Comparative Study of Basic Constellation Models for Regional Satellite Constellation Design

Transcription

1 2016 Sixth International Conference on Instrumentation & Measurement, Computer, Communication and Control Comparative Study of Basic Constellation Models for Regional Satellite Constellation Design Yu Mo, Dawei Yan, Peng You, Shaowei Yong Institute of Electronic Science and Engineering National University of Defense Technology Changsha, China Abstract For the regional coverage satellite constellation design optimization under the condition of multi-constrains, an optimization model with the orbital semi-major axis minimization and the percentage of coverage time maximization is built. Basic constellation models such as Walker and Flower are used to reduce the search space dimensions and NSGA-II is applied to solve the educed bi-objective optimization model. Moreover, 4+2N constellation model is proposed and compared with Walker and Flower models in terms of optimization performance by interconnecting Matlab and STK. Finally, some conclusions about how to choose the constellation models for regional constellation design are given. Keywords satellite constellation design; constellation model; regional coverage; NSGA- I. INTRODUCTION Compared to other altitude orbits constellation system, low earth orbit satellite constellation system has many advantages of low emission cost, short propagation delay, strong system survivability [1]. Low earth orbit satellite constellation system has received high attention from all over the world because it can effectively complete a variety of tasks which suit for modern military activities and civil activities. Determining constellation coverage area is the first step in the process of constellation design. Coverage area is generally divided into global coverage and regional coverage [2]. Global coverage satellite constellations require a larger number of satellites, the higher the cost of the system and it does not meet our current needs and economic strength. Construction of the constellation covered China to provide communication, navigation, maritime rescue and other services is more conforming to the situation of our country. Satellite constellation design is a prerequisite for the establishment of a satellite constellation system, and it is a multi-objective optimization problem in essence. The goal of constellation design is to get the number of satellites in the constellation with six orbital parameters of each satellite [3], and to achieve constellation system performance and cost integrated optimal. However, in the constellation design optimization process, the number of optimization variables and the parameter space are very large which result in very high computational complexity. Therefore, adopting a basic constellation model (such as Walker constellation model, Flower constellation model) to reduce the search space dimension is very necessary. For global coverage constellation, Walker constellation model and Flower constellation model have been widely used. But for regional coverage constellation, there is no good basic constellation model. So this paper proposes a 4+2N constellation model for regional coverage constellation design, and compares it with Walker and Flower constellation model. In the use of Walker constellation model, the author [4] got a better performance constellation configuration in the mid-latitude areas by mixed orbits. Flower constellation model is used [5] to design a constellation that provides navigation services in the northern hemisphere and in the case of the same number of satellites, the Flower constellation of navigation performance is more superior than the existing constellation about navigation performance. Mortari [6] designed a better performance constellation by setting up the coverage performance and the inter-satellite links as objective functions on the basis of the Flower constellation model than the Walker constellation model. A 3+4N [7] basic constellation model was established that the semi-major axis, inclination, eccentricity are the same, but the RAAN and the number of satellites in different orbital plane are different. This constellation model can effectively reduce the search space dimension, but still exist a large parameter space. In addition, the objective in constellation design is not unique, nonlinear, and discontinuous. So we usually use modern optimization algorithm because the traditional optimization method is difficult to solve such problems. Nondominated sorting genetic algorithm with elitist strategy (NSGA-II) [8] is good at solution efficiency and performance and it has been widely studied and applied. The paper is organized as follows. Firstly, a region optimization model including two objective functions is established, and different basic constellation model is used to reduce the searching space. Secondly, the performance of the constellation is simulated by interconnecting Matlab and Satellite Tool Kit (STK). Finally, NSGA-II algorithm is applied to compare the optimal constellation of different basic constellation models. II. REGIONAL COVERAGE CONSTELLATION MODEL A. Basic Constellation Model Constellation is composed of some satellites to complete a specific task. Satellite s position consists of six orbital parameters: semi-major axis r, eccentricity e, inclination i, argument of perigee w, right ascension of the ascending node(raan) and mean anomaly f. Wherein, the orbital /16 $ IEEE DOI /IMCCC

2 semi-major axis and eccentricity determine the orbital shape, argument of perigee determine the semi-major axis direction, inclination and RAAN determine the location of orbital plane, Table 1 Three models orbital parameters Basic constellation model Walker Flower 4+2N Same parameters r, e, i, w r, e, i, w r, e, i, w 360 Different ( P m 1) P Each satellite parameters Np i NdMi mod(2 ) f ( Nm 1) F( Pm 1) S N and f mean anomaly determine the satellite's position with respect to relative positions of different orbital plane, S represents the time. These parameters in the constellation design number of satellites in each orbital plane, m is the satellite optimization process will be used as optimization variables, so number, Pm is the satellite's orbital plane number, Nm is the the N satellites constellation have 6N optimization variables and its variable space is very large. Therefore, we need a basic number in the orbital plane, N p and N d are a relatively prime constellation model to reduce variable space dimension. m Currently widely used basic model mainly are Walker constellation model and Flower constellation model. Walker constellation model is proposed in 1971 [9]. Feature of the model is that all satellites have the same orbital altitude, and evenly distribute on the inclined circular orbit, each plane has the same inclination. Walker constellation in the case of determining the orbital altitude and orbital inclination requires only three parameters (T/P/F) to describe the distribution of the entire constellation which greatly simplifies the parameter space and reduces the amount of calculation where T represents the total number of satellite constellation, P represents the number of track surface and F represents the relative position of different satellite orbital plane in the range of (0~P-1). Flower constellation model is proposed by professor Mortari in 2003 [10]. Flower constellation model is characterized by a constellation orbits of all satellites have the same orbital shape that each satellite s the return days, the argument of perigee, perigee altitude and orbital inclination are the same. Flower constellation model requires eight parameters to describe the constellation distribution: The number of petals N p, the number of return days N d,the number of satellites N s, configuration parameters Fn F d, argument of perigee, inclination i and orbit perigee altitude h p. The constellation model adopts repeat ground track which effectively reduces the difficulty for the ground station to monitor and control satellites. Due to the coverage type of this paper is regional coverage, the traditional constellation model (Walker constellation model and Flower constellation model) may not be the optimal solution and we may need a larger search space. As a result, this paper applies a 4+2N constellation model to compare with the traditional constellation model. This constellation model refers the satellites with same orbital altitude and orbital inclination, different right ascension of the ascending node and mean anomaly, and argument of perigee and eccentricity zero. Now these three models are summarized in Table 1. In Table 1, T is the total number of satellite in constellation, P is the number of orbital plane, F is the integers. Namely S T / P, Pm 1, Nm m ( Pm 1) S. S B. Optimization Model Construction of LEO satellite system is intended to provide a good service to the target area. The premise of providing services is to cover the target area, and the situation of a single satellite coverage is shown in Figure 1. Wherein R is the radius of the Earth, h is the satellite orbit altitude, is the satellite viewing angle, is the minimum elevation that is essential to establish a communication link between the satellite and the ground user. The formula to calculate is: Fig. 1 single satellite coverage Re arcsin cos h R (1) e That is to say, in the case of determining the height of the satellite orbit, the elevation between the satellite and terrestrial users can be converted into satellite viewing angle. In general, coverage performance computing is divided into two categories: one is analytical method that establishes an approximate relationship between optimization model and configuration of satellite constellation. Then using geometric method to analysis the covering performance and so on; another is the grid method that calculate grids in every target area on the surface of the earth. Analytical method is usually applied to design global constellation, and grid method is suitable for any type of shape analysis covering the track and 172

3 complex sensors while used in complex application environments with high precision. Coverage area of this article is the China region, belonging to the regional coverage, and therefore calculation of coverage performance uses the grid method. Grid method set feature points based on a certain latitude and longitude intervals within the target area. Feature points in the satellite coverage area are statistically analyzed to obtain a constellation coverage performance. If and represent latitude and longitude interval respectively, jk, represents a grid point P jk,, so its latitude and longitude coordinates are: j 0 j Pjk, j, k 1,2, (2) k 0 k Currently, the latitude and longitude of the grid point is j, k, the satellite coverage area is G G1, G2, G3, if the grid point j, k belongs to the G range, the grid point is covered, if not, the grid point is not covered: 1 Pjk, G covered Pi 0 Pjk, G uncovered (3) The indicators of satellite constellation coverage performance are: total time of coverage, times of coverage, percentage of coverage, the maximum revisit time, average revisit time and average response time. The goal of this paper is to provide effective coverage for the target area, so setting percentage of coverage time which is defined as the sum of coverage duration time with the ratio of the total simulation time as performance objective. Formula is written as following: P Tdur / Ttotal (4) Wherein, P represents the percentage of coverage, Tdur represents the sum of coverage time, Ttotal represents the total simulation time. In the process of constellation design, establishing objective functions and constraints according to mission requirements is important. In LEO satellite constellation, satellite orbital altitude and elevation between satellite and terrestrial users will affect system performance. The height of LEO satellite orbit is generally less than 2000km, but when the satellite orbit height is less than 700km, the satellite will be affected by complex environment resulting in shorter life. What s more, when the satellite orbit height is greater than 1500km, the satellite will be within the Van Allen belt. Therefore LEO satellite orbit height is in the range between 700km-1500km, namely the satellite orbital semi-major axis is km-km. In order to effectively overcome the impact of multipath effects and shadow effects, and to ensure the system to provide users with excellent communication services, the elevation angle between satellite and terrestrial users requires at least 10 degrees. The model can be expressed mathematically as follows: Objective function: Constraint: max min km r km 10 o (6) The objective of this article is to design a constellation that maximize coverage and minimize the semi-major axis of the satellite orbits and target area is the China region whose grid resolution that was set to 0.5 is both latitude and longitude. III. EVOLUTIONARY ALGORITHM Constellation design is a multi-objective optimization problem with multiple restrictions which can be solved by modern optimization algorithm. The algorithm can find sets of Pareto optimal solutions that illustrate key tradeoffs. Nondominated Sorting Genetic Algorithm (NSGA) is a sort of modern optimization algorithm with the concept of Pareto optimality based upon the genetic algorithm which has two main characteristics: firstly, retaining good individuals of the population and sorting the population based on non-dominated into each front; secondly, using the fitness sharing function to maintain the individual variety and keeping off trapping in local minima to some extent. But there are some disadvantages in NSGA. One is that the algorithm suffers from high computational complexity, especially when population is large. Secondly, the lack of elitism can significantly lower the optimal speed and loss the search of good solutions. It is essential to be specified the sharing radius in order to improve algorithm performance. gen=gen+1 No Initial population gen=0 Non-dominated sort and genetic operator gen=2 Combine the parent and child Generate a new parent Yes Select,recombine, mutation Gen<Max Yes End P r No Fast nondominated sorting Assigned fitness Create a new population Fig.2 The flow chart of NSGA- To solve these shortages in the NSGA, Ded had proposed to improve the non-dominated sorting genetic algorithm with (5) 173

4 elitist strategy (NSGA-II) [8], and there are some great characteristics as follow: 1) Non-Dominated sort. By using fast non-dominated sorting method will improve computational speed complexity. For each individual p in main population P, np is the number of individual that is being dominated by p and S p contain all the individuals that is being dominated by p. Then the current non-dominated solutions (i.e. = 0) is archived in the set Q1 and initialized the front counter to one. For each individual i in the set Q 1 search its dominated set Si and decrease the domination count for individual i ( n n 1), n p p p if np 0 then the individual i is archived in the set Q2 and set rank of individual i to front 2. And so on, until all individuals are sorted. 2) Crowding Distance. It takes place of the sharing function to maintain population diversity and improve the robustness of the algorithm. Crowding distance is assigned front wise and comparing the crowding distance between two individuals in different front is meaning less. The crowding distance is calculated the local density between the same level adjacent dots in the target space (standardize each goals before calculating crowding distance), and the value of the crowding distance is the length of the adjacent cube. While the selection is carried out using a crowded-comparison-operator, nondomi satellites 4 satellites 5 satellites 3 satellites 4 satellites 5 satellites 3 satellites 4 satellites 5 satellites Fig. 3 Walker optimal solutions Fig. 4 Flower optimal solutions Fig N optimal solutions nation rank and local level crowding distance for each individual are calculated. If the non-domination rank of two random individuals is different, the individual is selected with high non-domination rank; if the non-domination rank of two random individuals is the same, the individual with greater crowding distance is selected, so that the population evolves toward the direction of non-dominated and uniform distribution. 3) Elitist Strategy. Because of employing elitist strategy there is no need external document to maintain excellent individuals. Parent population (scale N) and offspring population (scale N) merge into a new population (size of 2N), and then the local crowding distance of the individuals are calculated after the new population is sorted by nondomination rank. The selection is carried out according to the non-domination rank, N individuals are selected as a new parent population for the next generation. Repeating selection, crossover and mutation operations to generate a new offspring until the termination condition is reached. NSGA- flow chart is shown in Figure 2. Firstly, initial population with size N is randomly generated and then operated including selection, crossover and mutation after non-dominated sorting. So the first generation population is obtained and the generation is 1. Next step is combining the parent population and offspring population, and sorting the combined populations according to the non-dominated rank with calculation of partial crowding distance of each individual. Then getting offspring inherits from parent population by genetic operation and the number of iteration plus 1. The process is repeated until reaching the maximum number of iterations. IV. SIMULATION ANALYSIS In this paper, constellation design process simulation is conducted by interconnecting the Matlab and STK. Firstly, the constellation orbital parameters are transferred from Matlab to STK. Then simulation scenarios are built based on these parameters and constellation coverage performance of the target area is obtained by STK. Next step is that transferring calculation results in STK back to Matlab to determine the fitness of individuals in the population and apply NSGA- algorithm to generate offsprings. The process above mentioned will repeat until it reaches the maximum generation. The coverage area of this paper is the Chinese region. Simulation time is set 1800 seconds and the simulation step is 10s. The initial population size is set 50 and the maximum generation is 300. In the case of the same number of satellites, we compare optimization results obtained by Walker, Flower and 4+2N constellation models. Since the coverage area is small, we choose 3, 4 and 5 satellites to compare different constellation models. Figure 3-5 respectively show the optimization results conducted by Walker, Flower and 4+2N constellation models that aim at minimizing the orbital semi-major axis and maximizing the percentage of coverage time. Conclusions that increasing the number of satellites and the orbital semi-major axis can notably improve coverage performance may be drawn from figure 3 and figure 5 with Walker constellation model and 4+2N constellation model. However, the optimal solution 174

5 obtained by Flower constellation model show that orbital semi-major axis is a key factor affecting the constellation coverage (Fig.4). Walker Flower 4+2N Walker Flower 4+2N Walker Flower 4+2N Fig. 6 Comparison with 5 satellites Fig. 7 Comparison with 4 satellites Fig. 8 Comparison with 3 satellites Figure 6-8 provide comparison of the optimal solutions with 3, 4 and 5 satellites. Obviously, 4+2N constellation model performs better than traditional constellation models. For instance, the percentage of coverage time obtained by 4+2N constellation model can reach 100% with 5 satellites while the maximum of traditional constellation is approximately 40%. The reason is that 4+2N constellation model exists a larger search space and contains the traditional model constellation configurations. So it can search for constellation configuration of greater coverage with lower orbital altitude. However, increasing the search space needs greater computational expense. Walker 5 satellites Flower 5 satellites 4+2N 3 satellites V. CONCLUSION In order to optimize the process of regional coverage constellation design with limiting orbital altitude and elevation, this paper establishes an optimization model that minimizes the orbital semi-major axis and maximizes the percentage of coverage time in the target area. Walker constellation model, Flower constellation model and 4+2N constellation model are applied to reduce the search space dimension and NSGA- algorithm is used to get optimal solutions with different constellation model. Some conclusions are made when the three constellation models are used to design regional coverage constellation with the same number of satellites. Construction of the actual regional coverage constellation requires establishing optimization model which involves many problems. Except to consider the coverage performance and orbital height, the system cost and other factors are indispensably added to regional coverage constellation model. In addition, how to search the optimal solutions fast in large searching space and improve the efficiency of the algorithm are the directions of future work Fig. 9 Comparison with different number of satellites Meanwhile, with the reduction in the number of satellite, the optimal solutions of the three basic models approach. In addition, the computational expense used by Walker and Flower constellation models is lower than 4+2N constellation model. For example, Walker and Flower constellation models just need 100 generations to reach Pareto solutions with 300 generations for 4+2N constellation model. However, a further analysis is needed while the number of satellites is different. The solution distribution in figure 9 illustrates the black points obtained by 4+2N constellation model with 3 satellites and the red and blue points obtained by Walker and Flower constellation models with 5 satellites. The black points under the condition of lacking two satellites are better than red and blue points. As a result, 4+2N constellation model is a better choice to design regional coverage constellation when the number of satellites is 3, 4 and 5. In summary, to design a regional coverage constellation, 4+2N constellation model is a better choice. References [1] Chini, P., G. Giambene and S. Kota, A survey on mobile satellite systems [J]. International Journal of Satellite Communications, (1): p [2] Zhang, Y L, Fan, L, Zhang, Y. Theory and Design of Satellite Constellations[M]. Beijing: Science Press, 2008: [3] Dai, G M, Wang, M C. Multi-objective Algorithms in Satellite Constellation Design[M]. Wuhan: China University of China Press, [4] Yuan, F Y. LEO Constellations for Continuous Global Coverage[J]. Japan Society for Aeronautical and Space Sciences. 1994, 37(16): [5] Zeng, Y J. Satellite Constellation Design Based on Genetic Algorithm[D]. Wuhan: Huazhong University of Science and Technology, 2007: [6] Daniele, M, Mauro, D S, Marco, L. Design of Flower Constellations for Telecommunication Services[J]. Proceedings of the IEEE. 2011, 99(11): [7] Wang, R, Ma, X R, LI, M. Optimization of Regional Coverage Satellite Constellations by Genetic Algorithm[J]. Journal of Astronautics. 2002, 23(3): [8] Deb, K, Pratap, A, Agarwal, S et al. A fast and elitist multiobjective genetic algorithm: NSGA-II[J]. IEEE Transactions on Evolutionary Computation. 2002, 6(2):

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