Analytical long-term evolution and perturbation compensation models for BeiDou MEO satellites

Li FAN,Min HU,Cho JIANG,c

aSchool of Aerospace Engineering,Tsinghua University,Beijing 100084,China

bSpace Engineering University,Beijing 101416,China

cCollege of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

1.Introduction

The Chinese BeiDou Navigation Satellite System(BDS)completed regional deployment phase on December 27,2012,which comprises five Geostationary Earth Orbit(GEO),five Inclined Geosynchronous Satellite Orbit(IGSO),and four Medium Earth Orbit(MEO)satellites.1The eventual BDS constellation will consist of five GEO,twenty-seven MEO,and three IGSO satellites.2The long-term evolution of the relative motion among the satellites under the perturbations plays a crucial role for the constellation performance.Frequent station keeping maneuver will consume the fuel,reduce the satellite lifetime, and interrupt the service of navigation constellation.To maintain a stable configuration and provide a better navigation service,the long-term perturbation model and perturbation compensation method should be focused,which aims to reduce the station keeping frequency as few as possible.

The main perturbations of the BDS MEO satellites include the non-spherical perturbations,the luni-solar perturbations,and the solar radiation pressure perturbations.Considering the order of the perturbations,theJ2perturbation is the main factor,with the magnitude of 10-5–10-3.For a group of satellites with the same semi-major axis,eccentricity,and inclination,the long-term evolution law under theJ2perturbation is identical.When there are some deviations of the semimajor axis,eccentricity or inclination,the relative in-plane motion will drift in the long term,and the cross-track amplitude will increase.3,4For the MEO region,the luni-solar perturbations are important influence factors except for theJ2perturbation,with the magnitude of 10-7–10-5.5–7Ref.7shows that the luni-solar perturbations will lead to the long-periodic variations of the inclination,and the variations are relevant to the longitude of ascending node.For satellites in the same orbital plane,the long-term evolution law of the inclination is almost identical;for satellites in different orbital planes,the long-term evolution law of the inclination is evidently different.The magnitude of the solar radiation pressure perturbations for MEO region could achieve 10-7.8–10The solar radiation pressure perturbations usually cannot be exactly modeled due to the solar activity and the precision of the area-to-mass ratio.In the sight of the analysis method,for the effects of theJ2perturbation,the analytical approach is adopted,and for the effects of the perturbations on the constellation configuration,the semi-analytical method or numerical simulation method is usually adopted.

To achieve a ground track that closely repeats from day to day,the orbit period of the GPS satellite is commonly given as half a sidereal day.The GPS satellites experience deep resonance with the Earth gravity,which affects the long stability of the constellation.About two GPS satellites should be maneuvered per month to maintain the configuration,and about four hours should be needed to complete the orbit maneuver and recover the navigation service.5,11As to the Galileo constellation,the station keeping requirements are fulfilled by selecting adequate initial offsets of the orbital parameters for each satellite,and at most one maintenance control is needed for each satellite during the lifetime span.7,12Although the perturbation compensation approach is adopted for Galileo constellation control,little literature introduces the implementation algorithm in detail.The orbital deviations and the perturbations are the fundamental reasons for the secular drifts of the periodical relative motion.Based on the longterm evolution law of relative motion under various perturbations,and by actively offsetting the orbital deviations to compensate the perturbations,the long-term variations of relative motion can be eliminated or mitigated.Therefore,the constellation configuration with certain deviations could be set as the target configuration,and by eliminating or mitigating the relative drift velocity,the control frequency and control budget can be reduced.In current literature,the orbital deviations for perturbation compensation are usually calculated by using the numerical simulation method.Daniel designed the deviations of semi-major axis,inclination,and argument of latitude for Galileo constellation,and by offsetting the initial configuration deviations,the constellation configuration can be maintained within a stable boundary.Zhang et al.proposed the semi-analytical design method of offsetting the orbital elements,which describes the approximate linear relationship between the semi-major axis deviation Δa,the inclination deviation Δiand the longitude of ascending node offset ΔΩ,the mean argument of latitude offset Δλ.13,14The existing perturbation compensation approach demonstrated the effectiveness of offsetting the configuration deviations to maintain the constellation stability.However,these methods are usually semianalytical or numerical,they always need several iterations to obtain the satisfied results,and they cannot reveal the essential relationship between the orbital element offsets and the secular drifts of the constellation configuration.

The purpose of the current study is to develop analytical methods of describing the long-term evolution and perturbation compensation.The perturbation analysis models are established,which consider the initial configuration deviation,theJ2perturbation,and the luni-solar perturbations.An analytical method for calculating the offset of the orbit elements is proposed,which is applied to the constellation maintenance of the BDS MEO satellites.

2.Theoretical analysis of long-term evolution of BDS MEO satellites

2.1.Perturbation analysis model

The canonical conjugate variables are constructed based on the Delaunay variables15:

wherea,e,i,Ω,M,ω and λ correspond to the semi-major axis,eccentricity,inclination,right ascension of the ascending node,mean anomaly,argument of perigee,and mean argument of latitude,respectively;μEis the gravitational constant of the earth.

According to Eq.(1)and the Hamiltonian model for MEO orbit,the canonical motion equations caused by the perturbations can be expressed as

whereHfis Hamiltonian function for the perturbation motion of spacecraft.Thus,the long-term variations of the orbital elements due to theJ2perturbation and the luni-solar perturbations can be analyzed.

The secular linear drifts of the longitude of ascending node and the mean argument of latitude can be expressed as

the subscripts L and S represent the corresponding variables of the moon and the sun.oεis the inclination of the ecliptic of the Earth.The long-periodical perturbation terms related to Ω only affect orbital inclination.Based on Eq.(4),we can obtain

Based on the long-periodical motion of inclination,the secular linear drifts of the longitude of ascending node and the mean argument of latitude,we can find the long-term evolution law between the orbital elements and the relative motion.When the perturbation compensation is the secular linear drift of the reference satellites,the accelerations of the deviations of the longitude of ascending node and the mean argument of latitude can be expressed as

Therefore,we can see that the fundamental reason for the nonlinear variations of the longitude of ascending node and the mean argument of latitude in the MEO region is the long-periodical variation of inclination due to the luni-solar perturbations.

2.2.Long-term evolution law of MEO constellation

The BDS MEO satellites usually deploy on the same orbital altitude and inclination,and form a certain geometry structure by distributing the longitude of ascending node and the mean argument of latitude.The nominal constellation configuration satisfies the condition Δa= Δe= Δi=0 in the case of twobody assumption.

For any two satellites in the same orbital plane,the condition Δa= Δe= Δi= ΔΩ0=0 can be established,where Δa,Δeand Δimean the relative semi-major axis,the relative eccentricity,and the relative inclination,respectively.According to the long-term perturbation analysis model shown as Eq.(3),the secular drifts of the longitude of ascending node and the mean argument of latitude are almost identical.Moreover,the long-periodical motion of inclination is almost the same.Then,the main influences causing the secular relation drift are initial orbital injection error and the analytical model error.The secular relative drifts are approximately linear,which can be compensated by offsetting the orbital elements and on-orbit identification.

Based on the perturbation analysis of the luni-solar perturbations,with the assumption of small eccentricity,the following constraints can lead to the approximately linear variations of the inclination:

wherem,sare natural numbers,respectively.In fact,when the left parts of Eq.(7)are less than 0.03(°)/day,the long periodical variation of Δicannot be neglected,which may destroy the long-term stability of MEO constellation.19In this case,the method of offsetting the initial orbital elements cannot compensate the secular relative drifts completely;it only can maintain the configuration in a certain timescale.

For MEO orbits with the altitude above 20,000 km and the inclination within the range of 50–60°,we can always find the perturbation terms of the moon or the sun.To keep a stable constellation configuration,it is imperative to analyze and compensate the effects of the luni-solar perturbations.

3.Design method of offsetting orbital elements

3.1.Perturbation coefficients of offset value

When the semi-major axis,eccentricity,and inclination vary secularly under the perturbations,the deviations of the longitude of ascending node and the mean argument of latitude satisfy the following constraints:

where Δλ0and ΔΩ0represent the initial relative mean argument of latitude and the longitude of ascending node,respectively;Kλ1andKΩ1represent the linear variation velocities of the mean argument of latitude and the longitude of ascending node,respectively;Kλ2andKΩ2represent the variation accelerations of the mean argument of latitude and the longitude of ascending node,respectively.

The relationship amongKλ1,KΩ1,Kλ2andKΩ2,the initial semi-major axis,eccentricity,inclination deviation,and the rate-of-change of the semi-major axis,eccentricity,inclination deviation are given as follows:

whereKλ2andKΩ2are determined by the perturbations inducing Δ˙a,Δ˙eand Δ˙i;Kλ1andKΩ1are determined by the initial deviations of the orbital elements.

When theJ2perturbation is considered,the Hamiltonian functions of the motion can be expressed as

Thus,the secular drift rates of the longitude of ascending node and the mean argument of latitude can be calculated as

wherendenotes the angular velocity.

According to Eq.(11),the perturbation coefficients of the offsets are given as follows:

In view of the magnitude,the following equations can be obtained:

We can see that offsetting the semi-major axis to eliminate the secular relative drifts of the mean argument of latitude is the most efficient way to compensate the perturbations.

3.2.Calculation model for orbital offsets

whereKλ1andKΩ1are the design parameters,which are determined by the offsets of the orbital elements.To prolong the control time span of configuration maintenance,the initial mean argument of latitude Δλ0should be set as one boundary of the control range ［Δλb,Δλg］,and the extreme value should be set as another boundary of the control range ［Δλb,Δλg］.Then,we can obtain

In the same way,the initial longitude of ascending node ΔΩ0should be set as one boundary of the control range［ΔΩb,ΔΩg］,and the extreme value should be set as another boundary of the control range ［ΔΩb,ΔΩg］.Then,we can also obtain

By combining Eqs.(21)and(22),the initial offsets of the orbital elements can be calculated.

3.3.Perturbation compensation for luni-solar perturbations

The general method of offsetting the orbital elements to compensate the perturbations should be further improved,while the luni-solar perturbations are included.The magnitude of the secular perturbations of the luni-solar perturbations is one tenth of that of theJ2perturbation.Therefore,the lunisolar perturbations must be considered while the perturbation coefficients of the offsets are determined.

Then,the perturbation coefficients of the offsets can be expressed as17

where theJ2perturbation coefficients are shown in Eqs.(12)–(17);the subscript Z represents the coefficients of the zonal perturbations,the superscriptJ2represents theJ2perturbation,the subscripts L and S represent the perturbation coefficients of the luni-solar perturbations.17

Meanwhile,Δ˙ishould be determined according to the main perturbation terms of the luni-solar perturbations.Without loss of generality,according to the analysis about Eq.(7),the long-periodical perturbation terms of the luni-solar perturbations can be calculated by the following equations:

To increase the precision of perturbation compensation,the calculation formula for Δ˙ican be obtained as

To compare the analytical perturbation compensation approach with the traditional numerical approach,we introduce the principle of the numerical perturbation compensation approach.

According to theJ2perturbation,the following linear relationship between Δa,Δiand ΔΩ,Δλ exists.20

Based on Eq.(26),to eliminate the secular drifts of ΔΩ and Δλ,the offsets of Δaand Δican be obtained as follows:

According to Eq.(27),Fig.1 presents the calculation flow of using the numerical approach to compensate the perturbations of MEO constellation.

As seen in Fig.1,the secular drifts of ΔΩ and Δλ can be obtained after high precision numerical propagation,and the offsets of Δaand Δican be obtained by using Eq.(27).To compensate the influences of many perturbations and obtain the satisfied constellation stability,the calculation flow,as shown in Fig.1,should be iterated.

4.Case study for perturbation compensation of BDS MEO satellites

4.1.Requirements of constellation stability

The altitude of the BDS MEO constellation is 21528 km,the inclination is 55°,and the constellation configuration is Walker 24/3/1.15The orbital elements of the constellation are as shown in Table 1.

To ensure the system service performance,the requirements of the constellation configuration can be given as follows:

(1)The drift of the inclination should be smaller than ±2°;

(2)The relative drift of the longitude of ascending node should be smaller than ±2°;

(3)The relative drift of the mean argument of latitude should be smaller than ±5°.

4.2.Long-term evolution of BDS MEO constellation

A long-term evolution for the BDS MEO constellation is conducted to verify the effectiveness of the presented analytical method.A high precision orbital dynamical environment is built,which includes an EGM96 Earth’s gravity field with 20×20 order,the luni-solar perturbations,and the solar radiation pressure perturbations.Random orbital injection errors of semi-major axis with hundreds of meters are considered in the simulation.The calculation time is 10 years.The results of the long-term evolution before compensation are shown in Fig.2.

Fig.1 Calculation flow of using numerical perturbation compensation approach.

Fig.2 Long-term evolution of BDS MEO satellites before compensation within 10 years.

As shown in Table 1,there are eight satellites in each orbital plane,therefore,each legend in Fig.2(c)represent the secular relative drifts of the mean argument of latitude for eight satellites.Orbital injection errors lead to the secular relative drifts of the mean argument of latitude,which can be corrected by using the on-orbit identification method.

As shown in Fig.2,we can see that the long-term evolution law is consistent with the analytical results discussed in Section 2.2:

(1)For any two satellites in the same orbital plane,the secular drifts of the longitude of ascending node and the mean argument of latitude are almost identical;the long-periodical motion of inclination is almost the same.

(2)For any two satellites in different orbital planes,the long-periodical variations caused by the luni-solar perturbations are not identical anymore.

The long-term evolution of the BDS MEO nominal configuration exceeds the requirement configuration maintenance range,which needs to be controlled.

4.3.Maintenance control for BDS MEO constellation

4.3.1.Results of proposed analytical perturbation compensation approach

The analytical perturbation compensation approach of offsetting the orbital elements is adopted to control the BDS MEO constellation.Based on Eqs.(21)and(22),we calculated the initial deviations.The orbital elements after compensation are shown in Table 2.

The results of the long-term evolution after compensation are shown in Fig.3.

As shown in Fig.3,we can see that the proposed analytical perturbation compensation method is effective.For the BDS MEO constellation,the configuration does not require any control during 10 year mission lifetime.

4.3.2.Results of traditional numerical perturbation compensation approach

The traditional numerical perturbation compensation approach for offsetting the orbital elements is adopted to control the BDS MEO constellation.We did eight iterations to obtain the orbital offsets,and the results of perturbation compensation are shown in Fig.4.

4.3.3.Comparisons and discussions

Comparing Fig.3 with Fig.4,we can see that the long-term evolution performance of the inclinations,the longitude of ascending node and the mean argument of latitude of the proposed analytical perturbation compensation approach are better than those of the traditional numerical perturbation compensation approach.The advantages of the analytical model are shown in two aspects.One is that it needs no iteration for computing the offsets of the orbital elements,and the other is that the offsets of the orbital elements can be calculated with respect to the mean argument of latitude and the longitude of ascending node directly.The comparisons validate the effectiveness of superiority of the proposed analytical perturbation compensation approach.

Table 2 Orbital elements after compensation.

Fig.3 Long-term evolution of BDS MEO satellites within 10 years by using analytical perturbation compensation approach.

5.Conclusions

In this study,the analytical models for long-term evolution and perturbation compensation of BeiDou Navigation Satellite System(BDS)Medium Earth Orbit(MEO)constellation are proposed.The proposed methods are intuitive,and can reveal the essential relationship between the orbital element offsets and the secular drifts of the constellation configuration.Moreover,they do not need any iteration compared with the traditional semi-analytical or numerical methods.The presented analytical methods are validated by the case study of the BDS MEO constellation maintenance.The results show that the perturbation compensation approach is effective,and the BDS MEO constellation needs no control during 10 years by using the analytical methods.

Fig.4 Long-term evolution of BDS MEO satellites within 10 years by using traditional numerical perturbation compensation approach.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(No.61403416).

Appendix A.Supplementary material

Supplementary data associated with this article can be found,in the online version,at https://doi.org/10.1016/j.cja.2017.10.010.

1.China Satellite Navigation Office[Internet].BeiDou Navigation Satellite System open service performance standard(Version 1.0).[updated 2013 December;cited 2016 Oct 18].Available from:http://en.beidou.gov.cn/.

2.China Satellite Navigation Office[Internet].BeiDou Navigation Satellite System signal in space interface control document open service signal(Version 2.0).[updated 2013 December;cited 2016 Oct 18].Available from:http://en.beidou.gov.cn/.

3.Schaub H,Alfriend KT.J2-invariant relative orbits for spacecraft formations.Celestial Mech Dyn Astron2001;79(2):77–95.

4.Koon WS,Marsden JE.J2dynamics and formation flight.Reston:AIAA;2001 Aug 6–9.Report No.:AIAA-2001-4090.

5.Schutz BE,Craig DE.GPS orbit evolution:1998–2000.Reston:AIAA;2000 Aug 14–17.Report No.:AIAA-2000-4237.

6.Ricardo P,Belen MP,Miguel RM.The Galileo constellation design:a systematic approach.Proceedings of the 18th international technical meeting of the satellite division of the institute of navigation(ION GNSS 2005);2005 Sep 13–16;Long Beach,CA,USA,2005.p.1296–306.

7.Cambriles AP.Galileo station keeping strategy.20th international symposium on space flight dynamics;2007 Sep 24–28;Annapolis,Maryland,USA,2007.p.1–14.

8.Marek Z,Sima A,Ant S,Paul C.Taking the long view:the impact of spacecraft structural design and high precision force modeling on long-term orbit evolution.ION GPS/GNSS 2003;2003 Sep 9–12;Portland,OR,USA,2003.p.1002–8.

9.Rodriguez SC.Adjustable box-wing model for solar radiation pressure impacting GPS satellites[dissertation].Munchen:Technische Universitat Munchen;2012.

10.Stefanelli L,Metris G.Solar gravitational perturbations on the dynamics of MEO:increase of the eccentricity due to resonances.Adv Space Res2015;55(7):1855–67.

11.Chao C,Schmitt D.Eliminating GPS stationkeeping maneuvers by changing the orbit altitude.Proceedings of the AAS/AIAA astrodynamics conference;1989 Aug 7–10;San Diego,CA,USA,1990.p.623–43.

12.Peiro AM,Beech TW,Garcia AM,Merino MR.Galileo in-orbit control strategy.Proceedings of the IAIN world congress in association with the U.S.ION Annual Meeting;2000 Jun 26–28;San Diego,CA,USA,2000.p.469–80.

13.Zhang YL,Fan L,Zhang Y,Xiang JH.Theory and design of satellite constellation.Beijing:Science Press;2008[Chinese].

14.Xiang JH,Fan L,Zhang YL.Design and capability analysis of an aircraft with in flatable wing.Flight Dyn2007;25(4):81–5[Chinese].

15.Kamel A,Ekman D,Tibbitts R.East-west station keeping requirements of nearly synchronous satellites due to Earth’s triaxiality and luni-solar effects.Celestical Mech1973;8(1):129–48.

16.Jiang C.Perturbations of spacecraft relative motion and its compensation control[dissertation].Beijing:Tsinghua University;2015.

17.Kaula WM.Development of the Lunar and Solar disturbing functions for a close satellite.Astron J1962;67:300–3.

18.Delhaise F,Morbidelli A.Luni-solar effects of geosynchronous orbits at the critical inclination.Celestial Mech Dyn Astron1993;57(1–2):155–73.

19.Li HN,Li JS,Jiao WH.Analyzing perturbation motion and studying con figuration maintenance strategy for Compass-M navigation constellation.J Astron2010;31(7):1756–61[Chinese].

20.Qian S,Li HN,Wu SG.Perturbation compensation strategy for MEO non-resonant navigation constellation maintenance.J Natil Univ Defense Technol2014;36(2):53–60[Chinese].