An Optimal Design of Constellation of Multiple Regional Coverage

An Optimal Design of Constellation of Multiple Regional Coverage Based on NSGA-II 1 Xiaoqian Huang, 1,* Guangming Dai 1, School of Computer Science, China University of Geosciences, gmdai@cug.edu.cn * Corresponding Author Abstract With the continuous development of space technology, the optimal design of constellation have gained wide attention. Current constellation optimization mainly deal with a single target region. However, it is important to monitor multiple regions simultaneously for earth observation or other missions. We proposed a new optimal design of constellation for multiple regional coverage. This method is based on Non-dominated Sorted Genetic Algorithm-II and adopts mesh dividing to calculate the regional coverage. The validity of Non-dominated Sorted Genetic Algorithm-II is tested via the simulation experiment by applying this method to the optimal design of constellation of multiple regional coverage. 1. Introduction Keywords: Constellation Optimization, Multiple Regional Coverage, NSGA-II With the continuous development of space technology, the optimal design of constellation have gained wide attention [1-3]. In order to forecast earthquakes, cyclones, tsunamis and other natural disasters precisely and in plenty of time to minimize damage and loss of life, we need to monitor multiple regions simultaneously. Kai Sun et al. suggest to arrange the satellites reasonably to take images as many as possible with good quality and satisfy different users [4]. It is necessary for numerous satellites to work cooperatively as a constellation of satellites because of a spike in demand for the regional coverage rate. The design of satellite constellation is one of the important steps to carry out space missions. For the design of the regional coverage constellation model, there are many progress has been achieved. Mason (1998) presented a multi-objective optimization model for constellation design by using genetic algorithm and the Pareto optimality [5]. Deb et al. (2000) proposed a method based on the NSGA-II to optimize the constellation model [6-8]. However, those constellation optimization mainly deal with a single target region, which cannot satisfy the demand for the coverage rate of multiple regions. The optimal design of constellation of multiple regional coverage is a typical multiobjective optimization problem. Thus, we choose multi-objective evolutionary algorithm to solve this problem. This paper deeply analyzes the NSGA-II algorithm and the regional coverage constellation model, then proposes a optimal design of constellation model based on the NSGA-II with the Fast-Non- Dominated-Sorting strategy. According to the different importance of each individual target region, we select some better solutions obtained as the final solutions. This paper is organized as follows: The next section describes the Non-dominated Sorted Genetic Algorithm-II. Then our new proposed constellation optimization method is explained in the third section. The experiment results and analysis are presented in the forth section. The paper is finalized by conclusions and remarks for future developments. 2. NSGA-II NSGA (Non-dominated Sorting Genetic Algorithm) is a multi-objective evolutionary algorithm based on the Pareto optimality, which is put forward by Deb and Srinivas in 1995. The main difference between NSGA and simple genetic algorithm is that they have different selection operator while crossover operator and mutation operator are the same. Before the selection operator, we need to sort the current population according to each individual s non-inferiority and choose those optimal International Journal of Advancements in Computing Technology(IJACT) Volume4, Number21,November 2012 doi: 10.4156/ijact.vol4.issue21.35 295

individuals in terms of the non-inferiority from the current population to constitute the first optimal solutions layer in terms of non-inferiority. Then give it a large assumed fitness value. In order to guarantee the diversity of the population, those optimal solutions share the assumed fitness, and then the original individual s fitness divided by a numerical value which is proportional to the number of the surrounding individuals. According to the new fitness value, the algorithm executes selection operator to make sure that some of optimal solutions coexist in the same population. After sharing, we temporarily ignore these optimal solutions in the first optimal solutions layer, classify the rest individual of the population in the same way to find out the second non-inferiority optimal solutions layer. Assign this layer a new assumed fitness value, the value must be always smaller than the smallest of the shared assumed fitness of those upper layers. Repeat until the whole population dividing is finished. The first version of NSGA has several obvious shortcomings, such as huge computation, lacking of optimal preservation strategy and depending on shared parameters. In order to solve this problem, Deb, et al. put forward the NSGA-II in 2000 to improve the NSGA, and proposed a Fast-Non-Dominated- Sorting strategy, then defined the crowding distance [9] which represents the solution density around a given point to replace the fitness sharing, and defined a order relation n. It means: if non-inferiority layer of I is smaller than J s, or the non-inferiority layer of I is equal with J s while the crowding distance of I is bigger than J s, then I n J. Compared with NSGA, the computational complexity of NSGA-II obviously drops from O(mN 3 ) into O(mN 2 ), m is the number of the objective functions and N is the population scale. The NSGA-II don t need to select sharing parameters, just compare the father with child individual to find out optimal solutions and to realize the optimal preservation strategy finally. The basic idea of the improved NSGA-II is as follows: Firstly, randomly generate initial population with N individuals. We use three basic operations (selection, crossover and mutation) to get the first generation of offspring population after non-dominated sorting. Secondly, from the second generation to start, merge the father generation population with the offspring population, and execute fast-nondominated-sorting, then calculate crowding distance of each individual in non-dominated layer. According to the non-dominated relations and the individual s crowding distance to select some appropriate individuals to constitute a new father generation. Finally, generate a new offspring population by executing the basic operations of genetic algorithm. Repeat the process until the termination condition is met. 2.1. Non-dominated sorting For a population P, we need to compute two parameters named n p and s p of each individual p in population P. n p means the number of individuals which dominates p, and s p is a set of individuals dominated by p. The main steps of the algorithm include: (1) Find out individuals whose n p is zero and save in the current set F i. (2) For each individual i in set F i, those individuals dominated by i make up a set S i. Then go through all individuals in S i, and execute n k =n k -1. If n k =0, save k in set H. (3) Those individuals in F 1 are marked as the first floor non-dominated. H is the current set, repeat this process until the whole population are divided in different levels. 2.2. Crowding distance Crowding distance is the density of the individuals around the given individual. Intuitively, crowding distance means the length of the biggest rectangle which is composed of individuals around i. Mark it i d. In figure.1, we can see that individuals around i are crowded when i d is small. In order to keep the diversity of population, we need a method to compare crowding distance of individuals so as to converge to a equally distributed face based on Pareto optimization. Every individual i in the population have two attributes : the layer of non-dominated irank and the crowding distance id. Then a order relation < n is defined as: if i rank <j rank, or i rank =j rank and i d >j d, i < n j. In other words, if the layers 296

of two individuals are different, we select the individual with the smaller layer. If the layers of two individuals are the same, we select the individual with the smaller crowding distance. Figure 1. Crowding distance of the individual i 3. Proposed constellation design As mentioned above, the optimal design of constellation of multiple regional coverage is a typical multi-objective optimization problem. For the optimal design of the constellation which is based on NSGA-II, the main emphasis and difficulty lie in the constellation modeling, the selection of the target function, and the workflow design. 3.1. Constellation model In theory, satellites may locate in any orbit. A constellation is composed of a series of satellites which are able to accomplish a specific space mission. A satellite can be described by six rail parameters [10-12]. As showed in figure 2: Figure 2. The six rail parameters of a satellite orbit. The shape and the size of the satellite are fixed by the orbital eccentricity e and the rail semi-major axis a. The orbit intersection angle i and the ascending node Ω determine the position of the orbit plane in space, and the argument of perigee ω determines the orientation of the orbit, the parameter mean anomaly M fixes the satellite s position in orbit at a certain moment. For the design of constellation, it should have a better versatility and be able to get a larger interests with a relatively small investment. The constellation model in this paper is simplified. All the satellites of the constellation have the same rail semi-major axis and the eccentricity which is 0. Other parameters need to be optimized. Figure.3 is the simulation of the constellation structure. There are several parameters: the number of the rail surfaces (N), the number of the satellites in each rail surfaces (Q j ), the rail semi-major axis(a), the rail eccentricity(e), the rail inclination(i j ), the argument of perigee(ω j ), the ascending node(ω j ), the mean anomaly(m jk ). Here, k = 1, 2,, Q j. In actual design, the rail surface number and the number of satellites in every rail is certain. There are 10 rail surfaces in our constellation model, and the 297

number of satellites in every rail surface is 1. Here, the rail semi-major axis(a), the rail inclination(i j ), the ascending node(ω j ), the argument of perigee(ω j ), and the mean anomaly(m jk ) are the parameters assumed to be optimized. Figure 3. Constellation structure. 3.2. Objective function For the optimization of regional constellation, due to the asymmetry of the regional coverage problem, it usually adopts grid point statistics method to solve this problem. In order to select some representative observation points from the experimental region to evaluate the coverage performance, generally, we can adopt the minimum longitude, the minimum latitude, the maximum latitude and the maximum longitude to define a rectangle. And then divides the rectangle into grids and treats these grid points as the characteristic points. In order to make each point more representative, and also make its surface area be rough equal, the number of the grid points of the latitude belt should be proportional to the cosine of latitude. We can also give each point a different weight to pay different attention to the surface areas. According to the actual optimization goal, we can evaluate it based on the coverage percentage, the maximum coverage gap and the average coverage gap [13]. This paper adopts the coverage percentage. For example, there are three target areas A, B and C showed in table 1. The design requirements of the satellite constellation is that the covering percentage for the three ground feature points in one day are as large as possible at any running time. As these three goals are contradictory, they can t be the largest simultaneous, but just be as large as possible. Table 1. Earth station A B C longitude E10 ~25 E30 ~50 E65 ~80 latitude N115 ~130 N95 ~110 N60 ~80 In order to judge the coverage performance of the constellation for the region, grid point method is adopted to calculate the coverage. As showed in figure 4, the given area is divided into some grids according to a certain step length, and the grid points are the feature points. Here, the given area is divided base on longitude and latitude. For example, for the target area A, the total grid points N is 16 * 16 (includes the feature points at the border).then the total grid points of area B and area C can be calculated by analogy. If we assume the total number of feature points is n and the number of the feature points covered in the area is m, the constellation coverage of this region is m/n in a given time slot T. The optimization goal of the constellation design is to make m/n as big as possible to meet the practical requirements. 298

Figure 4. Grid point method. For the calculation of the regional coverage, the method is described in detail below: For any one satellite i in the constellation, its width is ω i, and its orbital altitude is h i, in order to calculate the coverage in timeframe T, T is divided into k time slots[t 0, t 1,, t k ], and then calculate the regional coverage for those feature points in every time slot. The value of k should be selected to guarantee that the satellite coverage area is continuous in every two continuous time slots, so that we can assure the calculation precision. GM e k T / wi / (1) Re hi Here, G is the constant of universal gravitation, M e is the earth quality and R e is the earth radius. As shown in figure 5, we can get the substellar point A of satellite at a certain time t j (0<j<k), and then find the nearest feature point B according to A s position. This moment the points can be covered by the satellite must be close to the point B. Therefore, we can expand along the eight directions from point B, then judge whether those extended points are covered by the current satellite or not. If the sphere distance between A and those expanded points is less than the half of the breadth of the satellite, then mark the point covered. Figure 5. Coverage area calculation. The calculation of the coverage of each grid is important for the regional coverage. Figure 6 shows how to calculate it: Figure 6. The coverage of each grid 299

S represents a satellite, B is the subsatellite point of satellite. A is a grid, E is the smallest observing angle. R is the radius of earth. h is the altitude between satellite and earth. means the cover angle of satellite. is the angle between A and B. As showed in the following equations: Rcos E arccos( ) E (2) R h If, it means that the grid A is covered by this satellite. 3.3. Constellation optimization workflow design According to the model design above, there are forty parameters to be optimized as follows: ten orbit intersection angles i 1 -i 10, ten ascending nodes Ω 1 -Ω 10, ten mean anomalies M 1 -M 10, and ten argument of perigees ω 1 -ω 10. In the implementation process of algorithm, a real number coding method is adopted with each chromosome representing a constellation model. Figure 7 shows the structure of chromosome [14-16]. i1 Ω 1 ω 1 M1 i10 Ω 10 ω 10 M10 Figure 7. Chromosome structure. Firstly, initialize constellation parameters and calculate fitness which is the value of the target regional coverage. Then perform selection, mutation, merging, sorting and calculating crowded distance operations repeatedly to generate the next generation population until the termination condition is met. In this algorithm, the coverage calculation of three regions will be translated into three objective functions to accomplish the optimization of the multi-objective function. The optimization process is as showed in table 2: 4. Example analysis Table 2. Optimization process Step Description 1 Chromosome initialization, generation=0. 2 Calculate satellite ephemeris at sampling time 3 Separately calculate the coverage of region A, B and C. 4 Perform the genetic operations of NSGA II. 5 Generate new population 6 Back to step 2 repeat until the termination criterion is met. In order to obtain a total coverage of three target regions A, B, C as big as possible, we need to optimize the parameters of the satellites based on the NSGA-II algorithm. Here, the cycle is one day, the breadth is 200km, and the population size is 30, the number of the iterations is 50, the crossover rate is 0.9, and the mutation rate is 0.01, the number of target regions is 3, the dimension is 40. And then the constellation parameters are showed in table 3. Table 4 shows the initial constellation parameters before optimization. Table 5 ~ 8 show some optimal results selected from the optimization results, and its parameters are converted into radians. Here, F1, F2 and F3 are the coverage rate for A, B and C. Table 3. The list of the constellation control parameters. parameter value parameter value rail face 10 ω [0,360 ] Every rail 1 i satellite [0,180 ] a 1000+R Ω [0,360 ] e 0 M [0,360 ] 300

In order to reduce the complexity of the satellites, our work is based on circular orbit. When the orbital altitude is under 700km, the influence of air damping and oxygen corrosion is pretty large for the satellite, so that the satellite may have a short life. when the orbital altitude is limited in 1500 ~ 5000km, there is a strong damage to the satellite as the existence of the Van Allen belts, thus the satellite orbit should far away from those altitudes [17]. In terms of application, we hope that the track for substellar point of satellite has periodic repeatability to facilitate the coverage analysis. When all of these problems are considered together, we choose orbital altitude to be at 1000km which is located at the bottom of the Van Allen belts. Table 4. Initial satellite constellation parameters (F1=44.92%,F2=59.82%,F3=32.44% ) 1.019602 0.759210 2.711437 0.275326 1.886221 1.748299 2.004178 0.592331 1.698097 0.277279 1.278260 0.166293 1.537317 0.124934 0.691798 0.100500 0.713423 0.552516 1.108932 0.976369 0.398767 2.848674 0.632584 1.283398 0.988344 1.001402 0.551363 0.105748 1.384065 2.663881 0.133667 2.353697 1.142246 1.285160 1.156845 0.853280 1.811185 1.559468 1.208518 2.936888 Table 5. Optimized constellation 1 (F1=96.15%,F2=99.77%,F3=100%) 0.872964 0.566714 6.023618 0.895363 1.078066 3.000481 5.596270 2.124106 1.664039 3.050462 2.261348 0.610072 1.299527 3.938685 3.194788 6.196441 0.935330 2.078213 3.673662 2.336288 0.977094 3.850719 6.152803 4.015298 1.716502 4.751864 6.208554 0.549372 1.298993 4.789619 4.342430 1.454301 1.769483 1.801673 3.784476 4.109620 1.332735 2.401296 2.305266 0.088592 Table 6. Optimized constellation 2 (F1=95.69%,F2=100%,F3=94.64%) 0.878035 5.867843 6.142066 5.330726 1.035226 3.967631 0.912528 1.289790 1.650836 0.930196 5.911079 0.610402 2.409557 3.938685 5.951092 4.853662 0.935330 2.075102 3.642708 2.378338 1.082317 2.509614 6.076089 2.087985 1.994743 4.805012 6.192553 5.459539 1.678740 4.814200 3.019551 1.454281 1.311030 1.693474 2.080464 2.312041 1.173713 2.401296 2.317141 2.515994 301

Table 7. Optimized constellation 3 (F1=100%,F2=96.60%,F3=77.38%) 0.690659 0.484795 5.781982 0.894900 1.062280 3.003871 5.745451 2.267800 1.663928 3.046846 2.096849 0.627889 0.611344 3.938684 0.301315 2.334521 0.520453 2.038031 3.675416 2.375248 0.977094 2.529074 6.146095 4.015298 1.994743 4.839830 6.258147 5.033885 1.646535 4.786975 4.344069 1.453217 2.182355 3.659031 2.123504 4.109620 2.666137 4.069484 2.305266 0.022929 Table 8. Optimized constellation 4 (F1=98.87%,F2=98.64%,F3=99.40%) 2.117215 0.484795 5.781982 0.855769 1.899392 2.994513 5.806041 5.940964 1.637035 3.050501 5.871573 5.565023 0.614646 3.936342 0.393761 2.205716 0.538998 0.045202 0.483585 5.987409 1.635162 2.536090 2.781910 2.340527 1.845319 3.708818 5.304605 5.920744 2.314591 4.786975 4.344839 0.739521 2.057875 1.753261 2.176724 4.090665 1.764172 2.939822 4.375971 2.506605 Figure 8. The coverage percentage of individuals in population. Results analysis: we can see that the constellation after optimization is much better than the initial constellation by comparing the coverage of table 4 to table 5 ~ 8. For example, if we need to focuse on the observation of region A without slacking the observation of B and C, the coverage performance for region A must be better and more important than B or C. Table 5 to 7 show some optimization results which have a better coverage for the most important region to observe without affecting other area s observation performance. The result of table 8 is a optimization result to balance the observation performance for these three regions. Figure 8 shows us that the coverage percentage of individuals in population are very efficient. The value of F1 is greater than 95%. The value of F2 is greater than 90%. The value of F3 is greater than 85%. You can always get satisfied results in the population finally. They all meet the performance requirements for observation. The values of F1, F2 and F3 are the results we need. As the NSGA-II algorithm is suitable for solving multi-objective problems like this, when emergency tasks appear, the algorithm is able to get desired data quickly. As there are many optimization data, you can select some suitable data according to the actual needs. 302

5. Conclusions This paper aims at the optimal design of satellite constellation for multiple regional coverage. We deeply analyzes the NSGA-II algorithm and the regional coverage constellation model, then proposes a optimal design of constellation model based on the NSGA-II with the Fast-Non-Dominated-Sorting strategy after multi-layer screening. Different decision makers can choose the corresponding solution according to the different importance of each target. Although the effectiveness of the method of this paper is ideal at present, our research is still preliminary, we need to find some more accurate methods for the calculation of the regional coverage. There are still a lot of research works, in the future, we will improve the algorithm step by step and do more practice. 6. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 60873107). We wish to thank our senior fellow apprentice Tan Yi for his kindly guidance and help. 7. References [1] Erie Frayssinhes, Alcatel Espace, Investigating New Satellite Constellation Geometries with Genetic Algorithms, AAS/AIAA Astrodynamics Specialist Conference, pp.582-588, 1996. [2] Mason William Janet, Coverstone Carroll Victoria, John Hartmann, Optimal earth orbiting satellite constellation via a pareto genetic algorithm, AAS/AIAA A Strodynam Ics Specialist Conference and Exhibit, pp.169-177, 1998. [3] Wei Zhan, Hanmin Liu, Guangming Dai, Low Earth Orbit Regional Satellite Constellation Design via Self Organization Feature Maps, IJACT, Vol. 4, No. 13, pp. 250-260, 2012. [4] Kai Sun, Zhengyu Yang, Pei Wang, Yingwu Chen, Multi-objective Planning for a Constellation of Agile Earth-Observing Satellites, AISS, Vol. 4, No. 13, pp. 356-364, 2012. [5] Deb Kalyanmoy, Agrawal Samir, Pratap Amrit, A fast elitist non-dominated sorting Genetic algorithm for multi-objective optimization: NSGA-II, Proc of the Parallel Problem Solving from Nature VI, pp.849-858, 2000. [6] Eckart Zitzler, Marco Laumanns, Lothar Thiele, SPEA2:improving the strength Pareto evolutionary algorithm, Computer Engineering and Networks Laboratory(TTK), Switzerland, pp.1-20, 2001. [7] Deb Kalyanmoy, Pratap Amrit, Agrawal Samir, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, vol.6, no.2, pp.182-197, 2002. [8] Williams Edwin, Crossley William, Lang Thomas, Average and maximum revisit time trade studies for satellite constellations using a multiobjective genetic algorithm, AAS /AIAA Space flight mechanics meeting, pp.385-400, 2001. [9] Deb Kalyanmoy, Multi-objective Function Optimization Using Nondominated Sorting Genetic Algorithms, Evolutionary Computation, vol.2, no.3, pp.221-248, 1995. [10] Crossley William, Williams Edwin, Satellite constellation design for zonal coverage using genetic algorithms, Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, pp.443-460, 1998. [11] Mason William Janet, Optimal earth orbiting satellite constellations via a Pareto genetic algorithm, AIAA, pp.169-177, 1998. [12] Ulybyshev Yuri, Geometric analysis of low-earth-orbit satellite communication systems:covering functions, Journal of Spacecraft and Rockets, vol.37, no.3, pp.385-391, 2000. [13] Ulybyshev Yuri, Satellite constellation design for complex coverage, Journal of Spacecraft and Rockets, vol.45, no.4, pp.843-849, 2008. [14] Chen Qifeng, Dai Jinhai, Zang Yukun, Evolutionary algorithm for simultaneously optimization of regional coverage satellite constellation structure and parameters, Systems Engineering and Electronics, vol.26, no.4, pp.549-553, 2004. [15] Wu Tingyong, Wu Shiqi, The Design of Optimized Common-Track Constellation for Regional Coverage, Journal of Electronics&Information Technology, vol.28, no.8, pp.1360-1363, 2008. [16] Chen Rongguang, Li Chunsheng, Chen Jie, Yu Ze, Optimization of near-space aerocraft track for regional coverage based on greedy algorithm, Journal of Beijing University of Aeronautics and Astronautics, vol.35, no.5, pp.547-550, 2009. [17] Dai Guangming, Wang Maocai, Multi-objective evolutionary algorithm and its applications in constellation design, University of Geosciences Press, China, 2009. 303