comparing satellite orbit determination by batch

Vol.:(0123456789)1 3

GPS Solutions (2019) 23:25 https://doi.org/10.1007/s10291-018-0816-9

ORIGINAL ARTICLE

Comparing satellite orbit determination by batch processing and extended Kalman filtering using inter-satellite link measurements of the next-generation BeiDou satellites

Xia Ren1,2,3 · Yuanxi Yang1,2 · Jun Zhu4,5 · Tianhe Xu6

Received: 17 December 2017 / Accepted: 18 December 2018 / Published online: 2 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019

AbstractThe next-generation BeiDou satellites are equipped with inter-satellite link (ISL) payloads to realize the ability of autonomous navigation (Auto-Nav). We focus on the application of batch processing and extended Kalman filtering (EKF) algorithm on Auto-Nav of BeiDou satellite system (BDS). The mathematical model of dual one-way measurements and principle of batch mode and EKF are introduced. Using real ISL measurements, Auto-Nav experiments are conducted with batch processing and EKF, respectively. For the sub-constellation, the EKF with a priori equipment delay constraints is proposed. The result shows that (1) with three ISLs and only one anchor station, the ISL measurements are sparsely distributed and the coverage of the whole arc is about 31%. The observations and dynamic models per epoch contribute more to satellite position and velocity parameters than to equipment delay parameters. (2) For batch processing, the overlap precision of precise orbit determination (POD) with ISLs and ground-satellite links (GSLs) is about 0.1 m in the radial direction and is better than 1 m three-dimensionally. The variation of the estimated equipment delays is within ± 0.6 ns. The observation residuals of ISLs behave such as a normal distribution, while the residual of GSLs show periodical variation due to uncorrected troposphere delay. (3) For the EKF, the ISL-only orbit determination is sensitive to the accuracy of the initial state. Compared with batch result, the precision of ISL-only orbit determination using EKF is better than 2 m given accurate initial states. The filtering does not show constellation drift or rotation within 8 days. However, with approximate initial states which position accuracy is 100 m, the precision of POD decreases to dozens of meters. (4) For the EKF, the accuracy of POD improves to 1.5 m in three dimensions with the support of an anchor station. The period for equipment delay parameters to converge is about 24 h. However, the convergence rate of equipment delay parameters is much slower than that of satellite state parameters. The possible reason is the unbalanced contribution of observation and dynamic model information on the estimated parameters. Thus, it is better to constrain the equipment delay parameters with a priori information while filtering.

Keywords Inter-satellite link · Autonomous navigation · Batch processing · Extended Kalman filtering

Introduction

An auto-navigation system (ANS) is to realize updates of the satellite ephemeris automatically with only inter-satellite link (ISL) measurements (Ananda et al. 1990). The concept of ANS was first proposed for the Global Positioning Sys-tem (GPS) in 1984 to decrease the reliance of navigation systems on ground stations. It is also verified that ISLs can help to improve navigation satellite orbit determination and time synchronization (Wolf 2000; Gill 1999). The earliest in-orbit test on ANS was carried out with GPS BLOCK IIR. The experiment proved the feasibility of the ANS, and the URE is better than 3 m after auto navigation for 75 days (Rajan 2002).

* Xia Ren [email protected]

1 State Key Laboratory of Geo-Information Engineering, Xi’an 710054, China

2 Xi’an Research Institute of Surveying and Mapping, No. 1 Mid-Yanta Road, Xi’an 710054, China

3 Information Engineering University, Zhengzhou 450001, China

4 State Key Laboratory of Astronautic Dynamics, Xi’an 710043, China

5 Xi’an Satellite Control Center, Xi’an 710043, China6 Institute of Space Science, Shandong University,

Weihai 264209, China

http://libyc.nudt.edu.cn:8000/rwt/BLYUN/http/N73GG4LEF3YYE3D/0000-0002-0848-2652

http://libyc.nudt.edu.cn:8000/rwt/BLYUN/http/MN3G865UNWRYE43PMN3G865UPJTXMLUQPJUB/dialog/?doi=10.1007/s10291-018-0816-9&domain=pdf

GPS Solutions (2019) 23:25

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In 2015 China started the construction of ANS. Five dem-onstration satellites of next-generation of BeiDou system (BDS) were launched with ISL payloads that included two Inclined Geosynchronous Orbit (IGSO) satellites and three medium earth orbit (MEO) satellites. Among them, one MEO is undergoing maintenance, and the others are in-orbit testing. To solve the constellation rotation, anchor stations are built to provide ground datum with ground-satellite links (GSLs) (Menn and Bemstein 1994; Rajan et al. 2003). The ISLs and GSLs of BDS are both based on Ka-band signals with time-division multiple addresses (TDMA) that is quite different from L-band signals (Ren et al. 2017). With this sub-con-stellation, the overlap precision of POD is about 1 m in three dimensions (Tang et al. 2016, 2017). The precision of L-band POD and time synchronization is improved with the joint of ISLs (Chen et al. 2016; Yang et al. 2017). In the long run, the ISLs may also contribute to optimizing observation geometry, and extend the tracking coverage to periods in which satellites are not visible from BDS regional stations. The results men-tioned above are achieved with post processing.

Real Auto-Nav will be conducted with onboard computers to realize real-time orbit determination and clock error calcu-lation. Principles and theory of real-time processing method have been studied with simulation experiments (Wolf 2000; Yiet al. 2011). However, few results are available on Auto-Nav using real measurements. Also, it is important to analyze the character of the real-time POD given the sub-constellation.

We mainly present the initial result of orbit determination for Auto-Nav using real ISL measurements. The batch pro-cessing method is applied for post processing, and EKF is applied for real-time processing. First, we introduce the math-ematical model of dual one-way measurements and the prin-ciple of ISL orbit determination based on batch processing and EKF. Then, the characteristics of ISL measurements are analyzed, and Auto-Nav is conducted with batch processing and EKF, respectively. For the current constellation of BDS, a POD with a priori equipment delay constraints is proposed for the real-time POD. Finally, some conclusions are drawn.

Dual one‑way measurement

The ISLs for BDS are dual one-way measurements at Ka-band and are generated sequentially over different satellite pairs. The principle can be briefly summarized

as follows. The forward signal is transmitted from satel-lite A at the time t1 (expressed as TA

C(t1) given by the

satellite clock), and is received by satellite B at the time t2 (expressed as TB

C(t2) given by the satellite clock). The

pseudorange is ��BA and contains transmitting equipment delay �A

tr , receiving equipment delay �B

rc , satellite clock

error �Bclk

(t2) and �A

clk

(t1) , and space delay �BA

sp (Ren et al.

2017). The transmitting and receiving equipment delays are similar to DCB in L-band orbit determination (Tang et al. 2017). Similarly, the backward signal is transmitted by satellite B at the time t3 (satellite time TB

C(t3) ) and is

received by satellite A at the time t4 (satellite time TAC(t4) ). The pseudorange is ��AB containing transmitting

equipment delay �Btr

, receiving equipment delay �Arc

, satel-lite clock error �A

clk

(t4) and �B

clk

(t3) , and space delay �AB

sp .

It can be seen that the dual one-way measurements con-sist of a pair of measurements and are free of clock errors.

The mathematical model of forward and backward meas-urements between satellite A and B can be expressed as (Ren et al. 2017):

and

where ��AB and ��BA are pseudorange observations received by satellite A and B, respectively, RA

(ti) and RB

(ti) are

positions of satellite A and B at the time ti , �clk are corre-sponding clock errors, �rc and �tr are receiving equipment delay and transmitting equipment delay, and �AB

cor and �BA

cor

are systematic errors such as antenna phase center offsets and relativistic effects. For Ka-band measurements at about 22.5 GHz, the ionosphere delay is about 3 cm which is less than the normal random noise of 10 cm, and is thus negli-gible for ISLs (Tang et al. 2017).

The reduction formula of the two measurements can be written as

and

(1)

��AB = 𝜏ABsp

+ 𝜏Arc+ 𝜏B

tr+ c ×

(𝛿Aclk

(t4)− 𝛿B

clk

(t3))

=|||RA

(t4)− RB

(t3)|||+ c ×

(𝛿Aclk

(t4)− 𝛿B

clk

(t3))

+ 𝜏Arc+ 𝜏B

tr+ 𝛿AB

cor

(2)

��BA = 𝜏BAsp

+ 𝜏Brc+ 𝜏A

tr+ c ×

(𝛿Bclk

(t2)− 𝛿A

clk

(t1))

=|||RB

(t2)− RA

(t1)|||+ c ×

(𝛿Bclk

(t2)− 𝛿A

clk

(t1))

+ 𝜏Brc+ 𝜏A

tr+ 𝛿BA

cor

(3)��AB + d𝜌AB =|||RA

(t0)− RB

(t0)|||+ c ×

(𝛿Aclk

(t0)− 𝛿B

clk

(t0))

+ 𝜏Arc+ 𝜏B

tr+ 𝛿AB

cor

(4)��BA + d𝜌BA =|||RB

(t0)− RA

(t0)|||+ c ×

(𝛿Bclk

(t0)− 𝛿A

clk

(t0))

+ 𝜏Brc+ 𝜏A

tr+ 𝛿BA

cor


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where

and

where RA

(t0) , RB

(t0) , RA

(t4) , RB

(t3) , RB

(t2) , and RA

(t1)

are satellite positions given by the broadcast ephemeris, and �Bclk

(t0) , �A

clk

(t0) , �A

clk

(t4) , �B

clk

(t3) , �B

clk

(t2) , and �A

clk

(t1) are

satellite clock errors calculated by predicted clock errors. It is clear that clock errors can be eliminated in this way. The dual one-way measurement can be combined as

it can be seen in (7) that ��AB(t0) equals to the ‘visible

measurement’ transmitted and received at the same time t0 without satellite clock parameters. The proposed pro-cessing method of dual one-way measurement ignores the satellite clock drift. Therefore, the time difference of the coupled dual one-way measurement should be within 3 s.

For BDS, the GSLs share the same communication and measurement system with the ISLs. Thus, the processing method mentioned above is suitable for GSL measure-ments as well. The difference is that the tropospheric delay should be corrected for GSL measurements (Tang et al. 2017). Only the wet component of tropospheric delay is corrected with the Saastamoinen model in the following computation.

For POD with ISLs and GSLs, the estimated parameters for each satellite are satellite position and velocity, dynamic parameters, and equipment delay parameters. The equipment

delay parameter for each satellite is the sum of transmitting equipment delay and receiving equipment delay which can-not be estimated separately (Ruan et al. 2014).

(5)

d�AB =|||RA

(t0)− RB

(t0)|||

−|||RA

(t4)− RB

(t3)|||

+ c ×(�Aclk

(t0)− �B

clk

(t0))

− c ×(�Aclk

(t4)− �B

clk

(t3))

(6)

d�BA =|||RB

(t0)− RA

(t0)|||−|||RB

(t2)− RA

(t1)|||

+ c ×(�Bclk

(t0)− �A

clk

(t0))

− c ×(�Bclk

(t2)− �A

clk

(t1))

(7)

��(t0)=

1

2

(��AB + d𝜌AB + ��BA + d𝜌BA

)

=|||RB

(t0)− RA

(t0)|||+

1

2

(𝜏Brc+ 𝜏A

tr+ 𝜏A

rc+ 𝜏B

tr+ 𝛿AB

cor+ 𝛿BA

cor

)

Principle of extended Kalman filtering and batch processing

For orbit determination of a navigation constellation, one has centralized and distributed data processing methods. Centralized processing is to achieve the most optimal result. All the measurements are gathered and processed together. However, the computation burden is heavy especially for real-time processing with on-board computers (Wolf 2000). The Auto-Nav of GPS IIR is based on distributed process-ing (Ananda et al. 1990). A navigation constellation is con-sidered as distributed satellite system when each satellite determines its own orbit using received measurements and the ephemeris of visible satellites. The ephemeris errors of other satellites might be accumulated and affect the ultimate result. Thus, the orbit determination results are sub-optimal. The principle of EKF and batch processing introduced in this subsection is based on centralized processing.

Extended Kalman filtering

EKF algorithm is wildly used in real-time processing with good accuracy and efficiency. The estimated parameter vec-tor at epoch ti is Xi including satellite position rsj

i , velocity

risj , dynamic parameters pSj

i , and equipment delay �Sj

i . The

correction of Xi referring to a priori value X∗

i is 𝛿Xi.

The EKF contains the measurement update and time update which can be expressed as (Yang 2006; Yang et al. 2011)

and

where Vi is the measurement residual, Ai is the normal matrix, li is the observation referring to 𝛿Xi and meets the equation li = Li − AiX

∗

i , Xi is predicted states, Wi is

the dynamic model error, and Φ(ti, ti−1

) is the state tran-

sition matrix from ti−1 to ti . Assume there are n satellites S1, S2,… Sn , and 𝛿Xi at each epoch can be expressed as

where 𝛿rSji

, 𝛿 rSji

, �pSji and ��Sj

i are the corrections of satellite

position, velocity, dynamic parameters, and equipment delay parameters, respectively. The state transition matrix contains the partial derivatives of the state Xi with respect to the state Xi−1 at the time ti−1 , and is given by

(8)Vi = Ai𝛿Xi − li

(9)Xi = Φ

(ti, ti−1

)Xi +Wi

(10)𝛿Xi =||| 𝛿r

S1i

𝛿 rS1i

𝛿pS1i

𝛿𝜏S1i

𝛿rS2i

𝛿 rS2i

𝛿pS2i

𝛿𝜏S2i

… 𝛿rSni

𝛿 rSni

𝛿pSni

𝛿𝜏Sni

|||

T

(11)Φ

(ti, ti−1

)= diag

|||Φ

S1(ti, ti−1

)Φ

S2(ti, ti−1

)… Φ

Sn(ti, ti−1

) |||


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and

where 𝜕rSj

i

𝜕rSj

i−1

, 𝜕rSj

i

𝜕 rSj

i−1

, 𝜕rSj

i

𝜕pSj

i−1

, 𝜕rSj

i

𝜕rSj

i−1

, 𝜕rSj

i

𝜕 rSj

i−1

, and 𝜕rSj

i

𝜕pSj

i−1

are partial deriva-

tives. The least square solution of state parameters at time ti can be expressed as (Yang et al. 2011)

where Σi is the covariance matrix of measurements, and ΣXi

is the covariance matrix of predicted state parameters given by

where ΣWi is the state noise compensation matrix. The covar-

iance matrix of the estimated parameters is given by

when the a priori state Xapr with its variance matrix Λ is involved, then (8) can be derived as

where �Xapr is the correction of Xapr referring to X∗

i , VXapr

is

the residual of �Xapr . Based on least square estimation, Eqs. (13) and (15) can be derived as

where �Xapr is the correction vector of the a priori state Xapr referring to X∗

i.

It can be seen that the EKF result is the combination of measurement information and dynamic model information. However, the EKF may appear to diverge due to the accumu-lated dynamic model errors. The function of the state noise

(12)ΦSj =

|||||||||||||

𝜕rSj

i

𝜕rSj

i−1

𝜕rSj

i

𝜕 rSj

i−1

𝜕rSj

i

𝜕pSj

i−1

0

𝜕 rSj

i

𝜕rSj

i−1

𝜕 rSj

i

𝜕 rSj

i−1

𝜕 rSj

i

𝜕pSj

i−1

0

0 0 I 0

0 0 0 I

|||||||||||||

(13)𝛿Xti=

(

ATtiΣ−1tiAti

+ Σ−1

Xti

)−1(ATiΣlilti + Σ

−1

Xi

Xi

)

(14)ΣXi= Φ

(ti, ti−1

)ΣXi−1

ΦT(ti, ti−1

)+ Σ

Wi

(15)ΣXi= ΣXi

− ΣXiATi

(AiΣXi

ATi+ Σi

)−1AiΣXi

(16)|||||

Vi

VXapr

|||||=

|||||

Ai

I

|||||𝛿Xi −

|||||

li𝛿Xapr

|||||

(17)𝛿Xi =

(Λ

−1+ AT

iΣ−1iAi + Σ

−1

Xi

)−1(ATiΣ−1ili + Λ

−1𝛿Xapr + Σ−1

Xi

Xi

)

(18)

ΣXi

= ΣXi−

(AiΣ

−1

Xi

ATi+ Λ

−1+ Σi

)−1(AiΣ

−1

Xi

ATi+ Λ

−1)ΣXi

compensation matrix is to adjust the covariance of dynamic model information to ensure the contribution of measure-ment information and dynamic information is balanced. Thus, it is important to give proper state noise compensation matrix to ensure the stability of the filtering.

Batch processing

For ground assessment, batch processing is used to achieve the optimal result. This method gathers all the measurements in a certain arc and computes initial satellite positions and velocities, and dynamic model parameters. Then, the orbit is propagated with an accurate initial satellite state.

For ISL orbit determination, the estimated parameters are initial satellite state X0 including satellite position and veloc-ity, dynamic parameters, and equipment delays. The normal equation between measurements Li and state vector Xi at ti is (Montenbruck and Gill 2001)

where Ai contains the partial derivatives of the measure-ments with respect to the state Xi , and Pi is the observation weight matrix. For orbit determination, the estimated state represents usually the satellite parameters at initial epoch. Then, the normal equation at ti can be rewritten as

where Φ(ti, t0

) is the transition matrix from time ti to time t0 .

It can be computed successively from the preceding transi-tion matrix by

accumulate all the measurements at different epoch and the final normal matrix can be expressed as

and

where H contains the final partial derivatives of all the measurements with respect to the state Xi.

(19)(ATiPiAi

)−1Xi = AT

iPiLi

(20)

((AiΦ

(ti, t0

))TPi

(AiΦ

(ti, t0

)))X0 =

(AiΦ

(ti, t0

))TPiLi

(21)Φ

(ti, t0

)= Φ

(ti, ti−1

)× Φ

(ti−1, ti−2

)×⋯ × Φ

(t1, t0

)

(22)(HTPH

)X0 = HTPL

(23)HTPH =

n∑

i=1

(AiΦ

(ti, t0

))TPi

(AiΦ

(ti, t0

))

(24)HTPL =

n∑

i=1

(AiΦ

(ti, t0

))TPiLi


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Computations

In this section, Auto-Nav is conducted with real ISL meas-urements and GSL measurements of the next-generation BeiDou satellites. Based on centralized processing, batch processing and EKF are applied and the results are analyzed and compared.

Character of ISL and GSL measurements

The ISL measurements are generated between satellites I1S, M1S, and M2S. The GSL measurements are between the three satellites and one anchor station. The total length of the observations is 8 days. The detailed information about each link is shown in Table 1, and the visibility of each link is shown in Fig. 1. In Table 1, the number of observations N for a certain link is the total number of observations within

the arc, and the number of observations Nall for all the links is the total number of epochs with measurements. The ratio of coverage � for each link can be calculated with the fol-lowing equation

where Tsimple is the interval of measurements which is 3 s for BDS ISLs and GSLs, Tarc is the arc length. The ratio of coverage �all for all the links can be calculated as

where Tsimple and Tarc have the same meaning as in (24). Most epochs have only one ISL measurement due to the TDMA system, and several epochs have two measurements includ-ing one ISL and one GSL. The coverage of measurements on the whole arc is about 31%, and the coverage of each link is less than 20%. The visible arc between an MEO satellite and an IGSO satellite is only several hours per day that is far less than the MEO–MEO link. The tracking of I2S from the anchor station is much longer than the two MEOs. The two MEOs achieve more measurements than I2S in the period.

It can be seen that the measurement series are not con-tinuous and sparsely distributed with the sub-constellation, which may have negative effect on the POD. For batch pro-cessing, all the observations in a certain arc can be used. The negative effect caused by weak observation geometry can be avoided by choosing the suitable length of the arc. For EKF, only the observation in the current epoch can be used, and the negative effect may be inevitable.

Contributions of observations and dynamic model on estimated parameters

Usually, the coefficient of the observation equation is one factor reflecting the contribution of the measurement on each estimated parameter. For the EKF, the predicted parameters extrapolated with dynamic models could be seen as a kind of measurements as well. Assuming the observation at epoch ti is I2S-M1S, then the observation equation with the sub-constellation can be expressed as (Yang 1989)

(25)� = N ×

Tsimple

Tarc

(26)�all = Nall ×

Tsimple

Tarc

(27)

||||||||||

VXrS1

VXrS2

VXrS3

Vl

||||||||||

=

|||||||||

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0𝜕l

𝜕rS1

𝜕l

𝜕rS20 1 1 0 0

|||||||||

||||||||||||||||

𝛿XrS1

𝛿XrS2

𝛿XrS3

𝛿XEDS1

𝛿XEDS2

𝛿XEDS3

𝛿XEDSTA

||||||||||||||||

−

|||||||||

𝛿XrS1

𝛿XrS2

𝛿XrS3

l

|||||||||

Table 1 Detailed information about ISLs and GSLs

ISL/GSL Satellite/station Ratio of coverage Number of observations

I2S—M1S I2S (IGSO)M1S (MEO)

0.0232 5353

I2S—M2S I2S(IGSO)M2S (MEO)

0.0515 11,868

M1S—M2S M1S (MEO)M2S (MEO)

0.1539 35,477

Station-I2S StationI2S(IGSO)

0.1063 24,494

Station-M1S StationM1S(IGSO)

0.0360 8303

Station-M2S StationM2S(IGSO)

0.0370 8529

All 0.3169 73,024

0 1 2 3 4 5 6 7 8

M1S-M2S

I2S-M2S

I2S-M1S

STA-M2S

STA-M1S

STA-I2S

Time (Day)

Fig. 1 Visibility of ISLs and GSLs with time


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25 of 12

where rSj = |||XSj YSj ZSj XSj YSj ZSj |||

T

is satellite state vec-

tor including the position and velocity, S1, S2, S3 represent satellites I2S, M1S, and M2S, respectively. VX

rSj is the resid-

ual referring to predicted state 𝛿XrSj of satellite Sj , Vl is the

residual of the ISL measurement between satellite S1 and S2 , �l

�rSj is the partial of the measurement with respect to the sat-

ellite Sj state parameters, 𝛿XrSj is the correction term to the

satellite Sj state parameters including satellite position and velocity, 𝛿XED

Sj and 𝛿XEDSTA are correction terms of satellite

equipment delays and anchor station equipment delay.The nonzero coefficients reflect the observation contributes

to the certain parameter. It is clear that the coefficient of the predicted parameter 𝛿X

rSj is nonzero only for its own state

parameter 𝛿XrSj . The coefficients of ISL measurement l are

nonzero only for parameters of related satellites. The coeffi-cients corresponding to 𝛿XEDS3 and 𝛿XEDSTA

are all zero. It fol-lows that the predicted information and ISL measurements at each epoch contributes to all the satellite position and velocity parameters and only to some equipment delay parameters.

Post POD conducted with batch processing

In this subsection, the Ka-measurements are all processed based on the batch mode. The POD strategy is shown in Table 2. Only the dry component of tropospheric delay

is corrected with the Saastamoinen model. Only PCO is

Table 2 Strategies for orbit determination using batch mode

Data length 8 daysArc length 3 daysMeasurements ISLs and GSLs (six links)Dynamic model Two-body motion, earth non-sphere gravity, third-body gravity, solar radiation, solid tideSolar radiation model ECOM model with five parameters (the constant term in D, Y and B direction and the

periodical term in B direction)Systematic error of measurements ISLs: satellite phase center offset, and relativistic effects

GSLs: satellite phase center offset, relativistic effects, and tropospheric delayPhase centre correction Only PCO are consideredEstimated parameters Initial satellite position and velocity, ECOM parameters, and equipment delay

R T N0

0.5

1

Over-Lap Precision

RM

S (m

)

I2S M1S M2S

Fig. 2 RMS of over-lap comparison of POD

1 2 3 4 5 6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (Day)

)sn(tlusergnissecorp

hctabfoecnairav

DE

I2SM1SM2SStation

Fig. 3 Variation of estimated equipment delays

0 1 2 3-0.5

0

0.5

Time (Day)

)m(laudise

R

I2S-M1SI2S-M2SM1S-M2S

Fig. 4 Residual series of ISLs of the first batch processing arc


1 3

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corrected for satellite antenna phase center correction. The orbit determination arc is 3 days with a sliding window of 1-day. The initial satellite position and velocity, Extended CODE Orbit Model (ECOM) parameters, and equipment delays are parameters to be estimated. The equipment delay is considered as constant over the arc. The precision is evalu-ated by the average root mean squares (RMS) of over-lap comparison of the six arcs shown in Fig. 2. The variation of the estimated equipment delays is presented in Fig. 3. The observation residuals of ISLs and GSLs of the first arc are shown in Figs. 4 and 5. With the results, several facts can be concluded.

1. The precision of the determined satellite orbit is about 0.1 m in the radial direction and is better than 1 m in three dimensions. The precision of M1S and M2S is better than that of I2S.

2. Figure 3 shows the residuals of equipment delays by removing the average value. It is clear that the estimated equipment delays vary within ± 0.6 ns which is quite stable.

3. As shown in Figs. 4 and 5, the observation residuals of GSLs are much larger than those of ISLs. The residuals of ISLs vary within 0.3 m and behave randomly. How-ever, the residuals of the GSLs vary within 0.5 m and show periodical variation due to the uncorrected part of the tropospheric delay.

Real‑time POD conducted with EKF algorithm

The orbit determination strategy for the following computa-tion is presented in Table 3. The systematic error correction method in EKF is the same as in batch processing. The orbit arc is 8 days. The position and velocity noise compensation are 1.0 × 10−11 m2 and 1.0 × 10−14 m2/s2, respectively. The initial values of ECOM parameters and equipment delay parameters are achieved by batch processing.

With overlap precision better than 1 m, the orbit param-eters achieved by batch processing are used as reference orbits to evaluate the precision of POD conducted with the EKF algorithm.

EKF with ISLs-only

In this computation, three schemes are designed with differ-ent POD conditions. As shown in Table 4, approximate and accurate position and velocity are used in schemes 1 and 2,

0 1 2 3-0.5

0

0.5

Time (Day)

)m(laudise

RI2S-StaM1S-StaM2S-Sta

Fig. 5 Residual series of GSLs of the first batch processing arc

Table 3 Detailed POD strategy using EKF

Length of arc 8 daysDynamic model Two-body motion, earth non-sphere gravity, third-body gravity, solar radiation, solid tideSolar radiation ECOM model with 5 parameters (the constant term in D, Y and B direction and the

periodical term in B direction)Systematic error for measurements ISLs: satellite phase center offset, and relativistic effects

GSLs: satellite phase center offset, relativistic effects, and tropospheric delayAntenna phase center correction Only PCO are considered

Table 4 Different schemes for ISL-only orbit determination using EKF

Scheme Precision of initial states Equipment delay

Position (3D) (m)

Velocity (3D) (m/s) Equipment delay (ns)

1 5 0.00001 Accurate Not estimated2 100 0.0001 Accurate Not estimated3 5 0.00001 3 Estimated


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respectively. The precision of initial states given in Table 4 is the difference between initial states and L-band orbits in three dimension. In schemes 1 and 2, the equipment delays are given by batch processing and not estimated in filter-ing. In scheme 3, the equipment delay is estimated, and the precision of initial values is 3 ns. The result of each scheme is compared with reference orbits and the residual series are drawn in Figs. 6, 7 and 8, the statistic results (RMS) of the converged orbits are listed in Table 5.

From the calculation above, we find that

1. With proper state noise compensation, the difference between ISL-only POD using EKF and reference orbit is smaller than 0.3 m in the radial direction and is smaller than 2 m in three dimensions. The orbit determination accuracy of M1S and M2S is better than I2S. The POD result does not show constellation drifting or rotation in at least 8 days.

2. In scheme 2, the precision of initial satellite position is about 100 m, and the accuracy of POD decreases to 100 m. It can be seen that ISL-only orbit determination is sensitive to the precision of the initial state due to the lacking of a ground datum. Thus, accurate initial states should be given in ISL-only orbit determination.

3. In scheme 5, the equipment delays are estimated, while the POD accuracy is no better than that in scheme 1. Initially, the contribution of observations and dynamic model is limited. In addition, the number of ISLs is only 3, which equals the number of estimated equip-ment delay parameters. Without redundant observations, it could not improve the orbit determination accuracy by estimating equipment delay under current sub-constel-lation.

EKF with ISLs and GSLs

In this computation, the POD is conducted with ISL and GSL measurements. The precision of the initial position and velocity are 5 m, and 0.00001 m/s, respectively. The POD results are compared with reference orbits and the residu-als are drawn in Fig. 9. The measurement residuals of ISLs and GSLs are drawn in Figs. 10 and 11, respectively. The variation of equipment delay parameters is shown in Fig. 12. The estimated equipment delays are also compared with the result from the batch mode and the differences are drawn in Fig. 13.

Several facts can be drawn from this commutation:

1. Comparing the result with batch processing, the preci-sion of POD is less than 0.2 m in the radial component, and the difference in three dimension is less than 1.5 m.

2. The differences in equipment delay estimated by batch processing and EKF are less than 1 ns on average.

0 1 2 3 4 50

5

10

15

20

Time (DAY)

)m(laudise

RI2SM1SM2S

Fig. 6 Residual series of ISL-only POD in scheme 1 relative to refer-ence orbits

0 1 2 3 4 50

50

100

150

200

Time (DAY)

)m(laudise

R

I2SM1SM2S


0 1 2 3 4 50

10

20

30

40

Time (DAY)

)m(laudise

R

I2SM1SM2S



1 3

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3. The estimated equipment delay fluctuates obviously near 24 o’clock. The possible reason is the joining of meas-urements of new links. After converging, the variation of equipment delay parameters is within 0.5 ns.

4. As shown in Figs. 10 and 11, the measurement residuals of ISLs behave randomly. However, the measurement residuals of GSLs appear periodical due to uncorrected tropospheric delay. This is the same as the result of batch processing.

Table 5 RMS of the residuals of EKF orbits with referenced to reference orbits

Scheme Satellite Radial (m) Tangential (m) Normal (m) Position (m)

1 I2S 0.231 0.923 1.541 1.808M1S 0.207 0.718 1.238 1.626M2S 0.217 0.831 1.151 1.565

2 I2S 1.052 76.195 99.992 125.719M1S 0.324 59.035 26.464 64.696M2S 0.512 62.607 26.568 68.013

3 I2S 2.080 4.934 7.569 9.272M1S 1.034 2.823 3.771 4.823M2S 0.889 4.474 3.257 5.605

0 1 2 3 4 50

10

20

30

40

Time (DAY)

otgnirrefe

RlaudiseRtibr

O R

efer

ence

Orb

it (m

)

I2SM1SM2S

Fig. 9 Residual series of POD relative to reference orbits

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Time (Day)

)m(laudise

R

I2S-M1SI2S-M2SM1S-M2S

Fig. 10 Measurement residual series of ISL links

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Time (Day)

)m(laudise

R

Sta-I2SSta-M1SSta-M2S

Fig. 11 Measurement residual series of GSL links

0 1 2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

Time (Day)

tnempiuq

Ede

lay

varia

tion

(ns)

I2SM1SM2SSta

Fig. 12 Variation of estimated equipment delays while filtering


1 3

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5. For EKF, the filtering needs a period to converge due to initial state errors. Usually, the converging of each parameter is synchronous. In this computation, the con-verging time for equipment delay parameters is about 24 h that is longer than that of positions and velocities. The possible reason is that observations contribute more to satellite orbit parameters than to equipment delay parameters.

POD with a priori equipment delay constraints

Considering that the equipment delay is caused by the equipment itself and is constant in theory, accurate equip-ment delays can be calibrated by the producer or achieved with historical measurements. Thus, it is possible to con-strain equipment delay parameters with a priori values to

implement the disadvantage of weak observability of equip-ment delay parameters.

In this computation, the POD is implemented with EKF and is constrained with a priori equipment delay according to (16) and (17). The precision of the initial satellite posi-tion, velocity and equipment delays is 5 m, 0.00001 m/s, and 10 ns, respectively. The a priori equipment delay is obtained by historical measurements using batch mode. The result of POD is compared with reference orbits, and the residual series are drawn in Fig. 14. The variation of equipment delay is shown in Fig. 15. The estimated equipment delays are compared with the result of the batch mode and the differ-ences are drawn in Fig. 16.

1 2 3 4 5 60

1

2

3

4

Time (Day)

neewte

Becnereffi

DD

EB

atch

Pro

cess

ing

and

EK

F (n

s)I2SM1SM2SSta

Fig. 13 Comparison of estimated equipment delay between EKF and batch processing

0 1 2 3 4 50

10

20

30

40

Time (DAY)

gnirrefeRlaudise

RtibrO to

Ref

eren

ce O

rbit

(m) I2S

M1SM2S

Fig. 14 Residual series of constrained POD relative to reference orbits

0 1 2 3 4 5 6 7 8-2

-1

0

1

2x 10

-3

Time (Day)

Equ

ipm

ent

dela

y va

riatio

n (n

s)

I2SM1SM2SSTA

Fig. 15 Variation of estimated equipment delay

1 2 3 4 5 60

1

2

3

4

5

6

Time (Day)

neewteb

ecnereffidD

E)sn(

FK

Edna

gnissecorphctab

I2SM1SM2SSTA

Fig. 16 Difference of estimated equipment delay relative to batch result


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With a priori constraints, the precision of satellite orbits is better than 0.3 m in radial component and is about 1.5 m three dimensionally. It takes about less than 1 h to improve the precision of estimated equipment delay parameters to 1 ns, and the converge for equipment delay is about 3 h. The variation of equipment delays after converging is relatively stable and varies within 0.001 ns. The differences of ISL/GSL POD between batch mode and EKF algorithm can be summarized and listed in Table 6.

Conclusion

The measurement model of dual one-way measurements at Ka-band is introduced. The principle of orbit determina-tion using batch processing and EKF is presented. With real measurements of the next-generation BDS satellites, POD with batch processing and EKF are presented and analyzed. The differences between batch processing and EKF are sum-marized. Several facts can be given.

1. With the given sub-constellation, the ISL measurements are sparsely distributed and the coverage on the whole arc is about 31%.

2. With 3 ISLs and 3 GSLs, the POD precision using batch processing is about 0.1 m in radial component and better than 1 m in three-dimensional. The variation of equip-ment delay is within ± 0.6 ns.

3. The precision of ISL-only orbit determination using EKF is about 0.3 m in the radial component and is bet-ter than 2 m in three dimensionally. The ISL-only POD using EKF is sensitive to the precision of initial state parameters. With an accurate initial state, the real-time POD result does not show constellation drifting within 8 days. With approximate initial states, the ISL-only orbit determination filtering cannot converge.

4. The precision of POD with both ISLs and GSLs is better than 0.3 m in the radial component and is about 1.5 m in three dimensionally. For EKF, the converging for equip-ment delay parameters is slower than those of satellite

position and velocity due to weak observation geometry for equipment delay parameters. Thus, the equipment delay should be constrained by a priori value while fil-tering under the current sub-constellation.

Acknowledgements The study was funded by the National Key Research and Development Program of China (2016YFBOS01700, 2016YFB0501701), National Natural Science Foundation of China (41374019, 11503096, 41574013), and China Postdoctoral Science Foundation (2015M572691).

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Table 6 Difference of ISL/GSL POD between batch mode and EKF algorithm

Batch mode EKF

Efficiency Post processing Real time processingOrbit determination ARC 3 days with 1 day sliding window 8 daysUtilization of observations Observations within the arc Only the observations at current epochEOP parameters Integrated result published by IERS predicted result published by IERSObservability of equipment

delay parametersWell Weak

Prior constraints Not necessary Is necessary for equipment delay parameters under the sub-constel-lation

Orbit precision 1 m (overlap precision) 1.5 m (compared with batch result)

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Xia Ren is a Ph.D. candidate in the Information Engineering University, China. She received the B.S. and M.S. degrees from Information Engineering Univer-sity. Her research interests include GNSS, satellite orbit determination, and autonomous navigation of BeiDou satellites.

Yuanxi Yang is currently a Profes-sor of Geodesy and Navigation at China National Administration of GNSS and Applications (CNAGA). He got his Ph.D. from Wuhan Institute of Geod-esy and Geophysics of Chinese Academy of Science. He was honored as an Academic Mem-ber of Chinese Academy of Sci-ence in 2007. His research inter-ests mainly include geodetic data processing, geodetic coordinate system, crustal deformation anal-ysis and integrated navigation.

Jun Zhu is currently an engineer at Xi’an Satellite Control Center. He received the B.S., M.S. and Ph. D. degrees from the National University of Defense Technol-ogy, China. His research interest is satellite orbit determination.

Tianhe Xu is currently a professor at Shandong University. He received the B.S., M.S. and Ph. D. degrees from Information Engineering University. His research interests include geo-detic data processing, geodetic coordinate system, and satellite orbit determination.

comparing satellite orbit determination by batch

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