The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008
884
signal due to sampling and quantizing of a continuous signal
with a finite word length conversion, (f) exponentially
correlated (Markov) noise characterized by an exponential
decaying function with a finite correlation time, and (g)
sinusoidal noise characterized by one or more distinct
frequencies (IEEE Std. 952-1997 and IEEE Std. 528-2001).
Generally, any combination of these processes can be present in
the data, and different noise terms may appear in different
regions of the time scale. In practical applications, the random
items above can be chosen selectively to establish the stochastic
error model. From the simplest scenario that only considered
bias instability (e. g. see Schwarz, et al., 1994) to moderately
complicated models that were augmented with scale factors and
axis misalignments (Grejner-Brzezinska, 2001; Cramer, 2001)
were used for the aerial photogrammetric applications. In this
paper, the random item for the bias in the gyroscope can be
considered as:
d - d b + d R + d m + w d
(i)
implemented in the KF. Inclusion of the navigation parameters
and other constant parameters (for example, the lever arm of the
GPS antenna relative to the INS navigation center) results in the
lineralized error dynamic equation of the KF given as a state-
vector-based linear differential Equation (9):
X/
>11
F\ 2
Fn
Fn
X
X/
Xf
0
F 22
0
0
x f
+
Wf
X œ
0
0
X
0
X co
1
X-
1
0
0
0
L
1
X
t**
1
1
1
(9)
Where denotes a 9-dimensional navigation error state sub
vector (3 for position, 3 for velocity and 3 for orientation), x f
denotes the accelerometer error state sub-vector (b h ), x w denotes
gyroscope error state sub-vector, x L denotes the lever arm, wy^
Wf, w (u and w L are noises. F n is standard INS navigation error
matrix, and
Where d denotes random bias, d b denotes bias instability, d R
denotes gyro rate random walk, d m denotes first-order Markov
process noise, w d denotes white noise that drives into the angle
random walk. A more detailed representation of Equation (1)
can be found in (Yi, 2007). The rates of d b , d R and d m are
expressed in Equation (2), (3) and (4) as:
d„= 0
(2)
d R — W dR
(3)
X = d m + W dm
(4)
Where w dR denotes white noise, a denotes the correlation time
of the process, w dm denotes white noise.
The random item for the bias in the accelerometer can be
written as:
=
Fx 3 =
0
>1
II
S3
■ a
0
<
o-
>
II
0
0
9x3
F
ea>j _
F -
’ r eco d
-K’
’ F £ (0 d - ’ F 22 — 0 3x3 ,
F 33 = 0 3x3 , F 44 is generally zero if the GPS is fixedly
mounted with respect to the IMU body; however, it can be
considered as a first order Markov process if the gimbal is used
for the attitude compensation of platform tilt in the aerial
photogrammetric applications. It should be mentioned that if the
noises Wf, w w are set as non-zero values, the Xf and x w will be
modeled as random walk d R and b R . In this case, the whole
structure of the KF will remain unchanged from the random
constant model, and just with different stochastic parameter
configurations.
b = b k +b R +b m +w b (5)
Where the meanings of the suffixes are the same as those in
Equation (1), and the rates of b b , b R and b m are expressed in
Equation (6), (7) and (8) as:
As mentioned above, the remaining error of the scale factor can
still be considered in the stochastic error model to improve the
navigation performance even if it has been calibrated by the
manufacturer in the factory. The scale factor error is typically
considered as a random constant (Yi, 2007) or random walk
(Feng, 1999). The random constant model is given here, and the
state equation of the system with 12 error states of the inertial
sensor of Equation (9) is modified to give Equation (10).
b m = A + W
m m
bm
(6)
(7)
(8)
F n
0
0
0
F'n Fn F H
F 21 0 0
0 F 33 0
0 0 F u
X„,c
w 0
rV£
rve
X f
+
Wf
x,.
vv
(O
Û)
- X L .
. W L .
(10)
Where w bR denotes white noise, |3 denotes the correlation time
of the process, w bm denotes white noise.
By ignoring d R , d m , b R and b m and consideration of only random
constants (d b and b h ), the 6-state error model can be