Journal of Global Positioning Systems (2006)
Vol. 5, No. 1-2:62-69
Adaptive Two-Stage EKF for INS-GPS Loosely Coupled System with
Unknown Fault Bias
Kwang Hoon Kim, Jang Gyu Lee
School of Electrical Engineering and Computer Science, Seoul National University,
133-dong, ASRI #610, San 56-1, Sillim-dong, Gwanak-gu, Seoul, 151-742, South Korea
Chan Gook Park
School of Mechanical and Aerospace Engineering, Seoul National University,
San 56-1, Sillim-dong, Gwanak-gu, Seoul, 151-744, South Korea
Abstract. This paper proposes an adaptive two-stage
extended Kalman filter (ATEKF) for estimation of
unknown fault bias in an INS-GPS loosely coupled
system. The Kalman filtering technique requires
complete specifications of both dynamical and statistical
model parameters of the system. However, in a number of
practical situations, these models may contain
parameters, which may deviate from their nominal values
by unknown random bias. This unknown random bias
may seriously degrade the performance of the filter or
cause a divergence of the filter. The two-stage extended
Kalman filter (TEKF), which considers this problem in
nonlinear system, has received considerable attention for
a long time. The TEKF suggested until now assumes that
the information of a random bias is known. But the
information of a random bias is unknown or partially
known in general. To solve this problem, this paper
firstly proposes a new adaptive fading extended Kalman
filter (AFEKF) that can be used for nonlinear system with
incomplete information. Secondly, it proposes the
ATEKF that can estimate unknown random bias by using
the AFEKF. The proposed ATEKF is more effective than
the TEKF for the estimation of the unknown random bias.
The ATEKF is applied to the INS-GPS loosely coupled
system with unknown fault bias.
Keywords. adaptive two-stage extended Kalman filter;
covariance rescaling; unknown fault bias
1 Introduction
The well-known Kalman filtering has been widely used
in many industrial areas. This Kalman filtering technique
requires complete specifications of both dynamical and
statistical model parameters of the system. However, in a
number of practical situations, these models contain
parameters, which may deviate from their nominal values
by unknown constant or unknown random bias. These
unknown biases may seriously degrade the performance
of the filter or even generate the divergence of the filter.
To solve this problem, a new procedure for estimating the
dynamic states of a linear stochastic system in the
presence of unknown constant bias vector was suggested
by Friedland (Friedland, 1969). This filter is called the
two-stage Kalman filter (TKF). Many researchers have
contributed to this problem. Recently, the TKF to
consider not only a constant bias but also a random bias
has been suggested through several papers (Ignagni,
1990; Alouani, 1993; Keller, 1997; Hsieh, 1999; Ignagni,
2000).
Several researchers performed the extension of the TKF
to nonlinear system (Shreve, 1974; Mendel, 1976;
Caglayan, 1983). Unknown random bias of nonlinear
system may also generate the large problem in the TEKF
because the TEKF may be diverged if the initial estimates
are not sufficiently good. So the TEKF for nonlinear
system with a random bias has to assume that the
dynamic equation and the noise covariance of unknown
random bias are known although these are unknown in
general. To solve these problems, an adaptive filter can
be used. This paper proposes an adaptive two-stage
extended Kalman filter (ATEKF) by using an adaptive
fading extended Kalman filter (AFEKF). As a result,
unknown random bias can be estimated by the ATEKF.

Kim et al.: Adaptive Two-Stage EKF for INS-GPS Loosely Coupled System with Unknown Fault Bias 63
To verify the performance of the ATEKF, the proposed
ATEKF is applied to the INS-GPS loosely coupled
navigation system with unknown fault bias. In general,
the INS-GPS loosely coupled system uses the EKF,
which cannot effectively track time-varying parameters
(Ljung, 1979). So when the navigation system has
unknown fault bias, the EKF cannot estimate unknown
fault bias and cannot eliminate this. This paper shows that
the ATEKF can estimate and eliminate unknown fault
bias in the INS-GPS loosely coupled system with
unknown fault bias.
2 Problem Formulation
Consider the following nonlinear discrete-time stochastic
system represented by
()
1,x kkkkk
x
fxb w
+=+ (1a)
b
kkkk
bAbw=+ (1b)
()
,
kkkk k
zhxbv=+ (1c)
where, k
x
is the 1n× state vector, k z is the 1m
×
measurement vector and k b is the 1p× bias vector of
unknown magnitude. All matrices have the appropriate
dimensions. The noise sequence x
k
w, b
k
w and k v are zero
mean uncorrelated Gaussian random sequences with
00
00
00
T
xx x
kk k
bb b
kkk kj
kk k
ww Q
Ew wQ
vv R
δ
⎡⎤
⎡⎤⎡⎤ ⎡⎤
⎢⎥
⎢⎥⎢⎥ ⎢⎥
=
⎢⎥
⎢⎥⎢⎥ ⎢⎥
⎢⎥
⎢⎥⎢⎥ ⎢⎥
⎣⎦⎣⎦ ⎣⎦
⎢⎥
⎣⎦
(1d)
where 0, 0, 0
xb
kkk
QQ R>>> and kj
δ
is the Kronecker
delta. The initial states 0
x
and 0 b are assumed to be
uncorrelated with the white noise processes x
k
w, b
k
w and
k
v. Assume that 0
x
and 0 b are Gaussian random
variables with
[
]
*
00
Ex x=,
()()
**
0000 0
0
Tx
Ex xx xP
⎡⎤
−−=>
⎣⎦
,
[
]
*
00
Eb b=,
()()
**
00 000
0
Tb
Eb bb bP
⎡⎤
−−=>
⎣⎦
and
()()
**
0000 0
Txb
Exx bbP
⎡⎤
−−=
⎣⎦
The function
()
,
kkk
f
xb and
()
,
kkk
hxb can be expanded
by Taylor series expansion as
(
)
(
)
(
)
(
)
()() () () ()
() () () () ()
()() ()
,,
,
,
,,
kkk kkk
kkkkk
kkkk k
fkkk
fxb fxb
Fxbx x
Bxbb b
xx b ϕ
=++
+
++ −+
+
++−+
+++
)
))
)
)
(2)
(
)
(
)
(
)
(
)
() () () () ()
() () () () ()
()() ()
,,
,
,
,,
kkkkkk
kkkk k
kkkkk
hkk k
hxb hxb
Hxbx x
Dxbbb
xx b ϕ
=−−
+
−−−−
+
−−−−
+−−
)
))
)
)
(3)
where
(
)
k
x
⋅
)
and
(
)
k
b
⋅
are the state estimate and the bias
estimate respectively,
f
ϕ
and h
ϕ
mean the higher order
terms and
()() ()
()
()
,
k
k
k
kk kxx
bb
f
Fx bx
=
+
=
+
∂
++=
∂)
)
,
()() ()
()
()
,
k
k
k
kk kxx
bb
f
Bx bb
=
+
=
+
∂
++=
∂)
)
,
() () ()
()
()
,
k
k
k
kk kxx
bb
h
Hx bx
=
−
=
−
∂
−−=
∂)
),
() () ()
()
()
,
k
k
k
kk kxx
bb
h
Dx bb
=
−
=
−
∂
−−=
∂)
)
.
The problem is to design an adaptive two-stage extended
Kalman filter (ATEKF) to give a solution for nonlinear
stochastic system with a random bias on the assumption
that the stochastic information of a random bias is
unknown or partially known.
3 Two-stage Extended Kalman Filter in the Presence
of a Random Bias
In this section, we extend the optimal TKF to nonlinear
stochastic system models in order to obtain a suboptimal
TEKF. Until now, the several researchers have proposed
the TEKF (Mendel, 1976; Caglayan, 1983). But the
proposed TEKF was based on the suboptimal TKF.
Recently, several researchers have proposed the optimal
TKF for linear stochastic system with unknown random
bias (Keller, 1997; Hsieh, 1999; Ignagni, 2000). Thus we
newly propose the TEKF based on the optimal TKF as
Lemma 3.1.
Lemma 3.1: A discrete-time two-stage extended Kalman
filter is given by the following coupled difference

64 Journal of Global Positioning Systems
equations when the information of nonlinear stochastic
system (1) is perfectly known.
()()() () () ()
11
,
kk kkkk
xxUxbb
−−
−= −+++−
)
) (4a)
() ()()
(
) () ()
,
kkkkkk
xxVxbb+= ++−−+
)
) (4b)
()()() ()
()
()
() () ()
11
11
,
,
xx b
kk kkkk
T
kk k
PPUxbP
Ux b
−−
−−
−= −+++−
++
)
) (5a)
()()() () () ()
() () ()
,
,
xx b
kkkkkk
T
kk k
PPVxbP
Vx b
+= ++−−+
−−
)
) (5b)
where k
A
and b
k
Q are known. The two-stage extended
Kalman filter can be decomposed into two filters such as
the modified bias free filter and the bias filter. The
modified bias free filter, which is designed on the
assumption that the biases are identically zero, gives the
state estimate
()
k
x⋅. On the other hand, the bias filter
gives the bias estimate
(
)
k
b⋅. Finally the corrected state
estimate
()
k
x⋅
)
of the TEKF is obtained from the
estimates of two filters and the coupling equation k U and
k
V. The modified bias free filter is
()() ()
()
()
11 111
,
kkkkkkk
x
Fx bxCu
−− −−−
−=+++++
)
(6a)
()() () () ()
() ()
()
11 11
11 11
,
,
xx
kkkkk
Tx
kk kk
PFxbP
Fx bQ
−− −−
−− −−
−=+ ++
+++
)
) (6b)
() ()
()
()() ()
()
() ()
()
()() ()
()
1
,,
,,
xxT
kk kkkk k
xT
kkkkk kkk
Kx bPHx b
Hx bPHx bR
−
−−=− −−
⎡⎤
−− −−−+
⎣⎦
)
)
))
(6c)
() () () (
)
() () (
)
()
,,
xx
kkkkkkk
x
k
PIKxbHxb
P
⎡⎤
+=−− −−−
⎣⎦
−
)
)
(6d)
() () () ()
,
x
kk kkkkk
zHx bxE η =−−−−−
)
(6e)
() ()
xx
kk kk
xxK
η
+= −+ (6f)
and the bias filter is
() ()
11kkk
bAb
−−
−= + (7a)
() ()
111 1
bbTb
kkkkk
PAPAQ
−−− −
−=++ (7b)
() () () ()
() () ()
()() ()
()
()
,
,,
bbT
kk kkk
xT
kk kkkk k
bT
kkk k
KxbP N
Hx bPHx b
RNP N
−−=−
⎡⎤
−−−−−
⎢⎥
⎢⎥
++ −
⎣⎦
)
))
(7c)
() () ()
(
) ()
,
bb b
kkkkkk
PIKxbNP
⎡⎤
+
=−− −−
⎣⎦
)
(7d)
(
)
1
bx
kk kk
Nb ηη
−
=
−− (7e)
(
)
(
)
1
bb
kk kk
bb K
η
−
+= −+ (7f)
with the coupling equations
()()
(
) () () ()
()() ()
11
,,
,
kkk kkkk
kk k
NHxbUxb
Dxb
−−
=
−−+ +
+−−
)
)
) (8a)
()()
(
)
()
1
11 1
,bb
kkkkk k
UxbUIQ P−
−− −
⎡
⎤
⎡⎤
++=− −
⎣⎦
⎣
⎦
) (8b)
()()
(
)
()()
(
)
() () ()
11 111 11
1
11 1
,,
,
kk kkk k
k k
kk k
Fxb Vxb
UA
Bxb
−− −−− −
−
−
−− −
⎛⎞
++ −−
⎜⎟
=⎜⎟
+++
⎝⎠
)
)
)(8c)
() ()
(
)
() ()
(
)
() () ()
11
,,
,
kk kkkk
x
kk kk
Vx bUxb
K
xbN
−−
−−=+ +
−−−
)
)
) (8d)
()()
(
)
(
)
()
11 , kkkk kkk
uUUxbAb
++
=
−+++
)
(8e)
11
xx b
kkkkk
QQUQU
+
+
=+ (8f)
Here, the initial conditions:
(
)
**
0000
x
xVb+= − ,
(
)
*
00
bb
+
= (9a)
(
)
00000
xxbT
PPVPV+= − ,
()
00
bb
PP+= (9b)
where
(
)
1
000
xb b
VPP
−
=.
The equations of the TEKF are similar to the equations of
the optimal TKF. But although the structure of this TEKF
is possible for nonlinear stochastic system, the coupling
between the modified bias free filter and the bias filter in
the nonlinear case exists and all linearizations about the
state and bias estimate have to be evaluated. Thus a
designer has to give attention to the linearization point
when the TEKF is used.
In general, the EKF may give biased estimates and be
diverged if the initial estimates are not sufficiently good.
So the incomplete information of nonlinear system may
generate the large problem in the EKF. The TEKF has
also the same problem. Thus the TEKF of Lemma 3.1
assumes that k
A
and b
k
Q are known.
However, in most case, these are unknown. If this
information is incomplete, the performance of the TEKF
may be degraded or the TEKF may be diverged. To solve
this problem, the TEKF has to be adapted to environment
of incomplete random bias information. Thus an adaptive
two-stage extended Kalman filter is needed

Kim et al.: Adaptive Two-Stage EKF for INS-GPS Loosely Coupled System with Unknown Fault Bias 65
4 Adaptive Two-stage Extended Kalman Filter in the
Presence of a Random Bias
4.1 Adaptive Fading Extended Kalman Filter using
Innovation Covariance
Consider the following nonlinear discrete-time stochastic
system represented by
()
11,kkk kk
x
fxw
+=+∆+
(10a)
()
2,kkk kk
zhx v=+∆+
(10b)
where k
x
is the 1n× state vector and k z is the 1m
×
measurement vector. Moreover k
w and k v denote
sequences of uncorrelated Gaussian random vectors with
zero means. Each covariance matrix is
, , 0
TT T
kjkkjkjkkjkj
EwwQEvvREwv δδ ⎡⎤⎡⎤⎡⎤== =
⎣⎦⎣⎦⎣⎦
(11)
where kj
δ
is the Kronecker delta. The variable 1,k
∆
means the uncertainty term in the dynamic equation that
is generated by incomplete process noise covariance,
unknown input bias or incomplete model coefficients. On
the other hand, the variable 2,k
∆
means the uncertainty
term in the measurement equation that is generated by
incomplete measurement noise covariance or unknown
measurement bias. Also 1,k ∆ and 2,k ∆ are independent of
k
w, k v and k
x
respectively. If the system information is
perfectly known, 1,k ∆ and 2,k ∆ are all zero.
The function
()
kk
f
x and
(
)
kk
hx can be expanded by
Taylor series expansion as
() () () () ()
(
) ()
,
kkkkkkkf kk
fxfxFx xxx ϕ =++−++ +
)
))
(12)
() ()
()
()
()
(
)
()
,
kkkkkk khkk
hxhxH xxxx ϕ =−+−−+ −
)
))
(13)
where
()
k
k
k
xx
f
Fx=+
∂
=∂) ,
()
k
k
k
xx
h
Hx=−
∂
=∂) . The variable
f
ϕ
and h
ϕ
mean the higher order terms.
Definition 4.1: Several equations related to the
innovation are arranged as follows.
() ()
kkkk
zhx η =− −
)
(14a)
()
TT
kkkkkkk
CEHPHR ηη ==−+ ⎡⎤
⎣⎦ (14b)
1
1
1
k
T
kii
ikM
CM
ηη
=− +
=−∑ (14c)
where k
η
is the innovation of filter, k C is the calculated
innovation covariance and k C is the estimated innovation
covariance. Here
M
is a window size.
(
)
k
P
−
is a
calculated error covariance of the EKF.
The equations of the EKF are similar to those of the
linear Kalman filter. Thus first we derive an optimal
property from the linear Kalman filter and secondly
expand this result in terms of nonlinear system. One of
the important properties of the optimal linear Kalman
filter is that the innovation is a white sequence when the
optimal filtering gain is used. Although the EKF is a
suboptimal filter, this property can be used to improve the
performance of the EKF in nonlinear system (10). The
following equation (15) is the auto-covariance of the
innovation in the EKF:
[]
()
111
111
, 1,2,3,
T
kj kkjkjkjkj
T
kkkkkkkk
EHFIKH
F
IKH FPHKC
j
ηη
++ +−+−+−
+++
⎡⎤ ⎡⎤=−
⎣⎦ ⎣⎦
−−− ⎡⎤
⎣⎦
∀=
L
L
(15)
(
)
T
kkk kk
SP HKC=− − (16)
The white sequence has to satisfy a condition that the
auto-covariance is zero because whiteness means
uncorrelated in time. If the common term (16) of the
auto-covariance (15) for all 1, 2, 3,j=L, is zero, then the
innovation sequence is a white sequence. If we insert the
error covariance and the Kalman gain of the EKF into
equation (15), the auto-covariance of the innovation is
identically zero since k
S is zero for all 1, 2, 3,j
=
L.
Actually, the Kalman gain of the EKF is easily obtained
when 0
k
S
=
in the equation (16). In case of the linear
Kalman filter, if the auto-covariance equation is zero, we
can say that the filter is optimal. But, in case of the EKF,
the real innovation sequence is not a white sequence due
to higher order terms to be neglected and the linearization
error. Nonetheless if the auto-covariance equation (15) of
the innovation in the EKF is zero, then the performance
of the EKF will be improved. When the system model is
incomplete, the real covariance of the innovation is
different from a theoretical one given in equation (14b).
Hence the real auto-covariance of the innovation may not
be identically zero although 0
k
S=. Under this
background, this section newly proposes a new AFEKF.
Definition 4.2: For a design of a new AFEKF, let’s
define several variables. Assume that the system
information is incomplete. From now,
(
)
k
P−, k
K
and
k
C are defined as an error covariance, a Kalman gain and
an innovation covariance for system with incomplete
information. Here, k C is calculated from the model
equation of system with incomplete information. Thus
k
C is also called as the calculated innovation covariance.
On the other hand,
(
)
k
P
−
, k
K
and k C are defined as an
error covariance, an optimal Kalman gain and an

66 Journal of Global Positioning Systems
innovation covariance for system with complete
information. We have to obtain these
()
k
P−, k
K
and k C
by the appropriate method because the information of
system to be considered is incomplete. Here, k C can be
obtained from (14c). Thus k C is also called as the
estimated innovation covariance. We assume
() ()
kkk
PP λ −=− where k
λ
is a forgetting factor. If
()
k
P− can be obtained byk
λ
, then the Kalman gain k
K
can be obtained from
(
)
k
P−.
Let’s think the case that the information of system is
unknown or partially known. If
(
)
k
P− can be obtained
by using the forgetting factor and k C can be exactly
estimated by (14c), then the Kalman gain k
K
makes that
0
k
S=, where
()
T
kkkkk
SP HKC=− − (17)
However, in many cases, the difficulty is to get
(
)
k
P
−
and k
K
. Therefore the filter is suboptimal for system
with incomplete information in general. As a result, the
problem is to seek how to exactly obtain
()
k
P− and k
K
.
Definition 4.3: Scalar variable k
α
is defined by
()
()
()
1
1
max1 ,
or max1 ,
kkk
k
k
k
traceC C
m
trace C
trace C
α
α
−
⎧⎫
=⎨⎬
⎩⎭
⎧⎫
⎪⎪
=⎨⎬
⎪⎪
⎩⎭
(18)
where the AFEKF uses the relationship kkk CC α =.
In general, the innovation covariance shows the effect of
the present error because the innovation is easily affected
by the error. Let’s think for system with incomplete
dynamic equation. The real innovation covariance of the
present is estimated as k C. Then k C is represented by
() ()
TT
kkkkkk kkkkk k
CCHP HRHP HR αλ == −+=−+ (19)
The increase of the innovation covariance by k
α
is due to
the increase of the error covariance by k
λ
because the
system has incomplete dynamic equation. On the other
hand k R remains unchanged. If the increased error
covariance is larger than k R, we can think that k
α
is
almost equal to k
λ
.
Remark 4.1: Sometimes, we only partially know the
dynamic equation of nonlinear stochastic system. Then,
the estimation error may be increased by the effect of
unknown information. This means that the error increase
is due to incomplete process noise covariance, unknown
input bias or incomplete model coefficients. The effects
of incomplete information in dynamic equation can be
compensated through the increase of the magnitude of
(
)
k
P
−
. If we assume the relationship kk
λ
α =, then the
error covariance is
(
)
(
)
kkk
PP α
−
=−.
Definition 4.4: A discrete-time adaptive fading extended
Kalman filter is given by the following coupled
difference equations when the information of nonlinear
stochastic system (10) is partially known.
(
)
(
)
(
)
11kkk
xfx
−−
−
=+
)
) (20a)
(
)
(
)
111 1
T
kkkkkk
PFPFQ λ
−−− −
−=+ +
⎡
⎤
⎣
⎦ (20b)
() ()
1
TT
kk kkkkk
KP HHP HR
−
=−−+
⎡
⎤
⎣
⎦ (20c)
(
)
(
)
(
)
kkkk
PIKHP
+
=− − (20d)
(
)
(
)
(
)
(
)
kkkk kk
xxKzhx
⎡
⎤
+
=−+−−
⎣
⎦
)
))
(20e)
where kk
λ
α
=
.
4.2 Adaptive Two-stage Extended Kalman Filter using
the AFEKF
The ATEKF can be easily designed by the proposed
AFEKF. This ATEKF is used when the information of
k
A
and b
k
Q are incomplete. For a compensation of the
effects of incomplete information in the bias filter of the
TEKF, the calculated innovation covariance and the
estimated innovation covariance are defined by (21b) and
(21c).
Definition 4.5: Several equations related to the
innovation are arranged as follows.
(
)
1
bx
kk kk
Nb ηη
−
=
−−
(21a)
() ()
(
)
()() ()
(
)
()
,,
bxT
kkkk kkkk
bT
kkk k
CHxbP Hxb
RNPN
=
−− −−−
++ −
)
)
(21b)
1
1
1
kT
bbb
kii
ikM
CM
ηη
=− +
=−∑ (21c)
The b
k
α
is equal to the forgetting factor b
k
λ
where
bbb
kkk
CC α =. By the forgetting factor calculated from
(21b) and (21c), the state error covariance equation (7b)
is changed into
(
)
(
)
111 1
bbbTb
kkkkkk
PAPAQ λ
−−− −
⎡
⎤−=+ +
⎣
⎦ (22)

Kim et al.: Adaptive Two-Stage EKF for INS-GPS Loosely Coupled System with Unknown Fault Bias 67
The modified bias free filter of the TEKF has k u and x
k
Q.
For convenience, the equation of (8e) and (8f) are
rewritten as (23a) and (23b). In (23a), k u is related to the
incomplete k
A
and b
k
Q. In (23b), x
k
Q is also related to the
incomplete b
k
Q.
()()
()
()
()
()
()
()
() ()
11
1
11 1
1
11
,
kkkk kkk
bb
kk kkkk
bb
kkk kk
uUUxbAb
UUIQP Ab
UQ PAb
++
−
++ +
−
++
=−+++
⎡⎤
⎡⎤=− −−+
⎣⎦
⎣⎦
⎡⎤ =−+
⎣⎦
)
(23a)
11
xx b
kkkkk
QQUQU
++
=+ (23b)
For a compensation of the effects of incomplete
information in the modified bias free filter of the TEKF,
the each innovation covariance is defined by (24b) and
(24c).
Definition 4.6: Several equations related to the
innovation are arranged as follows.
() ()
()
()
,
x
kk kkkkk
zHx bxE η =−−−−−
)
(24a)
() ()
()
()() ()
()
,,
xxT
kkkk kkkkk
CHxbPHxb R=−−− −−+
)
)
(24b)
1
1
1
kT
xxx
kii
ikM
CM
ηη
=− +
=−∑ (24c)
The x
k
α
is equal to the forgetting factor x
k
λ
where
xxx
kkk
CC α =. By the forgetting factor calculated from
(24b) and (24c), the state error covariance equation (6b)
is changed into
() () ()
()
()
() ()
()
11 11
11 11
,
,
x
kk kk
xx
kk
Tx
kk kk
Fx bP
P
Fx bQ
λ
−− −−
−− −−
⎡⎤
+++
⎢⎥
−= ⎢⎥
+++
⎣⎦
)
) (25)
The proposed ATEKF is applied to the INS-GPS loosely
coupled system with unknown fault bias in next section.
5 Application to the INS-GPS Loosely Coupled
System with Fault Bias
5.1 INS-GPS Loosely Coupled System
The dynamic model of the INS-GPS loosely coupled
system can be formed from the differential equations of
the navigation error. The time varying stochastic system
model is
()()()()()()
INSLCLC LC
x
tFtxtGtbtGtw(t)=++
& (26)
[]
T
NEDNED
xLlhVVV
δ
δδ δδδ
φφφ
= (27)
[]
T
LCaXaY aZ gXgYgZ
wwwwwww= (28)
()()() INS GPS
LC LC
INS GPS
PP
ztH xtvtVV
⎡
⎤⎡⎤
=+=−
⎢
⎥⎢⎥
⎣
⎦⎣⎦
(29)
13
13
13
13
13
13
1000000
0100000
0010000
0001000
0000100
0000010
LC
H
×
×
×
×
×
×
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
(30)
where the error state variable ()
x
t represents the
position, velocity and attitude errors of vehicle, the bias
term ()bt represents the inertial sensor bias errors, which
consist of the accelerometer bias error and the gyro bias
error,
(
)
~0,
LC LC
wNQis white noise of the inertial
sensor and
(
)
~0,
LC LC
vNR is white noise of the GPS.
(Hong, 2004; Titterton, 1997)
Remark 5.1: The accelerometer error consists of white
Gaussian noise and the accelerometer bias error, which is
assumed as a random constant. The gyroscope error
consists of white Gaussian noise and the gyroscope bias
error, which is assumed as a random constant.
To analyze the performance of the INS-GPS loosely
coupled system, experiments were carried on the campus
of the Seoul National University. The Differential GPS
(DGPS) trajectory is used as a reference trajectory. The
IMU (LP-81) and the GPS receiver (CMC Allstar) were
mounted on the test vehicle. LP-81 provides raw IMU
measurements at 100Hz, while CMC Allstar provides
GPS data at 1Hz. The specifications for LP-81 and CMC
Allstar are shown respectively in Table 5.1 and Table 5.2.
The initial alignment time is 310 sec and the total
navigation time is 1490 sec.
5.2 Fault Bias Estimation using the ATEKF
To verify the performance of the ATEKF for the INS-
GPS loosely coupled system with unknown fault bias, we
consider an accelerometer fault. We insert a fault bias
into data obtained from experiment of section 5.1.

68 Journal of Global Positioning Systems
Table 5.1 The specifications of LP-81
Sensor Accel.
(
)
1
σ
Gyro.
()
1
σ
Bias repeatability 500 g
µ
3 deg/h
Scale Factor 500
p
pm 500
p
pm
Misalignment 60 arcsec 60 arcsec
Noise 0.02 ft/sec 5 arcsec
Correlated Noise 40 g
µ
0.1 deg/h
Table 5.2 The specifications of CMC Allstar
Parameter Allatar ()
1
σ
Receive frequency L1 : 1575.42
M
Hz
Tracking code C/A code (SPS)
Position accuracy 16 CEPm
Velocity accuracy 0.2 ms
Table 5.3 The fault of the Accelerometer
Fault position Magnitude of fault bias Time
Accel. X-axis 10 mg, 30 mg, 50 mg, 100 mg 800 sec
As shown in Table 5.3, we insert fault biases of 10 mg,
30 mg, 50 mg and 100 mg into X-axis of an
accelerometer. The fault occurrence time is 800 sec.
When the accelerometer sensor has a fault, the position
error of the ATEKF is smaller than that of the TEKF. Fig
5.1 shows the bias estimation results of the TEKF and the
ATEKF for several accelerometer fault biases. The
tracking performance of the ATEKF is better than that of
the TEKF. The ATEKF quickly tracks unknown fault
bias. The estimate of fault bias is used for compensation
of the sensor fault, so the navigation error in the ATEKF
is more decreased than the TEKF. Finally the result of the
ATEKF for jumping fault bias is shown in Fig 5.2.
Totally, the ATEKF can effectively track unknown fault
bias.
0500 1000 1500
-2
0
2
4
6
8
10
12 x 104
TEKF : ACC Bias Estimate
Fault Signal
Time(sec)
100mg
50mg
30mg
10mg
(a) Bias estimation (TEKF)
0500 1000 1500
-2
0
2
4
6
8
10
12 x 10
4
ATEKF : ACC Bias Estimate
Fault Signal
Time(sec)
100mg
50mg
30mg
10mg
(b) Bias estimation (ATEKF)
Fig 5.1 The bias estimation results of the TEKF and the ATEKF for
several accelerometer fault biases
0500 1000 1500
-2
-1
0
1
2
3
4
5
6x 104
ATEKF : ACC Bias Estimate
Fault Signal
Time(sec)
50mg
10mg
Fig 5.2 The results of bias estimation in the ATEKF for jumping fault
bias
6 Conclusion
The problem of unknown random bias frequently
happens in a number of practical situations. The two-
stage Kalman filtering technique considers this problem.
The information of unknown random bias is unknown in
general. However until now, a designer assumed that this
information is known when two-stage extended Kalman
filter is designed. So, to solve this problem, this paper
proposes the adaptive two-stage extended Kalman filter
for nonlinear stochastic system with unknown random
bias. To design the adaptive two-stage extended Kalman
filter, the adaptive fading extended Kalman filter is firstly
designed. This AFEKF can be used for nonlinear
stochastic system when the system information is
partially known. The AFEKF can be applied to nonlinear
stochastic system with unknown random bias. But the
AFEKF can only estimate true state, so this cannot
eliminate unknown random bias. On the other hand, the

Kim et al.: Adaptive Two-Stage EKF for INS-GPS Loosely Coupled System with Unknown Fault Bias 69
ATEKF can estimate true state and unknown random bias
together, so this can eliminate unknown random bias.
The proposed ATEKF can be applied to the problem
related a fault compensation. The reliability of vehicle or
aircraft depends on integrated navigation system like the
INS-GPS loosely coupled system. Thus the fault
tolerance is very important in integrated navigation
system. The EKF is widely used for integrated navigation
system, but the EKF cannot effectively track time varying
parameters or unknown parameter. The TEKF has also
this problem. This paper applies the ATEKF to the INS-
GPS loosely coupled system with unknown fault bias.
Unknown fault biases in accelerometer can be effectively
eliminated by the ATEKF. As a result, this paper shows
that the performance of the ATEKF is better than that of
the TEKF in case of unknown fault bias.
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