localization is integrated with INS, like a ‘‘loosely-
coupled’’ model. The measurement equation is
Z
p
¼
ðL
I
L
C
ÞR
M
ðl
I
l
C
ÞR
N
cos L
¼
R
M
L þ V
N
R
N
cos Ll þ V
E
¼
R
M
R
N
cos L
l
L
þ
V
N
V
E
ð3Þ
where R
M
and R
N
are the curvature radius of the
reference ellipsoid in longitude and latitude circles of
the earth respectively; V
N
and V
E
are the localization
error of CNS in north and east directions respectively.
Improved tightly-coupled model for INS/CNS
integration
Without CNS’s iteration localization process in classic
framework, direct relationship between elevation
angle and INS error is established in tightly-coupled
INS/CNS integration. Given longitude l
I
and latitude
L
I
of vehicle by INS, the calculated value of elevation
angle h
I
satisfy
sin h
I
¼ sin L
I
sin
A
þ cos L
I
cos
A
cos l
I
þ t
G
ð4Þ
Considering the INS errors, so
L
I
¼ LþL , l
I
¼lþl ð5Þ
Substituting equation (5) into equation (2) and
considering Taylor expansion
sinh
I
¼ sinh þ l þ L ð6Þ
where the coefficients and are
¼cos
A
cos t
G
sin l cos L
cos
A
sin t
G
cos l cos L
¼ sin
A
cos L cos
A
cos t
G
cos l sin L
þ cos
A
sin t
G
sin l sin L
8
>
>
>
<
>
>
>
:
ð7Þ
Defining h ¼h
I
h and considering
sinh
I
¼ sinðh þ hÞsinh þ hcosh, then equation (6)
is transformed to
h ¼
sin h
I
sin h
cos h
¼
1
cos h
l þ LðÞ
ð8Þ
Equation (8) shows the error transfer from INS
error l and L to calculated elevation angle h
I
,
which is the basis of building tightly-coupled
INS/CNS filtering model.
The elevation angle observed by CNS can be
represented as
h
C
¼ h V
A
ð9Þ
where V
A
is the angle measurement error of star
sensor.
According to equations (9) and (8), the measure-
ment equation of single star is
Z
A
¼ h
I
h
C
¼ h þ V
A
¼
1
cosh
½
l
L
þV
A
ð10Þ
For synchronously observed multiple stars,
the simultaneous measurement equations of equa-
tion (10) are used
Z
A
¼
h
I1
h
C1
h
I2
h
C2
.
.
.
h
In
h
Cn
2
6
6
6
6
4
3
7
7
7
7
5
¼
1
=cosh
1
1
=cosh
1
2
=cosh
2
2
=cosh
2
.
.
.
.
.
.
n
=cosh
n
n
=cosh
n
2
6
6
6
6
4
3
7
7
7
7
5
l
L
þ
V
A1
V
A2
.
.
.
V
An
2
6
6
6
6
4
3
7
7
7
7
5
ð11Þ
Adaptive integrated algorithm with
weighted multi-stars observations
Scheme design
In this section, adaptive integrated algorithm is pro-
posed for evaluating of error levels and weighting of
multi-stars observations in INS/CNS integrated filter.
The designed system scheme is shown in Figure 2.
In Figure 2, INS measures the movement of the
vehicle by gyroscope and accelerometer, providing
preliminary solution of latitude and longitude by a
build-in resolution algorithm. CNS observes multiple
stars synchronously and measures their elevation
angles. Meanwhile, the corresponding declinations
and Greenwich hour angles of observed stars are
retrieved from a star ephemeris.
There are two paralleled measurements preprocess
route before INS/CNS fusion filter: One preprocess-
ing route is for preparing Z
k
and H
k
in tightly-
coupled INS/CNS model, which is corresponding to
current observed stars. The other preprocessing route
is for evaluating current error levels of stars’ observa-
tions, then prepares the qualities matrix Qua
k
for
weighting multi-stars observations. Finally, the eval-
uated qualities are used to tune the filter gain of each
star’s observation separately in tightly-coupled INS/
CNS integration filter.
System equations for tightly-coupled INS/CNS
integration
(1) System states equations
The navigation coordinate system is defined in the
ENU geography coordinate system. The INS
700 Proc IMechE Part G: J Aerospace Engineering 230(4)