravitational attraction of the Sun is also investigated. These problems involve specific constraints, i.e. the initial and final positions are given, but the initial and final velocities are unknown. These are boundary value problems that are here efficiently solved using the Theory of Functional Connections. This yields to the construction of a 3-body porkchop, where the V costs can be obtained in terms of the boundary values.
... trajectories to different types of orbits in the Earth-Moon system. The most often studied ones are the Earth-Moon transfer trajectories, from direct transfers (Mengali and Quarta, 2005) to low-energy ones (Belbruno and Miller, 1993;Koon et al., 2000;Parker and Anderson, 2013;Topputo, 2013), and free-return ones (Schwaniger, 1963;Miele and Mancuso, 2001;Hou et al., 2013). Except for the space close to the Moon, some researchers already noticed that libration point orbits (Zhang and Hou, 2015;Lei and Xu, 2018), distant retrograde orbits (Scott and Spencer, 2010;Tan et al., 2014;Capdevila et al., 2014;Welch et al., 2015), and resonant orbits (Vaquero and Howell, 2014) are also potential candidate orbits for future space missions. ...
China’s Chang’E-4 (CE-4) probe has started its exploration on the far side of the Moon thanks to the relay satellite---Queqiao---which is around the Earth-Moon collinear libration point L2 and provides continuous communication links between the lander and ground stations. The triangular libration points of the Earth-Moon system, L4 and L5, have long been considered as potential locations for future space applications. As a possible extension of Queqiao or other future missions around L2, the work presented here contributes to constructing transfers from the collinear libration point (L2) to the triangular libration points. Taking the L4 point as an example, two types of two-maneuver transfers from a planar Lyapunov orbit around L2 to the periodic orbit around L4 are investigated in the planar Earth-Moon system and compared with each other, including the type-I transfer utilizing invariant manifolds along with powered lunar gravity assist and the type-II transfer using the Jacobi integral. Our studies indicate that type-I transfer saves more Δv cost than type-II transfer. For the cases studied by us, the minimum Δv cost is about 0.1636 km/s for type-I transfer, with a corresponding transfer time of 169.84 days. For type-II transfer and the cases studied by us, the corresponding values are 0.4745 km/s and 137.19 days. At the end of the paper, the type-I transfer is generalized (with some modifications) to the ephemeris model and some example transfer trajectories are given. Although only the transfer from L2 to L4 is studied in this paper, the same orbit design strategy can be generalized to the transfer to the triangular libration point L5, and the transfers from the collinear libration point L1 to triangular libration points.
... days of transfer duration via the double twobody hypothesis [15]. Mengali and Quarta proposed a method for designing the two-impulse trans-lunar orbit based on CR3BP [16]. Assadian and Pourtakdoust calculated the Pareto set of the two-impulse delta-v and transfer duration of the trans-lunar orbit by applying the multi-objective genetic algorithm [17]. ...
... The paths of the two local optimal trans-lunar orbits in family A are shown in Fig. 8 (in the rotating frame) and Fig. 9 (in the Earth-centered inertial frame; the Appendix provides the transformation mentioned by Topputo [16], and it is also equally applicable to the families B and C in the next sections). The common feature of the paths of family A is that the path of the trans-lunar orbit does not fly around the Earth. ...
The solution set of the Sun-perturbed optimal two-impulse trans-lunar orbit is helpful for overall optimization of the lunar exploration mission. A model for computing the two-impulse trans-lunar orbit, which strictly satisfies the boundary constraints, is established. The solution set is computed first with a circular restricted three-body model using a generalized local gradient optimization algorithm and the strategy of design variable initial continuation. By taking the solution set of a circular restricted three-body model as the initial values of the design variables, the Sun-perturbed solution set is calculated based on the dynamic model continuation theory and traversal search methodology. A comparative analysis shows that the fuel cost may be reduced to some extent by considering the Sun’s perturbation and choosing an appropriate transfer window. Moreover, there are several optimal two-impulse trans-lunar methods for supporting a lunar mission to select a scenario with a certain ground measurement and to control the time cost. A fitted linear dependence relationship between the Sun’s befitting phase and the trans-lunar duration could thus provide a reference to select a low-fuel-cost trans-lunar injection window in an engineering project.
... Table 2 lists the locations and Jacobi integrals of the collinear points L i , i = 1, 2, 3. As shown in Fig. 1(a), for J < C 1 , motion between the Earth and the Moon is feasible (Mengali and Quarta 2005). Note that the state vectors and dynamics equation given above are discussed in the Earth-Moon rotating frame. ...
A cislunar cargo spacecraft with low-thrust propulsion traveling between the Earth and the Moon is essential for sustainable, long-term manned lunar exploration. In low-thrust Earth-Moon transfer (LTEMT), lunar capture is the primary prerequisite for spacecraft subject to the circular restricted three-body model. Therefore, this study identifies sufficient conditions for lunar capture, which are determined by the Jacobi integral and Hill's region. This paper proposes a guidance scheme that includes thrust direction, thrust efficiency, and a five-stage flight control sequence based on the variation of the Jacobi integral. The LTEMT problem is then converted to an initial value problem of a differential equation with three parameters. Lunar capture set theories (LCSTs), which are convenient for identifying lunar capture sets, are presented and proved according to the continuous properties of the ordinary differential equation. Finally, the solutions of the LTEMT trajectories departing from a geosynchronous orbit with an altitude of approximately 35,827 km are discussed for different thrust accelerations and cutoff values of the thrust efficiency. The ro-bustness is analyzed assuming that navigation and switching time errors are present to demonstrate the adaptability of this method. The results reveal that the proposed guidance scheme and LCSTs can provide technical support for future cislunar cargo missions.
... In addition, discussion on relaxation 100 of frozen conditions and behavior of the resulting quasi-frozen orbits (QFOs) along with quantification of station-keeping propulsive requirements is not presented in these works. These ideas can be extended to identify target orbits for mission design purposes using low-thrust electric [26,27,28] or chemical rocket propulsion systems [29] wherein cooperating with natural perturbed motions 105 might be expected to minimize station-keeping requirements [30,31]. Finding a set of theoretical QFOs is the first step for an envisioned polar imaging mission, however, for eLLO further exploration is desired to find the most stable subset of these orbits, including a high-fidelity force model. ...
Designing long-duration lunar orbiter missions is challenging due to the Moon’s highly nonlinear gravity field and the third-body perturbations induced by the Earth, Sun and other large bodies. The absence of a Lunar atmosphere has offered the possibility for mission designers to search for extremely low-altitude, quasi-stable lunar orbits. In addition to the reduced amount of propellant required for station-keeping maneuvers, these orbits present great opportunities
for unique scientific studies such as high resolution imaging and characterization of the polar ice deposits in deep craters. Prior to the GRAIL mission, mission planning for Lunar orbiters had suffered from inaccuracies, mainly due to the lack of an accurate Lunar gravity model, which resulted in severe deviations with respect to the spacecraft’s nominal orbit. We study station-keeping feasibility for spacecraft in near-polar and extremely low-altitude, quasi-frozen orbits
around the Moon, that are perturbed by a high-fidelity lunar gravity model and third-body effects from the Earth and Sun. For several candidate orbits, we compare the trade-space between mission duration and ∆V budget, considering impulsive maneuvers applied once every ‘N ∈ {2,6,10,14,18}’ orbits at periselene or aposelene. Additionally, we investigate the propulsive cost for different orbit insertion dates, the location of impulsive corrections for arresting
argument of periselene (ω) drift, and controlling periselene altitude.
... In addition to this, discussion on relaxation of frozen conditions and behavior of the resulting quasi-frozen orbits along with quantification of requisite station-keeping effort is lacking in these works. This analysis is also useful for identifying target orbits for mission design purposes using low-thrust electric [21][22][23] or chemical rocket propulsion systems [24,25]. ...
Design of long-duration lunar orbiter missions is challenging due to the Moon's highly non-linear gravity field and third-body perturbations induced by the Earth, Sun and other large bodies, on the orbiting spacecraft. The absence of a Lunar atmosphere, and hence the lack of orbital atmospheric drag, has encouraged mission designers to search for extremely low-altitude, stable, lunar orbits. In addition to the reduced amount of propellant required for station-keeping maneuvers, these orbits present great opportunities for unique scientific studies such as high resolution imaging and characterization of the polar ice deposits in deep craters. Mission planning for Lunar orbiters has historically suffered from inaccuracies, mainly due to the lack of an accurate Lunar gravity model, which resulted in severe deviations with respect to the spacecraft's nominal orbit. In 2012, JPL's Gravity Recovery and Interior Laboratory (GRAIL) mission mapped the Moon's gravity field with much improved accuracy, allowing future missions to be designed and flown with far better models. In this paper, we perform a station-keeping feasibility study for quasi-frozen, near-polar and extremely low-altitude orbits around the Moon with a high-fidelity lunar gravity model and when perturbations due to the Earth and Sun are taken into consideration. We study the trade-space between mission duration and ∆V budget considering impulsive maneuvers applied once every 3, 5 or 10 orbits.
