International Journal of Aerospace Engineering

Volume 2018, Article ID 4560173, 17 pages

https://doi.org/10.1155/2018/4560173

## Enhanced Hybrid Differential Evolution for Earth-Moon Low-Energy Transfer Trajectory Optimization

^{1}School of Computer, China University of Geosciences, No. 388 LuMo Road, Hongshan District, Wuhan, China^{2}Hubei Key Laboratory of Intelligent Geo-Information Processing, China University of Geosciences, Wuhan, China

Correspondence should be addressed to Lei Peng; nc.ude.guc@gnep.iel

Received 27 November 2017; Revised 4 February 2018; Accepted 5 March 2018; Published 3 May 2018

Academic Editor: Christian Circi

Copyright © 2018 Yanyun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is known that the optimization of the Earth-Moon low-energy transfer trajectory is extremely sensitive with the initial condition chosen to search. In order to find the proper initial parameter values of Earth-Moon low-energy transfer trajectory faster and obtain more accurate solutions with high stability, in this paper, an efficient hybridized differential evolution (DE) algorithm with a mix reinitialization strategy (DEMR) is presented. The mix reinitialization strategy is implemented based on a set of archived superior solutions to ensure both the search efficiency and the reliability for the optimization problem. And by using DE as the global optimizer, DEMR can optimize the Earth-Moon low-energy transfer trajectory without knowing an exact initial condition. To further validate the performance of DEMR, experiments on benchmark functions have also been done. Compared with peer algorithms on both the Earth-Moon low-energy transfer problem and benchmark functions, DEMR can obtain relatively better results in terms of the quality of the final solutions, robustness, and convergence speed.

#### 1. Introduction

Deep space exploration, as a very important hotspot in space exploration, has attracted attention of various countries, while as the first step to explore the deep space, the lunar exploration has been receiving renewed attention, such as the NASA’s LADEE, ARTEMIS, and GRAIL missions and CNSA’s Chang’e 3 mission. To carry out lunar exploration successfully, the first priority is the Earth-Moon transfer trajectory design. The traditional way to construct the transfer trajectory is Hohmann transfer [1], which utilizes the two-body system dynamics model, and it was later proven optimal analytically by Barrar in [2]. Short as the duration of the Hohmann transfer orbit is, the fuel consumption, however, is high. And for a long-term unmanned space mission, it is essential to consider the increase of scientific loads and the reduction of the energy consumptions at the same time. Low-cost alternative methods rather than conventional methods have been explored as the knowledge of the solution space is expanded, and algorithms that employ the dynamical relationships are developed.

With the discovery of new types of particular solutions in the three-body problem, there has been accelerated interest in missions utilizing trajectories near libration points, and a number of missions have already incorporated the periodic halo orbits and/or quasiperiodic Lissajous trajectories as a part of the trajectory design, such as ISEE-3 (1978 launch), WIND (1994 launch), SOHO (1995 launch), and ACE (1997 launch). And for the trajectory design of a liberation point mission, dynamical systems theory was applied by Barden and Howell to better understand the geometry of the phase space in the three-body problem via stable and unstable manifolds [3]. In later years, low-energy trajectory designing based on hyperbolic invariant manifolds has also been proposed in [4–6].

Besides, in 1987, as a different methodology, the notion of a weak stability boundary (WSB) was first introduced heuristically by Belbruno for designing fuel-efficient space missions [7]. After that, a low-energy Earth-Moon transfer trajectory from the Earth to the Moon leading to ballistic capture was constructed based on WSB by Belbruno and Miller in [8], which used approximately 18% less cost in total than the Hohmann transfer to insert a spacecraft into a circular orbit about the Moon. In their work, it is presented that in the framework of the circular restricted three-body problem (CR3BP), a WSB region exists in the phase space near the Moon in the Earth-Moon system, where the spacecraft will be captured by the Moon’s gravity into lunar orbit requiring little or no fuel (called ballistic capture). The use of this transfer was firstly demonstrated by Japan’s Hiten spacecraft [9], and thus, an important role of WSB is to design the similar ballistic capture for space missions. Thereafter, Belbruno and Carrico [10], Belló et al. [11], and Belbruno [12] did further research on WSB gradually, which enhanced the theoretical basis of this kind of low-energy transfer. What is more, it has also been suggested that the hyperbolic invariant manifold method can be used to explain the trajectories obtained through the WSB method in some cases, while WSB exists even in models where the hyperbolic invariant manifolds are no longer well defined. It seems possible that the WSB may turn out to provide a good substitute for the hyperbolic invariant manifolds in such models [13].

In the past few years, varieties of Earth-Moon low-energy trajectories incorporating new concepts have been developed, such as the low-energy trajectory only in the three-body Earth-Moon system, corresponding to the interior capture around the Moon [14], and the low-energy trajectory which considers the gravity of the Sun and is also based on the concept of the WSB theory, corresponding to the so-called exterior WSB transfer [8]. The Earth-Moon low-energy transfer trajectory discussed in this paper belongs to the exterior WSB transfer. This type of low-energy transfer trajectory is mainly achieved by taking advantage of the invariant manifold structures associated with the 3D halo orbits (or 2D Lyapunov orbits) in the vicinity of the libration points in the Sun-Earth system and Earth-Moon system and can be modelled as two coupled CR3BPs. In fact, the Japanese Hiten mission was a paradigmatic example for a class of low-energy Earth-to-Moon orbits obtained by considering the gravitational effects of the Earth, the Moon, and the Sun on the motion of the spacecraft simultaneously [8, 9, 14, 15] and was later found to be related to the hyperbolic invariant manifolds of the CR3BP by employing the patched three-body approximation [5, 16, 17]. Gómez et al. [18] had also presented the use of the restricted three-body problem’s invariant manifold structural to design the low-energy transfer orbit in detail. And de Sousa-Silva and Terra [19] presented a survey of preliminary Earth-to-Moon transfers in the patched three-body approach, using the planar case of the CR3BP.

In general, there are two kinds of numerical optimization methods to deal with the trajectory designing, namely, the deterministic methods and the stochastic methods [20]. For the deterministic methods, they are local methods in nature and a suitable initial solution in the region of convergence, which is difficult to obtain for the space trajectory problem in the real system, is also needed [21]. For example, for the design of the Earth-Moon low-energy transfer trajectory based on the patched three-body approximation, the setting of the initial integral point in the Poincaré section of the unstable manifold of the Lyapunov orbit around the Sun-Earth L2 and the stable manifold of the Lyapunov orbit around the Earth-Moon L2 is sensitive to the traditional deterministic methods, which may easily cause the inefficiency of the traditional deterministic methods. To overcome these difficulties, stochastic methods, which do not require initial solution and are global optimization methods, are usually adopted.

Recently, evolutionary algorithm- (EA-) based optimization techniques for space trajectory design have received significant attention. As a branch of EAs, genetic algorithm (GA) has been proposed by several authors [22–25] for trajectory optimization. However, drawbacks (e.g., low speed, premature convergence, and degradation for highly interactive fitness functions) have been observed for GAs. Topputo et al. [26] proposed a hybridization of EAs with sequential quadratic programming method to look for the best intersection between invariant manifolds associated to the departure and arrival Sun-Planet-Spacecraft systems. By extension of evolutionary neurocontrol (ENC) to the planetary case of an Earth-Moon transfer, Ohndorf et al. [27] showed that automatic optimization of low-thrust trajectories crossing the boundary of the spheres of influence between the involved celestial bodies was feasible with ENC. The particle swarm optimization (PSO) algorithm has also been applied in the field of space trajectory widely. In [28], PSO was applied to a variety of space trajectory optimization problem, including the determination of periodic orbits in the context of the CR3BP and the optimization of (impulsive and finite thrust) orbital transfers, which was proved to be quite effective in finding the optimal solution to all of the applications. While in [29], a further study about the optimization problem of impulsive orbital transfer was proposed. Zotes and Peñas [30] used the swarm algorithms to optimize the tuning parameters of the trajectory in both the single-criteria case and multicriteria cases. In the single-criteria case, only the fuel consumption was considered as a variable to be minimized, while in the multicriteria case, the fuel consumption and the total time of the mission were simultaneously taken into account. Although PSO is easy to implement, and the convergent speed is fast, it still easily traps in a local optimum [31].

Differential evolution (DE) [32] is a simple yet powerful evolutionary algorithm developed by Storn and Price for global continuous optimization and has been successfully used in a variety of domains [33, 34], as well as the trajectory optimization. In 2007, an improved DE algorithm was applied by Lei et al. [21] into two-impulse transfer trajectory problem. In 2011, Attia [35] presented an adaptive probability of crossover technique as a variation of the differential evolution algorithm, for optimal parameter estimation in the general curve-fitting problem. And the technique was successfully applied to the determination of orbital elements of a spectroscopic binary system (eta Bootis). In 2015, Lu et al. [36] used DE to optimize the parameters describing the patched manifolds of low-energy transfer orbit to Mars with multibody environment effectively. In 2016, Nath and Ramanan [37] developed alternative single-level schemes for the mission design to a halo orbit around the libration points from the Earth based on DE. And this DE-based scheme for the transfer trajectory identified the precise location on the halo orbit that needs minimum energy for insertion and avoided exploration of multiple points. In 2017, in [38], a multistart DE enhanced with a deflection strategy with strong global exploration and bypassing abilities was adopted as a search engine to find multiple globally optimal regions in which potential periodic orbits (POs) were located.

The motivations and contributions of this paper are as follows: (1)The Earth-Moon low-energy transfer trajectory is extremely sensitive with the changes of the initial condition in the Poincaré section, and with a small change of the initial condition (such as a small change in velocity at a fixed point), the destination of an orbit can be changed dramatically. So proper choices of the initial parameter values are key to the optimization of the Earth-Moon low-energy transfer trajectory.(2)As a powerful global search algorithm, DE possesses several unique features such as simple implementation and low complexity, while its performance is limited because of the slow converging at the late stage of evolution. Given this reason, the proposed algorithm combined DE with a local search sequential quadratic programming (SQP) to enhance the local search ability of DE and finally improve the search accuracy.(3)An extra storage is used in DEMR to collect the superior solutions, and it is associated with the reinitialization of the population later. The reinitialization procedure is triggered when the local search fails to find a better solution than DE. And the procedure includes both the random reinitialization and the biased reinitialization based on the extra storage.

To evaluate the performance of the proposed approach in the optimization of Earth-Moon low-energy transfer and the performance of the proposed algorithm itself, experiments on Earth-Moon low-energy transfer problem both in the two-dimensional space and three-dimensional space, as well as the experiments on the benchmark CEC2005 are conducted, and the results are compared with several peer algorithms.

The rest of this study is organized as follows. In Section 2, the Earth-Moon low-energy transfer trajectory model and objective function used in this work are briefly introduced. Section 3 presents the proposed DEMR method in detail, followed by the experimental results and analysis in Section 4. Lastly, in Section 5, this work is concluded, and several possible avenues of future work are discussed.

#### 2. Formula and Optimization Model

##### 2.1. Formula about CR3BP

###### 2.1.1. Formula in Two-Dimensional Space

The planar CR3BP describes the motion of a spacecraft under the gravitational attractions of two primaries and [39]. Choose a rotating coordinate system so that the origin is at the center of mass between and . See Figure 1.