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International Journal of
Geo-Information
Article
Kinematic Precise Point Positioning Using
Multi-Constellation Global Navigation Satellite
System (GNSS) Observations
Xidong Yu and Jingxiang Gao *
School of Environment Science and Spatial Informatics, China University of Mining and Technology,
Xuzhou 221116, China; sxjmyxd@126.com
* Correspondence: jxgao@cumt.edu.cn
Academic Editors: Zhao-Liang Li, Jose A. Sobrino, Chao Ren and Wolfgang Kainz
Received: 8 September 2016; Accepted: 26 December 2016; Published: 5 January 2017
Abstract: Multi-constellation global navigation satellite systems (GNSSs) are expected to enhance the
capability of precise point positioning (PPP) by improving the positioning accuracy and reducing the
convergence time because more satellites will be available. This paper discusses the performance
of multi-constellation kinematic PPP based on a multi-constellation kinematic PPP model, Kalman
filter and stochastic models. The experimental dataset was collected from the receivers on a vehicle
and processed using self-developed software. A comparison of the multi-constellation kinematic
PPP and real-time kinematic (RTK) results revealed that the availability, positioning accuracy and
convergence performance of the multi-constellation kinematic PPP were all better than those of both
global positioning system (GPS)-based PPP and dual-constellation PPP. Multi-constellation kinematic
PPP can provide a positioning service with centimetre-level accuracy for dynamic users.
Keywords: multi-constellation; PPP; convergence time; positioning accuracy
1. Introduction
Since 1994, the International Global Positioning System (GPS) Service (IGS) organization has
provided precise GPS satellite orbit and clock products, enabling the development of a novel
positioning methodology known as precise point positioning (PPP) [1,2]. Based on the processing of
un-differenced pseudo-range and carrier phase observations from a single GPS receiver, positioning
solutions with accuracies in the centimetre to decimetre range can be attained globally. Precise point
positioning (PPP) is one of the most popular techniques for carrier phase-based precise positioning.
PPP sometimes makes use of ionosphere-free linear combination to decrease the effect of ionospheric
delay. However, it is not based on integer coefficients, and currently, the state information does not
preserve the integer nature of ambiguities. Consequently, PPP cannot adequately resolve ambiguities
and access the full range of global navigation satellite system (GNSS) carrier-phase accuracies [3,4].
Moreover, long observation times are required for convergence [5]. Many researchers have attempted
to improve the performance of PPP by improving the precision of the satellite orbit and clock
products [2,6] and speeding up the ambiguity-resolution process [7–9]. Satellite positioning availability
and integrity can be significantly improved by using multiple GNSSs so that more satellites will be
available [10–12]. Li et al. tested the accuracy of multiple-constellation PPP and discussed the main
challenges associated with this process [13,14]. The use of multiple GNSSs is expected to enhance the
capability of PPP by improving the positioning accuracy and reducing the convergence time.
Since the International Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS)
Experiment (IGEX-98) and the follow-on GLONASS Service Pilot Project (IGLOS), the precise
GLONASS orbit and clock data have become available. A combined GPS and GLONASS PPP program
ISPRS Int. J. Geo-Inf. 2017, 6, 6; doi:10.3390/ijgi6010006
www.mdpi.com/journal/ijgi
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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was developed by Cai and Gao in 2007 [15]. Research on GPS- and GLONASS-based PPP has
been conducted continuously since then, and the results have demonstrated that the positioning
precision and convergence speed were improved by the dual-constellation signal [16–18]. The BeiDou
Navigation Satellite System (BDS) is a global satellite navigation system that was independently
developed, deployed, and operated by China and remains in operation today [19]. The BDS consists of
two separate satellite constellations: a limited test system that has been operating since 2000 and a
full-scale global navigation system that is currently under construction. Table 1 lists the satellites in
the BDS constellation at or before December 2012. In total, 23 satellites have been involved, 3 of which
are no longer operational. Twenty BDS satellites are currently in operation: 6 in geostationary orbits
(GEOs), 8 in 55-degree inclined geosynchronous orbits (IGSOs) and 6 in medium Earth orbits (MEOs).
The full constellation is planned to eventually comprise 35 satellites. According to its overall planning
schedule, the BDS will have global coverage by 2020 [17].
Table 1. Constellation of the BeiDou Navigation Satellite (BDS) regional system. GEO: geostationary
orbits; IGSO: inclined geosynchronous orbits; MEO: medium Earth orbits.
Common Name
Int. Satellite ID
Pseudo Random Noise (PRN)
Notes
BEIDOU M1
BEIDOU G2
BEIDOU G1
BEIDOU G3
BEIDOU G4
BEIDOU IGSO 1
BEIDOU IGSO 2
BEIDOU IGSO 3
BEIDOU IGSO 4
BEIDOU IGSO 5
BEIDOU G5
BEIDOU M3
BEIDOU M4
BEIDOU M5
BEIDOU M6
BEIDOU G6
BEIDOU I1-S
BDS M1-S
BDS M2-S
BDS I2-S
BDS M3-S
BEIDOU IGSO 6
BEIDOU G7
2007-011A
2009-018A
2010-001A
2010-024A
2010-057A
2010-036A
2010-068A
2011-013A
2011-038A
2011-073A
2012-008A
2012-018A
2012-018B
2012-050A
2012-050B
2012-059A
2015
2015
2015
2015
2016
2016
2016
C30
C02
C01
C03
C04
C06
C07
C08
C09
C10
C05
C11
C12
C13
C14
C02
C31
C34
C33
C32
C35
C15
C17
Not in operation
Not in operation
GEO 140.0◦ E
GEO 110.5◦ E
GEO 160.0◦ E
IGSO 120◦ E
IGSO 120◦ E
IGSO 120◦ E
IGSO 95◦ E
IGSO 95◦ E
GEO 58.75◦ E
MEO
MEO
Not in operation
MEO
GEO 80.3◦ E
IGSO
MEO
MEO
IGSO
MEO
IGSO
GEO
Researchers [20–22] have also developed a model that combines GPS- and BDS-based PPP. The test
results revealed that the combined GPS- and BDS-based PPP can decrease the convergence time
and improve the positioning precision. Because of improvements in the precision of the BDS and
Galileo satellite orbit and clock products, quad-constellation (GPS, BDS, GLONASS and Galileo)
PPP has become possible [23]. Tegedor et al. and Cai et al. [24,25] improved the performance of
quad-constellation PPP. However, most of the research mentioned above addressed PPP for a statistic
object. In this work, the performance of PPP for a kinematic user is addressed.
In this study, we assessed the performance of multi-constellation kinematic PPP in terms of the
positioning accuracy and convergence time using the measurements collected from receivers on a
vehicle. The multi-constellation kinematic PPP model, Kalman filter and stochastic models applied
here are introduced in the second section. Section three describes the multi-constellation kinematic
PPP data-processing strategy. The performance of multi-constellation kinematic PPP is elucidated by
applying it to real data in the fourth section, which is followed by the conclusions.
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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2. Multi-Constellation PPP Model
In this study, we developed a PPP software and conducted several tests of the precision and
stability of the data processing achieved using this PPP software. The results showed that the accuracy
of statistical positioning could reach the millimetre level in the horizontal direction and the centimetre
level in the vertical direction. The root mean square (RMS) value of the tropospheric delay was between
0.01 m and 0.02 m. The convergence time was primarily distributed from 10 min to 40 min. These tests
will be described in a future research paper. Compared with other open-source PPP software, this
software could batch process large amounts of data, and it used precision products provided by the
IGS, Multi-GNSS Experiment (MGEX) and International GNSS Monitoring & Assessment System
(iGMAS). The structure design of our software was simpler and had good stability. In this section,
the observation equation, Kalman filter model and stochastic model used in our software will be
discussed in detail.
2.1. Multi-Constellation PPP
Ionosphere-free combination observations are normally used in PPP to remove the first-order
ionospheric delay. Their code and carrier phase observation can be expressed as:
f 12
f2
ρ − f 2 −2 f 2 ρ2 = R + c(dt − dT ) + Ttrop + ε P
f 12 − f 22 1
2
1
f 12
f 22
λ
φ
−
λ φ = R + c(dt − dT ) + λ IF NIF
1
1
2
2
2
f1 − f2
f 1 − f 22 2 2
(1)
+ Ttrop + ε φ
where fi (i = 1, 2) are carrier-phase frequencies in Hertz; ρi is the code observation at the ith frequency
in metres; φi is the carrier phase observation in cycles; dt is the receiver clock offset in seconds; dT is
the satellite clock offset in seconds; c is the speed of light in metres per second; Ttrop is the tropospheric
delay in metres; NIF are the parameters of the float ambiguity after being redefined; R is the geometric
range in metres; ε ρ and ε φ are the code and carrier phase observation noise, respectively, including the
multipath in metres; and λIF is the wavelength after being redefined.
The linearization formula of Equation (1) is:
ρ IF = R + c(δt − δT ) + Ttrop + ε p
=
xi − x p
r dx p
+
yi − y p
r dy p
+
zi − z p
r dz p
+ cdt + MFTztd − cdT + R + ε p
λ IF Φ IF = R + c(δt − δT ) + λ IF NIF − Ttrop + ε φ
=
xi − x p
r dx p
+
yi − y p
r dy p
+
zi − z p
r dz p
+ cdt + MFTztd + λ IF NIF − cdT + R + ε φ
(2)
In Equation (2), r is the geometric range computed using the linearization point. MF is the
mapping function; Tztd is the zenith tropospheric wet delay; ρIF are the code observations; and ΦIF
are carrier observations. xi , yi , zi are the position of satellites, and xp , yp , zp are the coordinates of
the stations.
Then, the least squares residuals can be written as:
VρIF
VφIF
=
B
B
A
A
M
M
O
λ
dX p
dt
Tztd
NIF
−
LρIF
LφIF
(3)
where the parameters dXp , dt, Tztd and NIF are the receiver position, receiver clock offset, zenith
tropospheric wet delay and ambiguity of the ionosphere-free combination, respectively; B, A, M, andλ
are the corresponding coefficient matrices, respectively. VρIF and VφIF are the observation residuals.
For multi-constellation kinematic PPP, the code hardware delay biases should be absorbed into
the receiver clock and system time difference, and the carrier-phase biases related to the frequency
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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should be absorbed into the ambiguity. In addition, the system time difference and receiver clock offset
were linearly correlated. Generally, instead of estimating a receiver clock parameter for each satellite
system observation, we always introduce a system time difference parameter to reflect the difference
between the different system times. The multi-constellation kinematic PPP observation equations can
be expressed as follows:
VρIF
VφIF
VρIF1
VφIF1
=
B
B
B1
B1
A
A
O
O
O
O
A
A
M
M
M
M
O
λ
O
O
O
O
O
λother
dX p
dt gps
dtother
Tztd
NIF
NIF1
−
LρIF
LφIF
LρIF1
LφIF1
(4)
where VρIF and VΦIF are the GPS observation residuals. VρIF1 and VφIF1 are residuals of other navigation
satellites but not GPS satellites. LρIF and LρIF1 are ionosphere-free code observations of GPS and another
GNSS, respectively. VφIF and VφIF1 are the ionosphere-free carrier phases observable. For B1 , λother is
the corresponding coefficient matrix of other navigation satellites. dtgps is the clock offset of GPS, and
dtother is the system time difference parameter. In the tests performed in this study, the ambiguities
were treated as constants (assuming no cycle-slips), and other parameters were all epoch dependent,
as shown in Equation (4).
Combining multiple GNSSs is associated some issues, such as the integration of the time
reference system, the coordinate reference systems, and the inter-frequency biases. These issues
also represent challenges affecting multiple-constellation PPP. Because the individual GNSSs have
different frequencies and signal structures, the code bias values are different in a multiple-GNSS
receiver. These biases are the inter-system biases for the code observation. The phase delays are
also different, and their differences represent inter-system biases for phase observations. GLONASS
satellites emit their signals on individual frequencies, which will also lead to frequency-dependent
biases in the receivers. For the GLONASS satellites with different frequency factors, the receiver
code biases differ from the carrier phase bias. These differences are usually called inter-frequency
biases. Both the inter-system and inter-frequency biases must be considered in a combined analysis of
multi-GNSS data. The corresponding parameters should be estimated for all multi-GNSS receivers:
one bias for the code measurements of each system and each frequency for GLOANSS. However,
the inter-system/inter-frequency biases and obtained satellite clocks are fully correlated. Consequently,
when satellite clocks are used, the corresponding biases must also be estimated or corrected for these
GNSS receivers. It should be noted that such a receiver internal bias is relevant only when processing
the code data. Indeed, when analysing the phase measurements, the corresponding phase ambiguity
parameters will absorb the phase delays. These will only be relevant if ambiguities are resolved to
their integer values (i.e., mixed ambiguity resolution between different GNSSs, GLONASS ambiguity
resolution or un-differenced ambiguity resolution).
2.2. Kalman Filter Model
A Kalman filter was used for the multi-constellation kinematic PPP processing in this study.
The GNSS dynamic positioning system state equation and observation equation are:
Xi = Φi,i−1 Xi−1 + Wi
(5)
Li = Ai Xi + ei Pi
(6)
where i is the time of ti , and Xi and Xi−1 are the m × 1 state vectors at ti and ti−1 , respectively. m is
the number of parameters; Φi,i−1 is an m × m dimensional state transition matrix; and Wi is a system
noise vector, which is assumed to be drawn from a zero mean multivariate normal distribution with
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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covariance ΣWi . Li is the observation design vector matrix, and ei is the measurement error. Pi is a
weight matrix.
The steps involved in Kalman filtering are as follows [26,27]:
(1) Store Xˆ i−1 and ∑ Xˆ at ti−1
i −1
(2) Predicted (a priori) state estimate:
X i = Φi,i−1 Xˆ i−1
(7)
(3) Predicted (a priori) estimate covariance:
∑Xi = Φk,k−1 ∑Xˆ k−1 ΦT k,k−1 + ΦWi = PX−i1
(8)
(4) Innovation or measurement residual and covariance:
V i = Ai X k − Li
(9)
∑V i = Ai ∑Xi AiT + ∑
(10)
i
(5) Optimal Kalman gain:
−1
1 T
Ki = Φi,i−1 Ai ∑V = PX−
ˆ Ai Pi
i
i
(11)
(6) Updated (a posteriori) state estimate:
Xˆ i = X i − Ki V i
(12)
(7) Updated (a posteriori) estimate covariance:
∑Xˆ i = ( I − Ki Ai ) ∑Xi−1 ( I − AiT KiT ) + Ki ∑i KiT
(13)
(8) Let I = i + 1 and then return to the first step until the end of the data.
Appropriate stochastic models for the observations and dynamic models for the state vector
must be provided in the Kalman filter. The stochastic model is usually defined using an appropriate
covariance matrix that describes the statistical properties of the measurements [28]. In the following
section, the stochastic models used in this study will be introduced in detail.
The kinematic model used in the dynamic model is the constant acceleration:
r(t + 1) = r(t) + r(t) ∆t + r(t) ∆t2 /2
(14)
where r(t) is an unknown vector, r(t) is the velocity, and ∆t is the acceleration time length.
In our software, the coordinate, receiver clock, tropospheric delay and ambiguity parameters were
the unknown parameters. These parameters were estimated using the first-order Gauss–Markov (GM)
random process. If no cycle-slips occurred, we treated the ambiguity as a constant. In our software, we
only stored the information of the current epoch and the next epoch, and we estimated the parameters
of each epoch. The discussion regarding dynamic noise variance of the related parameters is as follows:
The discrete first-order GM process is:
x k +1 =
xk + ωk
(15)
where x is the state vector, xk = e−∆t/τ , τ is the correlation time, ω is the white noise sequence with
zero mean value, and t is the time interval.
Generally,β = τ1 , β is the damping coefficient. If the damping coefficient is too large, the current
and next epochs will have greater volatility. However, the two epochs have a strong time correlation.
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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The state transition matrix is:
Φk+1,k = e− β∆t
(16)
The dynamic noise variance matrix is:
Q=
q
(1 − e−2β∆t )
2β
(17)
where q is the spectral density or dynamic noise variance matrix. When the correlation time τ is zero,
Equation (15) represents the pure white noise model. If τ is infinite, the filtering process is a pure
random walk.
In general, pure random walks are suitable for simulating the three-dimensional coordinates and
ambiguities, where the tropospheric delay and the receiver can be modelled in two forms.
For static PPP without speed and acceleration, the state parameter vector is:
Xk = [X, Y, Z, clk, trop, N1 , · · · , Nn ] T
(18)
The corresponding state transition matrix, Φk+1,k , and the dynamic noise variance matrix of the
three-dimensional position coordinate are:
Q pos =
q ϕ ∆t
0
0
0
qλ ∆t
(Rn +h)2 cos2 ϕ
0
0
0
(Rm +h)
2
qh ∆t
where:
q ϕ : latitudinal spectral density;
qλ : longitudinal spectral density;
qh : elevation spectral density;
Rm : meridian radius of curvature;
Rn : radius of curvature on the dome circle; and
h: the height of the station.
The dynamic noise variance matrix of the receiver is as follows:
If it is a pure random walk:
Qclk = [qdt ∆t]
If it is pure white noise:
Qclk = [
qdt
]
β dt
(19)
(20)
(21)
where qdt is the receiver clock density, and β dt is the corresponding damping coefficient.
In general, the tropospheric zenith delay is expressed as a pure random walk, and thus, its dynamic
noise variance is:
Qtrop = [qtrop ∆t]
(22)
where qtrop is the spectral density of the tropospheric zenith wet delay. The ambiguity parameter can
be regarded as a constant: Q N = 0.
For dynamic PPP, the state parameters should include the speed and acceleration parameters,
and the corresponding matrix should change. Thus, the state vector parameters should be:
Xk = [X, Y, Z, Vx , Vy , Vz , a x , ay , az , clk, trop, N1 , · · · , Nn ] T
(23)
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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The corresponding state transition matrix is:
Φk+1,k =
where:
Φ pos
=
Φ pos
0
0
0
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
Φclk
0
0
∆t
0
0
1
0
0
0
0
0
0
0
0
0
0
ΦN
Φtrop
0
0
∆t
0
0
1
0
0
0
0
(11+n)×(11+n)
∆t2
2
0
0
∆t
0
0
1
0
0
0
(24)
0
0
∆t2
2
0
0
∆t
0
0
1
9×9
0
∆t2
2
0
0
∆t
0
0
1
0
0
0
0
∆t
0
0
1
0
Φclk = 1
ΦN
(25)
(26)
Φtrop = 1
1 ··· 0
= ... . . . ...
0 · · · 1 n×n
(27)
(28)
The state noise variance is:
Q=
Q pos
0
0
0
0
Qclk
0
0
0
0
Qtrop
0
0
0
0
QN
(29)
2.3. Stochastic Models
Error corrections affect the performance of PPP. Some corrections can be eliminated by using
function models, which are discussed in Section 3. However, function models are not sufficient to
improve the accuracy and convergence time of PPP. Consequently, an appropriate stochastic model
must be applied. Different stochastic models reflect different PPP results. Generally, three common
stochastic models exist: the equal-weight stochastic model, the carrier-noise rate-based stochastic
model and the elevation angle-based stochastic model. In this study, the last model was applied.
Most of the GNSS observation errors (i.e., the troposphere refraction delay, ionosphere refraction
delay, and multipath effect) are related to the elevation angles of satellites. To decrease these errors,
stochastic models based on the elevation angles of satellites can be established. Elevation angle-based
stochastic models mainly include trigonometric function models and exponential function models [29].
In this study, we used the sine function-based elevation angle stochastic model, which is described by
Equation (30):
σ02
(30)
σ2 =
sin2 θ
where θ is the elevation angle of the satellite, and σ02 is the prior variance of observations.
ISPRS Int. J. Geo-Inf. 2017, 6, 6
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Generally, the multipath and large observation noise exist at low elevation angles. We defined the
weight segment to reduce the weight of the observations at lower elevation angles. The corresponding
code and carrier phase variance matrices are:
σP2
=
2
σP,0
sin θ
2
σP,0
sin2
θ
θ>α
θ<α
, σφ2
=
2
σφ,0
sin θ
2
σφ,0
sin2
θ
θ>α
(31)
θ<α
2 and σ2 are the prior variances of code and carrier phase observations, respectively. α is
where σP,0
φ,0
the elevation angle threshold and is typically set to 30◦ . When adopting the code and carrier phase
observations simultaneously, the variance–covariance expression is:
σi2 =
2
σP,i
0
0
2
σφ,i
(32)
It should be noted that different GNSSs have different prior observation variances. For the GPS
and GLONASS code and carrier phase observations, the precisions are set to 0.3 m and 0.002 m,
respectively. Because the BDS satellite orbit and clock have relatively lower accuracies [30,31], their
measurements are down-weighted. That is, the phase observation precision is set to 0.004 m, and the
code observation precision is set to 0.6 m for the BDS [25].
3. Multi-Constellation Kinematic PPP Data Processing Strategy
In this study, a self-developed PPP software was used for the data processing. Here, the data
preprocessing directly affected the accuracy of the positioning results. The eliminated error and model
used in this study are shown in Table 2.
Table 2. Models used in multi-constellation kinematic precise point positioning (PPP) data
preprocessing. PCO: phase centre offsets; PCV: phase centre variations.
Parameters
Model and Mitigation Methods
Ionospheric delay
Tropospheric delay(dry part)
Receiver antenna PCO and PCV
Satellite antenna PCO and PCV
Solid earth tides
Ocean loading
Polar tides
Phase wind-up effect
Receiver clock errors
Ionosphere-free combined observables
UNB3m model [32]
IGS ANTEX
IGS ANTEX
IERS2010
IERS2010
IERS2010
Wu model [33,34]
Estimation, white noise
The first-order ionospheric delay errors were removed using the ionosphere-free combined
observables in our self-developed PPP software. The hydrostatic (dry) tropospheric delay was corrected
based on observations using the UNB3m model [32], whereas the non-hydrostatic (wet) part was
estimated as a parameter. The Neill mapping functions [35] were used for projection from the slant
delays to the zenith delay. The receiver and satellite antenna phase centre offsets (PCO) and phase
centre variations (PCV) were collected using the parameters provided by IGS ANTEX. In addition, the
MGEX of IGS investigated the new GNSSs [21]. In this study, we applied the multi-constellation GNSS
precise satellite orbit and clock products from MGEX to mitigate the satellite orbit and clock errors.
The main PPP parameters included the coordinates, the zenith tropospheric delay parameter,
multi-constellation system time difference parameters, and the ambiguities. To estimate both the static
and kinematic parameters, our software used the Kalman filtering model. The cut-off angle was set
at 10◦ . To decrease the convergence time and make full use of the satellite observation information,
we regarded the satellite as a new satellite when the cycle slip occurred. The cycle slip detection and
ISPRS Int. J. Geo-Inf. 2017, 6, 6
9 of 15
repair were conducted using the MW combination and geometry-free combination. The software
automatically calculated the scaling matrix based on satellite movements to improve the computational
efficiency. Simultaneously, the software automatically set the parameters according to the static or
dynamic mode.
4. Performance Analysis of Multi-Constellation Kinematic PPP
4.1. Experimental Data Description
To assess the availability of multi-constellation kinematic PPP applied to dynamic objects,
a multi-constellation kinematic PPP test was conducted in Huainan, China on 18 January 2015.
Kinematic GNSS observations were collected over 2 h from three receivers mounted on a vehicle.
The speed of the vehicle was approximately 20 km/h. The receivers were Hi-Target V8 (TRM59800.00
NONE) devices, which can collect GPS, BDS, and GLONASS observations. The sampling interval
was 1 s. Figure 1 shows the vehicle’s route, which started from the “start point” and went through
each section twice. To investigate the accuracy of the PPP result, we also set a base station with
the same type of receiver and antenna at the start point to determine the coordinates of the rover
station with cm-level accuracy using the double-difference real-time kinematic (RTK) approach.
The surrounding environment was generally good for observations, and the sky visibility was very
high. Multi-constellation mixed precise satellite orbit and clock products provided by the Deutsches
GeoForschungsZentrum (GFZ) (Germany) were adopted for PPP data processing. It should be noted
that no BDS receiver antenna PCO and PCV correction file was available for the antenna mentioned
above.
ISPRS Int.Consequently,
J. Geo-Inf. 2017, 6,the
6 BDS PPP positioning results contained a systemic error.
10 of 15
Figure 1. The vehicle’s route.
Figure 1. The vehicle’s route.
4.2. Results and Analysis
4.2. Results and Analysis
4.2.1. Availability
of Multi-Constellation
Multi-Constellation Kinematic
Kinematic PPP
PPP
4.2.1.
Availability of
To investigate
investigatethe
thebenefits
benefitsofofmulti-constellation
multi-constellation
PPP,
availability
analysed
in terms
of
To
PPP,
thethe
availability
waswas
analysed
in terms
of the
the numbers
of visible
satellites
and
the positional
of precision
(PDOP)
value,
which is
numbers
of visible
satellites
and the
positional
dilutiondilution
of precision
(PDOP) value,
which
is important
important
for
quantifying
the
GNSS
positioning
accuracy
[25].
for quantifying the GNSS positioning accuracy [25].
Figure 22 shows
values
forfor
each
processing
case.
G
Figure
shows the
thenumber
numberof
ofvisible
visiblesatellites
satellitesand
andthe
thePDOP
PDOP
values
each
processing
case.
represents
GPS
only,
G +G
R represents
the GPS/GLONASS
combination,
G + C represents
the GPS/BDS
G
represents
GPS
only,
+ R represents
the GPS/GLONASS
combination,
G + C represents
the
combination,
and
G
+
R
+
C
represents
the
GPS/GLONASS/BDS
combination
(as
mentioned
below).
GPS/BDS combination, and G + R + C represents the GPS/GLONASS/BDS combination (as mentioned
As shown
in thisinfigure,
the GPS-only
geometry
waswas
weaker
than
those
below).
As shown
this figure,
the GPS-only
geometry
weaker
than
thoseofofthe
thethree
three combined
combined
systems.
The
largest
PDOP
value
of
GPS
was
20,
whereas
the
PDOP
value
of the
systems. The largest PDOP value of GPS was 20, whereas the PDOP value of the GPS/GLONASS/BDS
GPS/GLONASS/BDS combination was always below 5. The numbers of satellites and the PDOP
values of the four systems varied very stably in the first 0.4 h and from 1.1 to 1.5 h. In contrast,
variation occurred very frequently during the rest of the experimental period. The reason for this
result was that the vehicle stayed in static mode for the first 0.4 h and the 1.1–1.5 h. Figure 3 also
explains this situation, as it shows the coordinate increments at the directions of north, east and up.
the numbers of visible satellites and the positional dilution of precision (PDOP) value, which is
important for quantifying the GNSS positioning accuracy [25].
Figure 2 shows the number of visible satellites and the PDOP values for each processing case. G
represents GPS only, G + R represents the GPS/GLONASS combination, G + C represents the GPS/BDS
combination, and G + R + C represents the GPS/GLONASS/BDS combination (as mentioned below).
ISPRS Int. J. Geo-Inf. 2017, 6, 6
10 of 15
As shown in this figure, the GPS-only geometry was weaker than those of the three combined
systems. The largest PDOP value of GPS was 20, whereas the PDOP value of the
combination
was always
below 5. The
satellites
andnumbers
the PDOP
of the
four
GPS/GLONASS/BDS
combination
wasnumbers
always of
below
5. The
ofvalues
satellites
and
thesystems
PDOP
varied
stably
the firstvaried
0.4 h and
from
1.1 toin1.5
h. first
In contrast,
variation
occurred
values very
of the
fourinsystems
very
stably
the
0.4 h and
from 1.1
to 1.5 very
h. Infrequently
contrast,
during
theoccurred
rest of the
experimental
period. the
Therest
reason
forexperimental
this result was
that the
vehicle
in
variation
very
frequently during
of the
period.
The
reasonstayed
for this
result mode
was that
vehicle
stayed
in static
for the
firstexplains
0.4 h and
thesituation,
1.1–1.5 h.asFigure
3 also
static
for the first
0.4 h
and the
1.1–1.5mode
h. Figure
3 also
this
it shows
the
explains this
situation, at
asthe
it shows
the coordinate
increments
coordinate
increments
directions
of north, east
and up. at the directions of north, east and up.
Figure 2.
satellites and
positional dilution
dilution of
of precision
precision (PDOP)
(PDOP) values
values of
of multi-constellation
multi-constellation
Figure
2. Numbers
Numbers of
of satellites
and positional
PPP.
BDS:
BeiDou
Navigation
Satellite
System;
GLONASS:
Globalnaya
Navigatsionnaya
Sputnikovaya
PPP. BDS: BeiDou Navigation Satellite System; GLONASS: Globalnaya Navigatsionnaya Sputnikovaya
Sistema;
GPS:
global
positioning
system;
G:
GPS
only;
G
+
R:
GPS/GLONASS
combination;
Sistema; GPS: global positioning system; G: GPS only; G + R: GPS/GLONASS combination;G
G ++ C:
C:
GPS/BDS
combination;
G
+
R
+
C:
GPS/GLONASS/BDS
combination.
combination;
ISPRSGPS/BDS
Int. J. Geo-Inf.
2017, 6, 6 G + R + C: GPS/GLONASS/BDS combination.
11 of 15
Figure 3.
3. Coordinate
Coordinate increments
increments in
in the
the following
following directions:
directions: north,
north, east
east and
and up.
up.
Figure
4.2.2. Positioning Accuracy and the RMS of the Kalman Filter
4.2.2. Positioning Accuracy and the RMS of the Kalman Filter
To assess the kinematic positioning accuracy of the multi-constellation PPP, the double difference
To assess the kinematic positioning accuracy of the multi-constellation PPP, the double difference
RTK results were regarded as the true coordinates. These results were computed using the
RTK results were regarded as the true coordinates. These results were computed using the GNSS/INS
GNSS/INS tightly coupled resolution of Inertial Explorer 8.60 software (IE 8.60), using the
tightly coupled resolution of Inertial Explorer 8.60 software (IE 8.60), using the multi-constellation
multi-constellation signals. The positioning accuracy of this software can reach 1–2 cm. Figure 4a
signals. The positioning accuracy of this software can reach 1–2 cm. Figure 4a shows the positioning
shows the positioning results of multi-constellation kinematic PPP and RTK. When we zoomed in it
results of multi-constellation kinematic PPP and RTK. When we zoomed in it in Figure 4b, we found
in Figure 4b, we found that the position of GPS only was very different from that of RTK. There was
that the position of GPS only was very different from that of RTK. There was also little error in the
also little error in the results of the GPS/GLONASS PPP. The positions of the GPS/BDS and
GPS/GLONASS/BDS were very similar to those of the RTK. The differences between these RTK
coordinates and the PPP results of the GPS only, GPS/GLONASS, GPS/BDS and GPS/GLONASS/BDS
were compared to assess the performance of the multi-constellation kinematic PPP in detail. The mean
square errors of these differences are shown in Table 3. From Table 3, we can see that the mean
square errors of GPS were 0.045 m and 0.085 m in the east and north directions, respectively, which
4.2.2. Positioning Accuracy and the RMS of the Kalman Filter
To assess the kinematic positioning accuracy of the multi-constellation PPP, the double difference
RTK results were regarded as the true coordinates. These results were computed using the
GNSS/INS tightly coupled resolution of Inertial Explorer 8.60 software (IE 8.60), using the
multi-constellation
The positioning accuracy of this software can reach 1–2 cm. Figure
4a
ISPRS
Int. J. Geo-Inf. 2017,signals.
6, 6
11 of 15
shows the positioning results of multi-constellation kinematic PPP and RTK. When we zoomed in it
in Figure 4b, we found that the position of GPS only was very different from that of RTK. There was
results
of the
GPS/GLONASS
of the GPS/BDS
GPS/GLONASS/BDS
also little
error
in the resultsPPP.
of The
the positions
GPS/GLONASS
PPP. Theand
positions
of the GPS/BDSwere
and
very
similar
to
those
of
the
RTK.
The
differences
between
these
RTK
coordinates
the PPP
results
GPS/GLONASS/BDS were very similar to those of the RTK. The differences and
between
these
RTK
of
the GPS only,
andGPS/GLONASS,
GPS/GLONASS/BDS
were
to assess
coordinates
and GPS/GLONASS,
the PPP results of GPS/BDS
the GPS only,
GPS/BDS
and compared
GPS/GLONASS/BDS
the
performance
of
the
multi-constellation
kinematic
PPP
in
detail.
The
mean
square
errors
of these
were compared to assess the performance of the multi-constellation kinematic PPP in detail. The
mean
differences
are
shown
in
Table
3.
From
Table
3,
we
can
see
that
the
mean
square
errors
of
GPS
square errors of these differences are shown in Table 3. From Table 3, we can see that the were
mean
0.045
m and
0.085
m in were
the east
andmnorth
much larger
than those
of
square
errors
of GPS
0.045
and directions,
0.085 m in respectively,
the east andwhich
north were
directions,
respectively,
which
GPS/BDS.
With
the
GPS/BDS
combination,
accuracy
improved
by
71.11%
and
41.18%
over
GPS
only
were much larger than those of GPS/BDS. With the GPS/BDS combination, accuracy improved by
in
the north
east directions,
However,
positioning
of GPS/GLONASS
71.11%
andand
41.18%
over GPS respectively.
only in the north
and the
east
directions,accuracy
respectively.
However, the
was
slightly
poorer
than
that
of
the
GPS/BDS.
The
reason
would
be
that
there
were
more
positioning accuracy of GPS/GLONASS was slightly poorer than that of the GPS/BDS.
Thevisible
reason
GPS/BDS
satellites
than
which
of
GPS/GLAONSS,
and
the
PDOP
of
GPS/BDS
was
better
than
of
would be that there were more visible GPS/BDS satellites than which of GPS/GLAONSS, and thethat
PDOP
GPS/GLAONSS)
(Figure
2).that
Theofaccuracies
of GPS/GLONASS/BDS
PPP were
better than those of
of GPS/BDS was better
than
GPS/GLAONSS)
(Figure 2). The accuracies
of GPS/GLONASS/BDS
GPS/BDS
by
30.77%
and
56.00%.
PPP were better than those of GPS/BDS by 30.77% and 56.00%.
(a) zoom out
(b) zoom in
Figure4.4.Multi-constellation
Multi-constellationkinematic
kinematicPPP
PPPand
andreal-time
real-timekinematic
kinematic(RTK)
(RTK)positions.
positions.
Figure
Table 3. The mean errors of the four processing cases (unit: m2 ). STD: standard deviation.
STD
GPS
GPS/GLONASS
GPS/BDS
GPS/GLONASS/BDS
N
E
0.045
0.040
0.013
0.009
0.085
0.080
0.050
0.022
The standard deviations (STDs) of these differences are shown in Table 4, and the reference
coordinates are the positioning results of the RTK. Smaller STDs indicate better results. As shown in
Table 4, the STD of the GPS-only system was relatively large because the number of visible satellites
was low (Figure 2). In contrast, the STD of the GPS/GLONASS/BDS PPP was clearly the best.
Table 4. STDs of the positioning errors for the four processing cases (unit: m).
GPS
GPS/GLONASS
GPS/BDS
GPS/GLONASS/BDS
N
E
0.067
0.062
0.056
0.042
0.091
0.081
0.073
0.060
ISPRS Int. J. Geo-Inf. 2017, 6, 6
12 of 15
4.2.3. Convergence Time
To evaluate the convergence performance of multi-constellation kinematic PPP, we started from
the data sets in five-minute intervals (i.e., 12 times an hour) until convergence was achieved. In this
study, the position filter was considered to be converged when the positioning errors reached ±0.1 m
and remained within that range. Table 5 presents the mean convergence time of RTK. Figures 5 and 6
show the kinematic PPP positioning errors of the four different processing cases. In these figures,
we only present the convergence performance of PPP based on six replicates because of the space
limitations. The convergence time was defined as the period from the first epoch to the converged
epoch (indicated as a red line in these figures). Based on these figures and the table, we found that the
GPS-only PPP required substantially more time to converge than the other processing cases. In contrast,
the convergence performance of the GPS/GLONASS/BDS system was the best for all three coordinate
components. Relative to that of GPS-only PPP, the convergence times of GPS/GLONASS and GPS/BDS
PPP improved, particularly in the east direction. This result was very similar to that of Cai et al. [22].
Table 5. Mean convergence time of RTK.
Mean convergence time (min)
ISPRS Int. J. Geo-Inf. 2017, 6, 6
GPS
GPS/BDS
GPS/GLONASS
GPS/GLONASS/BDS
64.2
50.9
52.4
47.5
13 of 15
Accuracy(m)
Figure 5. The difference between the PPP kinematic and RTK solutions (GPS and GPS/GLONASS).
Figure 5. The difference between the PPP kinematic and RTK solutions (GPS and GPS/GLONASS).
ISPRS Int. J. Geo-Inf. 2017, 6, 6
13 of 15
Accuracy(m)
Accuracy(m)
Figure 5. The difference between the PPP kinematic and RTK solutions (GPS and GPS/GLONASS).
Figure 6.6. The
kinematic and
and RTK
RTK solutions
solutions (GPS/BDS
(GPS/BDS and
and
Figure
The difference
difference between
between the
the PPP
PPP kinematic
GPS/GLONASS/BDS).
GPS/GLONASS/BDS).
5. Conclusions
Conclusions
5.
In this
this study,
study, the
the performance
performance of
of multi-constellation
multi-constellation kinematic
kinematic PPP
PPP was
was assessed
assessed in
in terms
terms of
of the
the
In
positioning accuracy
accuracy and
and convergence
convergence time.
time. A
A kinematic
kinematic experimental
experimental dataset
dataset was
was processed
processed using
using
positioning
a
self-developed
software
based
on
the
multi-constellation
kinematic
PPP
model,
Kalman
filter
and
a self-developed software based on the multi-constellation kinematic PPP model, Kalman filter and
stochastic
models.
Based
on
the
discussions
above,
we
drew
the
following
conclusions:
stochastic models. Based on the discussions above, we drew the following conclusions:
1.
2.
3.
4.
The availability of multi-constellation kinematic PPP was significantly better than that of GPS-only
PPP because more satellites were observable and because the PDOP was better.
The accuracy of multi-constellation kinematic PPP was greater than that of GPS-only, GPS/BDS
and GPS/GLONASS PPP. The positioning accuracy of GPS/GLONASS PPP was slightly lower
than that of GPS/BDS PPP.
The convergence performance of GPS/GLONASS/BDS kinematic PPP was the best for all three
coordinate components, particularly in the East direction.
Multi-constellation kinematic PPP can provide a positioning service with centimetre-level
accuracy for dynamic users.
Acknowledgments: This work has been partially supported by the National Nature Science Foundation of China
(Grant No. 41674008), partially sponsored by the Fundamental Research Funds for the Central Universities (Grant
No. 2014ZDPY29), partially sponsored by a Project Funded by the Priority Academic Program Development of
Jiangsu Higher Education Institutions (Grant No. SZBF2011-6-B35).
Author Contributions: Jingxiang Gao provided the initial idea for this study; Xidong Yu conceived and designed
the experiment; Xidong Yu and Jingxiang Gao analyzed the results of the experiment; Xidong Yu and Jingxiang
Gao wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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