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A general evaluation criterion for the coverage performance of LEO constellations
Aerospace Science and Technology 48 (2016) 94–101
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A general evaluation criterion for the coverage performance of LEO constellations Yongjun Li a,∗ , Shanghong Zhao a , Jili Wu b a b
School of Information Navigation, Air Force Engineering University, Xi’an 710077, China The Air Force Xi’an Flight Academy, Xi’an 710306, China
a r t i c l e
i n f o
Article history: Received 19 July 2015 Received in revised form 4 November 2015 Accepted 5 November 2015 Available online 10 November 2015
a b s t r a c t A general evaluation criterion for the coverage performance of LEO constellations that can be used for various constellation configurations is proposed. The method transforms the minimum elevation angle characteristics at the specified latitude limits into the standard number of satellites with the same coverage performance, and the ratio with respect to the actual number of satellites in a constellation is defined as the index of coverage performance. Because this index is actually the average coverage level of each satellite, it is suitable for different configurations. Therefore, the optimal configuration can be selected for global coverage or coverage of specific latitudes. Using the evaluation method, the coverage index for three typical Walker-δ constellations was computed; these indexes were: Globalstar, 0.51; Celestri, 0.46; and NeLS, 0.45. Using the general evaluation method, the optimum Walker-δ constellation suitable for China is determined. The final optimum configuration was selected as 46◦ :42/7/0, or an inclination angle of 46◦ , a total of 42 satellites in 7 orbital planes, and a phasing factor of 0. The coverage performance index is as high as 0.566, providing coverage with elevation angles greater than 13◦ in the targeted area at an orbital altitude of 1400 km. This configuration can continuously cover China with a minimum elevation of 20◦ , and the coverage range includes the latitudes from 24.5◦ to 53◦ , which is definitely suitable for China. © 2015 Elsevier Masson SAS. All rights reserved.
1. Introduction LEO satellites have the advantages of low cost, short time delays and low path losses. Constellations of LEO satellites will be an important part of the next generation of global mobile communication networks [1–3]. The Walker-δ constellation developed by J.G. Walker, was initially used for MEO constellations [4] but was subsequently applied to LEO satellites. Many LEO Walker-δ satellite constellations were created in the 1980s, such as Iridium and Globalstar. Teledesic, Celestri, NeLS, SkyBridge [5], and M-star are other examples that are used for multimedia services and integrated broadband applications. In the 2000s, because of the high costs resulting from suboptimal configurations [6] and the competition from terrestrial mobile communication networks, several of these well-known satellite constellations were abandoned. To satisfy the coverage demands, there are several types of satellite constellation configurations. Coverage performance is an important factor in determining the competitiveness of a new constellation. The way to
*
Corresponding author. Tel.: +8602984791632. E-mail address: [email protected] (Y. Li).
http://dx.doi.org/10.1016/j.ast.2015.11.003 1270-9638/© 2015 Elsevier Masson SAS. All rights reserved.
solve the problem of constellation coverage was outlined in the 1960s by way of dividing into continuous and periodic coverage. For continuous coverage of the entire earth or of polar regions extending to arbitrary latitude, optimum satellite configurations are designed. Satellite constellations which minimize the number of satellites required for continuous coverage are derived as a function of the angle subtended at the earth’s center by the coverage of a single satellite [7]. A multi-objective evolutionary optimization tool is used to characterize the design space surrounding Walker’s optimal constellations and tradeoff solutions for both continuous and intermittent global coverage are provided [8]. A geometric approach for complex coverage of a geographic region is proposed. The key idea of the method is a two-dimensional space application for maps of the satellite constellation and coverage requirements. The space dimensions are right ascension of ascending node and argument of latitude. Visibility requirements of each region would be presented as a polygon and satellite constellation as a uniform moving grid [9]. Satellite constellation design for periodic coverage is considered as a unique and separate problem. A new methodology for the earth periodic coverage is brought forward to optimize arbitrary constellations [10]. A novel constellation design method for multi-regional coverage based on Non-dominated Sorted Ge-
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
netic Algorithm is proposed [11]. A new universal method for analysis of maximum revisit times in discontinuous coverage satellite constellations in circular orbits is developed. It can be used for any type of satellite constellations and is applicable to arbitrary sets of satellites [12]. It is difficult, by using traditional estimated indexes, such as the minimum inter-satellite distance or the minimum elevation angle from the receiver, to evaluate and compare constellations with different numbers of satellites and orbital altitudes. It is impossible for traditional estimated indexes to compute the average utilization rate of every satellite in different constellation configurations. Furthermore, if orbital altitude variations, ascending and descending nodes and the total coverage of multiple satellites are taken into account simultaneously, then traditional evaluation methods for constellations are much more complex. Satellite coverage optimization has been subject of multiple studies in the last four decades [13]. However, primary efforts have been focused on constellation design. Constellation coverage evaluating methods vary with constellation configuration. Few general constellation coverage evaluating methods have been reported in the past years. In this paper, a general evaluation criterion for the coverage performance of LEO constellations is proposed. This method can be used for LEO constellations with various numbers of satellites and orbital altitudes. Several representative LEO constellations are evaluated using this method. According to this criterion, 234 different LEO constellations are categorized into 6 constellations with different number of orbits and 12 constellations with maximum single-satellite and dual-satellite coverage. Using the general evaluation method, the optimum Walker-δ constellation suitable for China was determined. 2. Coverage evaluated method of LEO constellations
2.1. Minimum elevation angle A well-chosen constellation with global coverage will cover all the land masses with a certain minimum elevation angle. This minimum elevation angle applies in the low-latitude regions for constellations with polar orbits. The value applies out of the lowlatitude region for constellations with inclined orbits. This minimum elevation is defined as the elevation angle threshold ε :
0≤ ϕ ≤ i
2.2. Defining the coverage zone The elevation angle of a constellation in a polar orbit as observed by a ground station increases with the latitude of the ground station. In polar regions, the elevation angle is maximal. If the elevation angle for a low-latitude zone is set to the minimal threshold, a constellation in polar orbit will achieve global coverage continuously. For a constellation in an inclined orbit, the elevation angle is lower in the equatorial zone, and the elevation angle increases gradually with increasing latitude. Generally, when the latitude is equal to the orbital inclination angle, the elevation angle is maximal. Then, as the latitude continuously increases, the elevation angle decreases rapidly. When the latitude of a region is greater than a certain value, a constellation in an inclined orbit cannot provide coverage for that region. The minimal elevation angle in Eq. (1) is considered as the threshold for which a constellation in an inclined orbit would cover the region between a given south latitude and a given north latitude. The area of this region is:
2π re2 cos(ϕ )dϕ
AG =
(2)
El(ϕ )>ε
where ϕ is the latitude, re is radius of the earth, and A G is the area of coverage for which the elevation angle is greater than the minimum. To include the global heterogeneous distribution of population within the coverage area of the constellation, A G can be weighted by the population density of the earth:
2π re2 cos(ϕ ) P (ϕ )dϕ
A GW =
(3)
El(ϕ )>ε
The evaluation method resembles that for rating a golf course. Every constellation configuration can be considered as a golf course, and the minimum number of satellites required for coverage of the same region is defined as the normal number of satellites, or the “par”. The ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index. The orbital altitude, the number of satellites, the coverage region and the minimum elevation can be unified by the index. Therefore, this index provides an evaluation criterion for the coverage performance of satellite constellations. The basic principles are explained in the following.
ε min El(ϕ )
95
(1)
where i is orbital inclined angle, and El(ϕ ) is the elevation angle at the latitude ϕ . The elevation angle is maximal when the latitude is equal to the orbital inclination angle in Walker-δ constellations. Therefore, the elevation angle threshold ε is the minimum elevation angle for the low-latitude region. If only ascending orbits are used, the actual threshold is appreciably higher than the chosen minimum elevation angle [14–16]. The reason is that when the ascending satellites define the minimum elevation angle, the descending satellites provide a higher minimum elevation angle. In this paper, “minimum elevation angle” is based on the ascending and descending satellites together.
where P (ϕ ) is the normalized population density of the earth, and A GW is the coverage area weighted by the population density, which integrates the population distribution into the coverage performance of the constellation. 2.3. Computing the coverage region of a single satellite The coverage zone of a single satellite can be represented by a spherical cap of the earth’s surface. Coverage zone overlapping is unavoidable for constellations with several satellites. Wellproportioned coverage is preferred, so the smaller the overlap, the better the coverage performance. Among all of the inscribed regular polygons for a circle, the maximal polygon that provides seamless splicing is the planar regular hexagon. The spherical cap representing the coverage of a single satellite can be considered to be a spherical regular hexagon, which is approximately depicted in Fig. 1. Obviously, unlike a planar regular hexagon, a spherical regular hexagon cannot provide seamless splicing. However, the coverage area of a LEO satellite is approximately ten percent of the earth’s surface. The corresponding internal angles of a spherical regular hexagon are appreciably greater than 120◦ , but the overlap between spherical regular hexagons is a small part of the entire coverage zone, so the overlapping parts are neglected. If the minimal elevation angle ε is a constraint, the radius of the coverage circle of a satellite with altitude h is:
θ = arccos
re cos ε
re + h
−ε
(4)
where θ is the spherical distance. The spherical area of an inscribed polygon of this spherical cap is determined by the internal angle of the polygon. As shown in Fig. 1, using the law of sines for a spherical triangle SBC, the arc BC can be expressed as:
BC = arcsin sin θ sin
π 6
(5)
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Table 1 Coverage parameters with single-satellite visibility for four LEO constellations.
ε (deg.) AG AS NS I cov
Iridium
Globalstar
Celestri
NeLS
7.80 1 0.026 38.8 0.59
12.64 0.920 0.038 24.5 0.51
16.45 0.866 0.030 29.0 0.46
20.24 0.951 0.018 53.9 0.45
3.1. Single-satellite visibility coverage performance
Fig. 1. Coverage zones of satellites.
According to the law of cosines for a spherical triangle, half of the internal angle of the inscribed spherical polygon Φ is:
Φ = arccos
cos(θ)[1 − cos(2BC )] sin(θ) sin(2BC )
(6)
The area of the spherical triangle SAC is:
A SAC = 2Φ −
2 3
π
(7)
The coverage area of a single satellite is:
A S = 12Φ − 4π
(8)
2.4. Defining the coverage performance index A baseline configuration is formed based on the coverage area of a single satellite, as defined in Section 2.3. In this baseline configuration, all the satellites are distributed above the earth uniformly and statically, and the spherical regular hexagons covered by adjacent satellites fit together seamlessly, as depicted in Fig. 1. The coverage area and the minimum elevation angle of this baseline configuration are the same as the actual constellation. The number of satellites in the baseline configuration is defined as the normal number of satellites in the actual constellation:
Ns =
AG
(9)
As
Finally, the ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index:
I cov =
Ns
(10)
N
where N is the actual number of satellites in the constellation. The coverage performance of single-satellite visibility can be computed from the given formulation, and the coverage performance of k satellites visibility in a constellation can be expressed as:
I cov (k) =
kN s N
(11)
3. Evaluation of coverage performance of representative LEO constellations Making use of the evaluation index defined in Section 2, the coverage performance values of representative LEO constellations including Iridium, Globalstar, NeLS and Celestri are computed.
The coverage performance parameters of four LEO constellations are given in Table 1, where ε is the elevation angle threshold of single-satellite visibility, which is a crucial parameter in constellation design. However, the actual value is not used here. Making use of simulations, the minimum elevation angle for single-satellite visibility is obtained by statistical methods. Then, according to equation (1), the elevation angle threshold can be obtained. The elevation angle threshold may be set in advance to satisfy various coverage demands. In Table 1, the minimum elevation angle of Iridium is 7.8◦ , and its corresponding geographical position is at the equator. If this minimum elevation angle is considered the elevation angle threshold, Iridium can cover the entire globe continuously, so its coverage area is 1. When the satellite orbital altitude is 778 km, the area of the inscribed regular spherical hexagon defining the single-satellite coverage is 0.026 (with an elevation angle threshold of 7.8◦ ). Therefore, the normal number of satellites for Iridium is 38.8, and its corresponding coverage index for single-satellite visibility is 0.59. Globalstar can cover the region within ±67◦ of latitude continuously with a minimum elevation angle of 12.64◦ , and its coverage area is 92 percent of surface area of the earth. The area of the inscribed regular spherical hexagon for single-satellite coverage is 0.0376. Therefore, the normal number of satellites is 24.5, and the coverage index for single-satellite visibility is 0.51. Celestri can cover the area within ±60◦ of latitude continuously with a minimum elevation angle of 16.45◦ , its corresponding A G is 0.866, and its I cov is 0.46. NeLS can cover the region within ±72◦ of latitude continuously with a minimum elevation angle of 20.24◦ with A G = 0.951, A S = 0.0176, and I cov = 0.45. 3.2. Dual-satellite visibility coverage performance The minimum elevation angle versus latitude with dual-satellite visibility is depicted in Fig. 2. The satellite density of a constellation in polar orbit above the polar regions is much greater than that in the equatorial region. Because of the unequal distribution of satellites above the earth’s surface, Iridium cannot provide dualsatellite visibility coverage for regions within ±50◦ latitude. The dual-satellite visibility coverage performance of Iridium will not be discussed in this paper. The coverage performance parameters with dual-satellite visibility for the other three LEO constellations are listed in Table 2. Globalstar can cover the region within ±55◦ latitude continuously with a minimum elevation angle of 5.01◦ , and its coverage area is 81.9 percent of surface area of the earth. The coverage index with dual-satellite visibility is 0.58. However, this minimum elevation angle is too small to provide reception diversity. If a slightly larger elevation angle such as 10◦ is chosen, the coverage area is 36 percent of surface area of the earth. The corresponding I cov is only 0.34. It is because of the small number of satellites and the higher orbital inclination that the coverage performance with dualsatellite visibility of Globalstar is poor; this will be discussed in detail later.
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Fig. 3. Coverage performance versus orbital inclination angle.
Fig. 2. Minimum elevation angle versus latitude with dual-satellite visibility. Table 2 Coverage parameters with dual-satellite visibility for three LEO constellations.
ε2 (deg.) A G2 A S2 N S2 I cov (2)
Globalstar
Celestri
NeLS
5.01 0.819 0.058 28 0.58
12.85 0.829 0.037 45 0.71
16.51 0.951 0.022 86 0.71
Celestri and NeLS have orbital altitudes and numbers of satellites that enable them to provide continuous dual-satellite visibility coverage for nearly all regions of the earth. Celestri can cover the region within ±56◦ latitude with a minimum elevation angle of 12.86◦ . NeLS can provide dual-satellite visibility coverage for the region within ±72◦ latitude with a minimum elevation angle of 16.51◦ . The I cov of both Celestri and NeLs is 0.71, which is larger than the 0.58 of Globalstar. 4. Improvement of representative LEO constellation configurations 4.1. Iridium constellation A polar-orbiting constellation can cover all the earth. Its coverage index lies on the minimum elevation angle merely. The Iridium constellation has been optimized for the minimum elevation angle, so the coverage index with single-satellite visibility is optimal, as depicted in Fig. 3 and Fig. 4. In the two figures, the constellation has the same number of orbits and number of satellites. Fig. 3 shows the coverage performance versus the orbital inclination, and Fig. 4 shows the coverage performance versus the phasing between two neighboring satellites. The coverage performance is practically constant with the orbital inclination, which varies from 80◦ to 88◦ in Fig. 3. In Fig. 4, when the phasing is 15◦ , the coverage performance index is optimal, which is approximately the same as the value for the actual Iridium system (16.4◦ ). Based on these two figures, the coverage performance of Iridium with single-satellite visibility is optimal. The coverage performance of Iridium with dual-satellite visibility will not be investigated for the reason mentioned previously. 4.2. Globalstar constellation The initial design goal of Globalstar was to service the entire North America region, including Alaska, so the orbital inclination was chosen to be 52◦ . The minimum elevation angle of Globalstar is greater than 10◦ for coverage of the region within ±72◦ latitude. Globalstar can service the densely populated region near 30◦ latitude with dual-satellite visibility. The coverage performance index
Fig. 4. Coverage performance versus phasing.
of Globalstar may be improved significantly. In Fig. 5, the coverage performance index is plotted, where key parameters including the number of orbits and the number of satellites are the same as Globalstar. The differences are in the orbital inclination and the phasing. The x-axis is the orbital inclination angle, and the y-axis is the coverage performance index for single-satellite visibility. The various curves correspond to different values of the phasing factor F . Even phasing factors are omitted because they would result in satellites colliding. The configuration denoted with a diamond is better than the other two configurations, so the optimal phasing factor is 1, which is consistent with the actual value for Globalstar. However, the optimal orbital inclination is less than 45◦ , unlike the actual value of 52◦ . The coverage performance index for inclinations between 40◦ and 48◦ and a phasing factor of 1 is magnified in the top right corner of Fig. 5. It can be observed that the optimal orbital inclination is 42◦ . The difference in the coverage performance index between 42◦ and 45◦ is less than 0.5%. Taking into account of the effect of computing precision, 45◦ is a reasonable choice because of its wider coverage area. The improved constellation configuration of Globalstar is: 45◦ :48/8/1. The minimum elevation angle versus latitude of the improved and initial constellations are plotted in Fig. 6, where the curves plotted with diamonds and circles represent the single-satellite and dual-satellite visibility coverage of the improved configuration, respectively, and the solid and dashed curves represent the single-satellite and dualsatellite visibility coverage of the initial configuration, respectively. The coverage performance of the improved configuration is better at latitudes less than 40◦ . The minimum elevation angle with single-satellite visibility increased from 12.64◦ to 17.21◦ . However, the cut-off latitude of the improved configuration for high-latitude regions decreased from 67◦ to 56◦ . Correspondingly, the coverage area of the improved configuration is 83 percent of entire surface area of the earth. For single-satellite visibility coverage only, the minimum elevation angle of the improved configuration increased by 5◦ with a loss of coverage area of approximately 10%. Because of the increase in the minimum elevation angle, the coverage area of a single satellite decreased to 2.89 percent of the
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Fig. 5. Coverage performance index of Globalstar configuration.
Fig. 6. Elevation angle characteristics of initial and improved Globalstar configurations.
surface area of the earth. At least 29 satellites would be required to provide equivalent coverage in the baseline configuration. The I cov of the improved constellation is 0.58. If the actual minimum elevation angle of Globalstar is used, the cut-off latitude of the improved configuration would be reduced from 70◦ to 63◦ . The corresponding increase in the coverage area would be 4.8 percent of the surface area of the earth; however, the additional population covered by this increased area is less than 1 percent. In summary, the improved configuration transforms the excess coverage in highlatitude regions with the initial constellation configuration to lowand medium-latitude regions, and the coverage performance is improved as a whole. 4.3. Celestri constellation Celestri is designed for broadband and multimedia services, which is different goal than those of Iridium and Globalstar. Therefore, the dual-satellite visibility coverage is one of the most dominant factors for Quality of Service. The coverage performance index versus latitude is depicted in Fig. 7, where the number of orbits and the number of satellites are the same as those for the actual Celestri constellation. In Fig. 7, the x-axis is the orbital inclination and the y-axis is the coverage performance index (I cov ), and the various curves are for different phasing factors. From Fig. 7(a), the optimal configuration for single-satellite visibility is 56◦ :63/7/0; i.e., the orbital inclination is 56◦ , the number of satellites is 63, the number of orbits is 7 and the phasing factor is 0. This configuration will be denoted by Celestri-A. The optimal constellation configuration for dual-satellite visibility is 48◦ :63/7/4, as can be found in Fig. 7(b). This configuration will be denoted by Celestri-B. The minimum elevation angle versus the latitude of the initial and the two improved Celestri configurations are depicted in Fig. 8. The detailed performance parameters of the three configurations with single-satellite and dual-satellite visibility are shown in Table 3 and Table 4, respectively. In addition, the cut-off latitude L cut was computed.
Fig. 7. Coverage performance index of Celestri configurations versus inclination.
As shown in Fig. 8(a) and Table 3, the cut-off latitude of Celestri-A increased from 60◦ to 70◦ and the coverage area increased from 0.866 to 0.945 when compared with the initial Celestri constellation. The coverage performance index I cov for singlesatellite visibility increased to 0.49, but the satellites are much more dense in the higher-latitude regions with Celestri-A. Consequently, the coverage for lower-latitude regions is reduced, which is clearly shown in Fig. 8(b) and in Table 4. The minimum elevation angle of Celestri-A in the lower-latitude regions was approximately 3◦ smaller. Furthermore, the minimum elevation angle with dualsatellite visibility was lower than that of the initial configuration in the region below 45◦ latitude. Although the coverage area of Celestri-A increased by 10 percent of the global surface area, the coverage performance index for dual-satellite visibility decreased by 0.68. The results for Celestri-B are opposite those for Celestri-A. The coverage performance with dual-satellite visibility improved in the lower-latitude regions. However, the coverage performance improved in the regions near 50◦ latitude. The elevation angle ε2 increased from 12.86◦ to 12.97◦ , and the index I cov(2) increased from 0.71 to 0.73, but the single-satellite visibility coverage performance of Celestri-B was worse than that of the other two configurations. Especially in the region of ±5◦ latitude, the minimum elevation angle was less than 15◦ , so the coverage performance index with single-satellite visibility increased to 0.4. In summary, the initial Celestri configuration is a tradeoff between the Celestri-A and Celestri-B configurations. 4.4. NeLS constellation The NeLS constellation was optimized in [17,18]. The orbital inclination was optimized to maximize the distance between two neighboring satellites to guarantee a uniform distribution of the satellites. Therefore, the coverage performance was improved. In
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
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Fig. 8. Minimum elevation angle versus latitude of initial and improved configurations of Celestri. Table 3 Single-satellite visibility coverage performance.
ε (◦ )
L cut (◦ ) AG AS NS I cov
Celestri
Celestri-A
Celestri-B
16.45 60 0.866 0.030 29.0 0.46
15.98 71 0.945 0.031 30.9 0.49
14.36 59 0.857 0.034 25.5 0.40
Celestri
Celestri-A
Celestri-B
16.45 60 0.866 0.030 29.0 0.46
15.98 71 0.945 0.031 30.9 0.49
14.36 59 0.857 0.034 25.5 0.40
Table 4 Dual-satellite visibility coverage performance.
ε (◦ )
L cut (◦ ) AG AS NS I cov
the optimization process, only ascending-orbit satellites are used. Introducing descending-orbit satellites can optimize the dualsatellite visibility coverage performance because of the symmetry of a 2π constellation. This change can provide seamless broadband and multimedia services. This idea may be feasible in large-scale constellations because each satellite can quickly meet another satellite traveling in the opposite direction in a short distance. When the ascending-orbit satellites are optimized, the entire satellite configuration is optimal for dual-satellite visibility coverage. The improved configuration of NeLS is close to the optimal configuration in the coverage performance index for both single-satellite and dual-satellite visibility, as depicted in Fig. 9, where F = 0, the inclination is 60◦ , and the configuration denoted by the diamonds is most similar to the actual NeLS configuration. 5. A LEO constellation design for China Because mobile satellite communication design and development is ongoing in China, the cost and the risk should be minimized as much as possible. The territory of China, including the ocean, is mostly located in the low and medium latitudes of the northern hemisphere. Therefore, the coverage demand in the highlatitude regions may be reduced, and a cutoff latitude of 60◦ is sufficient to cover all of the territory. Thus, the coverage region is limited to the range of ±60◦ latitude. The single-satellite coverage area can be computed based on the minimum elevation angle of this limited region. The coverage performance index may be computed to find the most suitable LEO constellation configuration for China. In simulations, the orbit number and the number of satel-
Fig. 9. Coverage performance index of NeLS configurations versus inclination.
lites ranged from 6 to 11, and the orbital inclination was varied from 40◦ to 60◦ with a step size of 1◦ . All possible phasing factors were used to obtain all of the possible configurations. The possible configurations are listed in Table 5. The corresponding coverage performance indexes are shown in Table 6 and Table 7. Based on the simulation results, the three configurations with the best coverage performance are, in descending order: P1: 47◦ :48/8/1 P2: 46◦ :42/7/0 P3: 48◦ :54/9/2 The coverage performance indexes of the three configurations are all greater than 0.55, which is more than those of other LEO constellations. The coverage performance index of P1 is the best. Its corresponding configuration is similar to that of Globalstar, except for an inclination difference of 5◦ . The second-best configuration, P2, has an inclination of 46◦ , an orbit number of 7, the
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Table 5 The most optimal constellation configurations (suitable for region with latitude of ±60◦ ). M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
10
F F F F F F
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 I = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Table 6 Single-satellite visibility coverage performance indexes of the most optimal configurations. M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
F F F F F F
10 F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 i = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Table 7 Dual-satellite visibility coverage performance indexes of the most optimal configurations. M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
number of satellites in each orbit is 6, and the phasing factor is 0. The single-satellite-visibility coverage performance index of P2 is slightly lower than that of P1. However, the dual-satellite-visibility coverage performance index of P1 is only 0.48, as is that of Globalstar. Therefore, the seamless handoff of access links and channel diversity would be affected. The dual-satellite-visibility coverage performance index of P2 is only 0.57, which is 20% greater than that of P1. However, the lower orbit number and fewer satellites would be important in reducing the construction and maintenance costs. P3 is not as good as P2, not only in the coverage performance but also in the number of satellites. In conclusion, P2 is a reasonable configuration for a mobile-communication satellite constellation for China. The minimum elevation angle of P2 is plotted in Fig. 10. This configuration can continuously cover China with a minimum elevation of 20◦ , and the coverage range is from 24.5◦ to 53◦ latitude. Taking the minimum elevation angle of 13◦ in the low-latitude region of P2 as the elevation threshold, P2 can provide continuous coverage for the region of ±60◦ latitude. The minimum elevation angle of 13◦ is greater than that of Globalstar in the equatorial region. Because Globalstar can achieve seamless handoffs of mobile users, it is possible for P2 to provide continuous communication and seamless handoffs for ground users. Compared with the global coverage of Iridium and the 92% coverage of Globalstar, P2 can cover only 86% of the earth. However, the coverage area of P2 includes 99% of the settled populations of the earth. Thus, P2 may not only satisfy the demand of China but also many foreign users. In summary, P2, with an orbital altitude of 1400 km, is recommended as a feasible mobile-communication satellite constellation for China.
F F F F F F
10 F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 i = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Fig. 10. Minimum elevation angle versus latitude of P2.
6. Conclusions To address the problem of evaluating potential LEO satellite constellations, a general evaluation criterion for the coverage performance of LEO constellations was proposed. The method resembles that of rating a golf course. Every constellation configuration can be considered as a golf course, where the minimum number of satellites required for coverage for the same region is defined as the normal number of satellites, or the “par”. The ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index. The orbital altitude, the number of satellites, the coverage zone and the minimum el-
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
evation angle can be unified by the index. Therefore, this index provides a standard evaluation criterion for the coverage performance of constellation configurations. Using this general evaluation method, the optimum Walker-δ constellation suitable for China was determined. The optimal configuration was selected as 46◦ :42/7/0, in which the orbital inclination is 46◦ , the number of satellites is 42, the orbit number is 7 and the phasing factor is 0. The coverage performance index of this configuration can reach 0.566. The minimum elevation angle is 20◦ between 24.5◦ and 53◦ latitude, which is suitable for China. Conflict of interest statement No conflict of interest. Acknowledgements This research was supported by the Natural Science Foundation of Shaanxi Province (No. 2015JM6360). References [1] I.F. Akyildiz, E. Ekici, G. Yue, A distributed multicast routing scheme for multiLayered satellite IP networks, Wirel. Netw. 9 (5) (2003) 535–544. [2] Wu Ting-yong, Wu Shi-qi, Performance analysis of the inter-layer inter-satellite link establishment strategies in two-tier LEO/MEO satellite networks, J. Electron. Inf. Technol. 30 (1) (2008) 67–71. [3] T. Taleb, D. Mshimo, A. Jamalipour, Explicit load balancing technique for NGEO satellite IP networks with on-board processing capabilities, IEEE/ACM Trans. Netw. 17 (1) (2009) 281–293. [4] John G. Walker, Continuous whole-Earth coverage by circular-orbit satellite patterns, Royal Aircraft Establishment technical report, 77044, Royal Aircraft Establishment, 1977.
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[5] Pierre Fraise, Bernard Coulomb, Bertrand Monteuuis, SkyBridge LEO satellites: optimized for broadband communications in the 21st century, in: IEEE Aerospace Conference [C], IEEE, Big Sky, Montana, 2000, pp. 241–251. [6] Xu Jia, Han Chao, Improved evaluation method of service availability for satellite constellation, Chin. Space Sci. Technol. 5 (2007) 44–49. [7] D.C. Beste, Design of satellite constellations for optimal continuous coverage, IEEE Trans. Aerosp. Electron. Syst. 14 (3) (1978) 466–473. [8] William R. Whittecar, Matthew P. Ferringer, Global coverage constellation design exploration using evolutionary algorithms, in: AIAA/AAS Astrodynamics Specialist Conference, AIAA, San Diego, CA, 2014, pp. 1–9. [9] Yuri Ulybyshev, Satellite constellation design for complex coverage, J. Spacecr. Rockets 45 (4) (2008) 843–848. [10] Yury N. Razoumny, New research methodology for earth periodic coverage and regularities in parametric localization of optimal low-earth-orbit satellite constellations, in: 2013 AAS/AIAA Astrodynamics Specialist Conference, AIAA, SC, United States, 2013, pp. 3117–3136. [11] Hung Xiaqian, Dai Guangming, An Optimal design of constellation of multiple regional coverage based on NSGA-II, Int. Adv. Comput. Technol. 4 (21) (2012) 313–317. [12] Yuri Ulybyshev, A general analysis method for discontinuous coverage satellite constellations, in: AIAA/AAS Astrodynamics Specialist Conference, AIAA, San Diego, CA, 2014, pp. 1–13. [13] Ignasi Lluch, Alessandro Golkar, Satellite-to-satellite coverage optimization approach for opportunistic inter-satellite links, in: 2014 IEEE Aerospace Conference, IEEE, Big Sky, Montana, 2014, pp. 1–13. [14] Li Yong-jun, Construction Designing and Key Technologies Research of Leo/Meo Optical Satellite Network, AFEU, Shaanxi Xi’an, 2009. [15] Ryutaro Suzuki, Yasuhiko Yasuda, Study on ISL network structure in LEO satellite communication systems, Acta Astronaut. 61 (2007) 648–658. [16] Li Yongjun, Wu Jili, Zhao Shanghong, et al., A novel two-layered optical satellite network of LEO/MEO with Zero phase factor, Sci. China, Ser. F, Inf. Sci. 53 (6) (2010) 1261–1276. [17] Ryutaro Suzuki, Yasuhiko Yasuda, Study on ISL network structure in LEO satellite communication systems, Acta Astronaut. 61 (2007) 648–658. [18] R. Suzuki, R. Nishiyama, S. Motoyoshi, A study of constellation for LEO satellite network, in: The 22nd International Communications Satellite Systems Conference and Exhibit, AIAA, Monterey, California, 2004, pp. 1–10.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
A general evaluation criterion for the coverage performance of LEO constellations Yongjun Li a,∗ , Shanghong Zhao a , Jili Wu b a b
School of Information Navigation, Air Force Engineering University, Xi’an 710077, China The Air Force Xi’an Flight Academy, Xi’an 710306, China
a r t i c l e
i n f o
Article history: Received 19 July 2015 Received in revised form 4 November 2015 Accepted 5 November 2015 Available online 10 November 2015
a b s t r a c t A general evaluation criterion for the coverage performance of LEO constellations that can be used for various constellation configurations is proposed. The method transforms the minimum elevation angle characteristics at the specified latitude limits into the standard number of satellites with the same coverage performance, and the ratio with respect to the actual number of satellites in a constellation is defined as the index of coverage performance. Because this index is actually the average coverage level of each satellite, it is suitable for different configurations. Therefore, the optimal configuration can be selected for global coverage or coverage of specific latitudes. Using the evaluation method, the coverage index for three typical Walker-δ constellations was computed; these indexes were: Globalstar, 0.51; Celestri, 0.46; and NeLS, 0.45. Using the general evaluation method, the optimum Walker-δ constellation suitable for China is determined. The final optimum configuration was selected as 46◦ :42/7/0, or an inclination angle of 46◦ , a total of 42 satellites in 7 orbital planes, and a phasing factor of 0. The coverage performance index is as high as 0.566, providing coverage with elevation angles greater than 13◦ in the targeted area at an orbital altitude of 1400 km. This configuration can continuously cover China with a minimum elevation of 20◦ , and the coverage range includes the latitudes from 24.5◦ to 53◦ , which is definitely suitable for China. © 2015 Elsevier Masson SAS. All rights reserved.
1. Introduction LEO satellites have the advantages of low cost, short time delays and low path losses. Constellations of LEO satellites will be an important part of the next generation of global mobile communication networks [1–3]. The Walker-δ constellation developed by J.G. Walker, was initially used for MEO constellations [4] but was subsequently applied to LEO satellites. Many LEO Walker-δ satellite constellations were created in the 1980s, such as Iridium and Globalstar. Teledesic, Celestri, NeLS, SkyBridge [5], and M-star are other examples that are used for multimedia services and integrated broadband applications. In the 2000s, because of the high costs resulting from suboptimal configurations [6] and the competition from terrestrial mobile communication networks, several of these well-known satellite constellations were abandoned. To satisfy the coverage demands, there are several types of satellite constellation configurations. Coverage performance is an important factor in determining the competitiveness of a new constellation. The way to
*
Corresponding author. Tel.: +8602984791632. E-mail address: [email protected] (Y. Li).
http://dx.doi.org/10.1016/j.ast.2015.11.003 1270-9638/© 2015 Elsevier Masson SAS. All rights reserved.
solve the problem of constellation coverage was outlined in the 1960s by way of dividing into continuous and periodic coverage. For continuous coverage of the entire earth or of polar regions extending to arbitrary latitude, optimum satellite configurations are designed. Satellite constellations which minimize the number of satellites required for continuous coverage are derived as a function of the angle subtended at the earth’s center by the coverage of a single satellite [7]. A multi-objective evolutionary optimization tool is used to characterize the design space surrounding Walker’s optimal constellations and tradeoff solutions for both continuous and intermittent global coverage are provided [8]. A geometric approach for complex coverage of a geographic region is proposed. The key idea of the method is a two-dimensional space application for maps of the satellite constellation and coverage requirements. The space dimensions are right ascension of ascending node and argument of latitude. Visibility requirements of each region would be presented as a polygon and satellite constellation as a uniform moving grid [9]. Satellite constellation design for periodic coverage is considered as a unique and separate problem. A new methodology for the earth periodic coverage is brought forward to optimize arbitrary constellations [10]. A novel constellation design method for multi-regional coverage based on Non-dominated Sorted Ge-
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
netic Algorithm is proposed [11]. A new universal method for analysis of maximum revisit times in discontinuous coverage satellite constellations in circular orbits is developed. It can be used for any type of satellite constellations and is applicable to arbitrary sets of satellites [12]. It is difficult, by using traditional estimated indexes, such as the minimum inter-satellite distance or the minimum elevation angle from the receiver, to evaluate and compare constellations with different numbers of satellites and orbital altitudes. It is impossible for traditional estimated indexes to compute the average utilization rate of every satellite in different constellation configurations. Furthermore, if orbital altitude variations, ascending and descending nodes and the total coverage of multiple satellites are taken into account simultaneously, then traditional evaluation methods for constellations are much more complex. Satellite coverage optimization has been subject of multiple studies in the last four decades [13]. However, primary efforts have been focused on constellation design. Constellation coverage evaluating methods vary with constellation configuration. Few general constellation coverage evaluating methods have been reported in the past years. In this paper, a general evaluation criterion for the coverage performance of LEO constellations is proposed. This method can be used for LEO constellations with various numbers of satellites and orbital altitudes. Several representative LEO constellations are evaluated using this method. According to this criterion, 234 different LEO constellations are categorized into 6 constellations with different number of orbits and 12 constellations with maximum single-satellite and dual-satellite coverage. Using the general evaluation method, the optimum Walker-δ constellation suitable for China was determined. 2. Coverage evaluated method of LEO constellations
2.1. Minimum elevation angle A well-chosen constellation with global coverage will cover all the land masses with a certain minimum elevation angle. This minimum elevation angle applies in the low-latitude regions for constellations with polar orbits. The value applies out of the lowlatitude region for constellations with inclined orbits. This minimum elevation is defined as the elevation angle threshold ε :
0≤ ϕ ≤ i
2.2. Defining the coverage zone The elevation angle of a constellation in a polar orbit as observed by a ground station increases with the latitude of the ground station. In polar regions, the elevation angle is maximal. If the elevation angle for a low-latitude zone is set to the minimal threshold, a constellation in polar orbit will achieve global coverage continuously. For a constellation in an inclined orbit, the elevation angle is lower in the equatorial zone, and the elevation angle increases gradually with increasing latitude. Generally, when the latitude is equal to the orbital inclination angle, the elevation angle is maximal. Then, as the latitude continuously increases, the elevation angle decreases rapidly. When the latitude of a region is greater than a certain value, a constellation in an inclined orbit cannot provide coverage for that region. The minimal elevation angle in Eq. (1) is considered as the threshold for which a constellation in an inclined orbit would cover the region between a given south latitude and a given north latitude. The area of this region is:
2π re2 cos(ϕ )dϕ
AG =
(2)
El(ϕ )>ε
where ϕ is the latitude, re is radius of the earth, and A G is the area of coverage for which the elevation angle is greater than the minimum. To include the global heterogeneous distribution of population within the coverage area of the constellation, A G can be weighted by the population density of the earth:
2π re2 cos(ϕ ) P (ϕ )dϕ
A GW =
(3)
El(ϕ )>ε
The evaluation method resembles that for rating a golf course. Every constellation configuration can be considered as a golf course, and the minimum number of satellites required for coverage of the same region is defined as the normal number of satellites, or the “par”. The ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index. The orbital altitude, the number of satellites, the coverage region and the minimum elevation can be unified by the index. Therefore, this index provides an evaluation criterion for the coverage performance of satellite constellations. The basic principles are explained in the following.
ε min El(ϕ )
95
(1)
where i is orbital inclined angle, and El(ϕ ) is the elevation angle at the latitude ϕ . The elevation angle is maximal when the latitude is equal to the orbital inclination angle in Walker-δ constellations. Therefore, the elevation angle threshold ε is the minimum elevation angle for the low-latitude region. If only ascending orbits are used, the actual threshold is appreciably higher than the chosen minimum elevation angle [14–16]. The reason is that when the ascending satellites define the minimum elevation angle, the descending satellites provide a higher minimum elevation angle. In this paper, “minimum elevation angle” is based on the ascending and descending satellites together.
where P (ϕ ) is the normalized population density of the earth, and A GW is the coverage area weighted by the population density, which integrates the population distribution into the coverage performance of the constellation. 2.3. Computing the coverage region of a single satellite The coverage zone of a single satellite can be represented by a spherical cap of the earth’s surface. Coverage zone overlapping is unavoidable for constellations with several satellites. Wellproportioned coverage is preferred, so the smaller the overlap, the better the coverage performance. Among all of the inscribed regular polygons for a circle, the maximal polygon that provides seamless splicing is the planar regular hexagon. The spherical cap representing the coverage of a single satellite can be considered to be a spherical regular hexagon, which is approximately depicted in Fig. 1. Obviously, unlike a planar regular hexagon, a spherical regular hexagon cannot provide seamless splicing. However, the coverage area of a LEO satellite is approximately ten percent of the earth’s surface. The corresponding internal angles of a spherical regular hexagon are appreciably greater than 120◦ , but the overlap between spherical regular hexagons is a small part of the entire coverage zone, so the overlapping parts are neglected. If the minimal elevation angle ε is a constraint, the radius of the coverage circle of a satellite with altitude h is:
θ = arccos
re cos ε
re + h
−ε
(4)
where θ is the spherical distance. The spherical area of an inscribed polygon of this spherical cap is determined by the internal angle of the polygon. As shown in Fig. 1, using the law of sines for a spherical triangle SBC, the arc BC can be expressed as:
BC = arcsin sin θ sin
π 6
(5)
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Table 1 Coverage parameters with single-satellite visibility for four LEO constellations.
ε (deg.) AG AS NS I cov
Iridium
Globalstar
Celestri
NeLS
7.80 1 0.026 38.8 0.59
12.64 0.920 0.038 24.5 0.51
16.45 0.866 0.030 29.0 0.46
20.24 0.951 0.018 53.9 0.45
3.1. Single-satellite visibility coverage performance
Fig. 1. Coverage zones of satellites.
According to the law of cosines for a spherical triangle, half of the internal angle of the inscribed spherical polygon Φ is:
Φ = arccos
cos(θ)[1 − cos(2BC )] sin(θ) sin(2BC )
(6)
The area of the spherical triangle SAC is:
A SAC = 2Φ −
2 3
π
(7)
The coverage area of a single satellite is:
A S = 12Φ − 4π
(8)
2.4. Defining the coverage performance index A baseline configuration is formed based on the coverage area of a single satellite, as defined in Section 2.3. In this baseline configuration, all the satellites are distributed above the earth uniformly and statically, and the spherical regular hexagons covered by adjacent satellites fit together seamlessly, as depicted in Fig. 1. The coverage area and the minimum elevation angle of this baseline configuration are the same as the actual constellation. The number of satellites in the baseline configuration is defined as the normal number of satellites in the actual constellation:
Ns =
AG
(9)
As
Finally, the ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index:
I cov =
Ns
(10)
N
where N is the actual number of satellites in the constellation. The coverage performance of single-satellite visibility can be computed from the given formulation, and the coverage performance of k satellites visibility in a constellation can be expressed as:
I cov (k) =
kN s N
(11)
3. Evaluation of coverage performance of representative LEO constellations Making use of the evaluation index defined in Section 2, the coverage performance values of representative LEO constellations including Iridium, Globalstar, NeLS and Celestri are computed.
The coverage performance parameters of four LEO constellations are given in Table 1, where ε is the elevation angle threshold of single-satellite visibility, which is a crucial parameter in constellation design. However, the actual value is not used here. Making use of simulations, the minimum elevation angle for single-satellite visibility is obtained by statistical methods. Then, according to equation (1), the elevation angle threshold can be obtained. The elevation angle threshold may be set in advance to satisfy various coverage demands. In Table 1, the minimum elevation angle of Iridium is 7.8◦ , and its corresponding geographical position is at the equator. If this minimum elevation angle is considered the elevation angle threshold, Iridium can cover the entire globe continuously, so its coverage area is 1. When the satellite orbital altitude is 778 km, the area of the inscribed regular spherical hexagon defining the single-satellite coverage is 0.026 (with an elevation angle threshold of 7.8◦ ). Therefore, the normal number of satellites for Iridium is 38.8, and its corresponding coverage index for single-satellite visibility is 0.59. Globalstar can cover the region within ±67◦ of latitude continuously with a minimum elevation angle of 12.64◦ , and its coverage area is 92 percent of surface area of the earth. The area of the inscribed regular spherical hexagon for single-satellite coverage is 0.0376. Therefore, the normal number of satellites is 24.5, and the coverage index for single-satellite visibility is 0.51. Celestri can cover the area within ±60◦ of latitude continuously with a minimum elevation angle of 16.45◦ , its corresponding A G is 0.866, and its I cov is 0.46. NeLS can cover the region within ±72◦ of latitude continuously with a minimum elevation angle of 20.24◦ with A G = 0.951, A S = 0.0176, and I cov = 0.45. 3.2. Dual-satellite visibility coverage performance The minimum elevation angle versus latitude with dual-satellite visibility is depicted in Fig. 2. The satellite density of a constellation in polar orbit above the polar regions is much greater than that in the equatorial region. Because of the unequal distribution of satellites above the earth’s surface, Iridium cannot provide dualsatellite visibility coverage for regions within ±50◦ latitude. The dual-satellite visibility coverage performance of Iridium will not be discussed in this paper. The coverage performance parameters with dual-satellite visibility for the other three LEO constellations are listed in Table 2. Globalstar can cover the region within ±55◦ latitude continuously with a minimum elevation angle of 5.01◦ , and its coverage area is 81.9 percent of surface area of the earth. The coverage index with dual-satellite visibility is 0.58. However, this minimum elevation angle is too small to provide reception diversity. If a slightly larger elevation angle such as 10◦ is chosen, the coverage area is 36 percent of surface area of the earth. The corresponding I cov is only 0.34. It is because of the small number of satellites and the higher orbital inclination that the coverage performance with dualsatellite visibility of Globalstar is poor; this will be discussed in detail later.
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Fig. 3. Coverage performance versus orbital inclination angle.
Fig. 2. Minimum elevation angle versus latitude with dual-satellite visibility. Table 2 Coverage parameters with dual-satellite visibility for three LEO constellations.
ε2 (deg.) A G2 A S2 N S2 I cov (2)
Globalstar
Celestri
NeLS
5.01 0.819 0.058 28 0.58
12.85 0.829 0.037 45 0.71
16.51 0.951 0.022 86 0.71
Celestri and NeLS have orbital altitudes and numbers of satellites that enable them to provide continuous dual-satellite visibility coverage for nearly all regions of the earth. Celestri can cover the region within ±56◦ latitude with a minimum elevation angle of 12.86◦ . NeLS can provide dual-satellite visibility coverage for the region within ±72◦ latitude with a minimum elevation angle of 16.51◦ . The I cov of both Celestri and NeLs is 0.71, which is larger than the 0.58 of Globalstar. 4. Improvement of representative LEO constellation configurations 4.1. Iridium constellation A polar-orbiting constellation can cover all the earth. Its coverage index lies on the minimum elevation angle merely. The Iridium constellation has been optimized for the minimum elevation angle, so the coverage index with single-satellite visibility is optimal, as depicted in Fig. 3 and Fig. 4. In the two figures, the constellation has the same number of orbits and number of satellites. Fig. 3 shows the coverage performance versus the orbital inclination, and Fig. 4 shows the coverage performance versus the phasing between two neighboring satellites. The coverage performance is practically constant with the orbital inclination, which varies from 80◦ to 88◦ in Fig. 3. In Fig. 4, when the phasing is 15◦ , the coverage performance index is optimal, which is approximately the same as the value for the actual Iridium system (16.4◦ ). Based on these two figures, the coverage performance of Iridium with single-satellite visibility is optimal. The coverage performance of Iridium with dual-satellite visibility will not be investigated for the reason mentioned previously. 4.2. Globalstar constellation The initial design goal of Globalstar was to service the entire North America region, including Alaska, so the orbital inclination was chosen to be 52◦ . The minimum elevation angle of Globalstar is greater than 10◦ for coverage of the region within ±72◦ latitude. Globalstar can service the densely populated region near 30◦ latitude with dual-satellite visibility. The coverage performance index
Fig. 4. Coverage performance versus phasing.
of Globalstar may be improved significantly. In Fig. 5, the coverage performance index is plotted, where key parameters including the number of orbits and the number of satellites are the same as Globalstar. The differences are in the orbital inclination and the phasing. The x-axis is the orbital inclination angle, and the y-axis is the coverage performance index for single-satellite visibility. The various curves correspond to different values of the phasing factor F . Even phasing factors are omitted because they would result in satellites colliding. The configuration denoted with a diamond is better than the other two configurations, so the optimal phasing factor is 1, which is consistent with the actual value for Globalstar. However, the optimal orbital inclination is less than 45◦ , unlike the actual value of 52◦ . The coverage performance index for inclinations between 40◦ and 48◦ and a phasing factor of 1 is magnified in the top right corner of Fig. 5. It can be observed that the optimal orbital inclination is 42◦ . The difference in the coverage performance index between 42◦ and 45◦ is less than 0.5%. Taking into account of the effect of computing precision, 45◦ is a reasonable choice because of its wider coverage area. The improved constellation configuration of Globalstar is: 45◦ :48/8/1. The minimum elevation angle versus latitude of the improved and initial constellations are plotted in Fig. 6, where the curves plotted with diamonds and circles represent the single-satellite and dual-satellite visibility coverage of the improved configuration, respectively, and the solid and dashed curves represent the single-satellite and dualsatellite visibility coverage of the initial configuration, respectively. The coverage performance of the improved configuration is better at latitudes less than 40◦ . The minimum elevation angle with single-satellite visibility increased from 12.64◦ to 17.21◦ . However, the cut-off latitude of the improved configuration for high-latitude regions decreased from 67◦ to 56◦ . Correspondingly, the coverage area of the improved configuration is 83 percent of entire surface area of the earth. For single-satellite visibility coverage only, the minimum elevation angle of the improved configuration increased by 5◦ with a loss of coverage area of approximately 10%. Because of the increase in the minimum elevation angle, the coverage area of a single satellite decreased to 2.89 percent of the
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Fig. 5. Coverage performance index of Globalstar configuration.
Fig. 6. Elevation angle characteristics of initial and improved Globalstar configurations.
surface area of the earth. At least 29 satellites would be required to provide equivalent coverage in the baseline configuration. The I cov of the improved constellation is 0.58. If the actual minimum elevation angle of Globalstar is used, the cut-off latitude of the improved configuration would be reduced from 70◦ to 63◦ . The corresponding increase in the coverage area would be 4.8 percent of the surface area of the earth; however, the additional population covered by this increased area is less than 1 percent. In summary, the improved configuration transforms the excess coverage in highlatitude regions with the initial constellation configuration to lowand medium-latitude regions, and the coverage performance is improved as a whole. 4.3. Celestri constellation Celestri is designed for broadband and multimedia services, which is different goal than those of Iridium and Globalstar. Therefore, the dual-satellite visibility coverage is one of the most dominant factors for Quality of Service. The coverage performance index versus latitude is depicted in Fig. 7, where the number of orbits and the number of satellites are the same as those for the actual Celestri constellation. In Fig. 7, the x-axis is the orbital inclination and the y-axis is the coverage performance index (I cov ), and the various curves are for different phasing factors. From Fig. 7(a), the optimal configuration for single-satellite visibility is 56◦ :63/7/0; i.e., the orbital inclination is 56◦ , the number of satellites is 63, the number of orbits is 7 and the phasing factor is 0. This configuration will be denoted by Celestri-A. The optimal constellation configuration for dual-satellite visibility is 48◦ :63/7/4, as can be found in Fig. 7(b). This configuration will be denoted by Celestri-B. The minimum elevation angle versus the latitude of the initial and the two improved Celestri configurations are depicted in Fig. 8. The detailed performance parameters of the three configurations with single-satellite and dual-satellite visibility are shown in Table 3 and Table 4, respectively. In addition, the cut-off latitude L cut was computed.
Fig. 7. Coverage performance index of Celestri configurations versus inclination.
As shown in Fig. 8(a) and Table 3, the cut-off latitude of Celestri-A increased from 60◦ to 70◦ and the coverage area increased from 0.866 to 0.945 when compared with the initial Celestri constellation. The coverage performance index I cov for singlesatellite visibility increased to 0.49, but the satellites are much more dense in the higher-latitude regions with Celestri-A. Consequently, the coverage for lower-latitude regions is reduced, which is clearly shown in Fig. 8(b) and in Table 4. The minimum elevation angle of Celestri-A in the lower-latitude regions was approximately 3◦ smaller. Furthermore, the minimum elevation angle with dualsatellite visibility was lower than that of the initial configuration in the region below 45◦ latitude. Although the coverage area of Celestri-A increased by 10 percent of the global surface area, the coverage performance index for dual-satellite visibility decreased by 0.68. The results for Celestri-B are opposite those for Celestri-A. The coverage performance with dual-satellite visibility improved in the lower-latitude regions. However, the coverage performance improved in the regions near 50◦ latitude. The elevation angle ε2 increased from 12.86◦ to 12.97◦ , and the index I cov(2) increased from 0.71 to 0.73, but the single-satellite visibility coverage performance of Celestri-B was worse than that of the other two configurations. Especially in the region of ±5◦ latitude, the minimum elevation angle was less than 15◦ , so the coverage performance index with single-satellite visibility increased to 0.4. In summary, the initial Celestri configuration is a tradeoff between the Celestri-A and Celestri-B configurations. 4.4. NeLS constellation The NeLS constellation was optimized in [17,18]. The orbital inclination was optimized to maximize the distance between two neighboring satellites to guarantee a uniform distribution of the satellites. Therefore, the coverage performance was improved. In
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
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Fig. 8. Minimum elevation angle versus latitude of initial and improved configurations of Celestri. Table 3 Single-satellite visibility coverage performance.
ε (◦ )
L cut (◦ ) AG AS NS I cov
Celestri
Celestri-A
Celestri-B
16.45 60 0.866 0.030 29.0 0.46
15.98 71 0.945 0.031 30.9 0.49
14.36 59 0.857 0.034 25.5 0.40
Celestri
Celestri-A
Celestri-B
16.45 60 0.866 0.030 29.0 0.46
15.98 71 0.945 0.031 30.9 0.49
14.36 59 0.857 0.034 25.5 0.40
Table 4 Dual-satellite visibility coverage performance.
ε (◦ )
L cut (◦ ) AG AS NS I cov
the optimization process, only ascending-orbit satellites are used. Introducing descending-orbit satellites can optimize the dualsatellite visibility coverage performance because of the symmetry of a 2π constellation. This change can provide seamless broadband and multimedia services. This idea may be feasible in large-scale constellations because each satellite can quickly meet another satellite traveling in the opposite direction in a short distance. When the ascending-orbit satellites are optimized, the entire satellite configuration is optimal for dual-satellite visibility coverage. The improved configuration of NeLS is close to the optimal configuration in the coverage performance index for both single-satellite and dual-satellite visibility, as depicted in Fig. 9, where F = 0, the inclination is 60◦ , and the configuration denoted by the diamonds is most similar to the actual NeLS configuration. 5. A LEO constellation design for China Because mobile satellite communication design and development is ongoing in China, the cost and the risk should be minimized as much as possible. The territory of China, including the ocean, is mostly located in the low and medium latitudes of the northern hemisphere. Therefore, the coverage demand in the highlatitude regions may be reduced, and a cutoff latitude of 60◦ is sufficient to cover all of the territory. Thus, the coverage region is limited to the range of ±60◦ latitude. The single-satellite coverage area can be computed based on the minimum elevation angle of this limited region. The coverage performance index may be computed to find the most suitable LEO constellation configuration for China. In simulations, the orbit number and the number of satel-
Fig. 9. Coverage performance index of NeLS configurations versus inclination.
lites ranged from 6 to 11, and the orbital inclination was varied from 40◦ to 60◦ with a step size of 1◦ . All possible phasing factors were used to obtain all of the possible configurations. The possible configurations are listed in Table 5. The corresponding coverage performance indexes are shown in Table 6 and Table 7. Based on the simulation results, the three configurations with the best coverage performance are, in descending order: P1: 47◦ :48/8/1 P2: 46◦ :42/7/0 P3: 48◦ :54/9/2 The coverage performance indexes of the three configurations are all greater than 0.55, which is more than those of other LEO constellations. The coverage performance index of P1 is the best. Its corresponding configuration is similar to that of Globalstar, except for an inclination difference of 5◦ . The second-best configuration, P2, has an inclination of 46◦ , an orbit number of 7, the
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Table 5 The most optimal constellation configurations (suitable for region with latitude of ±60◦ ). M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
10
F F F F F F
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
= 4, = 0, = 0, = 8, = 0, = 1,
i = 48◦ i = 50◦ i = 56◦ i = 48◦ i = 48◦ i = 48◦
F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 I = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Table 6 Single-satellite visibility coverage performance indexes of the most optimal configurations. M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
F F F F F F
10 F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 i = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Table 7 Dual-satellite visibility coverage performance indexes of the most optimal configurations. M
N 6
6 7 8 9 10 11
F F F F F F
7
= 5, = 0, = 1, = 2, = 3, = 4,
i = 48◦ i = 46◦ i = 47◦ i = 48◦ i = 50◦ i = 49◦
F F F F F F
8
= 0, = 6, = 0, = 1, = 2, = 0,
i = 55◦ i = 47◦ i = 47◦ i = 47◦ i = 48◦ i = 50◦
F F F F F F
9
= 5, = 6, = 7, = 0, = 1, = 2,
i = 49◦ i = 46◦ i = 48◦ i = 47◦ i = 48◦ i = 48◦
number of satellites in each orbit is 6, and the phasing factor is 0. The single-satellite-visibility coverage performance index of P2 is slightly lower than that of P1. However, the dual-satellite-visibility coverage performance index of P1 is only 0.48, as is that of Globalstar. Therefore, the seamless handoff of access links and channel diversity would be affected. The dual-satellite-visibility coverage performance index of P2 is only 0.57, which is 20% greater than that of P1. However, the lower orbit number and fewer satellites would be important in reducing the construction and maintenance costs. P3 is not as good as P2, not only in the coverage performance but also in the number of satellites. In conclusion, P2 is a reasonable configuration for a mobile-communication satellite constellation for China. The minimum elevation angle of P2 is plotted in Fig. 10. This configuration can continuously cover China with a minimum elevation of 20◦ , and the coverage range is from 24.5◦ to 53◦ latitude. Taking the minimum elevation angle of 13◦ in the low-latitude region of P2 as the elevation threshold, P2 can provide continuous coverage for the region of ±60◦ latitude. The minimum elevation angle of 13◦ is greater than that of Globalstar in the equatorial region. Because Globalstar can achieve seamless handoffs of mobile users, it is possible for P2 to provide continuous communication and seamless handoffs for ground users. Compared with the global coverage of Iridium and the 92% coverage of Globalstar, P2 can cover only 86% of the earth. However, the coverage area of P2 includes 99% of the settled populations of the earth. Thus, P2 may not only satisfy the demand of China but also many foreign users. In summary, P2, with an orbital altitude of 1400 km, is recommended as a feasible mobile-communication satellite constellation for China.
F F F F F F
10 F F F F F F
= 5, = 6, = 7, = 8, = 9, = 0,
11 i = 46◦ i = 49◦ i = 50◦ i = 49◦ i = 48◦ i = 48◦
F F F F F F
= 2, = 5, = 0, = 0, = 0, = 0,
i = 46◦ i = 49◦ i = 49◦ i = 54◦ i = 57◦ i = 48◦
Fig. 10. Minimum elevation angle versus latitude of P2.
6. Conclusions To address the problem of evaluating potential LEO satellite constellations, a general evaluation criterion for the coverage performance of LEO constellations was proposed. The method resembles that of rating a golf course. Every constellation configuration can be considered as a golf course, where the minimum number of satellites required for coverage for the same region is defined as the normal number of satellites, or the “par”. The ratio of the normal number of satellites to the actual number of satellites is defined as the coverage performance index. The orbital altitude, the number of satellites, the coverage zone and the minimum el-
Y. Li et al. / Aerospace Science and Technology 48 (2016) 94–101
evation angle can be unified by the index. Therefore, this index provides a standard evaluation criterion for the coverage performance of constellation configurations. Using this general evaluation method, the optimum Walker-δ constellation suitable for China was determined. The optimal configuration was selected as 46◦ :42/7/0, in which the orbital inclination is 46◦ , the number of satellites is 42, the orbit number is 7 and the phasing factor is 0. The coverage performance index of this configuration can reach 0.566. The minimum elevation angle is 20◦ between 24.5◦ and 53◦ latitude, which is suitable for China. Conflict of interest statement No conflict of interest. Acknowledgements This research was supported by the Natural Science Foundation of Shaanxi Province (No. 2015JM6360). References [1] I.F. Akyildiz, E. Ekici, G. Yue, A distributed multicast routing scheme for multiLayered satellite IP networks, Wirel. Netw. 9 (5) (2003) 535–544. [2] Wu Ting-yong, Wu Shi-qi, Performance analysis of the inter-layer inter-satellite link establishment strategies in two-tier LEO/MEO satellite networks, J. Electron. Inf. Technol. 30 (1) (2008) 67–71. [3] T. Taleb, D. Mshimo, A. Jamalipour, Explicit load balancing technique for NGEO satellite IP networks with on-board processing capabilities, IEEE/ACM Trans. Netw. 17 (1) (2009) 281–293. [4] John G. Walker, Continuous whole-Earth coverage by circular-orbit satellite patterns, Royal Aircraft Establishment technical report, 77044, Royal Aircraft Establishment, 1977.
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