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The weighted sum and genetic algorithm-based hybrid method (WSGA-based HM), which has been applied to multiobjective orbit optimizations, is negatively influenced by human factors through the artificial choice of the weight coefficients in weighted sum method and the slow convergence of GA. To address these two problems, a cluster and principal component analysis-based optimization method (CPC-based OM) is proposed, in which many candidate orbits are gradually randomly generated until the optimal orbit is obtained using a data mining method, that is, cluster analysis based on principal components. Then, the second cluster analysis of the orbital elements is introduced into CPC-based OM to improve the convergence, developing a novel double cluster and principal component analysis-based optimization method (DCPC-based OM). In DCPC-based OM, the cluster analysis based on principal components has the advantage of reducing the human influences, and the cluster analysis based on six orbital elements can reduce the search space to effectively accelerate convergence. The test results from a multiobjective numerical benchmark function and the orbit design results of an Earth observation satellite show that DCPC-based OM converges more efficiently than WSGA-based HM. And DCPC-based OM, to some degree, reduces the influence of human factors presented in WSGA-based HM.

Earth observation satellites provide essential information on ocean, land, and atmosphere, which are very important in the environment protection and resources management. The first step of satellite mission design is usually the determination of a suitable orbit. The objective of orbit design for Earth observation satellites is to ensure that all target sites are best visited, including observation sites and ground stations. The quality of an orbit can be measured with key orbit performance indices [

In the evolutionary optimization for multiobjective orbit design, multiobjective functions are usually transformed into a single-objective function using the weighted sum method, and then a mature single-objective optimization method, such as genetic algorithm (GA), is employed to optimize the single-objective function to obtain the optimal orbit [

To address these two disadvantages, this study proposes a population-based optimization method named CPC-based OM, in which candidate orbits are gradually randomly generated until the optimal orbit is obtained using a clustering via principal components based data mining method. A sufficient number of candidate orbits could ensure that the global optimal solution is obtained. In addition, the influence of the human factors from the weighted sum method is reduced in the optimization procedure because the candidate orbits are clustered based on the principal components rather than the weighted functions of the optimization objectives. Many methods have been investigated to reduce the influences of human factors of weighted sum method in multiobjective optimization [

Methods must be introduced to accelerate convergence because the search procedure to obtain the optimal solution in CPC-based OM was a nearly exhaustive search with inefficient convergence. The methods of reducing feasible region are popular approaches [

In this study, an orbit optimization model with constraints, six design variables, and eight optimization objectives is developed for Earth observation satellites. The process to obtain the optimal orbit using CPC-based OM is presented. To improve poor convergence of CPC-based OM, a more advanced DCPC-based OM is proposed by introducing cluster analysis based on six orbital elements. Finally, a test with numerical benchmark functions is conducted and the performances of DCPC-based OM, CPC-based OM, and WSGA-based HM on the orbit optimization of Earth observation satellites are compared.

Abdelkhalik and Mortari [

For the orbit design of an Earth observation satellite without maneuvering, the relevant orbit dynamics equations in a geocentric equatorial inertial system (GEI) are as follows:

Various key performance indices have been employed in the orbit design of the Earth observation satellites [

The orbit optimization model of Earth observation satellites is shown in

The optimization objective is to make TCT, FC, ATC, and AT-TT&C be the maximum, the MCG, ICG, and ACG be the minimum, and the ATI-TT&C be within an expected range. The constraint is

To reduce the influences of human factors in orbit optimizations [

Process flow of CPC-based OM for orbit design optimizations.

Randomly generate

Nondimensionalize the coverage and TT&C performance indices of candidate orbits.

Calculate the principal components of nondimensionalized orbit performance indices of candidate orbits, divide candidate orbits into classes by performing cluster analysis based on the principal components, and evaluate all class centers using the weighted sum function of key orbit performance indices to obtain the optimal class.

If the number of orbits in the optimal class is more than six, go to Step

If the number of orbits in the optimal class is not more than six, determine the temporary optimal orbit from the optimal class by using the weighted sum method. Randomly generate

Note that Steps 3 and 4 constitute “multilevel cluster analysis,” in which the optimal class with not more than six orbits is obtained. Steps 2~5 constitute “random exhaustive” process to obtain the optimal orbit. In addition, because the cluster analysis is based on the principal components of nondimensionalized key orbit performance indices rather than based on weighted function of the nondimensionalized key orbit performance indices in Step

When various candidate orbits are generated and orbit performance indices with different units are calculated, the performance indices are nondimensionalized using (

Supposing the dimensionless coefficients can be represented by vector

Next, the eigenvalues

The number of principal components is the minimum

Because

The candidate orbits could be clustered according to the principal components to obtain the optimal class [

The procedures of cluster analysis are as follows.

Assume

Select initial cluster centers. Calculate Euclidean distance

Calculate Euclidean distances from each of the remaining

Determine new cluster centers. Calculate the average value of principal components of

If one or more than one new cluster center is different from the corresponding last cluster center, reassign all remaining

Set

The number of candidate orbits is very large, after one cluster analysis, the number of orbits in the optimal class is usually more than six. Therefore, cluster analyses are repeatedly carried out until the number of orbits in the optimal class is less than six. This is known as multilevel cluster analysis.

The weighted sum method [

In CPC-based OM, principal component analysis, rather than weighted sum method, is adopted to cluster orbits, and therefore the negative influence of the artificially set weight coefficients in weighted sum method is reduced. The detailed analyses are illustrated in contours resulting from principal component analysis and weighted sum method in Figure

The global optimization capability and convergence efficiency of DCPC-based OM: (a) effect of principal component analysis, (b) CPC-based OM with initial orbits, (c) CPC-based OM with initial and additional orbits, (d) additional orbits generating strategy, and (e) DCPC-based OM.

In CPC-based OM, when enough candidate orbits are randomly generated, the global optimal orbit with high accuracy could be achieved. For a certain number of candidate orbits, as indicated by black stars in Figure

The method of reducing the feasible region will be employed to improve the convergence. It could be known from Figure

By introducing cluster analysis based on the six orbital elements, a novel population-based optimization method named DCPC-based OM is developed based on CPC-based OM. The processing flow is shown in Figure

Process flow of DCPC-based OM for orbit design optimizations.

Randomly generate

Nondimensionalize the coverage and TT&C performance indices of candidate orbits.

Calculate the principal components of nondimensionalized orbit performance indices, divide candidate orbits into classes by performing cluster analysis based on the principal components, and evaluate class centers using the weighted sum method to obtain the optimal class.

When the total number of candidate orbits in the optimal class is greater than six, the six orbital elements are nondimensionalized by the upper limit value of the relevant feasible region. Cluster analysis is conducted on the orbits belonging to the optimal class, by using the nondimensionalized values of the six orbital elements. The candidate number of classes

Choosing the orbits in the optimal class and

The characteristics of DCPC-based OM can be concluded as follows.

(

(

It can be seen from (

For orbit optimization, the calculation procedure of orbit performance indices is more time-consuming than optimization operation, because a large number of numerical computations are needed to calculate orbit performance indices. Therefore, the total optimization time approximately equals the total number of generated candidate orbits multiplied by the computation time required for the performance indices of one orbit, which means the time cost is nearly predictable. The proposed orbit design optimization method with predictable time cost is more convenient than other population-based optimization methods with unpredictable time cost for a scheduled project.

(

A four-objective benchmark function (as shown in (

The optimizations with dimensions

Convergence curve of test: (a) dimension is 5; (b) dimension is 10.

Figure

For a certain Earth observation satellite, five observation targets are with latitude and longitude coordinates of (25°N, 120°E), (10°N, 110°E), (40°N, 130°E), (15°N, 90°W), and (20°S, 130°E) and the same vision field angle of 25°. The minimum elevation angle, latitude, and longitude coordinates of TT&C station are 5°, 40°N, and 120°E, respectively. The expected range of ATI-TT&C is 8000 s~40000 s. The range of the semimajor axis is 400 km~600 km and

The orbit design optimization is conducted using DCPC-based OM, CPC-based OM, and WSGA-based HM [

Evaluation indices of three optimization methods.

The data in Figure

Details of orbit optimization results with DCPC-based OM are presented in this section. In the first clustering, principal component analysis was conducted after the performance indices of all candidate orbits are calculated and nondimensionalized. The contribution ratios of the first three principal components in the principal component analysis were 74.10%, 11.69%, and 8.38%, respectively, as shown in Figure

Principal component contribution value for the first clustering.

The data in Figure

According to total class distance criterion, the optimal number of classes in the first cluster was five. The principal components of the cluster centers in the five classes are shown in Table

Principal components in the cluster centers for the first clusters.

Principal component | Classification | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

Principal component 1 (CP1) | −4.72 | 3.34 | −1.46 | 3.58 | −2.48 |

Principal component 2 (CP2) | 0.46 | 7.38 | −3.37 | −2.19 | −0.87 |

Principal component 3 (CP3) | −5.18 | 0.18 | 5.80 | −0.83 | 4.14 |

The data in Figure

Eight dimensionless performance indices of the five clustering centers after the first clustering.

Performance indices | Classification | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

Frequency of coverage (FC) | 0.12 | 0.34 | 0.42 | 0.93 | 0.32 |

Total coverage time (TCT) | 0.31 | 0.27 | 0.34 | 0.83 | 0.23 |

Average time per coverage (ATC) | 0.01 | 0.43 | 0.86 | 0.89 | 0.77 |

Maximum coverage gap (MCG) | 0.04 | 0.26 | 0.72 | 0.63 | 0.33 |

Average coverage gap (ACG) | 0.94 | 0.92 | 0.54 | 0.96 | 0.48 |

Minimum coverage gap (ICG) | 0.02 | 0.05 | 0.5 | 0.9 | 0.11 |

Average time interval of TT&C (ATI-TT&C) | 0.30 | 0.15 | 0.79 | 0.65 | 0.84 |

Average time of each TT&C (AT-TT&C) | 0.59 | 0.04 | 0.62 | 0.57 | 0.61 |

Results of the first clustering: (a) principal components of the five clustering centers and (b) dimensionless performance indices of the five clustering centers.

The data in Figure

The orbits in Class 4 from the first clustering and

Results of the fifteenth clustering: (a) principal components of the five clustering centers and (b) dimensionless performance indices of the five clustering centers.

The eight dimensionless performance indices for all orbits of Class 3 are shown in Figure

Dimensionless performance indices for all orbits in Class 3 after the fifteenth clustering.

The evolution process of the maximum and minimum evaluation indices of the orbits in the optimal class is shown in Figure

Evaluation index trend.

A decrease in the maximum evaluation index from O1 to O2 indicates that, in the 8th clustering, the maximum evaluation index is not contained in the optimal class and is filtered out. The coverage and TT&C performance indices of orbits O1 and O2 are listed in Table

Coverage and TT&C indices of orbits O1 and O2.

Index | Orbit O1 | Orbit O2 |
---|---|---|

Frequency of coverage (FC) | 100 | 100 |

Total coverage time (TCT)/s | 8445 | 5970 |

Average time per coverage (ATC)/s | 86 | 60 |

Maximum coverage gap (MCG)/s | 4213.5 | 2842.5 |

Average coverage gap (ACG)/s | 1117.75 | 1162.5 |

Minimum coverage gap (ICG)/s | 486.25 | 270.75 |

Average time interval of TT&C (ATI-TT&C)/s | 3886 | 3930 |

Average time of each TT&C (AT-TT&C)/s | 31 | 38 |

This paper proposes a population-based optimization method named DCPC-based OM, which consists of the index nondimensionalization method, principal component analysis, double cluster analysis, and the weighted sum method. Tests using numerical benchmark functions were conducted, and an example of orbit optimization for Earth observation satellites was analyzed. Both optimization results show that the proposed method, with characteristics of a predictable time cost, has the advantages of reducing the influence of human factors that commonly exist in the weighted sum method and bring more efficient convergence than genetic algorithm.

This paper describes the results of a preliminary research study of the developed optimization method. Further study will be performed, such as determining the quantity of initial orbits to ensure the capability of global optimization, understanding how evolutionary rate influences the accuracy of the optimal solution, and determining whether a dynamic evolutionary rate is necessary. In addition, the optimal capability when the optimal solution is on the boundary of the feasible region will be further investigated.

The authors declare that they have no conflicts of interest.

The research work is supported by the National Defense 973 Program (Grant no. 613237).