Copyright © 2007 IFAC
The 5th IFAC Intl. WS DECOM-TT 2007, May 17-20, Cesme, Turkey
UKF BASED FILTERS IN INS/GPS INTEGRATED NAVIGATION SYSTEMS:
NOVEL APPLICATION RESULTS


Jia-he Xu, Yuan-wei Jing, Lian-dong Shi *, 1
Sasho Gelev **
Georgi M. Dimirovski ***, 2


*
Institute of Information Science & Engineering,
Northeastern University, Shenyang, Liaoning, 110004, P.R. of China
ellipsis@ 163.com; ywjjing@ mail.neu.edu.cn
**
MA “General Mihailo Apostolski”, Ministry of Defense, MK-1000 Skopje, Rep. of Macedonia.
***
Dogus University of Istanbul, Faculty of Engineering, TR-347222 Istanbul, Rep. of Turkey, and with SS
gdimirovski@ dogus.edu.tr




Abstract: The filtering problem in the INS/GPS integrated navigation system is
investigated in this study. Firstly, the nonlinear model of the INS/GPS integrated
navigation system is proposed. Then the applications of the EKF, the UKF and the
modified adaptive UKF are put in practice based on the model. A sample set of
computer simulation results for the three filter algorithms, obtained by using
MATLAB platform, are given to illustrate the effectiveness of the proposed
algorithm with regard to the convergence rate and the estimation precision.
Copyright © 2007 IFAC

Keywords: Adaptive signal processing, integrated navigation system, INS/GPS,
trace tracking, unscented Kalman filter, nonlinear control systems





1. INTRODUCTION 

Because of the complementarities and good march
(He You et al., 2000; Wang et al., 2004), the
combination of GPS (Global Positioning System) and
INS (Inertial Navigation System) has satisfied the
higher demand to the precision and the reliability of
the navigation performance (Lin, 1991; Siouris,
2004; Dimirovski et al., 2004). In the integrated
navigation multi-sensors information fusion system,
Kalman Filtering is the most successful information
1 This work is supported by the National Natural Science
Foundation of China, under grant 60274009, and Specialized
Research Fund for the Doctoral Program of Higher Education,
under grant 20020145007, and also by Dogus University Fund for
Science.
2 Georgi M. Dimirovski is also with Cyril and Methodius
University, Faculty of Electrical Eng. & Inform. Technologies,
Skopje, Rep. of Macedonia.
fusion disposing method.
The UKF (Unscented Kalman Filter) algorithm is
proposed in 1990s (Julier et al., 1995) as a kind of
filter method which aims at the nonlinear system
directly (Julier et al., 2000; Lefebvre et al., 2002).
The initial application of UKF is to navigation and
tracking. The simulation instance which Julier used
for the proposition of the UKF is the problem of
missile in reentry. The simulation results illustrate
UKF can deal with the strong nonlinearity well from
the (Julier and Uhlmann, 1997). Brunke applied UKF
to autonomous vehicle positioning in order to transact
the nonlinear transform (Brunke and Campbell, 2002).
Julier solved the strong nonlinearity of vehicle
navigation (Julier et al., 1995; Julier and Uhlmann,
2002). Chen used UKF to the tracking of elliptical
contour object (as faces) and got better tracking
effect than EKF (Chen et al., 2002). Otherwise, the
researchers of MITRE Company employed UKF to
the parameter estimation at the missile project
moment and obtained better estimation precision than
EKF (James and Van, 2002).
In Part of Theoretical Study, the UKF algorithm is
introduced in allusion to the nonlinear model of the
integrated navigation system. And the appropriate
modification on UKF which has strong tracking
capability is analyzed theoretically for tracking
applications. Then the extended application to the
integrated navigation system and the thorough
analysis of the UKF filtering performance by
simulations and experiments is also indispensable
(Battin, 1987; Rrzemieniecky, 1990; Malysehv et al.,
1992).
2. AN INNOVATED MATHEMATICAL SYSTEM
MODEL

The considered INS/GPS integrated navigation
system with the form (Xionga et al., 2006) as follow.
x(k )  f ( x(k  1),k  1)  w(k )
z ( k )  H ( k ) x( k )  v( k )
where， x(k )  Rn is the state of the state. z(k )  Rm is
the estimated output of the system. w(k )  L1[0, ) is
the processing noise sequence. v(k )  L2[0, ) is the
processing noise sequence.

2.1 System State Model

Firstly, we give the system equations (Chobotov,
1991) which describes the dynamic characteristics of
integrated navigation system. According to the
analyses to the performances and the error sources of
INS and GPS, the situation of stationary base is be
considered only in order to calculate easily. The
errors of the state variable in INS is chosen as
follows: speed errors (  vEI ,  vNI ,  vUI ), position
errors (  LI , I ,  hI )，attitude errors ( E , N , U ).
Then
X INS  [ vEI  vNI  vUI  LI I  hI E N U ]T
The errors of the state variable in INS is chosen as
follows: speed errors (  vEG ,  vNG ,  vUG ), position
errors
(  LG , G ,  hG ),
attitude
errors(  G , G ,  G ). Then
X GPS  [ vEG  vNG  vUG  LG G  hG  G G  G ]T
Then the error dynamics model is as follows:
v
v
 vE  f N U  f UN  ( N tgL  U ) vE
RM  h
RM  h
(2ie sin L 

vE
tgL) vN  (2ie cos L
RN  h
vE vN
v
sec2 L  2ie sin LvU ) L  (2ie cos L  E ) vU
RN  h
RN  h
 vN  f U E  f EU  (2ie sin L 

vE
tgL) vE
RN  h
vU
v
v
 vN )  N  vU  (2ie cos L  E sec2 L)vE L
RM  h
RM  h
RN  h
 vU  f E N  f NE  2(ie cos L 
2
 
vE
) vE
RN  h
vN
 vN  2ie sin LvE L
RM  h
 vN
L 
RM  h
 vE
vE
sec L 
sec LtgL L
RN  h
RN  h
 h   vU
E  
 vN
RM  h
 (ie sin L 
(ie cos L 
N  
 vE
RN  h
vE
tgL)N
RN  h
vE
)U
RN  h
 ie sin L L  (ie sin L
vN
vE
tgL)E 
U
RN  h
RM  h
 vE
vE
U 
tgL  (ie cos L 
sec2 L) L
RN  h
RN  h

vN
vE
)E 
N
RN  h
RM  h
where, the footnotes E, N and U in the equation
denote East, North ，and sky. RM  Re (1  2 f  3 f sin 2 L) .
RN  Re (1  f sin 2 L) . Re  6378137m . f  1 298.257 .
Combining the error model equations and dispersing
them, we can obtain the state equation of the system
as follow.
(ie cos L 
x(k )  f ( x(k ),k )  w( k )
x(k )  [E , N , U ,  vE ,  vN ,  vU ,  L,  ,  h]T
w(k )  [ wgx , wgy , wgz , wbx , wby , wbz , wax , way , waz ]T
2.2 Measurement Model
Measurement equations of the integrated navigation
system reflect the relations of measurement and state.
The measurement equations about speed, position is
represented as follows
z ( k )  H ( k ) x( k )  v( k )
 [ vEI   vEG , vNI   vNG , vUI   vUG ,
 LI   LG ,I  G , hI   hG ,]T
According to observations, we can obtain the
decomposed observation equation as follow.
 zv ( k ) 
z (k )  H (k ) x(k )  v(k )  

 z p (k ) 
 H v (k ) x(k )  vv (k ) 


 H p (k ) x(k )  v p (k ) 
where, zv (k ) is measurement equation when
observation variable is speed, z p (k ) is measurement
equation when observation variable is position.
According to the speed information of the inertial
navigation and the speed information of the GPS
receiver, we can obtain the measurement equation
when observation variable is speed.
  vEI   vEG 
zv (k )  H v (k ) x(k )  vv (k )   vNI   vNG 
 vUI   vUG 
 (vE   vE )  (vE  M E )   vE   M E 
 (vN   vN )  (vN  M N )    vN    M N 
 (vU   vU )  (vU  M U )   vU   M U 
where, vE , vN and vU are the real speed of vector
along each axis in geography. M E , M N and M U is
the measurement speed error of GPS receiver. Then
we can obtain
Hv (t )  [033 diag{1 1 1} 033 ]
According to the position information of the inertial
navigation and the position information of the GPS
receiver, we can obtain the measurement equation
when the observations concern velocity.
 LI   LG 
z p (k )  H p (k ) x(k )  v p (k )   I  G 
  hI   hG 
NN  
NN 

 ( Lt   L)  ( Lt  R )    L  R

M
M

 


NE  
NE 
 (t   )  (t 
)    

RN cos L  
RN cos L 

 (ht   h)  (ht  N h )    h  N h 

 


 

 RM L  N N   RM L   N N 
  RN cos L  N E    RN cos L    N E 
  h  N h
   h
  N h 
where, t , Lt and ht is real position, N E ， N N ，
N U is the position error of the GPS receiver along
east, north and sky.
H p (t )  [033 033 diag{ RM
RN cos L 1}]
By combining the speed measurement and the
position measurement vectors we can obtain the
measurement equations as follow.
random pseudo-range measurement error deflection
is 0.05m/s. Similarly, again assume that the following
data apply too: the initial error of the navigation
system is: a-clinic attitude angle error is 300 ;
position error angle is 300 ; position error is 50m;
speed error is 0.6m/s. When GPS constellations are
optimal 21 constellations, the flying lane includes the
states: flying-off, crawling, turning, accelerating,
decelerating, horizontal flight. Furthermore, assume
that initial course is 45 ; a-clinic speed is 300m/s;
initial position is north latitude 45 , east longitude
120 , altitude 1000m. Assume that the position error,
speed error and attitude error are one-order Markov
process. Where, the correlation times  v ,  v ,  v ,
EG
G
G
G
G
G
G
In order to find the differences of the three kinds of
Kalman Filter, the scene is assumed as follows. The
aircraft flies at inherent altitude. From 0s to 400s, the
aircraft move straight with inherent speed along y
axis, and the speed is 15m/s. The initial point of the
aim is (2000m, 10000m). From 400s to 600s, it turns
90 to x axis slowly, its acceleration is
u x  u y  0.075m / s 2 . Upon completion of the turning,
the acceleration of the aim decreases to 0. From 610s
the aim turn 90 quickly, the acceleration is
u x  u y  0.3m / s 2 . When it is 660s, the turning
completes, and the speed decreases to 0. The state
from x and y axis is observed separately. And the
standard deviation of the observation noise is 100m.
By using Matlab platform, we simulated INS/GPS
integrated navigation systems that employ,
respectively, EKF ， UKF, and the modified UKF
filtering methods, which are discussed in Part of
Theoretical Study. The simulation results are
depicted in Figures 1 and 2.
-300
-200
the error curve of east angle
-100
φE/(")
0
100
200
300
400
500
600
0
50
100
150
200
250
300
350
400
450
t/s
(a)
4. SIMULATION EXPERIMENTATION
600
the error curve of east angle
500
400
φE/(")
Apply the INS/GPS integrated navigation system
designed in front to some aircraft, and control INS
with low precision and GPS synthetically. Assume
that the following data apply: the equivalent gyro
drift of the inertial navigation system is 0.1( ) / h ; the
equivalent acceleration zero-deviation is 104 g ; GPS
receiver is C/A code receiver with SA error; GPS
data updating rate is 1Hz; the pseudo-range
measurement error deflection is 10m, random is 32m,
UG
100s to 200s. In the paper, 200s is chosen.
 zv (k )   H v (k ) x(k )  vv (k ) 
z (k )  


 z p ( k )   H p ( k ) x( k )  v p ( k ) 
 H ( k ) x( k )  v( k )
Every GPS attitude error component (course, pitching
and rolling) is modeling to independent random
process: random constant, first-order Markov process
and white noise. Generally, they represent installation
error, receiver error and multi-path error (Wang and
Wu, 2002), respectively.
NG
 L ,   ,  h ,  ,   ,   are chosen in the range of
300
200
100
0
-100
-200
-300
0
50
100
150
200
t/s
(b)
250
300
350
400
450
600
500
Subsequently, the simulation results about tracking
the aim with the three kinds of filters, EKF, UKF and
modified UKF, can be found in Figures 3, 4, and 5.
the error curve of east angle
400
φE/(")
300
12000
200
true trace
10000
observation samples
100
8000
estimated samples
0
6000
l/m
-100
-200
-300
0
4000
2000
50
100
150
200
250
300
350
400
450
t/s
0
(c)
-2000
Fig. 1 The error curve of E via the respective filters:
-4000
1995
(a) EKF; (b) UKF; and (c) the modified UKF
2000
2005
2010
2015
2020
2025
λ/m
1
(a)
the error curve of east velocity
12000
10000
observation samples
8000
0.6
6000
l/m
δVeg/(m/s)
0.8
0.4
4000
2000
0.2
0
0
0
-2000
50
100
150
200
250
300
350
400
450
-4000
1995
t/s
2000
2005
1
2015
2020
2025
(b)
the error curve of east velocity
12000
0.8
estimated trace
10000
8000
0.6
6000
l/m
δVeg/(m/s)
2010
λ/m
(a)
0.4
4000
2000
0.2
0
0
0
-2000
50
100
150
200
250
300
350
400
450
-4000
1995
t/s
(b)
δVeg/(m/s)
2005
2010
2015
2020
2025
(c)
Fig. 3 The curves via EKF:
(a) The comparison curve of trace tracking ;
(b) The curve of observation samples;
(c) The curve of estimated samples
the error curve of east velocity
0.8
2000
λ/m
1
0.6
10000
true trace
observation samples
estimated samples
0.4
8000
6000
0
0
50
100
150
200
250
300
350
400
450
l/m
0.2
4000
2000
t/s
(c)
Fig. 2 The error curve of  vEG via the filters, respectively:
0
-2000
(a) EKF; (b) UKF; and (c) the modified UKF
-4000
1950
Upon analyzing Fig.1 and Fig.2, the following
concluding results can be obtained:
2000
2050
2100
2150
2200
2250
λ/m
(a)
10000
observation samples
8000
6000
4000
l/m
When solving the integrated navigation filtering
questions, the EKF algorithm can reach the steady
state; however, but state oscillation still exists and the
steady-state performance of the EKF algorithm is not
satisfactory enough.
Compared with EKF, UKF which is fit for nonlinear
model has better steady-state performance and higher
navigation precision.
The modified UKF, which is proposed in this paper,
in addition, has much higher (perhaps ideal)
convergence rate and navigation precision.
2000
0
-2000
-4000
1950
2000
2050
2100
λ/m
(b)
2150
2200
2250
obviously, but its predicted trace has a little deviation
with real trace. While the modified UKF in this paper
has both almost ideal convergence rate and
navigation precision, it suppresses the oscillation
effectively and improves the tracking performance
significantly. Compared with the EKF and the UKF,
its predicted trace has less deviation with real trace
when the modified UKF is employed in the system.
10000
estimated trace
8000
6000
l/m
4000
2000
0
-2000
-4000
1950
2000
2050
2100
2150
2200
2250
λ/m
(c)
Fig. 4 The curves via UKF:
(a) The comparison curve of trace tracking;
(b) The curve of observation samples;
(c) The curve of estimated samples.
10000
true trace
8000
observation samples
estimated samples
l/m
6000
4000
2000
0
-2000
1500
2000
2500
3000
3500
4000
4500
λ/m
(a)
10000
observation samples
8000
l/m
6000
4000
2000
0
-2000
1500
2000
2500
3000
3500
4000
4500
λ/m
(b)
10000
λ/m
estimated trace
8000
l/m
6000
5. CONCLUSION
The nonlinear model of INS/GPS integrated
navigation system has been proposed. And the UKF
has been introduced into the model. In order to solve
the high dimensional problem of the integrated
navigation system state model, a modified, adaptive
UKF algorithm has been applied as an integral part of
the navigation system.
For comparison, the EKF, the UKF and the modified
UKF are applied into the model based on the same
conditions. From the simulation results, it is apparent
that the UKF solves the filter problem of INS/GPS
integrated navigation system rather well and
efficiently. Compared with conventional EKF, the
UKF makes calculation easier and faster thus
improving the navigation precision effectively. At the
same time, the application of the modified UKF
proved to be rather effective in solving the tracking
problem of high dimensional system. It makes the
system error convergent in a shorter time while the
high navigation precision is ensured in parallel. And
it has satisfied the tacking performance of the
INS/GPS integrated navigation system.
To summarize, the simulations and experiments
illustrate the better applicability of the UKF for the
INS/GPS integrated navigation system than the EKF;
and both the higher speed and higher precision
information of the strong tracking UKF algorithm for
the integrated navigation system have also been
proved to be effective.
4000
REFERENCES
2000
0
-2000
1500
2000
2500
3000
3500
4000
4500
λ/m
(c)
Fig. 5 The curves via the modified UKF:
(a) The comparison curve of trace tracking ;
(b) The curve of observation samples;
(c) The curve of estimated samples.
The simulation results for tracking the trace of the
INS/GPS integrated navigation system with EKF,
UKF and modified UKF are shown in Figures 3, 4,
and 5, respectively. EKF algorithm presences overt
oscillation; its steady-state performance is not
satisfied enough; its predicted trace has much
deviation with real trace; the stated divergence exists.
Compared with EKF, UKF which is fit for nonlinear
model has better steady-state performance and higher
navigation precision, and the oscillation is suppressed
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