International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 6396032, 15 pages

https://doi.org/10.1155/2017/6396032

## A Novel Double Cluster and Principal Component Analysis-Based Optimization Method for the Orbit Design of Earth Observation Satellites

^{1}School of Astronautics, Beihang University, Beijing 100191, China^{2}Department of Satellite Application System, North China Institute of Computing Technology, Beijing 100191, China^{3}Institute of Manned Space System Engineering, China Academy of Space Technology, Beijing 100094, China

Correspondence should be addressed to Fengrui Liu; nc.ude.aaub@uilrf

Received 23 December 2016; Revised 6 April 2017; Accepted 12 April 2017; Published 9 May 2017

Academic Editor: Christian Circi

Copyright © 2017 Yunfeng Dong 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

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.

#### 1. Introduction

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 [1]. The key orbit performance indices of an Earth observation satellite include the total coverage time, the frequency of coverage, the average time per coverage, the maximum coverage gap, the minimum coverage gap, and the average coverage gap [1, 2]. Thus, orbit design is a typical multiobjective optimization problem. Numerical methods for multiobjective orbit design optimization can be classified into three primary groups: indirect methods, direct methods, and evolutionary algorithms [3]. The last group is currently receiving research attention because of the capability of achieving global optima in very large search spaces.

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 [4–9]. Abdelkhalik and Mortari [4, 5] employed GA to optimize the weighted sum of squares of the distances between each target site and the satellite at the nearest ground track point, taking five orbital elements plus all visiting times as the design variables. Abdelkhalik and Gad [6] applied a weighted function of the total number of covered sites and the ground track repetition period as fitness function and adopted GA to optimize eccentricity, inclination, space-craft’s true anomaly above the first ground site, and the ground track repetition period, to design space orbits for Earth orbiting missions. Vtipil and Newman [7] and Vtipil [8] employed the sum of all time slot values of visiting as the cost function and adopted GA to conduct optimizations. The effect of population sizes was further researched. Zhang et al. [9] used a hybrid-encoding GA to optimize the sum of absolute value of velocity increment for long-duration rendezvous phasing missions. In weighted sum method [10–12], weight coefficients are utilized to transform the multiobjective function into the single-objective function. One disadvantage of the WSGA-based HM is that the artificially set values of the weight coefficients are unreasonable and subjective and depend significantly on human factors. In addition, the other disadvantage of the WSGA-based HM is the inefficient convergence of GA [13, 14].

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 [15–17]. Among them, the principal component analysis [18] is one of the most feasible methods, which transforms the original variables into a new set of variables, referred to as principal components, by using the eigenvalue-eigenvector method. The principal component analysis is thought to be the best way that explains the internal structure of the data [19, 20] and wildly applied by numerous researchers [21–26] to transform multiobjective functions for subsequent optimizations.

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 [27–29]. In the methods of reducing feasible region, parts of the feasible region that do not include the optimum solution are deleted, and the subsequent optimization is accelerated because the remaining search space (feasible region) is smaller. Cluster analysis with the capability of dividing the feasible region into different regions has been used to reduce feasible regions [30–32]. Therefore, the second cluster analysis is introduced to CPC-based OM to accelerate convergence, and a novel population-based optimization method named DCPC-based OM is presented.

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.

#### 2. Orbit Optimization Model for the Earth Observation Satellites

Abdelkhalik and Mortari [4, 5] explored the concept of developing an orbit based on target sites with no thrusters, in which design variables include five orbital elements and all the visiting times. The number of design variables increases with the increasing number of target sites. To avoid the increase of computational burden as the number of target sites increases, an optimization model with six orbital elements as design variables is employed. In addition, to deal with the increasing complexity of the observation mission, more orbit performance indices were taken into account than in prior studies [4–9].

##### 2.1. Orbital Dynamics Model and Six Orbit Elements

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:where , , and are the components of the satellite position vector; , , and are the components of the velocity vector; and , , and are the components of external force, including Earth’s gravity (considering Earth nonspherical shape perturbation forces), atmospheric drag perturbation forces, and solar radiation pressure as well as lunar and solar perturbations forces [1]. The positions of the satellite at each moment can be calculated using (1) and six orbital elements at the initial moment. The six orbital elements include the semimajor axis , the eccentricity , the inclination , the argument of the perigee , the longitude of the ascending node , and the true anomaly . The key orbit performance indices of an Earth observation satellite are calculated using the positions of the satellite at each moment, the longitude and latitude data of the observation sites, the longitude and latitude data of the ground stations, and the right ascension of Greenwich at the initial moment [1].

##### 2.2. Coverage and TT&C Performance Indices

Various key performance indices have been employed in the orbit design of the Earth observation satellites [4–9]. This paper adopts the key orbit performance indices which are systematically and comprehensively summarized by Wertz and Larson [1] and are applied by Wei et al. [2]. The key orbit performance indices of the Earth observation satellites can be separated into the coverage performance indices and the tracking telemetry and command (TT&C) performance indices. Among them, the coverage performance indices include the total coverage time (TCT), the frequency of coverage (FC), the average time per coverage (ATC), the maximum coverage gap (MCG), the minimum coverage gap (ICG), and the average coverage gap (ACG). And the TT&C performance indices include the average time interval of TT&C (ATI-TT&C) and the average time of each TT&C (AT-TT&C). The definitions of the eight orbit performance indices are as follows [1, 2].where is the total number of times of coverage in the simulation time , is the time of the th coverage, is the time of the th coverage gap, is the total number of coverage gaps, is the time of the th interval of TT&C, is the total number of the intervals of TT&C, is the time of the th TT&C, and is the total number of TT&C.

##### 2.3. Orbit Optimization Model

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 . The six orbital elements at the initial moment are the independent variables. The orbit optimization model in (3) is a typical multidimensional and multiobjective optimization problem. The purpose of this study is to provide an optimization method for orbit decision-making for an Earth observing satellite mission. The resulting optimal orbit may need to be refined in the case of additional system designs, including orbit stability, fuel consumption in orbital maneuvering, or launch site restrictions.

#### 3. CPC-Based OM for Orbit Optimizations

To reduce the influences of human factors in orbit optimizations [4–9], CPC-based OM is presented, and the accelerating convergence approach will be introduced in Section 4 to develop DCPC-based OM. The process flow of orbit design optimization with CPC-based OM is shown in Figure 1, including five steps.