Multiple Constellation Navigation Performance Using ...

Multiple Constellation Navigation Performance Using Long-Term Ephemeris Extension with Backward Error Representation Yihe Li, Zhixi Nie, Shaohua Chen, Yang Gao Department of Geomatics Engineering, University of Calgary, Canada BIOGRAPHY Yihe Li is PhD candidate in Department of Geomatics Engineering at University of Calgary from January, 2012. He received his bachelor and master degree in Geomatics Engineering from Tongji University in 2009 and 2011. His research focuses on GNSS orbit prediction, determination and PPP Ambiguity Resolution.

research focuses on high precision GPS positioning and multi-sensor integrated navigation systems. ABSTRACT Long-term

ephemeris

extension

is

needed

in

challenging environments in order to reduce the Time-To-First-Fixed (TTFF). It makes quick position fix possible for mobile devices while they are still acquiring the broadcast ephemeris. Based on the

Zhixi Nie is a PhD student in Department of

experiment results described in this paper, the median

Geomatics at China University of Petroleum and

User Range Error (URE) for the extended ephemeris

came to University of Calgary as a jonit training

after 3 and 7 days are 2.14m and 7.36m, respectively.

student in August, 2014. He received a bachelor

Both of the multi-constellation extended ephemeris

degree in Geomatics Engineering from China

and its accuracy represented with backward errors are

University of Petroleum in 2011.His research focuses

used

on multi-system GNSS precision positioning.

obstruction experiments are also conducted to assess

Shaohua Chen is a PhD student in the Department of Geomatics Engineering at the University of Calgary. He received his MSc in Electrical Engineering at Beij ing Institute of Technology in 2012 following the bac helor’s in Communication Engineering, at Yanshan U niversity in 2009. His research interests include High Sensitivity GNSS and digital signal processing.

for

urban

navigation.

Simulated

signal

the position fix accuracy, availability and reliability based on different mean square positional error (MSPE)

thresholds,

prediction

durations

and

obstruction scenarios. The results have demonstrated superior

performance

of

multiple

constellation

navigation system with backward error representation method compared to the GPS only without backward error representation. 92.9% navigation availability at

Dr. Yang Gao is a Professor in the Department of Geomatics engineering at the University of Calgary. His research expertise includes both theoretical aspects and practical applications of satellite-based positioning and navigation systems. His current

97.1% reliability with one obstruction and 91.2% navigation availability at 91.7% reliability with two obstructions are achieved based on a user-defined MSPE of 15m after 3 days’ prediction.

fitting a span of the previous ephemeris. The orbits

INTRODUCTION Broadcast ephemeris is required for a position fix while decoding broadcast ephemeris for each satellite in open sky normally takes 30s, but valid just for a short period of time (2-4 hours). For applications that require the lowest possible power consumption which typical GNSS chipsets cannot deliver, fast first-timeto-fix (TTFF) is crucial feature as it will greatly increases the battery life of today's portable consumer electronics.

However,

fast

TTFF

becomes

challenging in some tough environments such as urban canyon and indoor, where GNSS signals suffer from multipath effect, signal attenuation and building blockage etc, making the acquisition of the broadcast navigation messages much difficult (David et al. 2005; Zhang et al. 2008; Stacey et al. 2011). As a result, a position fix may become impossible or the positioning accuracy is significantly degraded due to inadequate number of available satellites. With longterm ephemeris extension, a GNSS receiver can totally skip downloading the broadcast ephemeris and process directly to calculate a position as soon as the receiver is turned on. The typical result of using Extended Ephemeris is that instead of taking 30-60 seconds to get a position, a receiver will now take 1-2 seconds to do the job. In addition, such ephemeris extension is particularly of value to mobile users when they only have limited access to wireless network over a long time period plus power consumption concerns. Ephemeris extension involves both GNSS satellite orbit prediction and clock prediction (Choi et al. 2012, Huang et al. 2014). The performance of the satellite orbit prediction depends on two major components. One is the satellite dynamic model and the other is the satellite initial condition. The most popular method is to determine a set of initial parameters by

are integrated with fully considering the dynamic model of GNSS satellites. As to the prediction of the satellite clock, many models have been developed such as the linear model, the quadratic and higherorder polynomial model and the gray model (Vernotte et al. 2001; Zheng et al. 2008; Heo et al. 2010). The IGU predicted clock model is currently the most widely used, stable and precise one, which uses a linear model combined with sinusoid terms for satellites with Cs and IIR Rb clocks and a quadratic model combined with sinusoid terms for satellites with II/IIA Rb clocks. Some numerical investigations have been conducted to assess the accuracy of the extended ephemeris. The median User Range Error (URE) of both the JPL GPS and GLONASS satellites (http://www.gdgps.net/index.html) can be better than 10m after 7 days of prediction and better than 100m after 28 days. A good mobile navigation system should not only satisfy precision requirement but also have good navigation availability. At the same time, it should also

have

a

reliable

decision

mechanism

to

accept/reject the positional solutions with small decision errors (Li et al. 2013a). Li et al. (2013b) has analyzed the navigation performance of a GPS only navigation system with extended ephemeris by simulating obstruction scenarios

with different

azimuths. With an increasing number of navigation satellites, the scenario of GNSS is speeding up on a no-return track towards a multi-constellation and multi-frequency GNSS. This gives chances for improving both of the availability and accuracy of GNSS positioning, but also creating challenges that need to be addressed to ensure the optimal use of all the available navigation signals and extended ephemerides by the end users. When the extended

ephemeris is used, it is necessary to know how big the extended ephemeris errors will be at any time, which is helpful for users to control positioning quality and decide how long the predicted orbit can be used under a certain positioning accuracy requirement. As mentioned above, the accuracy of extended ephemeris depends on both initial condition and dynamic model. Hence the accuracy of backward-extended ephemeris should be equivalent to that of forward-extended ephemeris at the same initial condition with assumption of the same error caused by dynamic model. In this paper, a novel combined extended ephemerides of GPS/ GLONASS (GLO)/Beidou (BDS) multiple GNSS constellation are generated. Besides, the accuracy of extended

ORBIT AND CLOCK PREDICTION MODEL If a GNSS satellite’s position and velocity at a reference epoch t0, denoted as r0 and v0, respectively, are known, they can then be used to derive/predict the satellite’s position rk and velocity vk at the subsequent epoch tk by employing the numerical integral with dynamic models driven by the adopted force models. Here the satellite position and velocity at epoch t0 are also referred to as the initial condition in the orbit prediction. Therefore, both the initial condition and dynamic models are required for orbit prediction. In the following, we first outline the dynamic models driven by the adopted force models, followed by showing how to solve the initial condition with an arc length using the least squares method.

ephemeris is represented by using the errors of backward extended ephemeris. Thus, the stochastic model of observation will be determined by considering both extended ephemeris errors and elevation. Finally, the extended ephemeris products (up to 7 days) are employed to completely and objectively

demonstrate

GNSS

mobile

device

The dynamic models in the orbit integration consist of a set of force models, namely a set of satellite accelerations introduced by various effects. Therefore, the lumped satellite acceleration is expressed as a   ai

(1)

i

navigation performance in various simulated signal

where the acceleration components ai include the

reception challenging scenarios. With the extended

gravity of the Earth, the gravity of other astronomical

ephemerides of the multiple GNSS constellation,

bodies, the solar radiation pressure as well as any

both of the TTFF, positioning accuracy and reliability

other independent forces acting on the satellite that

can be further improved compared to GPS alone.

can be modeled (Montenbruck et al. 2000). For the

This paper is organized as follows. In Section 2, the methodology of GNSS orbit and clock prediction is described. In Section 3, the methodology of orbit and clock prediction is described. The backward extended ephemeris error representation method is presented in Section 4. In Section 5, the position fix performance with extended ephemeris is assessed in various simulated

experiments.

Conclusions

experiments are summarized in Section 6.

from

the

Earth gravity, it is suggested to use the EGM2008 geopotential up to degree and order 12 with the corrections recommended in IERS Conventions 2010 (Dick et al. 2011). The sun radiation pressure (SRP) is one of the main error sources in orbit prediction and in this study the Extended CODE Orbit Model (ECOM) is applied with 9 parameters (Dach et al. 2007; Herring et al. 2010). Once the initial condition (which include 3 satellite position parameters denoted by a 3x1 vector r0 and 3

velocity parameters denoted by a 3x1 vector v0) and 9

to tk. With m epochs of observed orbit for each

SRP parameters (denoted by a 9x1 vector p) are

satellite, the initial condition and the SRP parameters,

known for a satellite, one can predict the orbit at

x0 , are iteratively computed as

epoch k as follows:

rk  h(tk ; a, r0 , v0 , p)

(2)

xˆ 0  x0apr +



m k 1

H kT Pk H k ,0



1

T k

Pk vk

  k 1 H kT,0 Pk  rk m

(5)

where h denotes the nonlinear integral function; rk is k m

the predicted satellite position with tilde denoting the

ˆ 

predicted quantity. It is obvious that the quality of the orbit prediction depends on the initial condition r0, v0 and SRPs p. In following, we will discuss the estimation of the initial condition as well as the SRP parameters based on the least-squares approach with

k 1

3m  15

(6)

where ˆ is a-posterior standard deviation for initial condition. For more details about orbit prediction, one refers to Li et al. (2014). We would like to emphasize that the satellite

an arc of observed orbits.

prediction and integration processing are done in the

T

Let x0  r0T v0T

v

pT  , we form the following error

inertial system. However the observed orbits are

equation at the epoch k,

usually in the earth-fixed system. Thus, one needs to

vk  rk  h(tk ; a, x0 )  rk  rk , Pk

(3)

first transform the satellite positions from the earthfixed frame into those in the inertial frame, then

where vk is the error (residual) vector of the predicted orbit relative to the observed one rk. Pk is the weight matrix of the observed equations which is determined based on the accuracy of the observed orbit zk. Linearizing h(tk ; a, x0 ) with respect to x0 with apr 0

conduct the estimation of the initial conditions and the prediction of the satellite positions in the inertial system. After that, the predicted satellite position is transformed from the inertial system back to the earth-fixed system.

approximate initial condition x , Eq (3) can then be

For satellite clock prediction, the application of a

rewritten as

linear or quadratic polynomial model is mainly

h(t ; a, x0 ) vk  rk  h(tk ; a , x )  k x0 apr 0

decided based on whether the frequency drift of the x0 =x0apr

 x0 = rk  H k  x0 , Pk

(4)

satellite clock is noticeable or not. In fact, different satellite clocks have different characteristics of

apr 0

where h(tk ; a, x ) is the predicted orbit vector based

frequency drift. In order to properly determine the

on

clock model, a summative method based on the clock

the

approximate

initial

condition.

 rk =rk  h(tk ; a , x0apr ) is the difference between the observed and predicted orbit with the approximate initial condition;  x0 denotes the initial condition correction Hk 

vector

h(tk ; a , x0 ) x0

x0 =x0apr

relative

to

x0apr

;

is the state transition matrix

showing how the state vector translates from epoch t0

types can be expressed as clktk =c0 +c1tk +

 c2tk2

2 ,GPS IIA(Rb) 1  0 ,GPS IIA(Cs)/IIR/IIRM/GLO/BDS/GAL

where, β is the selective factor.

(7)

EXTENDED

EPHEMERIS

ERROR

REPRESENTATION Based on the orbit initial condition and clock coefficients determined at the reference epoch t0, the ephemeris can be extended with the time period of k forwardly. The orbit and clock prediction errors in the extended ephemeris will be assessed using the User Range Error (URE) (Hauschild et al. 2009), which provides the average range error in the line-ofsight direction in a global scale. The computation of URE is related to maximum satellite’s coverage on earth’s surface. For a MEO satellite, the maximum satellite’s

coverage

on

earth’s

surface

is

approximately 13.88°.Thus its URE of the extended





2

UREtj0 k = Rt0j k -clkt0j k 



 

2 1 At0j k  Ct0j k 49

 2

(9)

In real situation, uncertainty between the forward and backward extended ephemeris errors could not be avoided

because

the

models

w.r.t

different

perturbation forces acting on a GNSS satellite would have different performances with time variation. These uncertainties have to be considered when the backward error representation is used. Hence, a modified backward error representation method is proposed. The main steps are as follows: First, based on the orbit initial condition and clock coefficients determined at the reference epoch t0, the ephemeris is extended backwardly with the same period as that

ephemeris error time t0+k can be approximately

forwardly. Second, the backward ephemeris errors

determined as follows which has also considered the

are computed based on the previous precise

satellite clock prediction error

ephemeris as reference. Third, all the backward







 



2 2 1 At0j  k  Ct0j  k (8) 49 In Eq. (8), the URE for a single satellite i is

UREtj0  k = Rt0j  k -clkt0j  k

2



determined by the cross-track and along-track orbit

ephemeris errors are sorted ascendingly, which is expressed as vector 𝒖𝒓𝒆𝑡0+𝑘 . The percentile for each backward ephemeris errors is determined, namely 𝑗

errors denoted as ∆𝐶 and ∆𝐴, respectively, and the

𝑝𝑒𝑟𝑐𝑡 . If the confidence level is a. The percentile of

combined radial orbit and clock error ∆𝑅 - ∆ clk.

backward ephemeris error for satellite j at predicted

Considering that the combined orbit and clock error

time t is 𝑝𝑒𝑟𝑐𝑡 , the corresponding confidence interval

has a larger impact on the user ranging error than the

is [𝑝𝑒𝑟𝑐𝑡 − 𝑎, 𝑝𝑒𝑟𝑐𝑡 + 𝑎 ]. a is set to 15% in this

orbit error in the cross-track and along-directions,

study. Based on the confidence interval, the

they are scaled by a factor of 1/49 for MEO.

ephemeris error 𝜎𝐸𝑝ℎ,𝑗 can be expressed as the STD

It is impossible to compute

𝑗 𝑈𝑅𝐸𝑡0+𝑘

based on the

true forward extended ephemeris errors. The accuracy of extended ephemeris depends on both initial condition and dynamic model. Hence the accuracy of backward-extended ephemeris can be used to represent that of forward-extended ephemeris with assumption of the same error caused by dynamic model. Figure 1 shows the sketch of backward extended ephemeris error representation. The formula of calculating URE can be expressed as

𝑗

𝑗

𝑗

of the n backward ephemeris errors fall into the confidence interval.

 Eph, j ,t  k = 0

i  pert j  a



i  pert  a j

ureti0  k / n

(10)

Based on the above ephemeris errors, the satellite elevation-dependent weighting strategy the n undifferenced observations, can be revised as follows

2 Ds  diag  12 + Eph ,1

  i2   02  2 2  i   0 (2 sin  )

2  n2 + Eph ,n 

  30o ,i  1   30o

University (WHU). Repeating this procedure for one (11)

n

where 𝐷𝑠 is variance matrix; 𝜃 is satellite elevation angle; 𝜎𝑖2 is

variance

of

the

undifferenced

observations w.r.t satellite i. 𝜎0 is the standard deviation of code measurement w.r.t IF combination, which is set to 0.9 m.

week (January 30th to February 5th of 2014), we obtain 7 predicted orbit and clock solutions for one weeks’ extension duration. The WUM precise ephemerides used as the reference for ephemeris extension

precision

assessment.

Single

point

positioning (SPP) with Ionosphere-free (IF) code measurements are conducted to assess the position fix performance using the extended ephemeris. UNB3

EXPERIMENT AND ANALYSIS

model is employed to correct the tropospheric delays.

In order to evaluate the precision of the multi-

Shown in Fig. 3, GPS, GLONASS and BDS data

constellation

collected in 8 globally distributed stations is used for

ephemeris

(GPS/GLONASS/BDS/GALILEO) extension

described

in

this

paper,

ephemerides are predicted based on initial condition

SPP

processing

with

obstruction

simulation

experiment. The data sampling interval is 30 seconds.

fitted with previous 24hr precise ephemerides released

by

GNSS

Research

Center,

Wuhan

Fig. 1 Median, 68th, 95th percentile Error of orbit prediction during 1.30-2.5, 2014 Extended Ephemeris Errors with Backward Error Representation

prediction, which are determined based on all

Fig. 2 represents the typical median and 68th

February 5th of 2014. For forward prediction, it can

percentile values of the satellite orbit, clock error and

be observed that the median URE reached 0.51m

ephemeris errors after 7 days’ backward and forward

after 1 days’ prediction and 7.36m after 7 days’

predictions carried out during January 23th to

prediction. The median orbit error became 0.08m

median orbit error became 0.03m after 1 days and

after 1 days and 2.26m after 7 days. The median

2.68m after 7 days. The median clock error reached

clock error reached 0.49m after 7 days and 6.52m

0.75m after 7 days and 10.90m after 7 days.

after 7 days. The overall trend of backward predicted

Therefore, the backward ephemeris extension errors

errors is similar with that of forward predicted errors.

can be used to represent the actual error for

The median URE reached 0.75m after 1 days’

ephemeris extension.

prediction and 10.67m after 7 days’ prediction. The 18 URE-Median URE-68th percentile Orb-Meidan Orb-68th percentile Clk-Meidan Clk-68th percentile

16 14

Meters

12 10 8 6 4 2 0 -7

-6

-5

-4

-3

-2 -1 0 1 2 Days of Ephmeris Extension

3

4

5

6

7

Fig. 2 Median, 68th, 95th percentile Error of multi-constellation ephemeris extension during 1.30-2.5, 2014 Table 1 The median of the backward and forward orbit and clock prediction precision of multi-constellation [m] Orbit Clock Constellation

Backward

Forward

Backward

Forward

1 day

7 days

1day

7days

1 day

7 days

1 day

7 days

GPS

0.027

1.417

0.044

1.972

0.259

9.916

0.224

6.278

GLONASS

0.027

1.721

0.044

2.345

0.502

5.312

0.337

7.807

BEDOU

0.058

2.571

0.244

2.941

0.444

52.337

0.256

56.880

GALILEO

0.029

3.317

0.104

2.144

0.412

50.263

0.556

24.452

All

0.028

1.633

0.050

2.181

0.361

9.168

0.278

7.657

Extended Ephemeris for Navigation in Simulated Signal Challenging Environment

When the extended ephemerides are used to bridge the outage period to reduce TTFF, a key concern

from users is about the position fix accuracy,

and elevation of satellite as  and h, then the

availability as well as reliability of the navigation

satellite is blocked if

solution

  h    ,    min ,  max  (14) arctan   d 1  tan 2    0    

in

a

signal

reception

challenging

environment. For accuracy analysis, the following MSPE can be applied (Leick, 2004):

ˆ MSPE  ˆ N2  ˆ E2

(12)

where ˆ N and ˆ E are the standard deviations for the

where the minimum and maximum azimuths formed by the antenna and obstruction are

 min   0  arctan  l / d   max   0  arctan  l / d 

(15)

computed position coordinates in the north and east

We set up 6 experiment scenarios with the

directions. Given a user-defined threshold 𝜎̂𝑀𝑆𝑃𝐸,0 ,

obstruction parameters, h = 50 m, d = 20 m, l = 20 m.

the probability that the positional solutions whose

For the first 4 experiment scenarios (s1 – s4), the

MSPE are smaller than 𝜎̂𝑀𝑆𝑃𝐸,0 can be determined

obstructions are simulated by applying different

and defined as the availability. As to reliability, it can

obstruction azimuths  0 = 0 ° , 90 ° , 180 ° and

be determined by one minus the probability of the

270 ° . For the remaining two (s5 and s6), the

decision errors that the positional solutions contain

obstructions are simulated using the following two

large errors, say that the true horizontal position error

azimuths combination:  0 = 0 ° , 180 ° and  0 =

is bigger than 𝜎̂𝑀𝑆𝑃𝐸,0 but whose MSPE values are smaller than 𝜎̂𝑀𝑆𝑃𝐸,0 (wrongly accepted positional solutions) as follows:

number of wrongly accepted solutions Perror  100% (13) number of accepted solutions The signal reception challenging environment will be simulated by introducing signal obstructions from different azimuths as illustrated in Fig. 3. For simulating an urban canyon, the obstruction azimuth

 0 is defined as the azimuth of the direction from the antenna vertically to obstruction. The distance of the obstruction to the antenna is d, its height above the antenna is h and its width is 2l symmetrically with respect to the azimuth direction. Denote the azimuth

90 ° , 270 ° . Both of the revised strategy with considering the ephemeris errors and traditional satellite elevation-dependent weighting strategy are used for SPP. For brevity, the 6 experiment scenarios are expressed as S1 to S6 while the 2 weighting strategies ae denoted as W1 and W2, respectively. Finally, the position fix accuray, availability and reliability (wrong decision) of GPS only and GPS+GLONASS+BDS by using the two weighting strategies

with

scenarios

are

those assessed

6

simulated for

experiment

demonstrating

the

superiority of multiple constellation navigation performance

using

ephemeris

backward error representation.

extension

with

Fig. 3 Distribution of eight user stations for simulated obstruction.

Fig. 4 The layout of simulated obstruction. The distance of obstruction to antenna is d and its height beyond antenna is h; the azimuth of direction from antenna vertically to obstruction is  0 . The width of obstruction is 2l symmetrically with respect to the azimuth direction. Table 2 The average MSPE [m] and possibility of position fix (Pf) [%] with simulated obstruction of different azimuths  0 (°) GPS

 0 (°)

0

W1

GPS+GLONASS+BDS W2

Pf

W1

W2

Pf

3d

7d

3d

7d

3d

7d

3d

7d

3d

7d

6.49

31.98

5.12

22.12

90.7

89.8

5.14

22.53

4.32

14.15

3d

7d

99.9

99.1

90

6.07

34.09

4.96

24.13

84.8

84.1

5.31

21.13

4.54

13.72

99.9

99.0

180

8.13

39.86

6.18

28.83

94.3

93.7

5.42

22.36

4.61

15.35

99.9

99.1

270

7.75

36.50

5.76

24.02

92.3

91.6

5.14

21.91

4.38

14.70

99.9

99.1

0 & 180

15.01

69.69

10.79

44.87

60.6

60.2

8.15

33.81

6.62

21.82

90.0

88.9

90& 270

9.08

43.39

7.23

30.23

72.8

73.3

5.93

23.95

5.12

15.93

99.6

98.8

Table 2 show that the average MSPE and of GPS

direction (𝛼0 = 90°) are significantly smaller than

only and multi-constellation systems with simulated

those with obstructions from the other direction.

obstruction of different azimuths. Higher availability

Combined results shown in Table 2 and Table 3, it

can be achieved if smaller MSPEs are computed

can be found that that for certain direction, the

under a given user-defined MSPE 𝜎̂𝑀𝑆𝑃𝐸,0 . As we

number of satellites is so few that make a significant

know, MSPE is determined by both of the posteriori

impact to the geometry strength (i.e., experiment

standard deviation and Horizontal Dilution Of

scenario with 𝛼0 = 0 ° and 180 ° ), subsequently

Precision (HDOP). The posteriori standard deviation

degrade the navigation solution. We further evaluate

is mainly affected by extended ephemeris errors

the availability of the position fix based on different

while HDOP reflects the geometry.

Compared to

user-defined MSPE 𝜎̂𝑀𝑆𝑃𝐸,0 . The results shown in Fig.

GPS only system, the multi-constellation system

4 and 5 reassure lower availabilities with increasing

significantly strengthen the geometry and thus

predicted time with a fixed 𝜎̂𝑀𝑆𝑃𝐸,0 . If 𝜎̂𝑀𝑆𝑃𝐸,0 is

decreases HDOP. It can be seen that the experiment

given as 15m, the average availability of GPS only

scenario with obstruction from north direction (𝛼0 =

with S1 to S4 using W1 and W2 strategies reaches

0 ° ) owns smallest MSPE while the experiment

77.8% and 85.0% after 3 days’ prediction while

scenario with obstruction from the east and west

decreases to 20.9% and 33.5% after 7 days’

direction (𝛼0 = 90° and 270°) have largest ones.

prediction,

Besides, the average horizontal positional RMSs are

availability of multi-constellation system with the

shown in Table 3. For the first four scenarios, the

same situation and strategies can achieve 97.3% and

impacts on the horizontal positional accuracy are

99.9% after 3 days’ prediction while decreases to

slightly different when having different azimuths of

36.9% and 64.5% after 7 days’ prediction. The

obstruction. The average horizontal positional RMSs

probabilities of wrongly accepted positional solutions

for 𝛼0 = 90 ° and 180 ° are larger than the other

are shown in Fig. 6 and Fig. 7. The result indicates

azimuths from the north and west direction (𝛼0 = 0°

that, in general, this probability is decreased with

and 270°). For the remaining 2 scenarios, both the mean and standard deviation of horizontal positional RMSs with obstructions from north and south

respectively.

While

the

average

increasing 𝜎̂𝑀𝑆𝑃𝐸,0 while increased with increasing predicted time. The average probabilities of wrongly accepted positional solutions of GPS only with S1 to S4 using W1 and W2 strategies are 8.4%, 7.5% and 70.4% and 57.1% after 3 and 7 days’ prediction.

Correspondingly, the reliabilities are 91.6%, 92.5%

availabilities increase to 85.6%, 91.2% and 27.6, 44.1%

and 29.6%, 42.9%. The average probabilities of

while the decision error probabilities are 19.4%, 16.5%

wrongly accepted positional solutions of multi-

and 67.7% and 59.5%, respectively. Considering both

constellation system with S1 to S4 using W1 and W2

the availability and the corresponding wrong decision

strategies are 2.6%, 2.9% and 52.7% and 41.3% after

error probability with different predicted times, it is

3 and 7 days’ prediction.

confirmed that long prediction leads to lower

For the last 2 experiment scenarios with two

availability and higher wrong decision probability

obstructions using 3 and 7 days’ extended ephemeris,

(lower reliability) while a larger 𝜎̂𝑀𝑆𝑃𝐸,0 leads to

the average availabilities of GPS only reach 47.7%,

higher availability with lower wrong decision

53.4% and 12.1, 19.4% while the decision error

probability.

probabilities are 19.4%, 16.5% and 67.7% and 59.5%, respectively. When the both GLONASSS and BeiDou are added for positioning, the average Table 3 The average RMS [m] of horizontal positional with simulated obstruction of different azimuths  0 (°). GPS

 0 (°)

GPS+GLONASS+BDS

W1 3d

W2

W1

W2

7d

3d

7d

3d

7d

3d

7d

0

8.20

29.10

7.33

24.92

6.57

22.77

6.45

16.71

90

10.28

35.18

9.86

32.55

6.97

23.48

6.89

18.26

180

8.44

28.84

7.37

25.09

6.49

22.75

6.18

17.43

270

8.59

30.46

7.93

28.78

6.40

22.54

6.25

16.69

0 & 180

10.95

35.96

9.81

32.15

9.52

32.21

9.12

28.13

90& 270

11.65

39.22

11.14

37.33

8.79

30.99

8.29

23.58

s1

s2

s3

s4

s5

100

80

80

60

60

40

40

Availibility [ % ]

100

20

20

GPS with W1

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

100

80

80

60

60

40

40

Availibility [ % ]

100

20 0

GPS+GLO+BDS with W1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

20

GPS with W2

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

s6

GPS+GLO+BDS with W2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

User-defined MSPE [ m ]

User-defined MSPE [ m ]

Fig. 5 The availability of navigation solutions as a function of user-defined MSPE (𝜎̂𝑀𝑆𝑃𝐸,0 ) with obstructions from different azimuths. The results from left-top to right-bottom are with respect to GPS with W1, GPS+GLO+BDS with W1, GPS with W2, GPS+GLO+BDS with W2 s1

s2

s3

Wrong Decision [ % ]

100 GPS with W1

80

60

60

40

40

20

20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

100 80

0

s6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

100 GPS with W2

GPS+GLO+BDS with W2 80

60

60

40

40

20

20

0

s5

GPS+GLO+BDS with W1

80

0

Wrong Decision [ % ]

s4 100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

User-defined MSPE

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

User-defined MSPE

Fig. 6 The probability of wrongly accepted positional solutions as a function of user-defined MSPE (𝜎̂𝑀𝑆𝑃𝐸,0 ) with obstructions from different azimuths. The results from left-top to right-bottom are with respect to GPS with W1, GPS+GLO+BDS with W1, GPS with W2, GPS+GLO+BDS with W2

Availibility [ % ]

s1

s2

s3

s4

100

100

80

80

60

60

40

40

20

s5

20

GPS+GLO+BDS with W1

GPS with W1

Availibility [ % ]

0 10 15 20 25 30 35 40 45 50 55 60 65 70

0 10 15 20 25 30 35 40 45 50 55 60 65 70

100

100

80

80

60

60

40

40

20

s6

20

GPS with W2

GPS+GLO+BDS with W2

0 10 15 20 25 30 35 40 45 50 55 60 65 70

0 10 15 20 25 30 35 40 45 50 55 60 65 70

User-defined MSPE [ m ]

User-defined MSPE [ m ]

Fig. 7 The availability of navigation solutions as a function of user-defined MSPE (𝜎̂𝑀𝑆𝑃𝐸,0 ) with obstructions from different azimuths. The results from left-top to right-bottom are with respect to GPS with W1, GPS+GLO+BDS with W1, GPS with W2, GPS+GLO+BDS with W2 s1

s2

Wrong Decision [ % ]

100

s4

GPS with W1 80

80

60

60

40

40

20

20

100 80

s5

s6

100

0 10 15 20 25 30 35 40 45 50 55 60 65 70

Wrong Decision [ % ]

s3

GPS+GLO+BDS with W1

0 10 15 20 25 30 35 40 45 50 55 60 65 70

100 GPS with W2

GPS+GLO+BDS with W2 80

60

60

40

40

20

20

0 10 15 20 25 30 35 40 45 50 55 60 65 70

0 10 15 20 25 30 35 40 45 50 55 60 65 70

User-defined MSPE

User-defined MSPE

Fig. 8 The probability of wrongly accepted positional solutions as a function of user-defined MSPE (𝜎̂𝑀𝑆𝑃𝐸,0 ) with one obstruction from different azimuths. The results from left-top to right-bottom are with respect to GPS with W1, GPS+GLO+BDS with W1, GPS with W2, GPS+GLO+BDS with W2

Choi K, Ray J, Griffiths J, Bae T (2012) Evaluation

CONCLUSIONS With a wide use of ephemeris extension to reduce TTFF in signal reception challenging environment, it is also important to understand the accuracy of the extended ephemeris with different predicted duration and how much impact on the position fix solutions in context of precision, availability and reliability. In this study, we propose a multiple constellation longterm ephemeris extension with backward error representation. The signal reception challenging environment is simulated by introducing signal

It is demonstrated that the average horizontal position RMSs after 3 and 7 days’ prediction with the proposed method reach 7.19m and 20.13m for obstruction

simulated

scenarios,

respectively. Compared to the GPS only system without the backward error representation method, both availability and reliability under a given userdefined MSPEs are significantly improved. This study can be useful for determining appropriate prediction interval for extended ephemeris generation with

given

user-defined

position

Ultra-rapid products, GPS Solut 1521-1886 Online, doi 10.1007/s10291-012-0288-2 Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese

GPS

Software

Version

5.0.

Astronomical Institute, University of Bern David L, Fank VD (2005) Assistance When There’s

No

Assistance,

GPS

World,

October2005, pp. 32-36 Dick WR, Richter B (2011) International earth rotation and reference systems service (IERS)

obstructions from different directions.

different

of GPS orbit prediction strategies for the IGS

precision,

availability and reliability requirements, which can be used by E911 system to keep the reliability of determining the position of caller as soon as possible in any signal challenging environment.

(2011) Annual Report 2008–09. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main Heo YJ, Cho J, Heo MB (2010) Improving prediction accuracy of GPS satellite clocks with periodic variation behavior. Meas SciTechnol 21:073001. doi:10.1088/09570233/21/7/073001 Herring TA, King RW, McClusky SC (2010) GAMIT Reference Manual Release 10.4, 28 October Huang G, Zhang Q, Xu G (2014) Real-time clock offset prediction with an improved model, GPS Solut 18:95–104 Leick, A. GPS Satellite Surveying, third ed John Wiley & Son, Inc, 2004.

ACKNOWLEDGMENTS The first author is supported by China Scholarship Council (CSC), NSERC and Tecterra, which are all acknowledged. REFERENCES Hauschild A, Montenbruck O (2009) Kalman-filterbased GPS clock estimation for near real-time positioning, GPS Solut 13:173–182, doi 10.1007/s10291-008-0110-3

Li B, Shen Y, Zhang X (2013a) Three frequency GNSS navigation prospect demonstrated with semi-simulated data. Advances in Space Research 51, 1175–1185. Li Y, Gao Y (2013b) Navigation Performance Using Long-Term Ephemeris Extension for Mobile Device (2013) Proceedings of ION GNSS 2013, Nashville, TN, September 2013, pp. 1642-1651

Li Y, Gao Y, Li B (2014) An Impact Analysis of Arc Length on Orbit Prediction and Clock Estimation for PPP Ambiguity Resolution. GPS Solut, doi 10.1007/s10291-014-0380-x (SCI) Montenbruck O, Grill E (2000). Satellite OrbitsModels, Methods and Applications, Springer Stacey P, Ziebart M (2011) Long-Term Extended Ephemeris Prediction for Mobile Devices, Proceedings of ION GNSS 2011, Portland OR, September, 2011, pp. 3235–3244 Vernotte F, Delporte J, Brunet M, Tournier T (2001) Uncertainties

of

drift

coefficients

and

extrapolation errors: application to clock error prediction. Metrologia 38:325–342 Zhang W, Venkatasubramanian V, Liu H, Phatak M, Han, S (2008) SIRF InstantFix II Technology, In ION GNSS, 2008, pages 1840–1847 Zheng ZY, Lu XS and Chen YQ et al (2008) Improved grey model and application in real-time GPS satellite clock bias prediction. Proceedings of 4th International Conference on Natural Computation. Jinan city, pp 419– 423