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The quasi-periodic halo orbit and halo orbit are depicted: (a) quasi-periodic halo orbit in the proposed model shown in GRC; (b) quasi-periodic halo orbit in the proposed model shown in LRC; (c) halo orbit in CRTP model shown in GRC.
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Quasi-periodic orbits in the Earth–Moon system are highly sensitive and even small errors in position and/or velocity have strong influence on the trajectory. These types of trajectories are unstable and must be controlled to maintain the corresponding orbits. This paper investigates the stationkeeping strategy for quasi-periodic orbits near the tr...
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... viewed in the time- dependent libration point centered rotating coordinate system, those orbits are still very similar to the orbits in restricted three-body problem. For example, in Figure 3(a), the trajectories are viewed from GRC. The line denotes the track of Moon's center and the curve denotes a quasi-periodic halo orbit. ...
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... Moon's center moves along the line connecting the Earth and the Moon and the halo orbit loses its peri- odicity under the influences of the Sun and the eccen- tricity of the Moon. However, the same halo orbit viewed from LRC, as shown in Figure 3(b), is still very similar to the orbit in restricted three-body prob- lem, as shown in Figure 3(c). This phenomenon indi- cates that the major reason for the orbit to lose its periodicity in x-z plane is due to the movement of the L 2 point which is no longer an inertial point but a time- dependent point moving along the line connecting the Earth and Moon. ...
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... Moon's center moves along the line connecting the Earth and the Moon and the halo orbit loses its peri- odicity under the influences of the Sun and the eccen- tricity of the Moon. However, the same halo orbit viewed from LRC, as shown in Figure 3(b), is still very similar to the orbit in restricted three-body prob- lem, as shown in Figure 3(c). This phenomenon indi- cates that the major reason for the orbit to lose its periodicity in x-z plane is due to the movement of the L 2 point which is no longer an inertial point but a time- dependent point moving along the line connecting the Earth and Moon. ...
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Citations
... Later, in the series of the papers Zotos [45][46][47][48][49][50], Zotos et al. [51], Zotos and Jung [52], Zotos and Nagler [53], Zotos et al. [54], Zotos et al. [55], Zotos et al. [56] and Zotos et al. [57] have studied the dynamics of the orbits of a test particle in the R3BP, where they have defined bounded, escape and collisional motion. Qian et al. [33] have investigated the station-keeping strategy for quasi-periodic orbits near the translunar libration point with full consideration of the contribution from the Sun's gravity and the Moon's eccentricity. Qian et al. [34] have analyzed the mechanism of the evolution of the probe orbital energy and developed a design method for the low-energy escaping orbits in the Earth-Moon system. ...
In this paper, we have drawn the periodic orbits in the restricted four-body problem. The fourth body of infinitesimal mass moves under the Newtonian gravitational law due to three mass points (known as primaries) in the collinear Euler’s configuration. We have studied the dynamics of the fourth body in the spatial collinear restricted four-body problem (SCR4BP) with non-spherical primaries. We have determined the periodic orbits by using the Fourier series method only around collinear libration points in SCR4BP with non-spherical primaries. The effects of the orbital parameter \(\varepsilon \) (\(0<\varepsilon <1\)), which controls the size of the orbit, on the periodic orbits and their period are investigated. In order to study the effect of this parameter, we have retained the terms up to third order with respect to the orbital dimensions. We have also drawn the variational graphs to determine the variation in the time period T of the periodic orbits. Moreover, we have unveiled that how oblateness and mass parameters influence the shape, size, and period of the periodic orbits.
... Since libration point orbits in cislunar space are highly unstable, a spacecraft moving on these orbits must use control input to remain close to their nominal orbits. A great variety of studies have investigated methods for stabilizing the unstable periodic orbits in the circular restricted three-body problem (CR3BP) (Gomez et al., 1998;Farquhar et al., 2004;Folta et al., 2014;Ulybyshev, 2014;Xu et al., 2016;Qian et al., 2018). Floquet mode control has been applied for stationkeeping of libration point orbits which eliminates the local unstable components by an impulsive maneuver (Simo et al., 1987). ...
This paper studies a control law to achieve formation flying in cislunar space. Utilizing the eigenstructure of the linearized flow around a libration point of the Earth-Moon circular restricted three-body problem, the fuel efficient formation flying controller based on the chattering attenuation sliding mode controller is designed and analyzed. Numerical studies are conducted for the Earth-Moon L 2 point and a halo orbit around it. The total velocity change required to achieve formation as well as to maintain the orbit is calculated. Simulation results show that the chattering attenuation sliding mode controller has good performance and robustness in the presence of unmodeled nonlinearity along the halo orbit with moderate fuel consumption.
... Folta et al. [11] summarized the results of ARTEMIS mission operations and used the optimal continuation strategy to obtain operational stationkeeping results, which were also compared to orbit stability information. Qian et al. [12] investigated a numerical method that constructs perturbed halo/Lissajous orbits in a full solar system model, which is based on an improved multiple-shooting strategy that utilizes the initial conditions with consideration of the instantaneous state of libration points for each integral arc. Qian et al. [13] also proposed a new continuous low-thrust-based stationkeeping strategy for quasiperiodic orbits around the Earth-Moon L 2 point in a high-fidelity model. ...
In this paper, a modified targeting strategy is developed for missions on libration point orbits (LPOs) in the real Earth-Moon system. In order to simulate a station-keeping procedure in a dynamic model as realistic as possible, LPOs generated in the circular restricted three-body problem (CRTBP) are discretized and reconverged in a geocentric inertial system for later simulations. After that, based on the dynamic property of the state transition matrix, a modified strategy of selecting target points for station-keeping is presented to reduce maneuver costs. By considering both the solar gravity and radiation pressure in a nominal LPO design, station-keeping simulations about fuel consumption for real LPOs around both collinear and triangular libration points are performed in a high-fidelity ephemeris model. Results show the effectivity of the modified strategy with total maneuver costs reduced by greater than 10% for maintaining triangular LPOs.
... Instead, perturbed Halo/Lissajous orbits may dominate considering the two center manifolds and strong external perturbations. [6][7][8][9][10] This research effort focuses on the development of an efficient construction method to best satisfy future mission requirements in these complex dynamical regimes. ...
An improved numerical method that can construct Halo/Lissajous orbits in the vicinity of collinear libration points in a full solar system model is investigated. A full solar system gravitational model in the geocentric rotating coordinate system with a clear presentation of the angular velocity relative to the inertial coordinate system is proposed. An alternative way to determine patch points in the multiple shooting method is provided based on a dynamical analysis with Poincaré sections. By employing the new patch points and sequential quadratic programming, Halo orbits for L1, L2, and L3 points as well as Lissajous orbits for L1 and L2 points in the Earth-Moon system are generated with the proposed full solar system gravitational model to verify the effectiveness of the proposed method.
... The most representative work was demonstrated by Richardson [18] who derived the third-order analytical solution of halo orbits around the collinear libration point in CRTBP with Lindstedt-Poincaré (L-P) method as well as Farquhar and Kamel [19]. The third-order analytical solutions of halo orbits are extensively used as initial condi-tions in mission trajectories designing numerical integration progress [20]. Erdi [21] and Zagouras [22] derived third-order and fourth-order solutions for the three-dimensional periodic orbits for triangular libration points with small parametric expansions, respectively. ...
... where ξ,ξ have been replaced by (u, v). Secondly, replacing η andη in (20) and (21) by equations in (15), including f 1 (u, η; v,η), the final equations with respect to u and v are obtained as ...
We propose a polynomial expansion method to investigate periodic motions around both collinear and triangular libration points in the planar circular restricted three-body problem. The polynomial expansion method focuses on the approximate nonlinear polynomial relations between the two directions in the motion plane, which provide an alternative way to inspect the periodic motions. Based on the nonlinear polynomial relations, the planar two-dimensional system is decoupled. As an example, the \(3\mathrm{rd}\) order analytical solutions for the periodic motion are determined by solving the decoupled system. The efficiency of the polynomial expansion method has been validated by numerical method.
... Although the analytical solutions are approximate, they provide the basis for further application in the mission trajectories design by more accurate numerical methods. [9][10][11] The traditional perturbation methods are based on the idea that assumed solutions of two or three directions are expressed in series expansions of some small parameters, such as frequency, amplitude, eccentricity, etc. To the authors' best knowledge, no periodic orbits investigations have focused on the nonlinear relations among the directions. ...
This paper revisits the planar periodic motions around libration points in circular restricted three-body problem based on invariant manifold technique. The invariant manifold technique is applied to construct the nonlinear polynomial relations between ξ-direction and η-direction of a small celestial body during its periodic motion. Such direct nonlinear relations reduce the dimension of the dynamical system and facilitate convenient approximate analytical solutions. The nonlinear directional relations also provide terminal constraints for computing periodic motions. The method to construst periodic orbits proposed in this study presents a new point of view to explore the orbital dynamics. As an application in numerical simulations, nonlinear relations are adopted as topological terminal constraints to construct the periodic orbits with differential correction procedure. Numerical examples verify the validity of the proposed method for both collinear and triangular libration cases.
... The analytical explicit solutions provide clear presentations of the period, phase and harmonics. The analytical techniques propose a way to survey the global dynamics of the trajectory and exhibit possible parameter design (Zhang et al. 2013;Qian et al. 2016). ...
Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The \(\zeta \)-component motion is treated as the dominant motion and the \(\xi\) and \(\eta \)-component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the \(\zeta \)-position and \(\zeta \)-velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on \(\zeta \)-direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.
... The most representative work was successfully demonstrated by Richardson [3] who derived the thirdorder analytical solution of halo orbits with small parametric expansions in CR3BP with Lindstedt-Poincaré (L-P) method. The third-order analytical solutions of halo orbits are extensively used as initial conditions in mission trajectories designing progress for numerical integration [4]. Similarly, Gómez and Marcote [5] solved a reduced case of the CR3BP, namely, the Clohessy-Wiltshire equation with L-P method. ...
The restricted three-body problem (R3BP) and restricted four-body problem (R4BP) are modeled based on the rotating frame. The conservative autonomous system for the R3BP and nonautonomous system with period parametric resonance due to the fourth body are derived. From the vibrational point of view, the methodology of polynomial series is proposed to solve for these problems analytically. By introducing the polynomial series relations among the three directions of motion, the three-degree-of-freedom coupled equations are transferred into one degree-of-freedom containing the full dynamics of the original autonomous system for the R3BP. As for the R4BP case, the methodology of polynomial series combined with the iterative approach is proposed. During the iterative approach, the nonautonomous system can be treated as pseudoautonomous equation and the final polynomial series relations and one-degree-of-freedom system can be derived iteratively.
... The design of a circumlunar station at Lagrange points (i.e. EML1/ EML2) [13] and development of technologies to reach and redirect asteroids for mining purposes [10] are currently favored strategies. ...
... In case of an emergency, it will be very difficult to implement a rescue mission at such a long distance (513,000 km). High radiation also adversely influences electronics and sensors, but, nevertheless there are already several scientific satellites that perform quite successfully in Sun/Earth L1 (WIND, SOHO, ACE, DSCOVR) [13]. ...
In recent years, a disruption-tolerant network (DTN) has attracted significant attention regarding lunar exploration, as they can be employed in various relay spacecraft, such as lunar orbiters, due to their excellent constant all-time coverage capability. An exclusive relay satellite with limited storage capacity and downlink communication bandwidth to Earth could be congested with huge amounts of return data from multiple sources on the lunar surface, resulting in low efficiency in the backhaul link. To address this issue, a network-coded forwarding scheme employing a Markov-decision-assisted bundle aggregation for a Halo-orbit Earth–lunar DTN is proposed in this article. Specifically, a network-coding-enabled store-and-forward scheme is designed for the downlink relay delivery in the L2-point Halo orbit using a two-priority bundle-aggregation mechanism. A size-optimal bundle slicing-aggregation algorithm is developed for the two-source network coding scheme using the Markov decision algorithm. Simulation results demonstrate that the proposed mechanism exhibits better performance than existing delivery schemes in reducing end-to-end delivery latency. Thus, the throughput can be improved, and the arrival rate of high-priority bundles can be guaranteed.
