199
1 INTRODUCTION
GNSS(Global Navigation Satellite System) and
INS(Inertial Navigation System) are widely used as
standalone navigation systems, respectively. The
integrationof GNSS andINS not only leads to high
accuracy of vehicleʹs position, velocity, andattitude,
butalsoprovidespositionerrorbound.Tomaximize
the performance of GNSS/INS integrated
systems,
time synchronization between GNSS and INS is an
importantissue.
There has been some research for time
synchronization of GNSS/INS measurements for
looselycoupledGNSS/INSsystems ortightlycoupled
GNSS/INSsystems.However,therearefewresearch
toconsiderboth.Thispapercomparesthetimedelay
effect of loosely coupled
GNSS/INS systems and
tightly coupled GNSS/INS systems. For loosely
coupledGNSS/INSsystems,weanalyzetheeffectof
time delay for the case when only position
information is used as Kalman filter measurement.
For tightly coupled GNSS/INS systems, we analyze
the effect of time delay for the case when only
pseudorange information
is used as Kalman filter
measurement.
Tocomparetheperformanceofthetwointegration
systems,itisanalyticallystudiedhowthetimedelay
betweenGNSSandINShaseffectontheKalmanfilter
innovation for each integration method. Then based
on the analysis results, computer simulations are
performedtocheck
howeachintegrationmethodhas
effectonthenavigationperformancesuchasposition,
velocity, and attitude of the vehicle. The two
GNSS/INS integration methods are reviewed briefly
insection2,andtheeffectsoftimedelayareanalyzed
insection3.Computersimulationsareperformedin
section4andconclusionsare
giveninsection5.
2 INTEGRATEDGNSS/INSNAVIGATION
SYSTEMS
Several approaches are possible for the integrated
GNSS/INSsystemsdependingonwhichinformation
is shared between GNSS and INS. Loosely coupled
andtightlycoupledintegratedsystemsareconsidered
Analysis of the Effect of Time Delay on the Integrated
GNSS/INS Navigation Systems
C.K.Yang&D.S.Shim
Chung‐AngUniversity,Seoul,Korea
ABSTRACT: The performance of tightly coupled GNSS
/
INS integration is known to be better than that of
looselycoupledGNSS/INSintegration.However,ifthetimesynchronizationerroroccursbetweentheGNSS
receiver and INS(Inertial Navigation System), the situation reverses. The performance of loosely coupled
GNSS/INS integration and tightly coupled GNSS/INS integration is analyzed and compared due to time
synchronizationerrorbycomputersimulation.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 2
June 2013
DOI:10.12716/1001.07.02.06
200
in this paper. The position and velocity calculated
from GNSS are used to update the INS filter in the
loosely coupled integration, while raw pseudorange
isusedinthetightlycoupledintegrationforKalman
filter.
2.1 LooselycoupledGNSS/INSnavigationsystem
TheblockdiagramofthelooselycoupledGNSS/INS
integration
is shown in Figure 1. The position and
velocity is calculatedin the GNSS receiver and then
usedasameasurementintheINSKalmanfilter.
As seen in Figure 1 the structure is simple and
modularization is possible. The computation time is
reducedsincethealgorithmintheGNSS
receivercan
be used. The Kalman filter in the loosely coupled
GNSS/INS integration has the error model with
dimensionof15.
x
z
,
INS INS
P
V
,
GNSS GNSS
PV
Figure1.LooselycoupledGNSS/INSintegration
, ~ (0, )
INS INS INS LC LC LC
xFxw wNQ
(1)
[]
nn
INS accel gyro
xLlhvBB
(2)
Thestatevariablecontainsthepositionerrorwith
latitude, longitude, and height
(,,)
L
lh
, and
velocityerror
()
n
v
,attitudeerror
()
n
expressedin
thenavigationframe,andbiaserrors
(,)
accel gyro
BB
of
accelerometerandgyroscopes(refertoKnight1997for
detailerrormodel).Themeasurementequationofthe
looselycoupledGNSS/INSintegrationcanbedenoted
asfollows:
,~(0,)
I
NS GNSS LC INS LC LC LC
zxvvNR
, (3)
where
33 39
0
LC
HI
.
Inthecaseoflooselycoupledintegrationthereare
some drawbacks. When there are short of visible
satellites,navigationinformationcannotbecalculated
in the GNSS receiver. Under the large dynamic
environment GNSS navigation solution becomes
inaccurate,thustheintegrationperformancebecomes
worse.
2.2 TightlycoupledGNSS/INSnavigationsystem
INS
Kalman
Filter
Range, Range rate
Prediction
GNSS
Receiver
Antenna
Inertial
Reference
Correction
Navigation Information
Aided Tracking
Loop
PR, PR rate
x
z
+
-
,
INS INS
P
V
Figure2.TightlycoupledGNSS/INSintegration
The block diagram of the tightly coupled
GNSS/INS integration is shown in Figure 2. The
GNSS pseudorange and pseudorange rate are used
directlyinthemeasurementofKalmanfilterandthe
clockerroroftheGNSSreceivershouldbeincluded
inthestatevariabletobeestimated.ThustheKalman
filter
error model contains position error, velocity
error, attitude error, accelerometer bias, gyroscope
bias, clock bias and clock drift of GNSS receiver,
resultingin17statevariablesasin(4).
15 2
215
0
,~(0,)
0
INS INS INS
TC TC TC
clock clock clock
xF x
ww NQ
xFx
(4)
clock bias drift
xcc
(5)
where
bias
c
,
drift
c
denotes for clock bias and
clock drift of GNSS receiver and
I
NS
F
,
clock
F
are
describedinKnight1997.
Themeasurementisgivenasin(6).
,~(0,)
I
NS GPS TC INS clock TC TC TC
zxxvvNR
(6)
InthecaseofGNSSpositioning,theformerGNSS
positionisusedasthe referenceoflinearizationand
thus the navigation error increases for large
acceleration and angular rate. However, for the
tightly coupled GNSS/INS integration, INS
informationwithgooddynamicperformanceisused
as the reference of linearization
of GNSS
measurement, thus the same problem does not
happen. Even though the visible GNSS satellites are
notenough,thenavigationcalculationisstillpossible.
Soitprovidesmoreefficientstructureintheusageof
informationthanthelooselycoupledintegration.The
drawback of tightly coupled integration is the
complexity of
the integration structure and thus
computing effort increases as the number of GNSS
satellitesincreases.
201
3 EFFECTSOFTHETIMEDELAYINTHE
INTEGRATEDGNSS/INSSYSTEM
Acceleromete
rs
Gyros
Inertial
Navigation
System
IMU
t=kT
s
Integrated
GNSS/INS
Kalman
Filter
t=kT
s
–T
d1
GNSS
Receiver
t= T
s
–T
d2
Antenna
kM
Estimated
Errors
(Navigation Errors,
Sensor Errors)
Figure3.StructureoftimesynchronizationerrorTd(=Td2‐
Td1)intheintegratedGNSS/INSnavigationsystem
For the GNSS/INS integration there exist time
delaybetweenGNSSreceiverandINSsincenotonly
samplingtime but also thesignal processingtime is
different as in Figure 3. For simplified analysis, the
GNSS receiver sampling period is assumed to be a
multipleMoftheIMUsamplingperiodT
s.Td1andTd2
are processing time delays of the inertial navigation
system(INS)andtheGNSSreceiver,respectivelyand
T
d=Td2‐Td1, i.e.,the difference of the INS and GNSS
processingtime.
3.1 EKFalgorithm
ConsidertheerrormodelofINSandEKFequationto
analyzetheeffectsoftimedelayinmeasurement.The
errormodelcanbedescribedasfollows:
1kkkkk
x
xGe
(7)
kkkk
zxw
(8)
Here,
k
istheerrorstatetransitionmatrix,
k
e
is the process noise,
k
G
is the process noise gain,
k
H
is the measurement matrix, and
k
w
is the
measurement noise. The subscript k implies k ‐th
sampling sequence. The process noise
k
e
and the
measurement noise
k
w
are considered as white
noise, uncorrelated with each other and the
covariance matrices are
{}
kk
Rww
and
{}
kk
Qee
, where,
denotes the expectation
operator.
TheEKFalgorithmfortheGNSS/INSintegrationis
giveninTable1.
Table1. EKFAlgorithmforIntegrationbetweentheGNSS
andtheINS
Kalmangainupdate
1
()
kkkkkk
R
Differenceofthetwo
measurements
,.
ˆˆ
kkINSkGNSS
zp p
Estimationofthestate
errors
ˆ
ˆ
ˆ
k
kkk
k
r
x
z
Errorcorrection
ˆ
ˆ
ˆ
ˆ
ˆ
k
k
k
k
k
r
r
x
Covarianceupdate
()
kkkk
Sensorerror
compensation
ˆ
ˆ
(,)
kkk
ugu
Navigationequation
update
1
ˆ
ˆ
ˆ
(, )
kkk
rfru
Sensorerrorupdate
1
ˆ
ˆ
kk
Covarianceprediction
1kkkkkk
GQG
3.2 Effectsoftimesynchronizationerror
Vectors
ˆ
,
k
r
ˆ
k
and
ˆ
k
u
in Table 1 denote the
estimated navigation state, the estimated sensor
errors, and the IMU measurements, respectively.
Effects of time synchronization error between GNSS
receiverandINScanbeanalyzedasfollows.
Letthetime differenceof sampling measurement
between GNSS receiver and INS be T
d and suppose
thatINSsamplinghastimesynchronizationerrorless
than 1sec. Then in the case that vehicle position is
used in loosely coupled integration, the estimated
positioninINSisgivenasin(9),where
()
p
t
means
truetrajectory.
2
,
ˆ
() 0.5
kINS S d k k d k d k
p
pk p v a w
(9)
where|T
d|≤1,
k
v
isthevelocityofthevehicleand
k
a
istheacceleration.
Suppose that the pseudorange, which is the
measurementoftightlycoupledintegration,hastime
synchronization error less than 1sec, then
pseudorange estimated in INS can be described as
(10),where
()t
istruepseudorange.
2
,
ˆ
() 0.5
kINS S d k k d k d k
kw
(10)
where the pseudorange rate is
,
()
kkGNSSkk
vvl
,
pseudorangeaccelerationis
,,
()
k INS k GNSS k k
al
,
k
l
is LOS(line of sight) vector between satellite and
vehicle, and
,kGNSS
v and
,kGNSS
are velocity and
accelerationofsatellite,respectively.
Thus the true observation
a
k
z
of two integration
systemsareobtainedasin(11).
202
Forlooselycoupledintegration
,,
ˆˆ
a
kkINSkGNSS
zp p
(11a)
2
,
ˆ
2
d
kkd k k kGNSS
pv a wp
Fortightlycoupledintegration
,,
ˆ
ˆ
a
kkINSkGNSS
z
(11b)
2
,
ˆ
2
d
kkdk kkGNSS
w
Let
,
ˆ
kk k kGNSS
xpp
or
,
ˆ
kk k kGNSS
x
,
then(11)becomes(12).
a
kkkkk
zxwd
(12)
In(12)theresidualerrorvectorofpositionorthe
residual error vector of pseudorange is introduced
anddefinedas
2
2
d
kkdk
dv a
or
2
2
d
kkdk
d
(13)
InSkog&Händel(2011),theerrors oftheclosed‐
loopsystemarefoundtobe
1
ˆ
()
kkkkkk
x
xxGe
(14)
kk k kk kk
x
zGe
Let
,pk k k
,then(14)becomes(15).
1, ,
()
kkpkkkpkkkk
x
xwGe
(15)
Here, by substituting
k
z
with
a
k
z
in (14) and
inserting (12) into this, the following difference
equationfortheerrorstatecanbeobtained:
1, , ,
()
k k pk k k pk k k k pk k
x
xwGed
(16)
If we introduce
,kkpkk
, then the
MSE(mean square error) of the navigation solution
canbeexpressedasin(17).
111
{()}
kkk
xx
(17)
,,kk k pk pk k k
RGQG
,, ,,
p
kkk pk kkk pk pkkk k
dd xd dx
wherethemeanerror
{}
kk
x
x
canbecalculatedas
in(18).
1,kkkpkk
x
xd
(18)
Noticetheresidualerrorvector
k
d
in(13).For the
loosely coupled integration, time delay T
d has little
effect on
k
d
in the case of small velocity and
acceleration, i.e.,
0
k
d
. Thus for small velocity
k
d
can be neglected in the loosely coupled integration
andMSEvaluein(17)becomes
1,,k k k k pk pk k k
RGQG
.
However, for the tightly coupled integration, the
residual error vector
k
d
does not become zero even
thoughthevelocityofvehicleisalmostzerobecause
ofhighspeedofGPSsatellitelike3.9km/sec,i.e.,
2
,,
()( )
2
d
kkGNSSkdkGNSSk
dl l
Thuswe can notice thattime delay inthe tightly
coupledintegrationhasmoreeffectonthenavigation
performance than in the loosely coupled integration
when velocity and acceleration of vehicle are not
large.
4 SIMULATIONS
In this section computer simulation is performed to
verifytheanalysisresultforthe
effectsoftimedelay.
Table2showsthespecificationofINSandGPSused
inthesimulation.Thevehicletrajectoryisassumedas
circle and run time is 200sec. Vehicle speed is
10km/sec, 200km/sec, and 400km/sec and the time
delayisassumedtobe1secinINSmeasurement.
Table2. Specification of error sources of the INS and the
GPS
_______________________________________________
ErrorSources1‐σvalue
_______________________________________________
INS initialpositionerror10m
initialvelocityerror1m/sec
initialhorizontalattitudeerror0.03deg
initialverticalattitudeerror5deg
accelerometerbias500μg
gyrobias3deg/hr
GPS clockbias10m
clockdrift1m/sec
_______________________________________________
Figure 4 shows the position errors in north
direction and pitch errors in the loosely coupled
GPS/INS according to the various vehicle speeds.
Generallythepositionerrorandattitudeerrorbecome
large as the vehicle speed increase. This happens
becausetheresidualerrorvectorin(13)becomeslarge
according
tothevehiclespeedandthustheoptimality
of Kalman filter gets worse. The oscillation of the
positionerrorcomesfromthecircletrajectory.
203
0 20 40 60 80 100 120 140 160 180 200
-150
-100
-50
0
50
100
150
position error[m]
time[sec]
North
no time-delay
1sec time-delay(10km/hr)
1sec time-delay(200km/hr)
1sec time-delay(400km/hr)
(a)thenorthpositionerrors
0 20 40 60 80 100 120 140 160 180 200
-0.2
-0.15
-0.1
-0.05
0
0.05
euler error[deg]
time[sec]
Pitch
no time-delay
1sec time-delay(10km/hr)
1sec time-delay(200km/hr)
1sec time-delay(400km/hr)
(b)thepitchangleerrors
Figure4. The position errors and attitude errors in the
loosely coupled GPS/INS according to the various vehicle
speeds(10km/hr,200km/hr,400km/hr)
0 20 40 60 80 100 120 140 160 180 200
-200
-100
0
100
200
300
400
500
600
700
position error[m]
time[sec]
North
no time-delay
1sec t ime-delay(10km/hr)
1sec t ime-delay(200km/hr)
1sec t ime-delay(400km/hr)
(a)thenorthpositionerrors
0 20 40 60 80 100 120 140 160 180 200
-4
-3
-2
-1
0
1
2
3
euler error[deg]
time[sec]
Pitch
no time-del ay
1sec t ime-delay(10km/hr)
1sec t ime-delay(200km/hr)
1sec t ime-delay(400km/hr)
(b)thepitchangleerrors
Figure5. the position errors and attitude errors in the
tightly coupled GPS/INS according to the various vehicle
speeds(10km/hr,200km/hr,400km/hr)
Figure 5 shows the position errors in north
direction and pitch errors in the tightly coupled
GPS/INSaccordingtothevariousvehiclespeeds.
Noticethepositionerroratfirst.Thepositionerror
islargelike300meventhoughthespeedofvehicleis
smalllike10km/hr.Theresidualerrorvectorin
(13)
is small, i.e.,
0
k
d
in loosely coupled integration
whenvehiclespeedissmall,whileintightlycoupled
integration the residual errorvector,
2
,,
()0.5( )
kkGPSkd kGPSkd
dl l
, is large since the
satellite speed is large like 3.9km/hr. Figure 5(b)
showsthattheattitudeerrorbecomeslargeaccording
tovehiclespeedandtheattitudeerrorismuchlarger
than that in loosely coupled integration since the
satellitespeedisthedominanttermind
k.
Figure4andFigure5showthattimedelayerror
between GPS and INS causes more effect on tightly
coupled integration than on loosely coupled
integration
5 CONCLUSIONS
The estimation performance of loosely coupled
GNSS/INSintegrationandtightlycoupledGNSS/INS
integrationiscompared duetotime synchronization
error between GNSS
receiver and INS. If the time
synchronization error occurs between two sensor
measurements, residual error exists. For loosely
coupledintegrationtheresidualerrorvectorincreases
accordingto vehiclespeed while for tightly coupled
integration the residual error vector increases
accordingtosatellitespeed aswellasvehiclespeed.
TheGPS
satellitespeedisabout3.9km/sec,which is
much larger than the vehicle speed. Thus residual
error vector in tightly coupled integration depends
mainlyonthesatellitespeed.
Simulations are performed to verify the analysis
resultforthetwoGPS/INSintegrationmethods.The
simulationshowsthattimesynchronizationerrorhas
effectmore
onthetightlycoupledintegrationthanthe
looselycoupledintegration
204
ACKNOWLEDGMENT
This research was a part of the project titled
ʺDevelopment of Wide Area Differential GNSSʺ
funded by the Ministry of Land, Transport and
MaritimeAffairs,Korea.
REFERENCES
Bar‐Itzhack,I.Y.&Vitek,Y.1985.Theenigmaoffalsebias
detection in a strapdown system during transfer
alignment.JournalofGuidanceandControl8(2):175‐180.
Bevly,D.,Ryu,J.&Gerdes,J.2006.IntegratingINSsensors
with GPS measurements for continuous estimation of
vehicle sideslip, roll, and tire
cornering stiffness. IEEE
Trans.Intell.Transp.Syst.7(4):483‐493.
Coelho,E.,Carvalhal,P.,Ferreira,M.,Silva,L.,Almeida,H.,
Santos,C.&Afonso,J.2006.ABluetooth‐basedwireless
distributeddataacquisitionandcontrolsystem.inProc.
IEEEInt.Conf.ROBIO:543‐548.
Ding, W., Wang, J., Li, Y.,
Mumford, P. & Rizos, C. 2008.
Time Synchronization Error and Calibration in
IntegratedGPS/INSSystems.ETRIJournal30:59‐67.
Fox, J. 2006. Synchronization of input and output
measurements using a Kalman filter. in Proc. 25th Int.
Conf.Model.,Identification,Control:435‐440.
Im, N. & Seo, J.‐H. 2010.
Ship Manoeuvring Performance
Experiments Using an Free Running Model Ship.
TransNav‐International Journal on Marine Navigation
andSafetyofSeaTransportation4(1):29‐33.
Knight,D.T.1997.RapidDevelopment of Tightly‐Coupled
GPS/INSSystem.IEEEAESSystemMagazine:14‐18.
Lee, H.K., Lee, J.G. & Jee, G.I. 2002. Calibration of
measurement delay in GPS/SDINS hybrid navigation.
JournalofGuidance,Control,andDynamics25(2):240‐247.
Li,B.,Rizos,C.,Lee,H.K.&Lee,H.K.2006.AGPS‐slaved
timesynchronizationsystemforhybridnavigation.GPS
solutions10(3):207‐217.
Li, W. & Leung, H. 2004. Simultaneous registration and
fusion of
multiple dissimilar sensors for cooperative
driving.IEEETrans.Intell.Transp.Syst.5(2):84‐98.
Nilsson, J.O., Skog, I. & Händel, P. 2010. Joint state and
measurement time‐delay estimation of nonlinear state
spacesystems.inProc.Int.Conf.ISSPA:324‐328. Kuala
Lumpur,Malaysia.
Skog,I.&Händel,P.2011.
TimeSynchronizationErrorsin
Loosely Coupled GPS‐Aided Inertial Navigation
Systems.IEEETrans.Intell.Transp.Syst.12(4):1014‐1023.
