# Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints
## 1 Introduction  
As the only natural satellite of the Earth, the Moon is an important object for studying the formation and evolution of the Earth. Exploring the Moon not only can help human beings to further understand cosmic phenomena but can also promote the rapid development of human scientific and technological progress and applications [1, 2]. Human exploration of the Moon began in the 1950s [3] as an important part of deep space exploration [4, 5, 6, 7, 8, 9, 10, 11, 12]. The second round of the lunar exploration climax was revived in the 1990s after a quiet period. The ESA, China, Japan, and India have carried out lunar exploration activities one after another since then [13, 14, 15, 16, 17]. Chinese lunar exploration has developed rapidly over the recent decades. Chang'E-2, launched on 1 October 2010, obtained 3D images of the lunar surface [18], completed the expansion mission of entering the orbit of the Sun-Earth L2 libration point [19, 20], and flew by asteroid 4179 Toutatis at an extremely close distance [18, 21]. Chang'E-5 was launched from the Wenchang Satellite Launch Site on 24 November 2020 [22] and successfully returned to the ground with lunar samples 23 days later [23, 24]. Follow-up lunar exploration schemes have also been demonstrated. The determination of the opportunity and trajectory of the Moon-to-Earth return is the premise for the implementation of lunar sampling return missions or manned exploration missions. The manned lunar return has higher requirements for reliability and emergency response compared with unmanned missions. It should be seriously noted that the Moon-to-Earth transfer is not a simple inverse process of the Earth-to-Moon process, for the selection of initial values, model establishment, and constraints are evidently different [25, 26]. For the Earth-to-Moon trajectory design, researchers have developed multiple trajectory design technologies. The traditional impulse transfer is the classic method adopted by the first Moon exploration missions [27; 28], while the small thrust transfer [29; 30; 31; 32; 33; 34; 35] and the weak stable boundary (WSB) transfer methods [36; 37; 38] appeared in the 1990s. In recent years, there have been the use of Moon flybys to achieve libration point transfers and even the deployment of Earth's high-orbit satellites [39; 40]. For the design of the Moon-to-Earth return for direct atmosphere re-entry, [PERSON] et al. [41] obtained the transfer trajectory with matching landing field through two-level differential correction. [PERSON] and [PERSON] [42] adopted the analytical method to optimize the fastest return trajectory according to the re-entry state, regardless of the fuel consumption. [PERSON] et al. [43] established the solution model of landing opportunity and point return trajectory from the perspective of space geometry. [PERSON] et al. [44] studied the return opportunity strategy departing from the lunar and provided the initial value of the transfer trajectory by applying the improved patched-conic method. [PERSON] et al. [45] proposed a Moon return opportunity search strategy based on the patched-conic method. The return opportunity is initially selected according to the minimum re-entry angle, and the transfer trajectory is further obtained, while it is difficult to guess the state of the probe at the sphere of influence (SOI) of the Moon. In general, what the above methods obtain is the transfer opportunity with a single impulse of alignment at the positions of the Earth and the Moon. In addition, the simplified transfer, such as the patched-conic method and space geometry strategy, causes a low accuracy of opportunity and trajectory because of the lack of consideration for third-body perturbation, and ignoring any celestial body will lead to a large deviation between the designed and the actual opportunity or trajectory. The improvement of the calculation efficiency while ensuring the accuracy of the opportunity and the transfer trajectory are the keys to lunar return missions.