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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
/ 
PERPENDICULAR 
TO GEOID 
SURFACE OF 
ELLIPSOID 
SURFACE 
OF GEOID 
7* CENTER OF THE EARTH 
CENTER OF THE ELLIPSOID 
Figure 1. Deflection of the vertical. 
> 
Inertial navigation follows Newton’s second law of motion 
defined in the inertial (nonrotating) frame (1): 
Ÿ=a+g(r) (1) 
z2 
g=8g +Ag and Ag=|-g,n (2) 
Ag 
where — X —the total acceleration vector 
the acceleration sensed by the accelerometer 
e| 8l 
I 
= the position vector 
g(x) = the total gravitational acceleration vector 
g, "the gravity model 
Ag - the difference between the actual gravity and 
the gravity model used 
£, 7 the nominal value of gravity 
& and 77 = north and east DOVs, respectively 
Ag = the gravity disturbance, which corresponds to 
the gravity error óg (equation 3) in inertial 
navigation, if only the normal gravity term is used 
for gravity compensation. 
Equation (3), expressing the navigation position errors to the 
first order due to errors in the system, is obtained by 
perturbing equation (1), i.e., by applying the differential 
operator, à. The solution of differential equation (3) provides 
expressions for the linearized error equations (Jekeli, 2001). 
y 
nd asm: (3) 
ox 
The primary observable provided by an accelerometer is the 
difference between kinematic inertial acceleration and mass 
gravitation; thus, errors in the observed accelerations are 
affected by errors in the gravity model used, translating to 
the sensor positioning errors, as seen in equation (3). These, 
in turn, translate into errors in the coordinates of objects and 
points extracted from the directly oriented imagery. if a 
GPS/INS system is used to support a camera or a LiDAR 
(Light Detection and Ranging) system. Several models, 
ranging from normal gravity to high-order spherical 
harmonic expansion, can be used to approximate the Earth's 
gravity field. Historically, normal gravity has been sufficient 
for inertial navigation, as already mentioned. However, 
modern mapping systems based on high-accuracy GPS/INS 
may require better representation of the Earth’s gravity in the 
navigation algorithm, especially during extended losses of 
GPS lock. 
The total error dynamics equation in matrix form is as 
follows (Jekeli, 2001): 
Ly =F" + Gn (4) 
dt 
where superscript # denotes the navigation frame 
& = vector of attitude, velocity and position errors 
u = vector of gyro, accelerometer and gravity 
errors, which can be estimated together in a 
GPS/INS filter (see, for example, Grejner- 
Brzezinska and Wang, 1998) 
F and G - free-inertial dynamics matrices of the 
system. 
A detailed analysis of (4) reveals coupling among the 
unknowns that, in general, may complicate the estimation 
process (see, Jekeli, 2001). The errors in DOVs enter directly 
into the horizontal velocity errors in linear combination with 
the attitude errors. This, generally speaking, makes the 
parameter separation difficult in the estimation procedure 
(Grejner-Brzezinska and Wang, 1998). Thus, using DOVs in 
gravity compensation, which introduce less tilt error, leads to 
less coupling of the horizontal accelerations into the vertical 
axis. Therefore, it can be expected that (high-accuracy) DOV 
compensation should decrease not only the positioning error, 
but also improve the attitude determination. 
2. PROCESSING STRATEGY AND TEST RESULTS 
2.1 Test data and processing software 
The GPS/INS system used in the analyses presented here is 
the OSU-developed AIMS™ system (see, for example, 
Grejner-Brzezinska and Wang 1998; Toth 1998). The 
positioning module of this system is based on a tight 
integration of dual frequency differential GPS carrier phases 
and raw velocity and angular rates provided by a medium- 
accuracy and high-reliability strapdown Litton LN-100 INS. 
LN-100 is based on Zero-lock™ Laser Gyro (ZLG™) and 
A-4 accelerometer triad (0.8 nmi/h CEP, gyro bias — 
0.003°/h, accelerometer bias — 25pgg) The land based 
GPS/INS data used in this study were collected on January 
31, 2001 near the OSU campus and the airborne data set was 
collected in Tucson, Arizona on May 6, 2002. The average 
DOVs along the land trajectory were about 6 arcsec (1) and 
below 0.5 arcsec (£); and 4-6 arcsec (1) and 3-4 arcsec (&) 
for the airborne test, with a sigma of | arcsec. Figure 2 
illustrates n together with the corresponding positioning 
error for the land-based test.