Information Fusion 55 (2020) 184–198

Contents lists available at ScienceDirect

Information Fusion

journal homepage: www.elsevier.com/locate/inffus

Full Length Article

Modeling of multi-sensor tightly aided BDS triple-frequency precise point

positioning and initial assessments

Zhouzheng Gao

a , b , Maorong Ge

b , You Li c , ∗ , Yuanjin Pan

d , Qijin Chen

e , Hongping Zhang

e

a School of Land Science and Technology, China University of Geosciences Beijing, 29 Xueyuan Road, Beijing 100083, China b German Research Centre for Geosciences (GFZ), Telegrafenberg, Potsdam 14473, Germany c Department of Geomatics Engineering, University of Calgary, 2500 University Dr. N.W., Calgary, AB T2N 1N4, Canada d State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, 430079, China e GNSS Research center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

a r t i c l e i n f o

Keywords:

BeiDou global navigation satellite system

(BDS)

Triple-frequency precise point positioning

(TF-PPP)

Inertial navigation system (INS)

Odometer and heading measurement

constraint

Relative measuring accuracy

a b s t r a c t

The BeiDou global navigation satellite system (BDS), which is the first satellite navigation system that provides

triple-frequency signals on B1, B2, and B3 for civil applications, has been applied widely around the Asian-

Pacifica region. The current BDS precise point positioning (PPP) approaches are mainly based on the B1&B2 dual-

frequency observations. To make full use of BDS’ triple-frequency observations, the motion sensors measurements,

and the platform motion information, this paper proposes an inertial sensor, odometer, and heading measurement

tightly aided BDS triple-frequency PPP model. In this model, inertial sensor biases, odometer scale factor, residuals

of slant ionospheric delays, and inter-frequency code biases are estimated simultaneously in a unique extended

Kalman filter. The Rauch-Tung-Striebel (RTS) smoother is further adopted to reduce the solution’s noises and

enhance the stability and relative measuring accuracy. To evaluate the capability of this method, a set of triple-

frequency BDS raw observations, inertial measurements, odometer data, and heading measurements collected

by a customized hardware system on Lanzhou-Urumqi high speed railway track in China, are processed and

analyzed. Results illustrated that both positioning accuracy and cycle slip detection capability are upgraded

significantly by applying B3 frequency observations in BDS PPP. About 13–55% position accuracy enhancements

from B3 observations, inertial sensors, and RTS smoother, and over 70% heading improvements from the aids

of heading measurements can be obtained. Moreover, such multi-sensor tight integration system can directly

provide millimeter-level positioning accuracy in term of repeatability and provide sub-millimeter-level accuracy

indirectly by transforming attitude solutions into distance solutions. Such accuracy is much higher than the state-

of-art GNSS and such method presents potential capability in 3D geometry measuring.

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. Introduction

To satisfy China’s social and economic development requirements,

he BeiDou Global Navigation Satellite System (BDS) was developed

ince last century. In BDS, the China Geodetic Coordinate System 2000

CGCS2000) [1] , BeiDou Time (BDT) [2] , and Code Division Multiple

ccess (CDMA) were adopted as the spatial datum, temporal system,

nd signal structure [3] , respectively. Contrast to the time datum of

merica’s Global Positioning System (GPST), the time datum differ-

nce between GPST and BDT is a constant (14 s) [3] , while difference

etween CGCS2000 and WGS-84 only leads to positioning difference

ithin 0.1 mm and thus it is negligible [4] . BDS can offer better anti-

hielding capability than GPS because there are more satellites in higher

rbits [5] . Additionally, BDS integrates the communication capability to

∗ Corresponding author.

E-mail address: liyou331@gmail.com (Y. Li).

ttps://doi.org/10.1016/j.inffus.2019.08.012

eceived 25 October 2018; Received in revised form 21 August 2019; Accepted 29 A

vailable online 29 August 2019

566-2535/© 2019 Elsevier B.V. All rights reserved.

rovide short message communication service, which has been proven

o be helpful in emergency rescue.

.1. Satellite constellation and signals

As described in [5] , BDS is built in three steps: the demonstration

ystem (BDS-1), the regional system (BDS-2), and the global system

BDS-3). The 35 satellites based BDS-3 is planned to be finished at

round 2020 [6–7] . Different from the constellation structure that is

dopted by the other Global navigation Satellite Systems (GNSS, e.g.,

PS, GLONASS, and Galileo), BDS satellite constellation consists of Geo-

tationary Earth Orbit (GEO) satellites, Inclined Geo-Synchronous Orbit

IGSO) satellites, and Medium Earth Orbit (MEO) satellites [8] . Wherein,

DS-1 was started in 1994 and it completed the mission to provide

ugust 2019

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Fig. 1. Sky plot of BDS-2 satellites in 2015 at Wuhan, China.

Fig. 2. BDS satellites visibility in one week in 2015 at Wuhan, China.

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Table 1

Information of in-orbit BDS satellite.

System Type PRN Satellite Launch Time

BDS-2 GEO C1 G1 17.01.2010

C2 G3 25.10.2012

C3 G6 02.06.2010

C4 G4 01.11.2010

C5 G5 25.02.2012

IGSO C6 I1 01.08.2010

C7 I2 18.12.2010

C8 I3 10.04.2011

C9 I4 27.07.2011

C10 I5 02.12.2011

C13 I6 30.03.2016

MEO C11 M3 30.04.2012

C12 M4

C14 M6 19.09.2012

BDS-3 (including test

satellites)

GEO C17 G7 12.06.2016

G59 G1 01.11.2018

IGSO C31 I1-S 30.03.2015

C18 I2-S 30.09.2015

MEO C57 M1-S 25.07.2015

C58 M2-S

– M3-S 01.02.2016

C19 M1 05.11.2017

C20 M2

C27 M7 12.01.2018

C28 M8

C21 M3 12.02.2018

C22 M4

C29 M9 30.03.2018

C30 M10

C23 M5 29.07.2018

C24 M6

C25 M12 25.08.2018

C26 M11

C32 M13 19.09.2018

C33 M14

C35 M15 15.10.2018

C34 M16

C36 M17 18.11.2018

C37 M18

Fig. 3. BDS satellites visibility in one week in 2018 at IGS HNIS (Horn Island)

station.

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ervices for China with the constellation of three GEO satellites in 2003.

DS-2, which was started in 2004, had the objective to provide ser-

ices for the Asia-Pacific region. Until December 27th, 2012, five GEOs

numbered as C1-C5), five IGSOs (numbered as C6-C10), and four MEOs

numbered as C11-C14) were launched to provide navigation services

or mass-market users in this region. Shown in Figs. 1 and 2 are the cor-

esponding sky-plots and visibility of available BDS satellites in 2015 at

uhan, China. It can be seen that GEOs are almost static above the equa-

or and have a perfect observing continuity. IGSOs are moving along the

8 ′ shape trajectory around the Asia-Pacific region with fixed untracked

eriods. MEOs are running with inverted ‘S’ shape with shorter-term

bserving continuity.

To provide global services, BDS-3 project was promoted in 2009.

owever, as shown in Table 1 , the BDS-3 test satellites were not

aunched until 2015. The new satellites numbered as C31-C34 are

navailable for mass-market receivers because of the unknown signal

tructures [9–10] . After 2016, more BDS-3 satellites were launched suc-

essfully, which can be found from http://www.beidou.gov.cn/xt/fsgl/ .

he basic information about these in-orbit satellites can be found

t http://mgex.igs.org/IGS_MGEX_Status_BDS.php . BDS-3 satel-

ites are transmitting signals centered at 1561.098 MHz (B1),

575.420 MHz (B1C), 1176.450 MHz (B2a), 1207.140 MHz (B2b),

nd 1268.520 MHz (B3), while the other BDS-2 satellites are

ending signals at 1561.098 MHz (B1), 1207.140 MHz (B2), and

268.520 MHz (B3) [10] . The corresponding Interface Control Docu-

ents (ICD) of open service signals for current BDS can be found at

ttp://www.beidou.gov.cn/xt/gfxz/ . As shown in Fig. 3 , several BDS-3

atellites can be tracked at International GNSS service (IGS) HNIS

tation (Horn Island) on March 6, 2018, where only B1 signal of these

DS-3 satellites can be captured.

.2. Previous works on BDS orbit/clock determination and precise

ositioning

After getting the initial service capability, many works have been

one on BDS-2. Researchers analyzed the BDS’ signals quality [9–

1] and found that its signal strength is weaker than that of GPS because

185

f the higher orbit altitude of BDS satellites, especially the IGSO and

EO satellites. Meanwhile, the BDS-2 signal strength expresses obvious

orrelations to satellite elevation angle and satellite motion. In general,

he measurement noises and thermal noises of BDS-2 MEO satellites are

lmost at the same level as that of GPS. In addition, the stability of

DS’ code-phase noise is even a little better than that of GPS. Some

esearchers have evaluated the impacts of BDS’s users-satellite geome-

ry structure strength [6,12] and satellite antenna Phase Center Offsets

PCO) [13] on BDS positioning. As shown by the zero-baseline test in

12] , there are system offsets among GEO, IGSO, and MEO satellites.

eanwhile, due to BDS-2 ′ special GEO + IGSO + MEO constellation, the

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

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verage Position Dilution of Precision (PDOP) of BDS-2 is bigger than

hat of GPS under the conditions with the same available satellites.

Additionally, the major works are focusing on BDS-2 ′ s clock and or-

it determination at present. As is proven in [14] , BDS-2 ′ special satellite

onstellation makes it difficult to calculate satellites’ orbits and clocks,

specially for GEO satellites. The conclusions in [15] indicate that the

ccuracy of BDS-2 broadcast ephemeris are around 1–30 m in orbit and

–7 ns in satellite clock offsets. For BDS precise orbit and clock products,

he overlap statistics in [14,16–18] illustrate that the RMS of orbit and

lock errors are approximately 3–200 cm and 0.1–0.8 ns. Moreover, dif-

erent types of BDS satellites (e.g., GEO, IGSO, and MEO) have different

rbit/clock accuracies for both broadcast ephemeris and precise prod-

cts, specifically, from Montenbruck et al. [19] , approximately 3–5 cm

or BDS MEO, 100–200 cm for GEO, and 1–20 cm for IGSO. Compared to

PS products, the current BDS satellite’s orbits and clocks have lower

recisions, especially for the GEO and IGSO satellites. By using these

atellite orbit/clock products and single-/dual-frequency data, BDS po-

itioning performance has been validated under the Single Point Posi-

ioning (SPP), Real-time Kinematic (RTK), and Precise Point Positioning

PPP) modes. The evaluation results indicate that BDS can provide po-

ition solutions at meter-level under the SPP mode [20,21] , centimeter-

evel under the RTK mode [14,22,23] , and decimeter-level under the

PP mode in the majority of the Asian-pacific region [24] .

Furthermore, because BDS can provide triple-frequency signals,

cholars have studied the effect of BDS B3 observation on improving

DS positioning performance. For example, as is proven in [25,26] ,

he Cycle Slip Detection (CSD) capability on detecting small slips in

eal time can be improved visibly by using BDS triple-frequency ob-

ervations. Similar solutions can be found in [27,28] . As studied in

29] , obvious enhancements on the dual-differenced ambiguity fixing

n 45–100 km length baselines can be obtained by introducing BDS’

hird frequency observations. Meanwhile, in [30] , the impact of code

ardware bias variations on ambiguity fixing is furtherly assessed by

rocessing 30-day’s BDS data with 500 ∼2600 km baselines. Results sug-

est that the BDS triple-frequency observation based Extra-Wide-Lane

EWL) ambiguity resolution is not susceptible the code bias variation,

hile the Wide-Lane (WL) ambiguity resolution is sensitive to it. Be-

ides, BDS triple-frequency PPP modes based on both ionosphere-free

ombination mode [31] and ionosphere-delay-constrained mode [31–

3] have also been validated. As is proven in [34] , the ionosphere-

elay-constrained PPP mode has more advantages than the ionosphere-

ree combination PPP mode because of the characteristic of ionosphere

elay slow-variation in the short term. However, these works on BDS

riple-frequency ionosphere-delay-constrained PPP have not estimated

he receiver code hardware time delays. Instead, they either correct the

eceiver code hardware time delays by using IGS code bias products

31] or merge them into the ionosphere parameters [32,33] . In this case,

f the data are not from IGS stations, no IGS code bias products can be

sed. Because the receiver code hardware time delays are temperature

nd environment dependent, merging he receiver code hardware time

elays into the ionosphere parameters may degrade the PPP’s conver-

ence performance significantly, as proven in [34] . Hence, in this pro-

osed triple-frequency PPP mode, we parameterize the receiver code

ardware time delays of B2&B1 and B3&B1, model the ionosphere de-

ay residuals (i.e., that after classic model correction) as random walk

rocesses, and estimate them as parameters to mitigate their effects on

ositioning.

.3. Previous works on improving dynamic dual-frequency PPP

Currently, BDS has been widely used in various applications in

hina, such as the SPP mode real-time navigation [35] , GEO satellites

ased time service [36] , BDS signal reflectometry based soil moisture

stimation [37] and seal level change [38] , train control system in low

ensity railway lines [39] , railway safety detection [40] , ionospheric

ariation monitoring [41] , zenith tropospheric delay calculation [42] ,

186

nd precise dynamic positioning [43] . Generally, centimeter-level to

eter-level positioning accuracy already satisfies the accuracy require-

ents in these applications. Nevertheless, the performance of BDS (e.g.,

ccuracy and continuity) at present is partially limited by the GEO and

GSO depended satellite-constellation and the limited number of avail-

ble satellites. Wherein, the satellite constellation will be improved at

he end of 2020 [5] . Therefore, the major problem is to solve the limited

vailable satellites caused by the users’ observing environments.

To mitigate the degradation caused by the limited available satellites

n dynamic applications, previous works (e.g., [44] ) have proposed to

ntegrate GNSS with Inertial Navigation System (INS). In such integra-

ion, the continuous INS solutions are utilized to bridge the discontinu-

us GNSS solutions under the poor satellite tracking conditions. To im-

rove the capability of detecting and rejecting of the poor quality data,

he research in [45] proposed data fusion process based on the multi-

ensor Kalman filter with the accelerometer measurements from Inertial

easurement Unit (IMU) sensors. References [46,47] suggested to use

he Input Delayed Neural Networks (IDNN), Strong Tracking Kalman

lter (STKF), and Wavelet Neural Network (WNN) to provide reliable

ositioning solutions during long GPS outages, and Li et al. [48] recom-

ended to utilize ensemble learning algorithm to upgrade the position-

ng accuracy of GPS/INS when GPS signals are blocked. In [49] , low-cost

MU and digital compass were integrated with GPS to furtherly improve

ositioning accuracy in complete GPS-outages environments. Due to the

dvantages of INS in improving positioning continuity and robustness,

NS based technologies have been adopted widely in multi-sensor inte-

ration system based applications, for example, integrating with mag-

etic and Zero velocity Update (ZUPT) in pedestrian navigation [50] ,

ith Wi-Fi and magnetometers for indoor navigation [51] , with GPS in

ehicle navigation [52] ,with odometer and GPS for personal positioning

ystems [53,54] , and with laser, Wi-Fi, compass, and camera in mobile

obot localization in crowded environments [55] .

Consequently, in order to upgrade the BDS’ PPP performance in

ynamic applications, reference [43] attempted to integrate BDS PPP

ith INS, by which BDS positioning accuracy and stability can be im-

roved, and centimeter-lever positioning solutions can be obtained.

uch GPS/INS integrated positioning accuracy has been validated in

56,57] . Besides, according to the work in [58] , the tight integration of

PS and INS performed more effective than the loose integration mode,

specially under the poor satellite tracking conditions. Similar solutions

ere obtained in [59] by processing BDS/GPS data and different grades

i.e., low-cost grade, tactical grade, and navigation grade) IMU measure-

ents in both tight and loose integration modes.

.4. Algorithms and contributions in this paper

Although the previous works above have indicated that the position-

ng accuracy of GPS or BDS upgrades significantly by applying external

ensors. However, only centimeter-level to meter-level positioning ac-

uracy can be obtained. For some specific applications such as railway

rack geometry structure monitoring [60,61] , submillimeter measuring-

ccuracy is needed. Actually, it is difficult to achieve such accuracy by

sing current BDS-only or GPS-only in dynamic PPP or RTK modes. The

eason for this fact is that the measuring accuracies of carrier-phase

bservation of BDS and GPS are about 2–3 mm [9–12] ; therefore, it is

ifficult to provide submillimeter absolute positioning accuracy. Fortu-

ately, in such high-accuracy applications, the solutions of positions and

ttitudes in the relative term instead of absolute term are adopted [60] .

herefore, in this contribution, we present a multi-sensor data tightly

ided BDS triple-frequency PPP (TF-PPP) mode, which integrate IMU ob-

ervations [44] , Heading Measurements Constraint (HMC) [62] , odome-

er data [54] , priori ionosphere delay model [34] , Non-Holonomic

onstraint (NHC) [62] , Zero velocity Updates (ZUPT) [50] , and Zero

ntegrated Heading Rate (ZIHR) [62] constraints with TF PPP tightly.

eanwhile, the Rauch-Tung-Striebel (RTS) smoother [63] is adopted to

educe solution noises and improve its stability and relative measuring

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Table 2

Previous BDS PPP methods and contributions in this paper.

Previous methods and technical features Improvements in this paper

DF PPP Using ionosphere-free (IF) combination or un-differenced

un-combined (UDUC) B1&B2 data [16,17,24,43] .

1 ○ Adopting B1, B2, and B3 observations and modeling

receiver DCBs.

2 ○ Estimating receiver DCBs between B1 and B2 and that

between B1 and B3 to separate receiver code hardware

time delays from ionosphere delays.

3 ○ Utilizing odometer and HMC measurements

augmentation, and applying NHC, ZUPT, and ZIHR motion

constraints.

4 ○ Applying RTS smoother to provide centimetre-level

absolute accuracy solutions and sub-millimeter relative

measuring accuracy.

DF PPP/INS Using INS tightly aided IF/UDUC PPP based on B1&B2

dual-frequency data [43,56,57] .

TF PPP Using IF PPP and UDUC PPP based on B1, B2, and B3 data,

and absorbing receiver DCB in ionosphere parameters

[22,26–28,30–33] .

BDS static tracking station

BDS TF PPP BDS receiver

IMU sensors

Odometer/NHC/ZUPT/ZIHR

Precise orbit and clock

Alignment & initialization

TF PPP/INS integration

Odometer/NHC/ZUPT/ZIHR tightly aided TF PPP/INS

MHC tightly aided TF PPP/INSHeading measurement

Rauch-Tung-Striebel

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Fig. 4. Mathematical model structure of multi-sensor tightly aided BDS triple-

frequency PPP; The red lines stand for the used sensors, measurements, and

smoother algorithm; the dashed lines denote the necessary procedures for the

initialization and alignment of the INS tightly aided triple-frequency BDS PPP,

which will stop when completed.

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ccuracy. As listed in Table 2 , compared to the previous works, in this

aper

1) To mitigate the performance degradations when absorbing receiver

code hardware time delays into ionosphere delays [32,33] or cor-

recting them by using IGS products [31] in the current BDS TF-PPP

mode, the receiver code hardware time delays on BDS B1, B2, and B3

frequencies are estimated in terms of Differential Code Biases (DCB)

between B2 and B1 and that between B3 and B1.

2) To improve BDS PPP’s stability and short-term positioning accuracy

[43] , the short-term high-accuracy characteristic of INS [60–62] is

utilized, and INS is firstly introduced into BDS TF PPP.

3) Because the weak observability of vertical gyroscope leads to low

heading accuracy [59,64] , the ZUPT, ZIHR, NHC, odometer, and

HMC constraints are utilized to increase the gyroscopes’ observabil-

ity to provide high-accuracy heading solutions in TF PPP/INS tight

integration model, as plotted in Fig. 4 . Meanwhile, the RTS smoother

is adopted to further weaken the influence of measuring noise and

to upgrade the system accuracy.

187

4) Compared to the absolute positioning accuracy provided by TF-PPP

by processing static data after long convergence time in [31–33] , the

proposed method can offer centimeter-level absolute positioning ac-

curacy after short convergence time and can provide solutions with

sub-millimeter-level relative measuring accuracy.

5) In previous works, the solutions from RTK based methods are

adopted to assess the absolute positioning of PPP based methods

[34,43,56,57] . Such strategies may lead to system offsets [43] be-

cause of different weight determination models, different priori vari-

ances of pseudo-ranges and carrier-phases, and different data pro-

cessing modes. To reduce the impact of such offsets, this paper

uses the repeatability-consistency measuring method to evaluate dy-

namic positioning performance.

In general, this paper is organized as: Section 2 introduces the math-

matic models of the proposed multi-sensor tightly aided TF-PPP in de-

ail. Sections 3 and 4 present the dynamic tests on China high-speed

ailway and data analysis. Finally, Section 5 draws the conclusions.

. Methodology

The method of this paper can be mainly divided into BDS TF-

PP, INS tightly aided TF PPP, odometer, NHC, ZUPT, ZIHR, and

MC tightly aided BDS TF-PPP, and RTS smoothed multi-sensor in-

egration models, respectively. The corresponding algorithms are de-

cribed in SubSections 2.1 - 2.5 . SubSection 2.6 provides details about

he general algorithm structure. As shown in Fig. 4 , before running BDS

F PPP, observations from BDS static tracking stations such as Multi-

NSS Experiment (MGEX) [19] are utilized to calculate BDS precise

rbits and clocks. Then, these products can be used in the proposed

ositioning algorithm. Specifically, the static IMU measurements and

DS TF PPP solutions are adopted to provide the initial attitude, ve-

ocity, position, ambiguities, ionosphere delay, receiver DCB, and wet

enith troposphere delay for Kalman filter. The initial IMU biases and

dometer bias are set to zero. Afterwards, the integration system can

ork.

.1. BDS triple-frequency PPP

According to Guo et al. [31] and Li et al. [33] , the raw observational

unction of BDS TF PPP can be given by

𝑗

1 = 𝜌𝑗

1 + 𝑇 𝑗 − 𝐼 𝑗

1 + 𝑡 𝑟 − 𝑡 𝑗 + 𝑑 𝑟, 1 − 𝑑 𝑗

1 + 𝜀 𝑗

𝑃 , 1

𝑗

2 = 𝜌𝑗

2 + 𝑇 𝑗 − 𝐼 𝑗

2 + 𝑡 𝑟 − 𝑡 𝑗 + 𝑑 𝑟, 2 − 𝑑 𝑗

2 + 𝜀 𝑗

𝑃 , 2

𝑗

3 = 𝜌𝑗

3 + 𝑇 𝑗 − 𝐼 𝑗

3 + 𝑡 𝑟 − 𝑡 𝑗 + 𝑑 𝑟, 3 − 𝑑 𝑗

3 + 𝜀 𝑗

𝑃 , 3 (1)

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

𝐿

𝐿

𝐿

w

𝐼

𝐼

w

p

a

f

t

l

t

o

u

o

b

c

(

a

b

a

m

t

r

f

e

v

t

i

w

c

a

d

c

r

n

s

𝐼

a

𝑑

𝑑

w

𝐼

c

𝑑

𝑑

𝑑

w

a

c

i

(

o

B

t

i

c

s

B

t

𝑥

w

t

c

l

B

c

2

f

a

s

c

b

𝒁

w

t

i

a

𝒁

w

𝒙

a

𝑯

w

t

i

d

I

o

s

i

b

o

H

𝑗

1 = 𝜌𝑗

1 + 𝑇 𝑗 + 𝐼 𝑗

1 + 𝑡 𝑟 − 𝑡 𝑗 + 𝜆1 𝑁

𝑗

1 + 𝑏 𝑟, 1 − 𝑏 𝑗

1 + 𝜀 𝑗

𝐿, 1

𝑗

2 = 𝜌𝑗

2 + 𝑇 𝑗 + 𝐼 𝑗

2 + 𝑡 𝑟 − 𝑡 𝑗 + 𝜆2 𝑁

𝑗

2 + 𝑏 𝑟, 2 − 𝑏 𝑗

2 + 𝜀 𝑗

𝐿, 2

𝑗

3 = 𝜌𝑗

3 + 𝑇 𝑗 + 𝐼 𝑗

3 + 𝑡 𝑟 − 𝑡 𝑗 + 𝜆3 𝑁

𝑗

3 + 𝑏 𝑟, 3 − 𝑏 𝑗

3 + 𝜀 𝑗

𝐿, 3 (2)

ith

𝑗

2 = 𝛾 ⋅ 𝐼 𝑗 1 , 𝛾 = 𝑓 2 1 ∕ 𝑓 2 2

𝑗

3 = 𝜅 ⋅ 𝐼 𝑗 1 , 𝜅 = 𝑓 2 1 ∕ 𝑓 2 3 (3)

here 𝑃 𝑗

𝑘 and 𝐿

𝑗

𝑘 (in meters) denote BDS pseudo-range and carrier-

hase observations at the k th frequency ( k = 1,2,3 denote BDS’ B1, B2,

nd B3 frequencies) of the j th satellite; 𝜌 is the geometrical distance

rom antenna phase centers of satellite and receiver; T j and 𝐼 𝑗

𝑘 refer

o the troposphere delay and the ionosphere delay along slant satel-

ite signal propagation path, where 𝐼 𝑗

𝑘 will be estimated as parame-

er; t r ( = c · 𝛿t r ) and t j ( c · 𝛿t s ) represent the clock offsets (in meters)

f receiver and satellite, where c, 𝛿t r , and 𝛿t s are light speed in vac-

um, receiver clock offset in second, and satellite clock offset in sec-

nd; 𝜆k and 𝑁

𝑗

𝑘 stand for carrier-phase wavelength and integer am-

iguity; d r,i and 𝑑 𝑗

𝑘 denote the hardware time delays of code on re-

eiver and satellite, wherein 𝑑 𝑗

𝑘 can be corrected by using IGS products

ftp://igs.ign.fr/pub/igs/products/mgex/dcb/ ), and d r,i will be modeled

s a parameter, which is different from the raw TF-PPP model in [33] ;

r,i and 𝑏 𝑗

𝑘 denote the hardware time delays of carrier-phase on receiver

nd satellite, which could be absorbed by ambiguity in the float PPP

ode [65] ; ɛ contains the un-modeled errors and observing noises.

Most existing works on un-differenced un-combined TF PPP lump

he receiver hardware time delays into the ionosphere parameters and

eceiver clock, instead of estimating them [33] . However, it has been

ound in the previous dual-frequency raw PPP work in [34] that the

stimation of the receiver hardware time delays can accelerate the con-

ergence speed of un-differenced un-combined PPP significantly. Hence,

he receiver hardware time delays on B1, B2, and B3 code are estimated

n the TF-PPP in this paper. Therefore, more parameters are estimated,

hich may lead to weaker parameter estimation in PPP’s parameter

alculation. Besides, as pointed out in [33] , it is also difficult to sep-

rate ionosphere delays from receiver hardware time delays in the un-

ifferenced un-combined PPP model due to the current satellite clock

alculation strategy. Hence, to strengthen TF-PPP solutions and sepa-

ate ionosphere parameters from receiver hardware time delays, exter-

al pseudo-observations of ionospheric delays are adopted, with the ob-

ervational function of

𝑗

1 = 𝐼 gim + 𝜀 gim (4)

nd

𝑟, 1−2 = 𝑑 𝑟, 1 − 𝑑 𝑟, 2

𝑟, 1−3 = 𝑑 𝑟, 1 − 𝑑 𝑟, 3 (5)

herein

0 =

(𝑑 𝑟, 1 ⋅ 𝑓

2 1 − 𝑑 𝑟, 2 ⋅ 𝑓

2 2 )∕( 𝑓 2 1 − 𝑓 2 2 )

gim = 40 . 28 ⋅ 𝑉 𝑇 𝐸𝐶∕ (𝑓 2 1 ⋅ cos ( 𝜃𝐼𝑃𝑃 )

)(6)

Based on (5) and (6) , the hardware time delay on each frequency

an be derived by

𝑟, 1 = − 𝑑 𝑟, 1−2 ⋅ 𝑓 2 2 ∕( 𝑓

2 1 − 𝑓 2 2 )

𝑟, 2 = − 𝑑 𝑟, 1−2 ⋅ 𝛾

𝑟, 3 = − 𝑑 𝑟, 1−2 ⋅ 𝑓 2 2 ∕( 𝑓

2 1 − 𝑓 2 2 ) − 𝑑 𝑟, 1−3 (7)

here d r ,1 –2 and d r ,1 –3 denote differential code biases (DCB) [34] of B2

nd B3 respect to B1; I gim

is the ionospheric delay along slant signal path

alculated from IGS’ Global Ionosphere Mapping (GIM) data [66] ; 𝜃IPP

s the zenith angle at ionospheric puncture point (IPP). From (3) and

188

7) , it can be known that two DCBs ( d r ,1 –2 and d r ,1 –3 ) and m (number

f available BDS satellite) ionospheric delays are parameterized in the

DS TF-PPP model. To estimate them in Kalman filter, both of the two

ypes of parameters are modeled as random walk processes [64] .

Other parameters are handled in the same way as those in the

onosphere-free combination PPP. That is, the position, velocity and re-

eiver clock offset are modeled as white noises, residual of wet tropo-

phere delay is estimated as random walk [62] , and the ambiguities on

1, B2, and B3 phase are estimated as random constants. In this case,

he parameter vector ( x ) of BDS TF-PPP can be written as

=

[𝑝 𝑟 , 𝑣 𝑟 , 𝑡 𝑟 , 𝑑 𝑡 𝑟 , 𝑇 𝑤 , 𝑑 𝑟, 1−2 , 𝑑 𝑟, 1−3 , 𝐼 1 , 𝑁 1 , 𝑁 2 , 𝑁 3

]𝑇 (8)

here p r and v r denote the receiver coordinate vector and velocity vec-

or; dt r = ( 𝑐 ⋅ 𝛿�� 𝑟 ) is the receiver clock drift in meter, where 𝛿�� 𝑟 is the re-

eiver clock offset rate; T w is wet component of tropospheric zenith de-

ay (WZTD), and others symbols are the same as those described above.

esides, the Doppler observations can also be utilized to calculate re-

eiver velocity [64] .

.2. INS tightly aided BDS triple-frequency PPP

The INS mathematic model in detail can be found in [62] . There-

ore, the detailed INS mechanization (e.g., update of position, velocity,

nd attitude, IMU biases modeling, and model correction of rotational,

culling, and coning motion errors) is not described in this paper. Ac-

ording to [67] , the measurement function of INS tightly aided PPP

ased on Extend Kalman Filter (EKF) can be expressed as

𝑘 = 𝑯 𝑘 𝒙 𝑘 + 𝜼𝑘 , 𝜼𝑘 ∼(0 , 𝑹 𝑘

)(9)

here 𝜼k is state noise with the apriori variance of R k that can be de-

ermined by the satellite elevation angle dependent model [65] ; Z is the

nnovation vector of TF-PPP/INS tight integration, which can be written

s

𝑘 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑷 m 1

𝑷 m 2

𝑷 m 3

𝑳

m 1

𝑳

m 2

𝑳

m 3

𝑰 m 1

𝑫

m 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑷 m 1 ,𝐼 𝑁 𝑆

𝑷 m 2 ,𝐼 𝑁 𝑆

𝑷 m 3 ,𝐼 𝑁 𝑆

𝑳

m 1 ,𝐼 𝑁 𝑆

𝑳

m 2 ,𝐼 𝑁 𝑆

𝑳

m 3 ,𝐼 𝑁 𝑆

𝑰 m 1 ,𝐼 𝑁 𝑆

𝑫

m 1 ,𝐼 𝑁 𝑆

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

|Δ𝒑 𝑟,𝜄||Δ𝒑 𝑟,𝜄||Δ𝒑 𝑟,𝜄||Δ𝒑 𝑟,𝜄||Δ𝒑 𝑟,𝜄||Δ𝒑 𝑟,𝜄|𝟎

|Δ𝒗 𝑟,𝜄|

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(10)

ith

𝑘 =

[ 𝛿𝒑 𝑟 , 𝛿𝒗 𝑟 , 𝛿𝜽, 𝛿𝑩 𝐼 𝑀 𝑈 , 𝛿𝑺 𝐼 𝑀 𝑈 , 𝑡 𝑟 , 𝑑 𝑡 𝑟 , 𝑇 𝑤 , 𝑑 𝑟, 1−2 , 𝑑 𝑟, 1−3 ,

𝑰 1 , 𝑵 1 , 𝑵 2 , 𝑵 3

] 𝑇 (11)

nd

𝑘 =

𝜕 𝒁 𝑘

𝜕 𝒙

||||𝒙 = 𝒙 𝑘 ∕ 𝑘 −1 (12)

here Z k is the innovation vector calculated by making difference be-

ween the BDS observed values and INS predicted ones [67] . Specif-

cally, it is calculated from BDS raw observations/pseudo-ionospheric

elays ( P , L , D , and I ) and INS predicted values ( P INS , L INS , D INS , and

INS ). m and 𝜄 are the number of the observed satellites and the lever-arm

ffset between BDS receiver and IMU, respectively. The corresponding

tate parameter vector will also include the parameters in TF-PPP model

n (8) and the INS related parameters (e.g., attitude corrections 𝛿𝜽, IMU

iases 𝛿B IMU and scale factors 𝛿S IMU ). After considering the lever-arm

ffsets between receiver antenna phase center and IMU center [51,64] ,

in (12) can be written as

k

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

𝑯

4

4

4

4

4

4

𝟎 𝟎

w

𝑯

𝑯

w

c

f

s

o

F

t

b

f

a

t

a

2

o

d

b

𝒗

a

h

𝒗

w

v

e

𝒁

w

𝜕

w

b

m

c

[

𝒗

𝜙

a

𝒁

𝒁

w

r

N

t

2

G

m

p

𝒁

Y

Γ

w

s

a

a

2

t

𝒙

w

𝒙

w

f

i

𝑘 =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝑯 3 𝟎 𝑯

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝑯 6 𝟎 𝑯

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝑯 3 𝐈 𝑯

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝟎 𝟎 𝑯

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝟎 𝟎 𝑯

𝑯 1 𝟎 𝑯 2 𝟎 𝟎 𝟎 𝟎 𝐈 𝟎 𝟎 𝟎 𝑯

𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑯 7 𝑯 8 𝑯 9 𝑯 10 𝟎 𝑯 11 𝟎 𝟎 𝐈 𝟎 𝟎

ith

𝑯 1 = 𝑨 𝑪 1

𝑯 2 = 𝑯 1 (𝑪

𝑛 𝑏 𝜾𝑏 𝐵𝐷𝑆

×)

𝑯 3 = 𝛽𝐈

𝑯 4 =

[𝑀

1 𝑤𝑒𝑡 , ⋯ , 𝑀

m 𝑤𝑒𝑡

]𝑇 𝑯 5 = dia 𝑔 ( [ 1 , 1 , ⋯ , 1 ] ) m×m

𝑯 6 = 𝛾𝑯 3

𝑯 7 = 𝑨 𝑫

−1 𝑪 2

𝑯 8 = 𝑨 𝑪

𝑒 𝑛

𝑯 9 = 𝑯 7 (𝑪

𝑛 𝑏 𝜾𝑏 𝐵𝐷𝑆

×)+ 𝑯 8 𝑯 𝜃

10 = − 𝑯 8 𝑪

𝑛 𝑏

(𝜾𝑏 𝐵𝐷𝑆

×)

11 = 𝑯 10 diag ( 𝝎

𝑏 𝑖𝑏 ) (14)

here C 1 is the rotation matrix to transform positions in the geodetic

oordinate system to the Earth Centered Earth Fixed (ECEF) frame ( e -

rame); C 2 is derived from 𝛿( 𝑪

𝑒 𝑛 𝑣 𝑛 𝑟 ) , wherein 𝑪

𝑒 𝑛

is used to transform po-

itions from navigation frame ( n -frame) to e-frame, and the expressions

f C 1 and C 2 can be found in [67] ; 𝑀

𝑚 𝑤𝑒𝑡

denotes the Global Mapping

unction (GMF) of WZTD components; I = [ 1 , 1 , ⋯ , 1 ] 𝑇 𝑚 × 1 is the unit vec-

or; 𝑪

𝑛 𝑏

is attitude transition matrix to transform lever-arm 𝜾𝑏 𝐵𝐷𝑆

from

ody frame ( b -frame) to the n-frame; 𝝎

𝑛 𝑖𝑛

denotes rotation rate of n -

rame respect to the inertial frame ( i -frame) projected in n -frame; 𝝎

𝑏 𝑖𝑏

nd ‘ × ’ refer to gyroscope ( G ) outputs and cross product computa-

ion, respectively. The other symbols are the same as these mentioned

bove.

.3. Odometer/NHC/ZUPT/ZIHR tightly enhanced TF-PPP/INS

When the platform moves with wheels pressing against road, there is

nly motion in forward direction and no motions in vertical and lateral

irections in b-frame. Such priori condition is named as NHC, which can

e given in vehicle frame (v-frame) as

v 𝑁 𝐻 𝐶

=

[ 𝑣 𝑅 𝑣 𝐷

] ≈[ 0 0

] (15)

nd if the forward velocity can be obtained (e.g., by using odometer), it

as

v 𝑂 =

[ 𝑣 𝐹 𝒗 v 𝑁 𝐻 𝐶

] ≈[ 𝑣 𝑂 𝟎

] (16)

here F, R, and D refer to forward-right-down directions; v O is platform

elocity measured by odometer ( O ). The corresponding measurement

quation of (16) can be written as

𝑂 = 𝒗 v 𝑂 − 𝑪

v 𝑏

(𝑪

𝑏 𝑛 𝒗 𝑟,𝐼 𝑁 𝑆 + 𝝎

𝑏 𝑖𝑏 × 𝜾𝑏 𝑂

)(17)

ith the differential equation of

𝒁 𝑂 = 𝒗 v 𝑂 𝑆 𝑂 − 𝑪

v 𝑏

(𝑪

𝑏 𝑛 𝛿𝒗 𝑛 𝑟,𝐼 𝑁 𝑆

− 𝑪

𝑏 𝑛 ( 𝒗 𝑛 𝑟,𝐼 𝑁 𝑆

×) 𝛿𝜽 − ( 𝜾𝑏 𝑂 ×) 𝛿𝝎

𝑏 𝑖𝑏

)(18)

here S O and 𝜾𝑏 𝑂

denote the scale factor of odometer and the lever-arm

etween odometer and IMU respectively; 𝑪

v 𝑏

is to correct the misalign-

ent angle between b -frame and v-frame.

189

𝟎 𝟎 𝟎 − 𝑯 5

𝟎 𝟎 𝟎 − 𝛾𝑯 5

𝟎 𝟎 𝟎 − 𝜅𝑯 5

𝑯 5 𝟎 𝟎 𝑯 5

𝟎 𝑯 5 𝟎 𝛾𝑯 5

𝟎 𝟎 𝑯 5 𝜅𝑯 5

𝟎 𝟎 𝟎 𝑯 5

𝟎 𝟎 𝟎 𝟎

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(13)

Moreover, when the platform keeps static, velocity and attitude

hanges should be zeros. In this case, ZUPT and ZIHR can be applied

62] , and it has

𝑛 ≡

[0 0 0

]𝑇 (19)

𝑘 ≡ 𝜙𝑘 −1 (20)

nd the corresponding measurement functions can be written as

𝑍𝑈𝑃𝑇 = 𝒗 𝑛 − 𝒗 𝑟,𝐼 𝑁 𝑆 with 𝜕 𝒁 𝑍𝑈𝑃𝑇 = 𝛿𝒗 𝑛 𝑟,𝐼 𝑁 𝑆

(21)

𝑍𝐼 𝐻 𝑅 = 𝜙𝑘 − 𝜙𝑘 −1 with 𝜕 𝒁 𝑍𝐼 𝐻 𝑅 = ℏ 𝜙𝛿𝑩 𝑔 (22)

here ℏ 𝜙 = Δ𝑡 𝑘 [ 0 sin 𝜗 ∕ cos 𝜑 cos 𝜗 ∕ cos 𝜑 ] , wherein ϑ, 𝜑 , and 𝜙

efer to roll, pitch, and heading angle; Δt k is time interval of IMU data.

Then, from (18) , (21) , and (22) , the designed coefficient matrix of

HC/odometer, ZUPT, and ZIHR tightly enhanced BDS TF-PPP/INS in-

egration can be obtained.

.4. HMC tightly enhanced TF-PPP/INS

Currently, attitude (especially the heading angle) can be given by

NSS, magnetometers, or specifically designed schemes. If the heading

easurements exist, according to Shin [62] , the HMC model can be ap-

lied, with the corresponding function of

𝑀 𝐻 𝐶 = 𝜙𝑘 − ��𝑘 and 𝜕 𝒁 𝑀 𝐻 𝐶 = Y 𝛿𝜽 (23)

= Γ⎡ ⎢ ⎢ ⎢ ⎣

(𝑪

𝑛 𝑏 𝑪

𝑏 v )( 1 , 1 )

(𝑪

v 𝑏

)(1 , ∶)

(𝑪

𝑛 𝑏

)(3 , ∶) (

𝑪

𝑛 𝑏 𝑪

𝑏 v )( 2 , 1 )

(𝑪

v 𝑏

)(1 , ∶)

(𝑪

𝑛 𝑏

)(3 , ∶)

−

(𝑪

𝑛 𝑏 𝑪

𝑏 v )( 1 , 1 )

(𝑪

v 𝑏

)(1 , ∶)

(𝑪

𝑛 𝑏

)(1 , ∶) −

(𝑪

𝑛 𝑏 𝑪

𝑏 v )( 2 , 1 )

(𝑪

v 𝑏

)(1 , ∶)

(𝑪

𝑛 𝑏

)(2 , ∶)

⎤ ⎥ ⎥ ⎥ ⎦ (24)

=

1 (𝑪

𝑛 𝑏 𝑪

𝑏 v )2 (1 , 1) +

(𝑪

𝑛 𝑏 𝑪

𝑏 v )2 (2 , 1)

(25)

here �� denotes the priori measured heading angle; 𝑪

𝑏 v is the transpo-

ition of 𝑪

v 𝑏 ; () ( k 1, k 2) refers to the element of matrix () at the k 1th row

nd the k 2th column, and () ( k 1, : ) refers to all the elements of matrix ()

t the k 1th row.

.5. Extend Kalman filter and RTS smoother

The state function of EKF corresponding to the measurement func-

ion in (9) can be written as

𝑘 = Φ𝑘 ∕ 𝑘 −1 𝒙 𝑘 −1 + 𝜂𝑘 , 𝜂𝑘 −1 ∼ 𝑵 (0 , 𝑸 𝑘 ) (26)

ith the corresponding adjustment solutions of

𝑘 = �� 𝑘 −1 + 𝑲 𝑘 ( 𝒁 𝑘 − 𝑯 𝑘 Φ𝑘 ∕ 𝑘 −1 𝒙 𝑘 −1 ) (27)

here 𝚽 denotes the state transition matrix, which can be obtained

rom the state models (e.g., the psi-angle error model for position, veloc-

ty, and attitude; first-order Gauss-Markov process for IMU biases and

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Initialization and alignment

Rover receiver

B1/B2/B3 Code

B1/B2/B3 Phase

B1/B2/B3 Doppler

GPS+GLONASS RTK

Wuhan University BDS analysis center

BDS precise orbit and clock based IF combination

Positions and velocity

Satellites code bias and ionosphere products

BDS/GPS/GLONASS

PANDA

Base receiver

Error correction

Error correction models

Specific force

Angular rate

Hardware error compensation

Motion error compensatio

n

EKF prediction INS Update

IMU

Time synchronization

Kalman update

Yes

Kalman predicted and updated state storage

RTS smootherNHC/ZUPT/ZIHR/Odo

meter/HMC

IMU errors feedback

Positions and Attitude

BDS TF-PPP

GPS+GLONASS

BDS

Fig. 5. Algorithm architecture of multisource data

tightly aided BDS triple-frequency PPP.

o

s

c

v

m

i

m

t

f

I

s

t

𝒙

𝐏

w

n

2

F

a

t

p

i

a

i

I

[

i

u

d

o

p

s

d

t

p

t

p

c

a

H

b

f

e

Fig. 6. Trajectory (top) and platform (bottom) of test area along Lanzhou-

Urumqi high-speed railway on Mach 9th, 2014.

p

p

s

d

a

b

l

r

3

t

t

o

t

dometer scale factor; random walk processes for the receiver clock off-

et, drift, and DCBs, tropospheric delay and ionospheric delays; random

onstants for ambiguities) in [64,67] ; and 𝝁k is the state noise with a

ariance of Q k .

As the EKF solutions at epoch k are computed based on the measure-

ents before epoch k , the parameters estimation accuracy at epoch k

s theoretically higher than those before epoch k . To improve the esti-

ation accuracy before epoch k , the backward filter is adopted. Then,

he smoothed solutions are calculated by combining solutions from both

orward and backward filters according to the corresponding variances.

n this paper, the Rauch-Tung-Striebel (RTS) backward fixed-interval

moother [60–62,68] is applied to improve the performance of the in-

egration solutions. The RTS smoother can be described by

𝑘 ∕ 𝑁 = 𝒙 𝑘 + 𝑱 𝑘 (𝒙 𝑘 +1∕ 𝑁 − 𝒙 𝑘 +1

)(28)

𝑘 ∕ 𝑁 = 𝐏 𝑘 + 𝑱 𝑘 (𝐏 𝑘 +1∕ 𝑁 − 𝐏 𝑘 +1

)𝑱 𝑇 𝑘

(29)

here 𝑱 𝑘 = 𝐏 𝑘 𝚽𝑇 𝑘 +1∕ 𝑘 𝐏 −1 𝑘 +1 is the RTS smoother gain matrix; N is the

umber of total IMU epochs;

.6. Algorithm architecture and hardware platform

The mathematical models described above is compactly expressed in

ig. 5 . According to the time synchronization results between BDS data

nd IMU measurements, the whole system can be divided into two parts:

he TF PPP part and multi-sensor tightly aided TF PPP part. In the TF PPP

art, BDS’ precise orbit/clock products based on BDS dual-frequency

onosphere-free combination are provided by the BDS analysis center

t Wuhan University, China. These products are calculated by process-

ng BDS tracking station observations (collected by GNSS receivers from

GS) in the PANDA software (developed by Wuhan University, China)

14] . Afterwards, these products as well as BDS satellites code biases,

onosphere products, satellite antenna phase data from IGS center are

sed in BDS TF PPP. The TF PPP mode should work until there are IMU

ata. Then, the position and velocity from TF PPP as well as increments

f velocity and angle rate output from IMU are adopted to initialize the

arameters (e.g., initial position, velocity, attitude, wet zenith tropo-

pheric delay, ionospheric delays, ambiguities, receiver clock offset and

rift, receiver DCBs of B1&B2 and B1&B3) needed in the TF PPP/INS

ightly integration Kalman filter. After the initialization and alignment

hase, the INS mechanization, which consists of IMU error compensa-

ion and INS update [62] , starts to work. Then, the Kalman filter is ap-

lied to estimate the parameters. Here, if the platform motion meets the

onditions of NHC, ZUPT, and ZIHR, the augmentations in (15) , (21) ,

nd (22) should be applied. Besides, if there are available odometer or

MC measurements (in this paper, the HMC observations are provided

y the construction contractor of Lanzhou-Urumqi high-speed railway),

unctions of (17) and (23) can work. During this procedure, the param-

ter vectors and the corresponding variance matrices from both Kalman

190

rediction and Kalman update are saved, which will be used as the in-

ut for the RTS smoother. Afterwards, the final position and attitude

olutions can be obtained.

To validate the performance of the proposed method, a specialized

ata collection platform is built, in which a Trimble NetR9 receiver,

navigation grade IMU, two GNSS antennas, and odometer, a lithium

attery, and several feeders were installed on a customized track trol-

ey (more details will be described below). Meanwhile, another NetR9

eceiver is used as the base station.

. Test description and data processing schemes

As shown in Fig. 6 (top), a test (about 2 km length) was arranged on

he Lanzhou-Urumqi high-speed railway (China) on Mach 9th, 2014,

o validate the absolute accuracy and relative measuring accuracy

f positioning and attitude determination of the multi-sensor data

ightly aided BDS triple-frequency PPP. In this test, a navigation-grade

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Table 3

Parameters of navigation grade IMU sensors.

Rate

(Hz)

Mass

(kg)

Power

(V/W)

Volume

(mm)

Gyroscope Accelerometer

Bias ( ◦∕h ) ARW ( ◦∕ √h ) Scale (ppm) Bias (mGal) VRW ( 𝑚 ∕ 𝑠 ∕

√h ) Scale (ppm)

200 8 28/60 190 × 191 × 183 0.005 0.002 10 25 0.00075 10

∗ ∗ ∗ ARW stands for angle random walk and VRW denotes velocity random walk.

I

m

h

a

w

T

a

r

d

o

d

d

T

T

t

c

a

r

c

t

−

3

f

6

v

N

m

i

t

a

w

d

p

l

S

l

F

r

l

c

3

a

Fig. 7. Velocity (top), position (middle), and attitude (bottom) changes over

time.

MU (POS-830, http://www.whmpst.com/en/page.php?cid = 26 ), a

ulti-system and multi-frequency GNSS receiver (Trimble NetR9,

ttps://www.trimble.com/Infrastructure/Trimble-NetR9.aspx ), and

n odometer (SICK-DFS60E, https://www.sick.com/cn/en/encoders/ )

ere rigidly fixed on the railway trolley, as shown in Fig. 6 (bottom).

he IMU consists of three laser gyroscopes and three quartz flexible

ccelerometers with the parameters listed in Table 3 . The GNSS

eceiver can track BDS triple-frequency signals (B1, B2, and B3) and

ual-frequency signals of GPS and GLONASS with 1 Hz sample rate. The

dometer provided trolley velocity every 0.1 s. These sensors collected

ata respect to different measuring reference points. Therefore, in the

ata processing phase, data from various sensors were first aligned.

he IMU center was defined as the computation reference point.

hus, the measurements and solutions from the other sensors (e.g.,

he BDS observations and odometer data) were transformed to IMU

enter by using lever-arm correction models in (10) and (17) . Before

pplying lever-arm correction, the lever-arm values from the GNSS

eceiver antenna phase center and the odometer center to the IMU

enter were measured precisely. In the b -frame (Forward-Right-Down),

he lever-arms between GNSS receiver and IMU were 55.5, 0.0, and

27.5 cm, while that between odometer and IMU were 2.9, 133.9, and

5.7 cm.

To keep the trolley pasting together with the tracks tightly, the plat-

orm was pushed by an electric motorcycle with a velocity of within

m/s, as illustrated in Fig. 7 (top). Such design is used to mitigate the

ertical and lateral platform motion, so as to meet the requirement of

HC. The trolley moved along tracks periodically (about 5 min), which

akes it possible for performance assessment in terms of repeatabil-

ty consistency and relative measuring accuracy. Generally, the tested

racks were measured for four times, including twice forward direction

nd the other twice in the reversed direction. Meanwhile, the platform

as kept in static for seconds between each adjacent measuring sets,

uring which the ZUPT and ZIHR constraints can be used.

The sky-plot of the observed GNSS satellites and the test site are

lotted in Fig. 8 . Compared to over ten observed GPS or GLONASS satel-

ites, only eight BDS satellites were tracked. Besides, only GEOs and IG-

Os of BDS were observed during the mission. The observed BDS satel-

ite number and corresponding PDOP time series are depicted in Fig. 9 .

igs. 7 and 9 shown that the number of available BDS satellites changed

egularly during the test; meanwhile, such variation seems to be corre-

ated with the platform motion. The available BDS satellite number is

learly reduced when the platform begins to move at the time of around

9, 45, and 54 min.

To evaluate the performance of the proposed method, the test data

re processed via five schemes.

Scheme a): The BDS data are processed in both dual-frequency PPP

and triple-frequency PPP modes using raw code and phase obser-

vations. The receiver DCBs of B1&B2 and B1&B3 are estimated.

This scheme is used to investigate the impact of BDS third fre-

quency observations on dynamic PPP;

Scheme b): BDS triple-frequency observations and IMU data are pro-

cessed in PPP/INS tight integration mode. This scheme is utilized

to present the improvement of INS on BDS TF PPP;

Scheme c): using RTS to smooth the solutions from TF PPP/INS tight

integration;

191

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Fig. 8. Observed GPS (blue), BDS (red), and GLONASS (green) satellites in the

test; the pentagram with green line and red filling shows the test area.

Fig. 9. Number of observed BDS satellites (bottom) and PDOP (top) during the

test.

G

f

4

c

t

H

t

4

[

l

f

t

i

p

o

m

f

h

Fig. 10. Position differences of dual-frequency PPP (red line), triple-frequency

PPP (green line), and triple-frequency PPP/INS tight integration (blue line)

modes in n-frame compared to the reference values.

Table 4

RMS of position and attitude of DF PPP (A), TF PPP (B), TF PPP/INS

(C), TF PPP/INS/RTS (D), TF PPP/INS/odometer/RTS (E), and TF

PPP/INS/odometer/HMC/RTS (F) in n -frame compared to the ref-

erence values.

Items A B C D E F

North [cm] 11.8 10.6 10.3 9.4 9.8 9.6

East [cm] 14.2 13.7 11.8 10.9 10.8 10.8

Down [cm] 10.0 10.9 4.6 4.1 4.3 4.0

Roll [°] – – 0.006 0.007 0.007 0.007

Pitch [°] – – 0.005 0.006 0.004 0.005

Heading [°] – – 0.084 0.064 0.063 0.014

m

p

a

a

i

c

d

d

o

5

b

s

a

a

C

w

t

o

a

d

M

r

p

r

m

v

p

r

c

Scheme d): applying odometer, NHC, ZUPT, and ZIHR augmenta-

tion in scheme c);

Scheme e): adding the designed HMC constraints to scheme d).

The positions and attitudes calculated from a commercial software’s

PS/GLONASS RTK/INS tight integration are adopted as the reference

or absolute accuracy assessment.

. Results and discussions

In this section, the absolute accuracy, measuring repeatability-

onsistency, relative measuring accuracy of the proposed method and

he impacts of BDS B3 data, IMU measurements, odometer, NHC, ZUPT,

MC, and RTS smoother on improving multi-sensor tightly aided BDS

riple-frequency PPP are evaluated and discussed in detail.

.1. BDS B3 observation and INS on improving PPP positioning accuracy

As proven by Lou et al. [24] , BDS PPP needs long time (over 30 min

43] ) to convergence. Hence, in the accuracy assessment phase, the so-

utions in the first 30 min are not used. Shown in Fig. 10 are the n -

rame position differences of BDS DF PPP, TF PPP, and the TF PPP/INS

ight integration, which are compared to the reference values. Accord-

ngly, there are visible impacts of using B3 observations on BDS PPP

erformance. First, TF PPP provides better position solutions than that

f DF PPP, especially in term of lower standard deviation (Std), which

akes position solutions smoother and more stable. Besides, the third

requency observations can help detect carrier-phase observations that

ave low measuring accuracy, small cycle slip, or affected by strong

192

ultipath effects. Such assistance can be reflected significantly from the

osition differences (especially in the vertical component) in Fig. 10 at

round 43 and 51 min. The mutations in the DF PPP results almost dis-

ppear when using the BDS triple-frequency observations. This results

s because the third frequency observation can aid the detection of cy-

le slip detections that may not be detected under the dual-frequency

ata case. The effectiveness of using B3 observations in aiding cycle slip

etection are also mentioned in previous works that used high-quality

bservations collected in IGS center’s static tracking stations [25,26,28] .

As plotted in Fig. 11 , several burrs (around 34, 40, 43, 44, 51, and

6 min) appear in B1 and B2 carrier-phase residuals. This outcome is

ecause carrier-phase observation with low-accuracy, undetected cycle

lips, or strong multi-path will result in offsets in parameter estimation

nd larger residuals. However, when comparing the residuals of DF PPP

nd TF PPP, larger carrier-phase B1 residuals are arisen on BDS satellites

5 and C6 in DF PPP at around 43 and 51 min, which are the time points

hen the DF PPP solutions are degraded. This phenomenon indicates

hat the BDS third frequency observation can detect some unexpected

bserving errors and help improve PPP performance.

As a comparison, the TF PPP/INS tight integration position solutions

re also illustrated in Fig. 10 and the corresponding phase residuals are

rawn in Fig. 11 . According to the statistics in Table 4 , the position Root

eans Square (RMS) values of TF PPP/INS tight integration position er-

ors are reduced to 10.4, 11.8, and 4.6 cm with accuracy improvement

ercentages of 12.5, 17.1, and 54.3% in north, east, and down directions

espectively, compared to DF PPP solutions. Particularly, such enhance-

ents from INS on the east and vertical components are much more

isible (in Fig. 10 ). Besides, from Fig. 11 , it can be seen that the big

hase residual burrs existed in both DF PPP and TF PPP modes are also

educed in the TF PPP/INS tight integration mode. In general, this out-

ome is because INS can improve PPP performance [67] .

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Fig. 11. BDS phase residuals calculated in DF PPP,

TF PPP, and TF PPP/INS integration modes.

Fig. 12. Position differences of the tight integration modes of TF PPP/INS/RTS

(marked as RTS), TF PPP/INS/RTS/odometer/NHC/ZUPT/ZIHR (marked

as RTS/odometer), and TF PPP/INS/RTS /odometer/NHC/ZUPT/ZIHR/HMC

(marked as RTS/odometer/HMC) in n-frame compared to the reference values.

4

d

a

p

a

a

o

a

0

a

p

a

Z

t

(

e

t

e

w

r

f

b

c

p

o

Fig. 13. Statistics values (in terms of Mean, Std, and RMS) of DF PPP (A), TF

PPP (B), TF PPP/INS (C), TF PPP/INS/RTS (D), TF PPP/INS/odometer/RTS (E),

and TF PPP/INS/odometer/HMC/RTS (F) in n-frame compared to the reference

values.

Fig. 14. Attitude differences of the tight integration modes of TF PPP/INS,

TF PPP/INS/RTS (marked as RTS), TF PPP/INS/RTS/odometer/NHC/ZUPT/

ZIHR (marked as RTS/odometer), and TF PPP/INS/RTS/odometer/NHC/ZUPT/

ZIHR/HMC (marked as RTS/odometer/HMC).

c

t

p

T

.2. Odometer/NHC/ZUPT/HMC/RTS on positioning and attitude

etermination

The position solutions of schemes c ), d ), and e ) are given in Fig. 12 ,

nd the corresponding accuracy in terms of Mean, Std, and RMS are de-

icted in Fig. 13 . Compared with the solutions from scheme b ), there

re about 7% − 9.7% position accuracy improvements (within 1.0 cm)

fter applying RTS smoother. Meanwhile, as listed in Table 4 , the usage

f the other sensors and constraints (e.g., odometer, NHC, ZUPT, ZIHR,

nd HMC) has slight contributions on the positioning accuracy (within

.5 cm enhancements). However, these augmentations are helpful for

ttitude determination. From Fig. 14 , it can be seen that RTS smoother

lays an important role in improving heading accuracy, which provides

heading improvement of around 24%. In contrast, the odometer, NHC,

UPT, and ZIHR updates have little impacts on both attitude and posi-

ion accuracy upgrading. This result is due to the fact that the high-grade

navigation-grade) IMU was used in the test, therefore, the impacts from

xternal measurements are rather weak. Such phenomenon is different

o the conclusions obtained in [69] when using a low-cost IMU. How-

ver, the influences of these sensors and constraints might be significant

hen there are long GNSS signal outages. During the GNSS outage pe-

iods, these sensors and measurements can provide accurate relative in-

ormation (e.g., relative distance changes and instantaneous velocity in

-frame) to constrain platform motion during the short term. The related

onstraints can restrain the divergent caused by IMU biases. While ap-

lying the heading measurement constraint, attitude error RMS values

f about 0.007, 0.005, and 0.014° in roll, pitch, and heading components

193

an be obtained, with about 77% heading enhancements, compared to

he attitude solutions without HMC ( Table 4 ). Besides, about 0.1–6.7%

osition accuracy improvements due to the use of HMC can be found in

able 4 . Therefore, the solutions from scheme e) are adopted to assess

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Fig. 15. Repeatability of attitudes and positions

along with track mileages; to make figures clear, atti-

tude shifts of 0.1, 0.2, and 0.3° are added to pitch of

measuring sets S2, S3, and S4, and shifts of 0.3, 0.6,

and 0.9° are added to roll and heading of measuring

sets S2, S3, and S4; position shifts of 10, 20, and 30 m

are added to north and east components of measur-

ing sets S2, S3, and S4, and shifts of 0.05, 0.10, and

0.15 m are added to down direction of measuring sets

S2, S3, and S4.

t

t

B

m

v

c

v

a

s

B

s

v

a

t

h

m

t

a

a

g

4

r

t

d

t

t

t

fi

t

i

a

s

i

o

t

l

e

fi

f

t

r

f

t

e

c

R

i

t

t

t

t

i

d

w

n

b

a

t

m

t

p

t

t

b

b

u

m

s

4

r

t

c

s

he measuring accuracy of using multi-sensors and multi-measurements

ightly aided BDS triple-frequency PPP.

In general, according to this validation, both the enhancements from

DS third frequency and the augmentations from odometer, heading

easurements, and RTS on positioning and attitude determination are

isible. On the one hand, the PPP performance depends highly on

arrier-phase continuity and convergent ambiguity. However, the con-

entional cycle-slip detection methods for dual-frequency observations

re only valuable for big cycle slips and difficult to discriminate the

mall ones, particularly when there are half or even quarter cycle slips.

y applying the third frequency observation, the possibility of detecting

mall cycle slips increases. However, using the triple-frequency obser-

ations still has missed the detection of small cycle slips, such as that at

round 40 min on B2 in Fig. 11 . Such weakness can be overcome by in-

roducing INS, especially while using high-grade ones, because INS has

igh short-term relative accuracy and can reduce the high-frequency

easuring noises [70] . On the other hand, measurements like odome-

er data and heading measurements can enhance the heading estimation

ccuracy by improving the observability of vertical gyroscope. But, such

ids from NHC, ZUPT, and ZIHR might be weaker when using of high-

rade IMU sensors.

.3. BDS B3 observation, IMU measurements, and RTS on dynamic

epeatability measuring accuracy

Besides using reference values from commercial software to assess

he absolute position and attitude accuracy, it is feasible to calculate the

ifferences of positions and attitudes among different observation-set at

he same location to evaluate the repeatability consistency and the rela-

ive measuring accuracy when the carrier-platform moves along railway

racks periodically. Because the platform moved along the tracks that are

xed on the ground, the differences among different observation-sets at

he same location should be theoretically zero. However, the limitation

n real measuring capability leads to nonzero differences in the results.,

nd such offsets are different for different data processing schemes. The

tatistics on these differences is the repeatability accuracy. Such method

s similar to the “overlap in time ” method [71] used in assessing satellite

rbit accuracy.

Depicted in Fig. 15 are the four measuring-set solutions along with

rack mileage. Both positions and attitudes of the four-set are interpo-

ated to the same location based on track mileages. As the distance of

194

ach adjacent track sleepers is 0.625 m, the start point is treated as the

rst track sleeper and the mileage location of each sleeper will obtain

our group solutions of positon and attitude. Because the platform hugs

he tracks while moving, the irregularities (i.e., rectangular area with

ed sideline in Fig. 15 ) of track surface or structure can be seen clearly

rom the solutions of different measuring-sets, particularly from the at-

itudes.

Shown in Fig. 16 are the positions’ repeatability consistency between

ach two measuring-set. As a comparison, solutions from four data pro-

essing schemes (i.e., DF PPP, TF PPP, TF PPP/INS tight integration, and

TS smoothed and the proposed multi-sensor tightly aided TF PPP/INS

ntegration) are given together. The proposed method performs better

han the other methods. The proposed method provides repeatable posi-

ion error RMS on average of 2.7, 2.2, and 3.5 cm in north, east, and ver-

ical components, which are 71.7, 79.2, and 10.1% more accurate than

he corresponding absolute position RMS values in Table 4 , especially

n horizontal directions. Similar conclusions can also be got from other

ata processing modes. Averagely, by comparing RMS values in Table 5

ith that in Table 4 , about 71.6% − 74.5% percentage improvements in

orth and east directions can be found. Such differences may be caused

y different data processing strategies (e.g., the methods deal with low-

ccuracy observations and cycle slips, weight determination rule, and

ime synchronization algorithm between sensors) between the proposed

ethod and the used commercial software. However, BDS B3 observa-

ions can help improve PPP’s performance and the solutions from the

roposed method have better accuracy. Such trend is similar to that ob-

ained from Fig. 13 and Table 4 . Meanwhile, from Figs. 10, 12 , and 16 ,

here are systematic offsets between the position differences calculated

y comparing with reference values and those computed from repeata-

ility consistency. For some high-accuracy applications (e.g., track irreg-

larity detection [60] ), such systematic offsets do not affect the relative

easuring accuracy. Hence, the relative measuring accuracy without

ystematic offsets are furtherly assessed in the following subsection.

.4. Dynamic relative measuring accuracy and its potential applications

As plotted in Fig. 15 , the tested tracks’ geometric structure can be

eflected by the variations of positions and attitudes, especially the atti-

udes. Therefore, multi-sensor data tightly enhanced PPP/INS solutions

an be used for track 3D geometry structure detection [60] . However, in

uch application, sub-millimeter to millimeter level relative measuring

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Table 5

Position repeatability RMS of TF PPP, RTS PPP/INS tight integration, and RTS smoothed TF

PPP/INS/odometer/HMC tight integration, unit [mm].

Items DF PPP TF PPP Scheme b) Scheme e)

North East Down North East Down North East Down North East Down

S12 26.8 49.7 78.4 21.5 41.0 65.0 27.6 36.7 25.5 24.4 28.0 52.7

S13 38.3 26.1 174.4 30.8 16.4 115.4 33.1 13.7 35.8 33.3 15.4 43.0

S14 19.6 45.9 88.9 25.1 15.9 133.9 59.9 15.6 48.7 47.7 6.1 35.9

S23 37.0 59.8 186.3 27.7 46.1 63.8 18.1 41.0 39.3 13.9 41.9 28.0

S24 26.5 91.9 138.9 24.5 46.2 78.7 44.0 37.9 47.1 27.6 29.8 34.7

S34 45.2 42.4 233.2 20.0 13.6 39.5 30.9 14.1 33.7 16.1 13.3 21.3

Fig. 16. Position repeatability accuracy of DF PPP, TF PPP, RTS PPP/INS tight

integration, and RTS smoothed TF PPP/INS/odometer/HMC tight integration;

S12, S13, …, S34 denotes the differences between measuring sets 1/2/3 and

2/3/4.

a

t

t

b

a

r

i

s

a

a

c

a

s

e

a

o

P

Fig. 17. Position relative measuring time-series of DF PPP, TF PPP, RTS

PPP/INS tight integration, and RTS smoothed TF PPP/INS/odometer/HMC tight

integration. S12, S13, …, S34 denotes the differences between measuring sets

1/2/3 and 2/3/4.

s

P

m

o

s

d

p

w

T

t

T

r

t

t

a

T

ccuracy is required [61] . Here, the relative measuring accuracy is not

he relative positioning accuracy (which means the baseline calculation

hat is used in GNSS RTK). Such measuring accuracy can be obtained

y making differences between solutions from each two measuring-sets

fter removing the systematic offsets. Because, such relative measuring

esults can be used to detect the irregularities of railway tracks. Depicted

n Fig. 17 and Table 6 are the computed results and the corresponding

tatistics in term of RMS. Accordingly, the proposed method can provide

bout 8.7, 5.5, and 21.6 mm on average in term of relative measuring

ccuracy in north, east, and vertical components, respectively. For verti-

al comparison, such measuring accuracy is also higher than its absolute

ccuracy (96, 108, and 40 mm, listed in Table 4 ) and repeatability con-

istency accuracy (27, 22, and 36 mm, listed in Table 5 ). Making a lat-

ral comparison on 3D position solutions, the proposed method provides

bout 62–80%, 41–61%, and 31–55% position accuracy improvements

n relative measuring accuracy than that of DF PPP, TF PPP, and TF

PP/INS tight integration modes, respectively.

195

Theoretically, position time-series in Fig. 17 should obey the Gaus-

ian distribution. However, they are not. For position time-series of DF

PP and TF PPP, it may be due to the comprehensive impact from un-

odeled multipath (within 0.25 wavelength) on BDS carrier-phases,

bserving noises, and residuals of incompletely modeled errors (e.g.,

atellite orbit and clock errors). After integrating INS with PPP, the ran-

om noise parts in the position time-series are reduced due to the low

ass filtering characteristic of the INS algorithm. Then, the trend terms

ith short correlation time are mitigated by adopting RTS smoother.

herefore, only long-term trend errors (e.g., IMU biases) remain in

he smoothed TF PPP/INS/odometer/HMC tight integration’s solutions.

herefore, further works (e.g., using new external data or refining cur-

ent model) should be on improving the relative measuring accuracy in

he long term.

As is proven above, attitude solutions are slightly affected by odome-

er, NHC, and ZUPT when using a high-grade IMU without GNSS out-

ges. In contrast, RTS and HMC improve attitude accuracy significantly.

herefore, in this part, only the relative measuring accuracy of attitude

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

Table 6

Position relative measuring RMS of TF PPP, RTS PPP/INS tight integration, and RTS smoothed TF

PPP/INS/odometer/HMC tight integration, unit [mm].

Items DF PPP TF PPP Scheme b) Scheme e)

North East Down North East Down North East Down North East Down

S12 25.6 15.8 73.6 21.1 15.7 32.6 21.2 13.8 23.5 12.8 8.1 14.3

S13 27.7 16.6 137.1 16.9 15.1 38.3 15.5 12.3 34.1 7.1 4.4 30.5

S14 15.9 10.4 40.1 18.4 14.6 42.9 15.7 15.4 40.1 8.8 5.6 29.4

S23 25.3 14.8 168.0 17.6 13.1 36.1 13.9 9.4 33.2 7.3 4.5 19.2

S24 23.4 14.4 89.2 20.8 13.3 34.8 18.1 10.1 28.6 8.7 5.8 17.0

S34 28.0 17.1 139.1 18.0 13.6 35.2 12.0 13.7 29.4 7.4 4.8 18.9

Fig. 18. Attitude time-series of repeatability (left) and relative measuring

(right) of RTS smoothed TF PPP/INS/odometer/HMC tight integration. S12,

S13, …, S34 denotes the differences between measuring sets 1/2/3 and 2/3/4.

Fig. 19. Track cant measuring by using roll angle (top) and the corresponding

measuring errors (bottom).

s

a

s

R

c

a

a

v

g

t

e

f

Table 7

Repeatability and relative measuring accuracy in term of attitudes.

Items Repeatability [°] Relative measuring [°]

Roll Pitch Heading Roll Pitch Heading

S12 0.0057 0.0062 0.0158 0.0064 0.0048 0.0088

S13 0.0047 0.0049 0.0250 0.0058 0.0048 0.0059

S14 0.0059 0.0067 0.0375 0.0064 0.0052 0.0092

S23 0.0052 0.0055 0.0138 0.0052 0.0045 0.0081

S24 0.0041 0.0044 0.0237 0.0042 0.0046 0.0044

S34 0.0051 0.0057 0.0144 0.0053 0.0047 0.0079

m

c

s

(

a

i

5

c

i

f

i

s

f

s

L

i

a

m

o

t

i

t

o

d

t

d

i

t

f

c

h

s

s

o

i

olutions from the proposed method are assessed. Fig. 18 shows the

ttitude repeatability consistency results and relative measuring time-

eries, and Table 7 demonstrates the corresponding statistics in term of

MS. Accordingly, the RMS values on average of attitude repeatability

onsistency are 0.0051, 0.0056, and 0.0217° in roll, pitch, and heading

ngles, respectively, and that of relative measuring are 0.0056, 0.0048,

nd 0.0074° The RMS differences are small in roll and pitch angles, but

isible in heading.

Such a high relative measuring accuracy is helpful in fixed route

eometry detection. For example, the roll angle can be used to shown

he cant irregularity of railway tracks [60] . As the track cant can be

xpressed by 𝑐𝑎𝑛𝑡 = 𝑔 𝑎𝑢𝑔 𝑒 ∗ 𝑠𝑖𝑛 ( 𝑟𝑜𝑙𝑙 ) , roll angles in Fig. 15 can be trans-

ormed to track cant (track gage is 1.435 m in China). Then, the relative

196

easuring accuracy of roll angle in Fig. 18 can be transformed to track

ant measuring accuracy. The statistics indicate that the track cant mea-

uring accuracy by such indirect method can reach up to 0.10–0.15 mm

as shown in Fig. 19 ). Such accuracy is much higher than that GNSS

bsolute positioning accuracy and can meet the accuracy requirements

n track geometry structure measuring [61] .

. Conclusions

This paper has investigated a multi-sensor integration data pro-

essing system, in which BDS triple-frequency observations, high-grade

nertial measurements, odometer data, heading measurements, plat-

orm motion constraints (e.g., non-holonomic constraint, zero veloc-

ty update, and zero integrated heading rate), and Rauch-Tung-Striebel

moother were adopted to enhance the BDS triple-frequency PPP’s per-

ormance. After detailed descriptions on mathematical algorithms of

uch integration system, dynamic multi-sensor data collected on China’

anzhou-Urumqi high-speed railway were processed and analyzed to

ndicate the absolute accuracy, repeatability consistency accuracy, rel-

tive measuring accuracy, and possibility application of the proposed

ethod. According to the evaluation results, several conclusions can be

btained.

BDS PPP enhancements benefit from BDS’ third frequency observa-

ion (B3) are significant in both detecting low-quality data and improv-

ng positioning accuracy. Compared to BDS dual-frequency PPP, the in-

roduction of B3 observations can discriminate small cycle slips or phase

bservations with low-accuracy. Because the PPP performance is highly

ependent on the accuracy of discriminating cycle slips on ambiguity,

he use of triple-frequency observation can improve PPP performance

irectly.

The use of inertial measurements and Rauch-Tung-Striebel smoother

n BDS triple-frequency PPP can furtherly increase the capability of de-

ecting low-accuracy observations and decreasing the impacts of high-

requency errors; this phenomenon is due to the INS’ low pass filter

haracteristic and its high short-term relative accuracy when using a

igh-grade IMU. Moreover, the enhancements from non-holonomic con-

traint, zero velocity updates, and zero integrated heading rate are not

ignificant when a high-grade IMU is used. However, the improvement

n heading is significant because of the enhancement of the observabil-

ty of the vertical gyroscope.

Z. Gao, M. Ge and Y. Li et al. Information Fusion 55 (2020) 184–198

g

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In conclusion, the proposed method, which is based on the tight inte-

ration of BDS triple-frequency PPP, high-grade inertial sensors, odome-

er, heading measurement constraint, and RTS smoother, can provide

elative position measuring accuracy at around 0.9, 0.6, and 2.2 cm in

orth, east, and vertical components and attitude relative measuring ac-

uracy at around 0.0056, 0.0048, and 0.0074° in roll, pitch, and head-

ng, respectively. While using such method in track cant structure eval-

ation, it can reach to about 0.10–0.15 mm measuring accuracy. Such

igh measuring accuracy makes the proposed method competitive in

recise applications, such as high-speed railway and subway track geo-

etric structure monitoring.

cknowledgments

Many thanks to GNSS Research Center at Wuhan University China

or providing the track measuring data and BDS’s precise orbit and

lock products. This paper is partially supported by the National Nat-

ral Science Foundation of China (NSFC) for Young Scientists (grant

o. 41804027 ), the Fundamental Research Funds for the Central Uni-

ersities (grant no. 2652018026 ), and the National Key Research and

evelopment Program of China ( 2016YFB0501804 ).

upplementary materials

Supplementary material associated with this article can be found, in

he online version, at doi: 10.1016/j.inffus.2019.08.012 .

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