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Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the luni-solar perturbations
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ScienceDirect Advances in Space Research xxx (2016) xxx–xxx www.elsevier.com/locate/asr
Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the luni-solar perturbations Li Fan a, Chao Jiang b, Min Hu c,⇑ b
a School of Aerospace Engineering, Tsinghua University, Beijing 100084, China College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China c Equipment Academy, Beijing 101416, China
Received 20 April 2016; received in revised form 11 September 2016; accepted 13 September 2016
Abstract Eight inclined geosynchronous satellite orbit (IGSO) satellites in the Chinese BeiDou Navigation Satellite System (BDS) have been put in orbit until now. IGSO is a special class of geosynchronous circular orbit, with the inclination not equal to zero. It can provide high elevation angle coverage to high-latitude areas. The geography longitude of the ground track cross node is the main factor to affect the ground coverage areas of the IGSO satellites. In order to ensure the navigation performance of the IGSO satellites, the maintenance control of the ground track cross node is required. Considering the tesseral resonances and the luni-solar perturbations, a control approach is proposed to maintain the ground track for the long-term evolution. The drifts of the ground track cross node of the IGSO satellites are analyzed, which is formulated as a function of the bias of the orbit elements and time. Based on the derived function, a method by offsetting the semi-major axis is put forward to maintain the longitude of the ground track cross node, and the offset calculation equation is presented as well. Moreover, the orbit inclination is adjusted to maintain the location angle intervals between each two IGSO satellites. Finally, the precision of the offset calculation equation is analyzed to achieve the operational deployment. Simulation results show that the semi-major axis offset method is effective, and its calculation equation is accurate. The proposed approach has been applied to the maintenance control of BeiDou IGSO satellites. Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Inclined geosynchronous satellite orbit (IGSO); BeiDou Navigation Satellite System (BDS); Ground track maintenance; Tesseral resonances; Luni-solar perturbations
1. Introduction The constellation of Chinese BeiDou Navigation Satellite System (BDS) includes Geostationary Earth Orbit (GEO) satellites, Inclined Geosynchronous Satellite Orbit (IGSO) satellites, and Medium Earth Orbit (MEO) satellites (Tian et al., 2016). Nowadays, the BDS has been constructed covering the Asia-Pacific region, which includes five GEO satellites, five IGSO satellites, and four MEO ⇑ Corresponding author.
E-mail addresses: [email protected] (L. Fan), [email protected] (C. Jiang), [email protected] (M. Hu).
satellites (Jin et al., 2016). IGSO is a special class of geosynchronous circular orbit, with the inclination not equal to zero. Its ground tracks don’t always stay at the same place on the equator, but form an 8-shaped curve symmetrically about the equator, and could provide high elevation angle coverage to high-latitude areas. In recent years, eight IGSO satellites with the inclination of about 55° have been successfully deployed in the BDS (http://www.beidou.gov. cn/ xtjs.html accessed on 2016/04/18), three IGSO satellites’ ground tracks coincide at 118°E, while the other five IGSO satellites’ ground tracks coincide at 95°E. Performance of the navigation satellite system is largely influenced by its ground coverage. It’s essential for the
http://dx.doi.org/10.1016/j.asr.2016.09.014 0273-1177/Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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geosynchronous satellites to maintain their ground coverage areas. The station-keeping approach of the GEO satellites has been studied for a long time, and some classical and effective methods have been obtained (Gedeon, 1969; Kame et al., 1973; Chao and Schmitt, 1989; Liu et al., 2016). Ely (1996) has investigated the method for GEO satellite’s east-west station-keeping, which is named as ‘nodal control’. To maintain the ground track of a GEO satellite within a pre-defined dead-band region around a mean node, the control strategy forces the trajectory to drift from one node boundary, and then graze the other boundary back towards the starting node, where the cycle is repeated. However, the maintenance control method cannot be applied directly to the IGSO satellite for two basic reasons. As the inclinations of the IGSO satellites are not equal to zero, the ground tracks of the IGSO satellites are more complicated than those of the GEO satellites, which would absolutely cause some differences for the maintenance control approaches. On the other hand, the control requirements of the IGSO and GEO satellites are also different. According to the International Telecommunications Union’s regulation, the station-keeping deadband width of the GEO satellites is normally about 0.1°, which avoids the interferences of the orbital slots and communication frequency among the GEO satellites (John et al., 2002). The IGSO satellites don’t have such special limitations; their maintenance control dead-band region is determined by the actual applications, and its width possibly varies from several degrees to tens of degrees. Different from the sub-satellite track longitude variation of the GEO satellites, the sub-satellite longitudes and in-orbit phase angles of the IGSO satellites must be maintained in a certain range for the PDOP requirements of BDS’s Positioning, Navigation and Timing (PNT) services. Moreover, the orbital maneuver frequency of the IGSO satellites should be reduced, so as to reduce the interruption of PNT services caused by the maintenance control. Therefore, it is imperative to investigate of the ground tracks of the IGSO satellites and their maintenance control methods. A number of analytical methods of the dynamical evolution of the IGSO satellites have been reported in recent literature. Liu (1984) proposed a calculation method of the orbital variation for 24-h synchronous satellite. Mao et al. (1999) analyzed the IGSO orbital evolution caused by earth’s non-spherical perturbation. By analyzing the influences caused by the perturbations, Xiang and Zhang (2007,2008) proposed a coordination control strategy based on the bias of the orbit elements. However, the control method requires large amount of calculation and high control precision. Zhu and Hu (2008) studied the cross node location evolution and keeping issues of the ground tracks of the IGSO satellites, and discussed a cross node maintenance strategy just like the ‘nodal control’ method for the station-keeping of the GEO satellites. Li et al. (2010) conducted research on the perturbation motions of the BeiDou-M navigation constellation caused by the J2
perturbation, the luni-solar perturbations as well as the radiation force. However, the study just compared the existing absolute control method and the relative control method instead of putting forward a new control strategy. Zhao et al. (2015) analyzed the long-term dynamical evolution of the IGSO satellites, and proposed a possible mitigation strategy to reduce the orbital lifetime of the IGSO satellites after the end-of-mission. To advance the research on the ground track maintenance of the IGSO satellites, a general analysis of the ground track evolution of the IGSO satellites and its maintenance control is investigated by both theoretical analysis and numerical computation methods in this paper, which aims at the practical application to the BDS. This paper is organized as follows. In Section 2, the characteristics of the ground track of the IGSO satellites are analyzed. Section 3 establishes the dynamical model based on the Earth’s tesseral and zonal perturbations and the dynamical model under the tesseral resonances and the luni-solar perturbations. Section 4 presents the ground track maintenance approach subject to the tesseral resonances and the luni-solar perturbations. Simulations are provided, and the analyses are discussed in Section 5. The conclusion is given in Section 6. 2. Characteristics of the ground track of the IGSO satellites IGSO satellite ground tracks can be described by the geographic latitude / and geographic longitude k. In the geocentric spherical fixed coordinate system, the geography latitude / and longitude k can be expressed by the classical orbital elements as follows. / ¼ arcsinðsin i sin uÞ
ð1Þ
k ¼ arctanðcos i tan uÞ þ X hE
ð2Þ
where hE is the Greenwich sidereal angle with h_ E as its associated rate, u ¼ x þ f is the argument of latitude, and i, X, x and f are the inclination, right ascension of the ascending node, argument of perigee, and true anomaly, respectively. Considering the two-body dynamics model, and according to Eqs. (1) and (2), the ground track of the IGSO satellites is shown in Fig. 1. Ideally, the ground track of the IGSO satellites is an 8shaped curve, which is symmetrical about the equator. The relationship of the boundaries is governed as follows: /max /min ¼ 2i kmax kmin ¼ 2Dki where Dki ¼ arctan
ð3Þ
1 pffiffiffiffiffiffi cos i
ð4Þ pffiffiffiffiffiffiffiffiffi arctanð cos iÞ.
Note that, kA ¼ M þ x þ X hE
ð5Þ
where M is the mean anomaly, kA is the longitude of the stroboscopic mean ascending node of 24-h orbit that given by Gedeon (1969). As kA equals to the longitude of the
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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For the IGSO satellites in deep 1:1 resonance, the eastwest station libration caused by the Earth’s geopotential should be discussed firstly. 3. Dynamical model 3.1. Dynamical model based on the Earth’s tesseral and zonal perturbations Considering the Hamiltonian with a set of action/angle variables (I, h), which is convenient for investigating the mean motion resonances (Ely, 1996), the following equation can be obtained: HE ¼
X l2 h_ E I 1 þ V JZ2 þ V Tlmpq 2 2I 1 l4 R2E J 2
Fig. 1. The ground track of the IGSO satellites under the two-body dynamics model.
cross node when sin u ¼ 0, it could represent the east–west station of the IGSO satellite s as well. For a group of ideal orbital elements of the IGSO satellites, we can see that i and kA determine the shape size and the longitude station of the ground track, respectively, while the argument of latitude u determines its location point on the ground track. Therefore, the IGSO satellites with the same parameters of i and kA could have the same ground track. Their location points vary on the ground track, with different values of u. In fact, the eccentricity of the IGSO satellites is always small, where e ¼ Oð103 Þ holds. Thus, the effect of the shape size caused by a small eccentricity could be evaluated by the following equation: kmax kmin ¼ 2ðDki þ Dke Þ
pffiffiffiffiffiffiffiffiffiffi i 2e 1cos . sin i
ð6Þ
where the maximum of Dke equals to emax Therefore, DkDk ¼ OðeÞ can be obtained for the IGSO i satellites with i ¼ 55 . The effects caused by the eccentricity could be ignored for BDS IGSO satellites because of a very small eccentricity. Since the IGSO satellites are in deep 1:1 resonance, they exhibit complex motions. Numerous studies of the resonant orbital dynamics have shown that the Earth’s tesseral harmonics will cause the longitude-dependent libration for the synchronous satellites. Therefore, the east-west stationkeeping is imperative for the IGSO satellites. Moreover, the IGSO satellites have the advantages in high-latitude coverage, and several satellites are always deployed in the same ground track with different location points to obtain the continuous high-latitude coverage. For the BDS, three IGSO satellites are deployed in the same ground track with a differential argument of latitude Du of 120 . Therefore, it is also required to keep the differential argument of latitude among each satellite.
where V JZ2 ¼ 4I 3 ðI 1
1 þI 2 Þ
5
ð7Þ
f3ðI 1 þ I 3 Þ2 ðI 1 þ I 2 Þ2 g, and the
action/angle pairs are defined as follows: pffiffiffiffiffiffi I 1 ¼ la; h1 ¼ kA ¼ M þ x þ X hE ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi I 2 ¼ lað1 e2 Þ la; h2 ¼ x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi I 3 ¼ lað1 e2 Þ cos i la; h3 ¼ X P T The tesseral expansion V lmpq can be expressed as follows: l 1 X l X l X 1 X X lJ lm RE V Tlmpq ¼ F lmp ðiÞGlpq ðeÞ a a l¼2 m¼1 p¼0 q¼1 cos½mðkA klm Þ qx
ð8Þ
where RE is the Earth’s radius, Flmp(i) and Glpq(e) are the inclination and eccentricity functions, which are defined by Kaula (1966); Jlm and klm are the coefficient and angle corresponding to the tesseral harmonic of degree l and order m. According to Eq. (8), we can see that J22, J31, J32 and J33 are significant while Jlm ðl P 4Þ are negligibly small. Their values for l 6 3 are shown in Table 1. P T Thus, selecting the expansion of V lmpq with l 6 3, with the resonance condition l 2p þ q m ¼ 0, the effective terms of lmpq and the corresponding Flmp(i) and Glpq(e) are shown in Table 2, where ci denotes cos(i) and si denotes sin(i). For the IGSO satellites with e ¼ Oð103 Þ, the tesseral terms of q – 0 are negligibly small compared with these terms of q ¼ 0. Then, Eq. (7) can by expressed as:
Table 1 Values of J lm and klm for l 6 3 . l
m
J lm
klm (degs)
2 3 3 3
2 1 2 3
1:766 106 2:111 106 0:311 106 0:239 106
14.79 7.82 18.12 23.60
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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Table 2 Values of effective terms of lmpq and the corresponding Flmp(i) and Glpq(e) for l 6 3. lmpq
Flmp(i)
Glpq(e)
2200
3ð1þciÞ2 4
1
3110
15si ð1þ3ciÞ 16 2
15ð1þciÞ 8
3300
3si 2
2212
HE ¼
3ð1þciÞ 4
5 2 2e
þ
1 þ 2e2 þ
3
1 6e2 þ
2
9 2 4e
lmpq
Flmp(i)
Glpq(e)
310-2
2 ð1þciÞ 15si 16
1 2 8e
3122
15si ð13ciÞ 16
11 2 8 e
2
15sið12ci3ci2 Þ
3211
þ 74 e4 þ
18 45si ð1þciÞ 8 2
3312
l2 h_ E I 1 þ V JZ2 þ V T2200 ðI 1 ; I 2 ; I 3 ; kA Þ 2I 21
V L: ¼
3ð1ciÞ 4
1 4 þ 48 e þ 4 þ 49 16 e þ
3 3e þ 11 4 e þ 53 2 8 e
4 þ 39 16 e þ
2 X 2 X 2 X mþs hL: fU 2m;s ðoe Þ 2msp ða; e; i; aL: ; iL: Þð1Þ m¼0 s¼0 p¼0
þ V T3110 ðI 1 ; I 2 ; I 3 ; kA Þ þ V T3300 ðI 1 ; I 2 ; I 3 ; kA Þ
ð9Þ
cosðmþ hÞ þ U 2m;þs ðoe Þ cosðm hÞg ð14Þ
where V
T 2200
V T3110 V T3300
2 lJ 22 RE ¼ F 220 ðiÞG200 ðeÞ cos½2ðkA k22 Þ; a a 3 lJ 31 RE ¼ F 311 ðiÞG310 ðeÞ cos½3ðkA k31 Þ; a a 3 lJ 33 RE ¼ F 330 ðiÞG300 ðeÞ cos½3ðkA k33 Þ: a a
V S: ¼
þ mðX XS: Þ
@H E @V @V 3110 @V 3300 I_ 1 ¼ ¼ 2200 @kA @kA @kA @kA T
T
ð10Þ
@V J 2 @V T2200 @V T3110 @V T3300 @H E l2 _ k_ A ¼ ¼ 3 hE þ þ þ þ @I 1 @I 1 @I 1 @I 1 @I 1 I1 Z
l € kA ¼ 3 4 I_ 1 þ I_ 1 I1 2
"
@ 2 V JZ2 @I 21
þ
@ 2 V T2200 @ 2 V T3110 @ 2 V T3300 þ þ @I 21 @I 21 @I 21
hS: lmsp ða; e; i; aS: ; iS: Þ cos½ðl 2pÞx
l¼2 m¼0 s¼0 p¼0
Then, the system reduces to 1-DOF with action/angle pair (I 1 , kA ). According to Eq. (9), the following equations can be obtained: T
1 X l X l X l X
ð11Þ #
ð12Þ
ð15Þ
where the index L: and S: refer to the Moon’s parameters and the Sun’s parameters, respectively. And k 1 em es ð2sÞ! a 2 L: h2msp ¼ lL: ð1Þ 2aL: ð2þmÞ! ðaL: Þ F 2mp ðiÞF 2s1 ðiL: ÞH 2pð2p2Þ ðeÞ, 1 m¼0 while k 1 is the largest integer part of m2 , em ¼ 2 m–0 1 s¼0 and es ¼ . m h ¼ ð2 2pÞx þ mX ðXL: p2Þ 2 s–0 0 seven lS: ðlsÞ! a l bs p, while bs ¼ . hS: lmsp ¼ k m aS: ðlþmÞ! ðaS: Þ 1=2 sodd 1 m¼0 F lmp ðiÞF lms ðiS: ÞH lpð2plÞ ðeÞ and k m ¼ . 2 m–0 According to Eqs. (13)–(15), we can see that the lunisolar perturbations are greatly affected by the initial values of x and X. Therefore, the effects on the IGSO satellites in the same ground track are different, as the parameters of x and X are different. 4. Ground track maintenance approach subject to tesseral resonances and the luni-solar perturbations
3.2. Dynamical model under tesseral resonances and the lunisolar perturbations
4.1. East-west station-keeping method under the Earth’s tesseral and zonal perturbations
Generally, the luni-solar perturbations can be of the same order of magnitude as the J2 perturbation at the altitudes of the geosynchronous orbits. The luni-solar perturbations effects should also be analyzed in the process of the ground track maintenance for the IGSO satellites. Considering the luni-solar perturbations, the Hamiltonian as shown in Eq. (7) can be written as follows:
Based on the dynamical equations shown in Section 3, the east-west station libration could be solved for the IGSO satellites with i ¼ 55 . The east-west station libration of the IGSO satellites with i ¼ 55 and e ¼ 0 is shown in Fig. 2. As shown in Fig. 2, we can see that the cross node longitude vibrates around two stable points along the east-west direction. According to the characteristics of the variations, the limitations of the vibrations for each cross node longitude exist. We can also see that kA could be kept in a dead-band region DkA . The traditional method can utilize the libration cycles to maximize the drift time between maneuvers. In Fig. 2, the black trajectory in the dead-band region is the natural motion caused by the tesseral resonances. Firstly, kA moves from the start control
H ¼ H E þ V L: þ V S:
ð13Þ
The approximate expansions of the lunar potential and solar potential are governed as follows (Delhaise and Morbidelli, 1993; Ely, 1996):
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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where the variables with superscript represent the nominal parameters of the IGSO satellites’ mean elements. Considering the Earth’s tesseral and zonal perturbations only, we can see that e and i don’t have the secular drifts, and the secular drift rates of X should be the same for each IGSO satellites with the same parameters of a, e, i, kAb and kAg . Moreover, the IGSO satellites in the same ground track have the same drift rate of Du. 4.2. Analysis of the effects caused by the luni-solar perturbations
Fig. 2. The east-west station libration of the IGSO satellites with i ¼ 55 and e ¼ 0.
boundary to the natural boundary. Then, it moves from the natural boundary back to the arrival control boundary. The start control boundary and the arrival control boundary have the same value of kA , while their semi-major axes are different from each other. In order to repeat the natural libration cycle, the semi-major axis should be adjusted as the red line shown in Fig. 2. The adjustment amount of semi-major axis Da and the time between maneuvers DT could be calculated according to Eqs. (10) and (11) with the condition k_ Ag ¼ 0 for the natural boundary. Since the dead-band of kA is always small and e ¼ Oð103 Þ holds, the following equation is obtained kA kA ¼ kA0 þ k_ A0 ðt t0 Þ þ € where € kA
mean
ðt t0 Þ 2
2
ð16Þ
3 lI 4 I_ 1 ða ; e ; i ; k A Þ is almost constant. 2
mean
1
2 _ k_ A0 lI 3 h_ E 3hE ða2a0 a Þ is only depended on the semi1
major axis of the start control boundary a0 . The variables with superscript represent the nominal parameters of the IGSO satellites’ mean elements. According to Eq. (16), and considering the condition k_ Ag ¼ 0 for the natural boundary, a0 could be easily solved and so does the adjustment amount of semi-major axis Da ¼ 2ða0 a Þ. Traditionally, the maneuver is performed at the perigee and the eccentricity and semi-major axis are changed simultaneously, the other elements are not changed. In order to obtain the required Da, the impulsive maneuver cost DV and eccentricity change De are given by the following equations: rffiffiffiffiffiffiffiffiffiffiffiffiffi n Da 1 e DV ¼ ð17Þ 2 1 þ e ð1 e ÞDa De ¼ a
ð18Þ
A numerical example is carried out to assess the effects of the luni-solar perturbations on the maintenance control of three IGSO satellites during their life time of about 10 years. Considering the operational performance of BDS IGSO satellites, the nominal value of kA is 118° with the dead-band of ±2°. Based on the initial orbit elements as shown in Table 3, all the IGSO satellites have the same initial parameters of a, e and i. Firstly, the target semi-major axis for stationkeeping is solved based on the Earth’s geopotential, it will be executed once kA meets the arrival control boundary. Then, the station-keeping cycle is simulated in two different conditions, one considers the Earth’s geopotential only and the other considers the Earth’s geopotential and the lunisolar perturbations. The Earth’s gravity field adopts EGM96 model. Daniel et al. (2015) analyzed the station keeping strategies for Galileo and proposed that a truncation of the Earth potential down to 12 12 is acceptable to retain enough accuracy for 10-year orbit propagation. Therefore, the gravitational order of 20 20 is adopted, which is enough for the purpose of the ground track maintenance for BeiDou IGSO satellites. Fig. 3 shows the comparisons of the station-keeping results. From Fig. 3(a), we can see that the offsets of the semimajor axis for three IGSO satellites are the same when the luni-solar perturbations are not considered. The drift rates of semi-major axis are affected by the luni-solar perturbations, and are different from each other. The proposed semi-major axis offset approach is effective, and the magnitude of the offsets of the semi-major axis is about 8.2 km. From Fig. 3(b), we can see that the maximum difference is up to 0.6°, compared with the nominal natural boundary, which may increase the risk of collision. Thus, the luni-solar perturbations cannot be neglected for the maintenance control of the ground track cross node.
Table 3 The initial parameters for the IGSO satellites with i ¼ 55 and e ¼ 0:005. Sat. ID
X/°
x/°
M/°
kA /°
1 2 3
90 330 210
128 128 128
0 120 240
118 118 118
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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Fig. 3. The station-keeping cycle in 10 years.
Orbital elements of the IGSO satellites affected by the luni-solar perturbations are shown in Fig. 4. From Fig. 4, we can see that e, i and X have different drift rates while considering the luni-solar perturbations. In particular, the maximum drift of i is up to about 5°, which is harmful for the stable coverage of BDS. At the same time, the drift rates of Du become different, leading to the secular drifts among the IGSO satellites. At the end of the station-keeping cycle of 10 years, the maximum drift of Du is up to about 9°, which will also disrupt the continuous high-latitude coverage of BDS.
4.3. Ground track maintenance approach including the effects of the luni-solar perturbations The analytical results of the effects caused by the luni-solar perturbations indicate that the luni-solar
perturbations affect not only the east-west station and the shape size of the ground track of each IGSO satellite, but also the location angle intervals among the IGSO satellites on the same ground track. In view of the above considerations, the ground track maintenance approach should be investigated considering the effects of the lunisolar perturbations. Note that the eccentricity of BDS IGSO satellite is small; the following equation can be obtained: u kA X þ hE
ð19Þ
Since kA is maintained in its dead-band region, the drift rates differences of Du will decrease with the decrease of the drift rates differences of X. Then, the ground track maintenance of the IGSO satellite can be considered to maintain kA and i in the dead-band region while eliminating the drift rates differences of X for the IGSO satellites.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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7
Fig. 4. Orbital elements of the IGSO satellites affected by the luni-solar perturbations.
Based on the corresponding relationships of each action/angle pair, a should be adjusted to maintain kA , while, the drift rates differences of X could be eliminated by adjusting any one of a, e and i. The adjustment amounts could be calculated according to Eqs. (16)–(18) using the Lie method (Ely, 1996). However, the calculation process is very complex and requires many mathematical transformations. In this paper, a semi-analytical approach is proposed to solve these problems. Firstly, Da for each maintenance should be determined to keep kA in its dead-band region. Based on the effects caused by the luni-solar perturbations, we can obtain the following equation: ðt t0 Þ k0A mean kA ¼ kA0 þ k_ A0 ðt t0 Þ þ € 2 _ where k_ A0 ¼ 3hE ða2a0 a Þ, € k0A
mean
@ X_ E Di þ DX_ 0 @i while i þ Di 2 ði Dimax ; i þ Dimax Þ min DX_ ¼
_E
2
ð20Þ
2 2 ¼ 3 lI 4 I_ 1 3 lI 4 I_ 01 . I_ 1 is the 1
turbations. I_ 01 could be estimated numerically, then a0 could be easily solved and Da ¼ 2ða0 a Þ. Since Da is determined to maintain kA , and e is required to keep very small, Di is adjusted at the same time just to maintain the location angle intervals among each two IGSO satellites. Moreover, during the life time of the IGSO satellite, i should be in its dead-band region to keep the shape size of the ground track. Therefore, Di could be calculated as follows:
1
drift rate caused by the Earth’s geopotential only, while I_ 01 is the additional drift rate caused by the luni-solar per-
ð21Þ
2
X 32 n J 2 a2E a 2 ð1 e 2 Þ sin i is caused by the where @@i Earth’s geopotential only. DX_ 0 ¼ X_ 0 X_ 0mean , X_ 0 is the additional drift rate caused by the luni-solar perturbations of one IGSO satellite while X_ 0mean is the mean additional drift rate of all the IGSO satellites in the same ground track. After obtaining X_ 0 of each IGSO satellite and X_ 0mean numerically, Di could be solved easily.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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Fig. 5. Ground track maintenance cycle in 10 years considering the tesseral resonances and luni-solar perturbations.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
L. Fan et al. / Advances in Space Research xxx (2016) xxx–xxx
5. Simulation example The ground track maintenance approach has been studied above to adapt the effects of the luni-solar perturbations by adjusting the semi-major axis and inclination. Since the adjustment amounts are solved approximately, the numerical analysis should be carried out to test and verify the semi-analytical approach. For three IGSO satellites with the initial orbit elements as shown in Table 3, their ground track maintenance were simulated for 10 years. The nominal value of kA is 118° with the dead-band of ±2°, while the nominal value of i is 55° with the dead-band of ±2°. Results are shown as illustrated in Fig. 5. Including the effects of the luni-solar perturbations, the target semi-major axes of each IGSO satellites are different, while their natural boundary of the east-west stationkeeping is almost the same. Compared with the nominal natural boundary, the maximum difference is less than 0.1°. Based on the dead-band region of inclination, Di of each maintenance is almost less than 0.2°, except one adjustment of about 2°. Meanwhile, the secular drifts of Du among each two IGSO satellites were well improved while comparing to Fig. 4. As seen in Fig. 5, the IGSO satellites maneuvered about 21 times during the ten year period, the control frequency decreased to a quarter of the conventional methods, which controls the cross node longitude to the nominal value at the boundary of both sides. The control period is almost the same as that of real IGSO data. The simulations indicate that the semianalytical approach of ground track maintenance is effective. 6. Conclusions In this paper, the orbital evolution characteristics and maintenance control approaches for BDS IGSO satellites are investigated. This work shows that the evolutions of the orbital cross node longitude are dominated by Earth’s non-spherical gravitational perturbation and the lunisolar tidal perturbations. As result of these perturbations, drifts of the longitude of the ground track cross node are demonstrated as Taylor expansion of time. Both the proposed semi-major axis offset approach and the offset calculation equation presented in this paper can provide coequal precise results for different dead-band widths. Simulation examples are given based on real BDS IGSO orbits, which provide references for the orbital maneuver operations of BDS. The numerical results have successfully verified that the method proposed in this paper can satisfy the IGSO satellites’ maintenance control demand.
9
The authors are grateful to the helpful discussions by Pro. Junhua Xiang in National University of Defense Technology. The authors also would like to thank all the referees and editors for their valuable suggestions to improve the contents of the paper. References Chao, C., Schmitt, D., 1989. Eliminating GPS stationkeeping maneuvers by changing the orbit altitude. In: AAS/AIAA Astrodynamics Specialist Conference, Stowe, VT, Paper No. 89-407. Delhaise, F., Morbidelli, A., 1993. Luni-solar effects of geosynchronous orbits at the critical inclination. Celes. Mech. Dyn. Astro. 57, 155–173. Daniel, N.R., Anna, N., Giacomo, T., 2015. Galileo Constellation: Evaluation of Station Keeping Strategies.. Ely, T.A., 1996. Dynamics and Control of Artificial Satellite Orbits with Multiple Tesseral Resonances. Ph.D. thesis, Purdue University, West Lafayette, IN. Gedeon, 1969. Tesseral resonance effects on satellite orbits. Celes. Mech. 1, 167–189. Jin, S.G., Jin, R., Li, D., 2016. Assessment of BeiDou differential code bias variations from multi-GNSS network observations. Ann. Geophys. 34 (2), 259–269. http://dx.doi.org/10.5194/angeo-34-259-2016. John, E.D., Paul, J.C., Richard, I., et al., 2002. Beyond GEO – using elliptical orbit constellations to multiply the space real estate. Acta Astron. 9, 467–489. Kame, A., Ekman, D., Tibbitts, R., 1973. East-west stationkeeping requirements of nearly synchronous satellites due to Earth’s triaxiality and luni-solar effects. Celes. Mech. 8, 129–148. Kaula, W.M., 1966. Theory of Satellite Geodesy. Blaisdell Publishing Co, Waltham, Mass. Li, H.N., Li, J.S., Jiao, W.H., 2010. Analyzing perturbation motion and studying configuration maintenance strategy for Compass-M navigation constellation. J. Astronaut. 37 (5), 3418–3422 (in Chinese). Liu, J.H., Gu, D.F., Ju, B., Shen, Z., Lai, Y.W., Yi, D.Y., 2016. A new empirical solar radiation pressure model for BeiDou GEO satellites. Adv. Space Res. 57, 234–244. Liu, L., 1984. The orbital variation of 24h synchronous satellite and its calculating method. Acta Astron. Sinica 3, 255–264. Mao, Y., Song, X.Y., Jia, X.L., 1999. The evolvement of satellite orbit and constellation configuration. Sci. Surv. Map. 1, 119–121 (in Chinese). Tian, Y.J., Zhao, D.Q., Chai, H.Z., Wang, S., 2016. BDS ambiguity resolution with the modified TCAR method for medium-long baseline. Adv. Space Res. http://dx.doi.org/10.1016/j.asr.2016.04.010. Xiang, J.H., Zhang, Y.L., 2007. Study on configuration optimization and coordination control strategy for the regional coverage satellite constellation. Flight Dyna. 25 (2), 87–91 (in Chinese). Xiang, J.H., Zhang, Y.L., 2008. Study on self-stabilization design of configuration for elliptical orbit satellite constellation. Flight Dyna. 20, 337–340 (in Chinese). Zhao, C.Y., Zhang, M.J., Wang, H.B., Xiong, J.N., Zhu, T.L., Zhang, W., 2015. Analysis on the long-term dynamical evolution of the inclined geosynchronous orbits in the Chinese BeiDou navigation system. Adv. Space Res. 56, 377–387. Zhu, M.C., Hu, S.J., 2008. Location evolution and keeping of IGSO cross node. Chin. Space Sci. Technol. 1, 41–44.
Acknowledgment This work is supported by the National Natural Science Foundation (NNSF) of China through Grant 61403416.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
ScienceDirect Advances in Space Research xxx (2016) xxx–xxx www.elsevier.com/locate/asr
Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the luni-solar perturbations Li Fan a, Chao Jiang b, Min Hu c,⇑ b
a School of Aerospace Engineering, Tsinghua University, Beijing 100084, China College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China c Equipment Academy, Beijing 101416, China
Received 20 April 2016; received in revised form 11 September 2016; accepted 13 September 2016
Abstract Eight inclined geosynchronous satellite orbit (IGSO) satellites in the Chinese BeiDou Navigation Satellite System (BDS) have been put in orbit until now. IGSO is a special class of geosynchronous circular orbit, with the inclination not equal to zero. It can provide high elevation angle coverage to high-latitude areas. The geography longitude of the ground track cross node is the main factor to affect the ground coverage areas of the IGSO satellites. In order to ensure the navigation performance of the IGSO satellites, the maintenance control of the ground track cross node is required. Considering the tesseral resonances and the luni-solar perturbations, a control approach is proposed to maintain the ground track for the long-term evolution. The drifts of the ground track cross node of the IGSO satellites are analyzed, which is formulated as a function of the bias of the orbit elements and time. Based on the derived function, a method by offsetting the semi-major axis is put forward to maintain the longitude of the ground track cross node, and the offset calculation equation is presented as well. Moreover, the orbit inclination is adjusted to maintain the location angle intervals between each two IGSO satellites. Finally, the precision of the offset calculation equation is analyzed to achieve the operational deployment. Simulation results show that the semi-major axis offset method is effective, and its calculation equation is accurate. The proposed approach has been applied to the maintenance control of BeiDou IGSO satellites. Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Inclined geosynchronous satellite orbit (IGSO); BeiDou Navigation Satellite System (BDS); Ground track maintenance; Tesseral resonances; Luni-solar perturbations
1. Introduction The constellation of Chinese BeiDou Navigation Satellite System (BDS) includes Geostationary Earth Orbit (GEO) satellites, Inclined Geosynchronous Satellite Orbit (IGSO) satellites, and Medium Earth Orbit (MEO) satellites (Tian et al., 2016). Nowadays, the BDS has been constructed covering the Asia-Pacific region, which includes five GEO satellites, five IGSO satellites, and four MEO ⇑ Corresponding author.
E-mail addresses: [email protected] (L. Fan), [email protected] (C. Jiang), [email protected] (M. Hu).
satellites (Jin et al., 2016). IGSO is a special class of geosynchronous circular orbit, with the inclination not equal to zero. Its ground tracks don’t always stay at the same place on the equator, but form an 8-shaped curve symmetrically about the equator, and could provide high elevation angle coverage to high-latitude areas. In recent years, eight IGSO satellites with the inclination of about 55° have been successfully deployed in the BDS (http://www.beidou.gov. cn/ xtjs.html accessed on 2016/04/18), three IGSO satellites’ ground tracks coincide at 118°E, while the other five IGSO satellites’ ground tracks coincide at 95°E. Performance of the navigation satellite system is largely influenced by its ground coverage. It’s essential for the
http://dx.doi.org/10.1016/j.asr.2016.09.014 0273-1177/Ó 2016 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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geosynchronous satellites to maintain their ground coverage areas. The station-keeping approach of the GEO satellites has been studied for a long time, and some classical and effective methods have been obtained (Gedeon, 1969; Kame et al., 1973; Chao and Schmitt, 1989; Liu et al., 2016). Ely (1996) has investigated the method for GEO satellite’s east-west station-keeping, which is named as ‘nodal control’. To maintain the ground track of a GEO satellite within a pre-defined dead-band region around a mean node, the control strategy forces the trajectory to drift from one node boundary, and then graze the other boundary back towards the starting node, where the cycle is repeated. However, the maintenance control method cannot be applied directly to the IGSO satellite for two basic reasons. As the inclinations of the IGSO satellites are not equal to zero, the ground tracks of the IGSO satellites are more complicated than those of the GEO satellites, which would absolutely cause some differences for the maintenance control approaches. On the other hand, the control requirements of the IGSO and GEO satellites are also different. According to the International Telecommunications Union’s regulation, the station-keeping deadband width of the GEO satellites is normally about 0.1°, which avoids the interferences of the orbital slots and communication frequency among the GEO satellites (John et al., 2002). The IGSO satellites don’t have such special limitations; their maintenance control dead-band region is determined by the actual applications, and its width possibly varies from several degrees to tens of degrees. Different from the sub-satellite track longitude variation of the GEO satellites, the sub-satellite longitudes and in-orbit phase angles of the IGSO satellites must be maintained in a certain range for the PDOP requirements of BDS’s Positioning, Navigation and Timing (PNT) services. Moreover, the orbital maneuver frequency of the IGSO satellites should be reduced, so as to reduce the interruption of PNT services caused by the maintenance control. Therefore, it is imperative to investigate of the ground tracks of the IGSO satellites and their maintenance control methods. A number of analytical methods of the dynamical evolution of the IGSO satellites have been reported in recent literature. Liu (1984) proposed a calculation method of the orbital variation for 24-h synchronous satellite. Mao et al. (1999) analyzed the IGSO orbital evolution caused by earth’s non-spherical perturbation. By analyzing the influences caused by the perturbations, Xiang and Zhang (2007,2008) proposed a coordination control strategy based on the bias of the orbit elements. However, the control method requires large amount of calculation and high control precision. Zhu and Hu (2008) studied the cross node location evolution and keeping issues of the ground tracks of the IGSO satellites, and discussed a cross node maintenance strategy just like the ‘nodal control’ method for the station-keeping of the GEO satellites. Li et al. (2010) conducted research on the perturbation motions of the BeiDou-M navigation constellation caused by the J2
perturbation, the luni-solar perturbations as well as the radiation force. However, the study just compared the existing absolute control method and the relative control method instead of putting forward a new control strategy. Zhao et al. (2015) analyzed the long-term dynamical evolution of the IGSO satellites, and proposed a possible mitigation strategy to reduce the orbital lifetime of the IGSO satellites after the end-of-mission. To advance the research on the ground track maintenance of the IGSO satellites, a general analysis of the ground track evolution of the IGSO satellites and its maintenance control is investigated by both theoretical analysis and numerical computation methods in this paper, which aims at the practical application to the BDS. This paper is organized as follows. In Section 2, the characteristics of the ground track of the IGSO satellites are analyzed. Section 3 establishes the dynamical model based on the Earth’s tesseral and zonal perturbations and the dynamical model under the tesseral resonances and the luni-solar perturbations. Section 4 presents the ground track maintenance approach subject to the tesseral resonances and the luni-solar perturbations. Simulations are provided, and the analyses are discussed in Section 5. The conclusion is given in Section 6. 2. Characteristics of the ground track of the IGSO satellites IGSO satellite ground tracks can be described by the geographic latitude / and geographic longitude k. In the geocentric spherical fixed coordinate system, the geography latitude / and longitude k can be expressed by the classical orbital elements as follows. / ¼ arcsinðsin i sin uÞ
ð1Þ
k ¼ arctanðcos i tan uÞ þ X hE
ð2Þ
where hE is the Greenwich sidereal angle with h_ E as its associated rate, u ¼ x þ f is the argument of latitude, and i, X, x and f are the inclination, right ascension of the ascending node, argument of perigee, and true anomaly, respectively. Considering the two-body dynamics model, and according to Eqs. (1) and (2), the ground track of the IGSO satellites is shown in Fig. 1. Ideally, the ground track of the IGSO satellites is an 8shaped curve, which is symmetrical about the equator. The relationship of the boundaries is governed as follows: /max /min ¼ 2i kmax kmin ¼ 2Dki where Dki ¼ arctan
ð3Þ
1 pffiffiffiffiffiffi cos i
ð4Þ pffiffiffiffiffiffiffiffiffi arctanð cos iÞ.
Note that, kA ¼ M þ x þ X hE
ð5Þ
where M is the mean anomaly, kA is the longitude of the stroboscopic mean ascending node of 24-h orbit that given by Gedeon (1969). As kA equals to the longitude of the
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For the IGSO satellites in deep 1:1 resonance, the eastwest station libration caused by the Earth’s geopotential should be discussed firstly. 3. Dynamical model 3.1. Dynamical model based on the Earth’s tesseral and zonal perturbations Considering the Hamiltonian with a set of action/angle variables (I, h), which is convenient for investigating the mean motion resonances (Ely, 1996), the following equation can be obtained: HE ¼
X l2 h_ E I 1 þ V JZ2 þ V Tlmpq 2 2I 1 l4 R2E J 2
Fig. 1. The ground track of the IGSO satellites under the two-body dynamics model.
cross node when sin u ¼ 0, it could represent the east–west station of the IGSO satellite s as well. For a group of ideal orbital elements of the IGSO satellites, we can see that i and kA determine the shape size and the longitude station of the ground track, respectively, while the argument of latitude u determines its location point on the ground track. Therefore, the IGSO satellites with the same parameters of i and kA could have the same ground track. Their location points vary on the ground track, with different values of u. In fact, the eccentricity of the IGSO satellites is always small, where e ¼ Oð103 Þ holds. Thus, the effect of the shape size caused by a small eccentricity could be evaluated by the following equation: kmax kmin ¼ 2ðDki þ Dke Þ
pffiffiffiffiffiffiffiffiffiffi i 2e 1cos . sin i
ð6Þ
where the maximum of Dke equals to emax Therefore, DkDk ¼ OðeÞ can be obtained for the IGSO i satellites with i ¼ 55 . The effects caused by the eccentricity could be ignored for BDS IGSO satellites because of a very small eccentricity. Since the IGSO satellites are in deep 1:1 resonance, they exhibit complex motions. Numerous studies of the resonant orbital dynamics have shown that the Earth’s tesseral harmonics will cause the longitude-dependent libration for the synchronous satellites. Therefore, the east-west stationkeeping is imperative for the IGSO satellites. Moreover, the IGSO satellites have the advantages in high-latitude coverage, and several satellites are always deployed in the same ground track with different location points to obtain the continuous high-latitude coverage. For the BDS, three IGSO satellites are deployed in the same ground track with a differential argument of latitude Du of 120 . Therefore, it is also required to keep the differential argument of latitude among each satellite.
where V JZ2 ¼ 4I 3 ðI 1
1 þI 2 Þ
5
ð7Þ
f3ðI 1 þ I 3 Þ2 ðI 1 þ I 2 Þ2 g, and the
action/angle pairs are defined as follows: pffiffiffiffiffiffi I 1 ¼ la; h1 ¼ kA ¼ M þ x þ X hE ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi I 2 ¼ lað1 e2 Þ la; h2 ¼ x; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi I 3 ¼ lað1 e2 Þ cos i la; h3 ¼ X P T The tesseral expansion V lmpq can be expressed as follows: l 1 X l X l X 1 X X lJ lm RE V Tlmpq ¼ F lmp ðiÞGlpq ðeÞ a a l¼2 m¼1 p¼0 q¼1 cos½mðkA klm Þ qx
ð8Þ
where RE is the Earth’s radius, Flmp(i) and Glpq(e) are the inclination and eccentricity functions, which are defined by Kaula (1966); Jlm and klm are the coefficient and angle corresponding to the tesseral harmonic of degree l and order m. According to Eq. (8), we can see that J22, J31, J32 and J33 are significant while Jlm ðl P 4Þ are negligibly small. Their values for l 6 3 are shown in Table 1. P T Thus, selecting the expansion of V lmpq with l 6 3, with the resonance condition l 2p þ q m ¼ 0, the effective terms of lmpq and the corresponding Flmp(i) and Glpq(e) are shown in Table 2, where ci denotes cos(i) and si denotes sin(i). For the IGSO satellites with e ¼ Oð103 Þ, the tesseral terms of q – 0 are negligibly small compared with these terms of q ¼ 0. Then, Eq. (7) can by expressed as:
Table 1 Values of J lm and klm for l 6 3 . l
m
J lm
klm (degs)
2 3 3 3
2 1 2 3
1:766 106 2:111 106 0:311 106 0:239 106
14.79 7.82 18.12 23.60
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L. Fan et al. / Advances in Space Research xxx (2016) xxx–xxx
Table 2 Values of effective terms of lmpq and the corresponding Flmp(i) and Glpq(e) for l 6 3. lmpq
Flmp(i)
Glpq(e)
2200
3ð1þciÞ2 4
1
3110
15si ð1þ3ciÞ 16 2
15ð1þciÞ 8
3300
3si 2
2212
HE ¼
3ð1þciÞ 4
5 2 2e
þ
1 þ 2e2 þ
3
1 6e2 þ
2
9 2 4e
lmpq
Flmp(i)
Glpq(e)
310-2
2 ð1þciÞ 15si 16
1 2 8e
3122
15si ð13ciÞ 16
11 2 8 e
2
15sið12ci3ci2 Þ
3211
þ 74 e4 þ
18 45si ð1þciÞ 8 2
3312
l2 h_ E I 1 þ V JZ2 þ V T2200 ðI 1 ; I 2 ; I 3 ; kA Þ 2I 21
V L: ¼
3ð1ciÞ 4
1 4 þ 48 e þ 4 þ 49 16 e þ
3 3e þ 11 4 e þ 53 2 8 e
4 þ 39 16 e þ
2 X 2 X 2 X mþs hL: fU 2m;s ðoe Þ 2msp ða; e; i; aL: ; iL: Þð1Þ m¼0 s¼0 p¼0
þ V T3110 ðI 1 ; I 2 ; I 3 ; kA Þ þ V T3300 ðI 1 ; I 2 ; I 3 ; kA Þ
ð9Þ
cosðmþ hÞ þ U 2m;þs ðoe Þ cosðm hÞg ð14Þ
where V
T 2200
V T3110 V T3300
2 lJ 22 RE ¼ F 220 ðiÞG200 ðeÞ cos½2ðkA k22 Þ; a a 3 lJ 31 RE ¼ F 311 ðiÞG310 ðeÞ cos½3ðkA k31 Þ; a a 3 lJ 33 RE ¼ F 330 ðiÞG300 ðeÞ cos½3ðkA k33 Þ: a a
V S: ¼
þ mðX XS: Þ
@H E @V @V 3110 @V 3300 I_ 1 ¼ ¼ 2200 @kA @kA @kA @kA T
T
ð10Þ
@V J 2 @V T2200 @V T3110 @V T3300 @H E l2 _ k_ A ¼ ¼ 3 hE þ þ þ þ @I 1 @I 1 @I 1 @I 1 @I 1 I1 Z
l € kA ¼ 3 4 I_ 1 þ I_ 1 I1 2
"
@ 2 V JZ2 @I 21
þ
@ 2 V T2200 @ 2 V T3110 @ 2 V T3300 þ þ @I 21 @I 21 @I 21
hS: lmsp ða; e; i; aS: ; iS: Þ cos½ðl 2pÞx
l¼2 m¼0 s¼0 p¼0
Then, the system reduces to 1-DOF with action/angle pair (I 1 , kA ). According to Eq. (9), the following equations can be obtained: T
1 X l X l X l X
ð11Þ #
ð12Þ
ð15Þ
where the index L: and S: refer to the Moon’s parameters and the Sun’s parameters, respectively. And k 1 em es ð2sÞ! a 2 L: h2msp ¼ lL: ð1Þ 2aL: ð2þmÞ! ðaL: Þ F 2mp ðiÞF 2s1 ðiL: ÞH 2pð2p2Þ ðeÞ, 1 m¼0 while k 1 is the largest integer part of m2 , em ¼ 2 m–0 1 s¼0 and es ¼ . m h ¼ ð2 2pÞx þ mX ðXL: p2Þ 2 s–0 0 seven lS: ðlsÞ! a l bs p, while bs ¼ . hS: lmsp ¼ k m aS: ðlþmÞ! ðaS: Þ 1=2 sodd 1 m¼0 F lmp ðiÞF lms ðiS: ÞH lpð2plÞ ðeÞ and k m ¼ . 2 m–0 According to Eqs. (13)–(15), we can see that the lunisolar perturbations are greatly affected by the initial values of x and X. Therefore, the effects on the IGSO satellites in the same ground track are different, as the parameters of x and X are different. 4. Ground track maintenance approach subject to tesseral resonances and the luni-solar perturbations
3.2. Dynamical model under tesseral resonances and the lunisolar perturbations
4.1. East-west station-keeping method under the Earth’s tesseral and zonal perturbations
Generally, the luni-solar perturbations can be of the same order of magnitude as the J2 perturbation at the altitudes of the geosynchronous orbits. The luni-solar perturbations effects should also be analyzed in the process of the ground track maintenance for the IGSO satellites. Considering the luni-solar perturbations, the Hamiltonian as shown in Eq. (7) can be written as follows:
Based on the dynamical equations shown in Section 3, the east-west station libration could be solved for the IGSO satellites with i ¼ 55 . The east-west station libration of the IGSO satellites with i ¼ 55 and e ¼ 0 is shown in Fig. 2. As shown in Fig. 2, we can see that the cross node longitude vibrates around two stable points along the east-west direction. According to the characteristics of the variations, the limitations of the vibrations for each cross node longitude exist. We can also see that kA could be kept in a dead-band region DkA . The traditional method can utilize the libration cycles to maximize the drift time between maneuvers. In Fig. 2, the black trajectory in the dead-band region is the natural motion caused by the tesseral resonances. Firstly, kA moves from the start control
H ¼ H E þ V L: þ V S:
ð13Þ
The approximate expansions of the lunar potential and solar potential are governed as follows (Delhaise and Morbidelli, 1993; Ely, 1996):
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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5
where the variables with superscript represent the nominal parameters of the IGSO satellites’ mean elements. Considering the Earth’s tesseral and zonal perturbations only, we can see that e and i don’t have the secular drifts, and the secular drift rates of X should be the same for each IGSO satellites with the same parameters of a, e, i, kAb and kAg . Moreover, the IGSO satellites in the same ground track have the same drift rate of Du. 4.2. Analysis of the effects caused by the luni-solar perturbations
Fig. 2. The east-west station libration of the IGSO satellites with i ¼ 55 and e ¼ 0.
boundary to the natural boundary. Then, it moves from the natural boundary back to the arrival control boundary. The start control boundary and the arrival control boundary have the same value of kA , while their semi-major axes are different from each other. In order to repeat the natural libration cycle, the semi-major axis should be adjusted as the red line shown in Fig. 2. The adjustment amount of semi-major axis Da and the time between maneuvers DT could be calculated according to Eqs. (10) and (11) with the condition k_ Ag ¼ 0 for the natural boundary. Since the dead-band of kA is always small and e ¼ Oð103 Þ holds, the following equation is obtained kA kA ¼ kA0 þ k_ A0 ðt t0 Þ þ € where € kA
mean
ðt t0 Þ 2
2
ð16Þ
3 lI 4 I_ 1 ða ; e ; i ; k A Þ is almost constant. 2
mean
1
2 _ k_ A0 lI 3 h_ E 3hE ða2a0 a Þ is only depended on the semi1
major axis of the start control boundary a0 . The variables with superscript represent the nominal parameters of the IGSO satellites’ mean elements. According to Eq. (16), and considering the condition k_ Ag ¼ 0 for the natural boundary, a0 could be easily solved and so does the adjustment amount of semi-major axis Da ¼ 2ða0 a Þ. Traditionally, the maneuver is performed at the perigee and the eccentricity and semi-major axis are changed simultaneously, the other elements are not changed. In order to obtain the required Da, the impulsive maneuver cost DV and eccentricity change De are given by the following equations: rffiffiffiffiffiffiffiffiffiffiffiffiffi n Da 1 e DV ¼ ð17Þ 2 1 þ e ð1 e ÞDa De ¼ a
ð18Þ
A numerical example is carried out to assess the effects of the luni-solar perturbations on the maintenance control of three IGSO satellites during their life time of about 10 years. Considering the operational performance of BDS IGSO satellites, the nominal value of kA is 118° with the dead-band of ±2°. Based on the initial orbit elements as shown in Table 3, all the IGSO satellites have the same initial parameters of a, e and i. Firstly, the target semi-major axis for stationkeeping is solved based on the Earth’s geopotential, it will be executed once kA meets the arrival control boundary. Then, the station-keeping cycle is simulated in two different conditions, one considers the Earth’s geopotential only and the other considers the Earth’s geopotential and the lunisolar perturbations. The Earth’s gravity field adopts EGM96 model. Daniel et al. (2015) analyzed the station keeping strategies for Galileo and proposed that a truncation of the Earth potential down to 12 12 is acceptable to retain enough accuracy for 10-year orbit propagation. Therefore, the gravitational order of 20 20 is adopted, which is enough for the purpose of the ground track maintenance for BeiDou IGSO satellites. Fig. 3 shows the comparisons of the station-keeping results. From Fig. 3(a), we can see that the offsets of the semimajor axis for three IGSO satellites are the same when the luni-solar perturbations are not considered. The drift rates of semi-major axis are affected by the luni-solar perturbations, and are different from each other. The proposed semi-major axis offset approach is effective, and the magnitude of the offsets of the semi-major axis is about 8.2 km. From Fig. 3(b), we can see that the maximum difference is up to 0.6°, compared with the nominal natural boundary, which may increase the risk of collision. Thus, the luni-solar perturbations cannot be neglected for the maintenance control of the ground track cross node.
Table 3 The initial parameters for the IGSO satellites with i ¼ 55 and e ¼ 0:005. Sat. ID
X/°
x/°
M/°
kA /°
1 2 3
90 330 210
128 128 128
0 120 240
118 118 118
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Fig. 3. The station-keeping cycle in 10 years.
Orbital elements of the IGSO satellites affected by the luni-solar perturbations are shown in Fig. 4. From Fig. 4, we can see that e, i and X have different drift rates while considering the luni-solar perturbations. In particular, the maximum drift of i is up to about 5°, which is harmful for the stable coverage of BDS. At the same time, the drift rates of Du become different, leading to the secular drifts among the IGSO satellites. At the end of the station-keeping cycle of 10 years, the maximum drift of Du is up to about 9°, which will also disrupt the continuous high-latitude coverage of BDS.
4.3. Ground track maintenance approach including the effects of the luni-solar perturbations The analytical results of the effects caused by the luni-solar perturbations indicate that the luni-solar
perturbations affect not only the east-west station and the shape size of the ground track of each IGSO satellite, but also the location angle intervals among the IGSO satellites on the same ground track. In view of the above considerations, the ground track maintenance approach should be investigated considering the effects of the lunisolar perturbations. Note that the eccentricity of BDS IGSO satellite is small; the following equation can be obtained: u kA X þ hE
ð19Þ
Since kA is maintained in its dead-band region, the drift rates differences of Du will decrease with the decrease of the drift rates differences of X. Then, the ground track maintenance of the IGSO satellite can be considered to maintain kA and i in the dead-band region while eliminating the drift rates differences of X for the IGSO satellites.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
L. Fan et al. / Advances in Space Research xxx (2016) xxx–xxx
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Fig. 4. Orbital elements of the IGSO satellites affected by the luni-solar perturbations.
Based on the corresponding relationships of each action/angle pair, a should be adjusted to maintain kA , while, the drift rates differences of X could be eliminated by adjusting any one of a, e and i. The adjustment amounts could be calculated according to Eqs. (16)–(18) using the Lie method (Ely, 1996). However, the calculation process is very complex and requires many mathematical transformations. In this paper, a semi-analytical approach is proposed to solve these problems. Firstly, Da for each maintenance should be determined to keep kA in its dead-band region. Based on the effects caused by the luni-solar perturbations, we can obtain the following equation: ðt t0 Þ k0A mean kA ¼ kA0 þ k_ A0 ðt t0 Þ þ € 2 _ where k_ A0 ¼ 3hE ða2a0 a Þ, € k0A
mean
@ X_ E Di þ DX_ 0 @i while i þ Di 2 ði Dimax ; i þ Dimax Þ min DX_ ¼
_E
2
ð20Þ
2 2 ¼ 3 lI 4 I_ 1 3 lI 4 I_ 01 . I_ 1 is the 1
turbations. I_ 01 could be estimated numerically, then a0 could be easily solved and Da ¼ 2ða0 a Þ. Since Da is determined to maintain kA , and e is required to keep very small, Di is adjusted at the same time just to maintain the location angle intervals among each two IGSO satellites. Moreover, during the life time of the IGSO satellite, i should be in its dead-band region to keep the shape size of the ground track. Therefore, Di could be calculated as follows:
1
drift rate caused by the Earth’s geopotential only, while I_ 01 is the additional drift rate caused by the luni-solar per-
ð21Þ
2
X 32 n J 2 a2E a 2 ð1 e 2 Þ sin i is caused by the where @@i Earth’s geopotential only. DX_ 0 ¼ X_ 0 X_ 0mean , X_ 0 is the additional drift rate caused by the luni-solar perturbations of one IGSO satellite while X_ 0mean is the mean additional drift rate of all the IGSO satellites in the same ground track. After obtaining X_ 0 of each IGSO satellite and X_ 0mean numerically, Di could be solved easily.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
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L. Fan et al. / Advances in Space Research xxx (2016) xxx–xxx
Fig. 5. Ground track maintenance cycle in 10 years considering the tesseral resonances and luni-solar perturbations.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014
L. Fan et al. / Advances in Space Research xxx (2016) xxx–xxx
5. Simulation example The ground track maintenance approach has been studied above to adapt the effects of the luni-solar perturbations by adjusting the semi-major axis and inclination. Since the adjustment amounts are solved approximately, the numerical analysis should be carried out to test and verify the semi-analytical approach. For three IGSO satellites with the initial orbit elements as shown in Table 3, their ground track maintenance were simulated for 10 years. The nominal value of kA is 118° with the dead-band of ±2°, while the nominal value of i is 55° with the dead-band of ±2°. Results are shown as illustrated in Fig. 5. Including the effects of the luni-solar perturbations, the target semi-major axes of each IGSO satellites are different, while their natural boundary of the east-west stationkeeping is almost the same. Compared with the nominal natural boundary, the maximum difference is less than 0.1°. Based on the dead-band region of inclination, Di of each maintenance is almost less than 0.2°, except one adjustment of about 2°. Meanwhile, the secular drifts of Du among each two IGSO satellites were well improved while comparing to Fig. 4. As seen in Fig. 5, the IGSO satellites maneuvered about 21 times during the ten year period, the control frequency decreased to a quarter of the conventional methods, which controls the cross node longitude to the nominal value at the boundary of both sides. The control period is almost the same as that of real IGSO data. The simulations indicate that the semianalytical approach of ground track maintenance is effective. 6. Conclusions In this paper, the orbital evolution characteristics and maintenance control approaches for BDS IGSO satellites are investigated. This work shows that the evolutions of the orbital cross node longitude are dominated by Earth’s non-spherical gravitational perturbation and the lunisolar tidal perturbations. As result of these perturbations, drifts of the longitude of the ground track cross node are demonstrated as Taylor expansion of time. Both the proposed semi-major axis offset approach and the offset calculation equation presented in this paper can provide coequal precise results for different dead-band widths. Simulation examples are given based on real BDS IGSO orbits, which provide references for the orbital maneuver operations of BDS. The numerical results have successfully verified that the method proposed in this paper can satisfy the IGSO satellites’ maintenance control demand.
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The authors are grateful to the helpful discussions by Pro. Junhua Xiang in National University of Defense Technology. The authors also would like to thank all the referees and editors for their valuable suggestions to improve the contents of the paper. References Chao, C., Schmitt, D., 1989. Eliminating GPS stationkeeping maneuvers by changing the orbit altitude. In: AAS/AIAA Astrodynamics Specialist Conference, Stowe, VT, Paper No. 89-407. Delhaise, F., Morbidelli, A., 1993. Luni-solar effects of geosynchronous orbits at the critical inclination. Celes. Mech. Dyn. Astro. 57, 155–173. Daniel, N.R., Anna, N., Giacomo, T., 2015. Galileo Constellation: Evaluation of Station Keeping Strategies.
Acknowledgment This work is supported by the National Natural Science Foundation (NNSF) of China through Grant 61403416.
Please cite this article in press as: Fan, L., et al. Ground track maintenance for BeiDou IGSO satellites subject to tesseral resonances and the lunisolar perturbations. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.09.014