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