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.
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 [
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 [
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 [
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 [
In general, there are two kinds of numerical optimization methods to deal with the trajectory designing, namely, the deterministic methods and the stochastic methods [
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 [
Differential evolution (DE) [
The motivations and contributions of this paper are as follows:
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. 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. 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
The planar CR3BP describes the motion of a spacecraft under the gravitational attractions of two primaries
The planar CR3BP in the rotating frame.
Let
Extending the planar problem to a three-dimensional situation, the above equations can be expanded as follows:
The motion equations of the spacecraft in three-dimensional case with normalized coordinates:
The effective potential in three-dimensional case:
and
The design of the low-energy transition orbit on the two-dimensional plane is a problem in the four-dimensional phase space From the initial point, make the reverse integration along the unstable manifold associated with the Lyapunov orbits around the Sun-Earth L2 to the Earth parking orbit, completing the orbit design of the launch section. Adjust the current energy of the initial point and finally the energy of the initial point is the same as that of the current lunar Moon system. From the initial point, make the positive integration along the stable manifold associated with the Lyapunov orbits around the Earth-Moon L2 to the parking orbit of the Moon, completing the design of the capture section.
Figure
The manifolds and the low-energy transfer trajectory in the rotating frame.
To get the initial patched point, the position of Poincaré section is selected firstly. At present, the location of the Earth
Given both the location and speed in the
Similarly, the decision variable can also be expanded in the three-dimensional Earth-Moon problem as follows:
From the above discussion and the illustration of Figure Impulsive vector Impulsive vector Impulsive vector
The optimization objective is to optimize the total cost:
Differential evolution is a population-based optimization algorithm. In general, DE is used to optimize certain properties of a system by pertinently choosing the system parameters. For convenience, these parameters are usually described as a vector
DE works on a population of
1: Generate the initial population 2: 3: Evaluate the fitness 4: 5: 6: 7: Select 8: 9: 10: 11: 12: 13: 14: 15: 16: Evaluate the trial vector 17: 18: 19: 20: 21: 22:
The new algorithm proposed in this paper is called DEMR. DEMR employs standard DE as a global exploration heuristic in combination with SQP as a local search mechanism and a reinitialization scheme for enhancement.
In DEMR, local search is triggered according to two contraction criteria. One is the roughness in objective space, defined as
The other is the maximum distance in decision space:
1: Initialize population 2: 3: Evaluate the fitness 4: 5: 6: 7: Select 8: 9: Using DE/rand/1/bin to generate 10: 11: Using DE/best/1/exp to generate 12: 13: Evaluate the trial vector 14: 15: 16: 17: 18: Update 19: 20: Calculate the contraction criteria 21: 22: Pick up the 23: Update 24: 25: Re-Initialize the whole population as Algorithm 26: 27: 28:
1: 2: 3: Re-initialize the whole population randomly 4: 5: Calculate the average value 6: Re-initialize the population as: 7:
SQP methods represent the state of the art in nonlinear programming methods for constrained optimization, which were developed mainly by Han [
As DEMR advances, SQP will be triggered to do the further exploitation when the whole population is relatively concentrated in the decision space or the objective space. And DEMR enables the reinitialization of the population only when the local search does not improve the best individual in the population, which helps DEMR jump out from the attraction of local optimal solutions. In order to guarantee the diversity of the population as well as the convergence speed at the same time, a mixed reinitialization, shown in Algorithm
In this section, two groups of experiments are conducted, including the experiments with the two-dimensional planar CR3BP model and experiments in the three-dimensional space. In order to clarify the performance of DEMR in the Earth-Moon low-energy transfer trajectory optimization, comparisons have been made with peer algorithms, such as CMA-ES [
The low-energy transfer trajectory is from the 200 km Earth parking orbit to the 100 km Moon parking orbit in this work. The amplitude of Sun-Earth L2 Lyapunov orbit is 201,000 km and Earth-Moon L2 Lyapunov orbit is 15,000 km. The parameters of DEMR are set as follows (Table
Parameters in DEMR | Value |
---|---|
Crossover rate | |
Scale factor | |
Dimension | |
Population size | |
Contraction criteria | |
Reinitialization control | |
Maximum evaluations | MaxNFEs = 10,000 |
Storn and Price [
According to the analysis in [
As for the other algorithms used in the comparison, the corresponding parameter settings are the same in the original papers describing the algorithms.
According to some of our previous work, the ranges of decision variables set as below are more conducive to the optimization (Table
Decision variables | Range |
---|---|
Position in |
|
Velocity in |
|
Conversion angle |
Table
Results for Earth-Moon low-energy transfer trajectory optimization in two-dimensional space.
Algorithms | Best (m/s) | Rank | Worst (m/s) | Rank | Median (m/s) | Rank | Mean (m/s) | Rank | Std | Rank |
---|---|---|---|---|---|---|---|---|---|---|
SaDE | 3910.7 | 5 | 3919.4 | 2 | 3914.9 | 2 | 3914.5 | 2 | 2.2993 | 2 |
CMA-ES | 3908.4 | 2 | 4080.1 | 7 | 4013.9 | 7 | 4002.3 | 7 | 42.3711 | 7 |
GL-25 | 3908.9 | 3 | 3922.2 | 3 | 3915.2 | 3 | 3915.5 | 3 | 3.0661 | 3 |
LBBO | 3908.4 | 2 | 4007.0 | 6 | 3931.8 | 6 | 3933.0 | 6 | 26.3087 | 6 |
MPEDE | 3913.4 | 6 | 3934.8 | 5 | 3928.3 | 5 | 3926.3 | 5 | 5.7536 | 5 |
SHADE | 3910.4 | 4 | 3931.4 | 4 | 3919.8 | 4 | 3921.0 | 4 | 5.3871 | 4 |
DEMR |
The best value illustrates the maximum search accuracy that the algorithm can obtain on the Earth-Moon low-energy transfer trajectory designing to a certain extent. From Table
The worst value can briefly reflect the maximum deviation from the optimal solution. In this case, the local search algorithm CMA-ES shows great disadvantages since it is more likely to trap in the local optimal solution, while the algorithms SaDE, GL-25, SHADE, and DEMR have relatively better performance. It can be inferred that a simple local search algorithm does not necessarily have an advantage, even though the exploitation occupies a very important position for the Earth-Moon low-energy transfer optimizing. And the outstanding advantage of DEMR on the worst value also proves its reliability, when comparing with other algorithms. This is not only because DEMR adopts a powerful global optimizer, DE, just as SaDE and SHADE does, and more importantly, the mix reinitialization strategy employed by DEMR plays an important role in balancing the exploration and exploitation.
The good performances on both the mean value and standard deviation indicate DEMR can obtain a better value easily and owns a more stable optimization effect in Earth-Moon low-energy transfer trajectory optimization. And compared to the mean value, the median value excludes the influence of the extreme cases, such as the best value and the worst value, and shows the performance of an algorithm in the usual case. Normally, DEMR can achieve a greater advantage over other algorithms. According to test results for all evaluation indexes, DEMR achieves overall best optimization performance among seven algorithms. Hence, it is a good alternative for the optimizing of the Earth-Moon low-energy transfer trajectory.
To describe the convergence speeds of these 7 algorithms, 14 record points are set during the optimizing process for each algorithm, and when the evaluation number reaches any record point, the lowest total cost
The convergence graph of the algorithms on the Earth-Moon low-energy transfer trajectory optimization problem.
From Figure
Figure
The best Earth-Moon low-energy transfer trajectory optimized by DEMR.
According to the discussions above, DEMR can get a relatively better solution with a high stability. However, all of these results are obtained within a relatively small search space, which has high possibility to get a relatively good solution. So, to further argue the stability of the proposed algorithm on the design of Earth-Moon low-energy transfer trajectory, an extra group of experiments on a wider setting of the decision variables has been done. The widened ranges are as follows (Table
Decision variables | Range |
---|---|
Position in |
|
Velocity in |
|
Conversion angle |
The results with the widened ranges are shown in Table
Results for Earth-Moon low-energy transfer trajectory optimization in two-dimensional space with widened ranges.
Algorithms | Best (m/s) | Rank | Worst (m/s) | Rank | Median (m/s) | Rank | Mean (m/s) | Rank | Std | Rank |
---|---|---|---|---|---|---|---|---|---|---|
SaDE | 3912.0 | 3 | 3930.4 | 2 | 3917.8 | 2 | 3918.5 | 2 | 4.6163 | 2 |
CMA-ES | 3908.1 | 1 | 4416.8 | 6 | 3961.3 | 4 | 3997.0 | 6 | 125.0366 | 6 |
GL-25 | 3908.1 | 1 | 4190.5 | 5 | 3918.6 | 3 | 3947.3 | 3 | 62.4762 | 4 |
LBBO | 3908.6 | 2 | 4640.2 | 7 | 3984.5 | 6 | 4002.8 | 7 | 139.6852 | 7 |
MPEDE | 3934.0 | 5 | 4075.9 | 3 | 3992.5 | 7 | 3993.2 | 5 | 33.2864 | 3 |
SHADE | 3923.8 | 4 | 4185.6 | 4 | 3968.3 | 5 | 3990.8 | 4 | 63.9009 | 5 |
DEMR |
In this section, the optimization of the design for Earth-Moon low-energy transfer trajectory is extended to the three-dimensional space. To reflect the stability of different algorithms, the widened ranges of the decision variables in last section are used here and the ranges of the location and speed in the
Decision variables | Range |
---|---|
Position in |
|
Velocity in |
|
Position in |
|
Velocity in |
|
Conversion angle |
From Table
Results for Earth-Moon low-energy transfer trajectory optimization in three-dimensional space with widened ranges.
Algorithms | Best (m/s) | Rank | Worst (m/s) | Rank | Median (m/s) | Rank | Mean (m/s) | Rank | Std | Rank |
---|---|---|---|---|---|---|---|---|---|---|
SaDE | 3903.3 | 4 | 3904.4 | 2 | 3904.3 | 2 | ||||
CMA-ES | 147694.8 | 6 | 18581.0.0 | 7 | 4.0468 |
6 | ||||
GL-25 | 3903.6 | 5 | 12876.1 | 4 | 3905.2 | 3 | 4269.3 | 4 | 1.7931 |
4 |
LBBO | 3903.2 | 3 | 25788.9 | 6 | 3916.6 | 5 | 6679.8 | 6 | 4.9129 |
7 |
MPEDE | 3907.0 | 6 | 12606.4 | 3 | 3921.6 | 6 | 4469.8 | 5 | 1.9637 |
3 |
SHADE | 3903.6 | 5 | 3934.1 | 2 | 3911.0 | 4 | 3912.0 | 3 | 7.3546 | 5 |
DEMR | 3903.0 | 2 | 0.6650 | 2 |
According to the analysis above, although CMA-ES, GL-25, LBBO, and MPEDE may be ineffective, their performances with the best values and median values are acceptable. So, the success rate of each algorithm is calculated with the results of the 25 independent tests. Firstly, to determine if an optimization is successful, we set a threshold. If the result is below the threshold, the optimization process will be considered successful and vice versa. According to the total cost of Hohmann transfer,
Simple investigation of the success rates for all of the algorithms in three-dimensional space.
Algorithm | SaDE | CMA-ES | GL-25 | LBBO | MPEDE | SHADE | DEMR |
---|---|---|---|---|---|---|---|
Success time(s) | 25 | 19 | 24 | 16 | 23 | 25 | 25 |
Success rate | 100% | 76% | 96% | 64% | 92% | 100% | 100% |
To further confirm the effectiveness of DEMR, another group of experiments between DEMR and the same algorithms used in experiments for Earth-Moon low-energy transfer trajectory optimization (CMA-ES, LBBO, GL-25, SaDE, MPEDE, and SHADE) is conducted on 25 benchmark functions from CEC2005 special session [
The performance of the algorithms is evaluated in terms of function error value (FEV), defined as
To provide a correct interpretation of the results, the nonparametric statistical tests, Wilcoxon rank-sum test and Friedman test, are employed in this section as similarly done in [
The mean error and standard deviation of all the algorithms for solving each function over 25 independent runs when dimension is 10 are shown in Table
Comparisons of mean error and standard deviation between DEMR and other six algorithms on CEC2005 functions.
Func. | DEMR | CMA-ES | LBBO | GL-25 | SaDE | MPEDE | SHADE |
---|---|---|---|---|---|---|---|
F1 |
|||||||
F2 | |||||||
F3 | 3 |
||||||
F4 | |||||||
F5 | |||||||
F6 | |||||||
F7 | |||||||
F8 | |||||||
F9 |
|||||||
F10 | |||||||
F11 | |||||||
F12 | |||||||
F13 | |||||||
F14 | |||||||
F15 | |||||||
F16 | 9 | ||||||
F17 | 4 |
||||||
F18 | |||||||
F19 | |||||||
F20 | |||||||
F21 | |||||||
F22 | |||||||
F23 | |||||||
F24 | |||||||
F25 | |||||||
+ | 6 | 7 | 6 | 10 | 8 | 13 | |
− | 15 | 11 | 16 | 12 | 10 | 9 | |
= | 4 | 7 | 3 | 3 | 7 | 3 |
Comparison between DEMR and other algorithms on different types of functions.
F1–F5: Unimodal functions
Algorithms | CMA-ES | LBBO | GL-25 | SaDE | MPEDE | SHADE |
---|---|---|---|---|---|---|
4 | 3 | 3 | 3 | 4 | 4 | |
1 | 1 | 2 | 2 | 1 | 1 | |
= | 0 | 1 | 0 | 0 | 0 | 0 |
F6–F12: Multimodal-basic functions
Algorithms | CMA-ES | LBBO | GL-25 | SaDE | MPEDE | SHADE |
---|---|---|---|---|---|---|
2 | 2 | 1 | 2 | 1 | 4 | |
3 | 2 | 5 | 4 | 4 | 3 | |
= | 2 | 3 | 1 | 1 | 2 | 0 |
F13-F14: Multimodal-expanded functions
Algorithms | CMA-ES | LBBO | GL-25 | SaDE | MPEDE | SHADE |
---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 0 | 1 | |
2 | 1 | 2 | 1 | 1 | 0 | |
= | 0 | 1 | 0 | 0 | 1 | 1 |
F15–F25: Multimodal-hybrid composition functions
Algorithms | CMA-ES | LBBO | GL-25 | SaDE | MPEDE | SHADE |
---|---|---|---|---|---|---|
0 | 2 | 2 | 4 | 3 | 4 | |
9 | 7 | 7 | 5 | 4 | 5 | |
= | 2 | 2 | 2 | 2 | 4 | 2 |
The Wilcoxon results of the multiple-problem statistical analysis are listed in Table
Results obtained by the multiple-problem Wilcoxon test for DEMR and other six algorithms on CEC2005.
VS | |||||
---|---|---|---|---|---|
CMA-ES | 274 | 26 | 0.000375 | ||
LBBO | 238.5 | 86.5 | 0.039554 | ||
GL-25 | 261 | 64 | 0.007727 | ||
SaDE | 183 | 117 | 0.338495 | = | = |
MPEDE | 216 | 84 | 0.057433 | = | |
SHADE | 183 | 117 | 0.338495 | = | = |
Eight functions are chosen from the 4 types of functions in CEC2005 benchmark to get a brief look at the convergence speed, including 2 unimodal functions, 2 basic functions (multimodal), 1 expanded functions (multimodal), and 3 hybrid composition functions (multimodal). The chosen functions are
The convergence graphs on different types of CEC2005 benchmark functions.
The convergence graph on the unimodal function f1
The convergence graph on the unimodal function f5
The convergence graph on the basic function (multimodal) f9
The convergence graph on the basic function (multimodal) f12
The convergence graph on the expanded function (multimodal) f13
The convergence graph on the hybrid composition function (multimodal) f18
The convergence graph on the hybrid composition function (multimodal) f21
The convergence graph on the hybrid composition function (multimodal) f23
In order to get a more intuitive comparison for all of the algorithms from the overall performance, the results of the Friedman statistical test are shown in Figure
Average rankings for all algorithms on 25 benchmark functions.
DEMR introduces its own parameters
As shown in Table
Results obtained by the Friedman test for contraction criteria combinations on CEC2005.
Friedman | ||||||
---|---|---|---|---|---|---|
1 | 1.5 | 2 | 2.5 | 3 | ||
1 | 11.08 | 13.28 | 12.32 | |||
1.5 | 11.14 | 11.74 | 13.98 | 12.24 | 13.90 | |
2 | 15.04 | 14.66 | 13.08 | 15.46 | 16.84 | |
2.5 | 11.14 | 12.78 | 14.10 | 14.48 | 14.66 | |
3 | 12.34 | 13.90 | 14.94 | 13.32 |
Average Friedman rankings for contraction criteria combinations on 25 benchmark functions.
Average Friedman rankings for the maximum number of random reinitialization.
Average Friedman rankings for the maximum number of random reinitialization on 25 benchmark functions
Friedman rankings | |
---|---|
5 | 3.16 |
10 | 3.52 |
15 | 2.72 |
25 | 3.02 |
Performances for the maximum number of random reinitialization on the Earth-Moon low-energy transfer problem
Average | |
---|---|
10 | 3909.8 |
15 | 3909.4 |
20 | 3909.3 |
25 | 3909.7 |
To optimize the Earth-Moon low-energy transfer trajectory, an hybridized differential evolution algorithm with an efficient reinitialization strategy, DEMR, is proposed in this paper. The DEMR method firstly hybridizes DE with local search algorithm SQP to balance the exploration and exploitation and then employs a mix reinitialization procedure to further avoid falling into local optimum. The effectiveness and efficiency of DEMR are validated by comparing the other six peer algorithms, which represent the state of art in ES, BBO, GA, and DE, respectively, on the Earth-Moon low-energy transfer problem in both the two-dimensional space and three-dimensional space. With an overall consideration for all the results, including the achieved best optimization performance, the success rate, and the stability, DEMR is a relatively better choice when comparing with those selected algorithms. The superiority of DEMR is also verified after comparing it with these six algorithms on benchmark problems. Although simple sensitive tests of the extra parameters have been done, a further more comprehensive analysis of the parameters’ features is necessary, and alternative contraction criteria will be developed based on the analysis.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors sincerely thank the anonymous reviewers for their constructive and helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China under Grant no. 61472375, the 13th Five-year Pre-research Project of Civil Aerospace in China, Joint Funds of Equipment Pre-Research, and Ministry of Education of China under Grant no. 6141A02022320, and Fundamental Research Funds for the Central Universities under Grant no. CUG160207 and no. CUG2017G01.