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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
If instead of a frame camera, either a pushbroom scanner or a
Lidar system are modeled, the only change necessary is in the
term. 7 . It will have the form for :
0
B A pushbroom scanner = a
o + 2 p
—d:sina
B A LIDAR scanner = 0 where,
—-d:cosa
d Raw laser range (distance) in laser frame
o Scanner angle from nadir along y-axis of
laser frame, see Figure 5
Figure 5: LIDAR Scanner Angle and Distance Measurement
[n addition, the misalignment matrix R c and the offset vectors
2 GPS : ; :
ES and r;ye have to be determined by calibration. A
detailed discussion of implementation aspects can be found in
El-Sheimy (1996) for land-vehicle applications and in Mostafa
and Schwarz (2000) for airborne applications.
Using Newton’s second law of motion in a rotating coordinate
frame, the inertial sensor measurements - specific force t? and
measured in the b-frame — can be
; b
angular velocity @; »
transformed into an Earth-fixed frame, say the Conventional
Terrestrial Coordinate Frame(e). The resulting system of
differential equations is of the form
i | v^
® | 7 |Rif*-2Q ve +g° | (2)
RC RS -0^)
b. . .
where e is the skew-symmetrical form of the angular
: b
velocity vector Qi, and the dot above a vector (bold lower
case) or a matrix (bold capital) indicates differentiation. Note
that the m-frame in the geo-referencing formula (1) has been
replaced by the e-frame. This system is integrated to yield the
parameters on the left-hand side, namely position, velocity, and
; ‘| e
the orthogonal rotation matrix Ry between the b-frame and
the e-frame. The determination of this time-variable matrix is
one of the central tasks of geo-referencing.
The integration of the system of differential equation is started
by initializing the parameters on the left-hand side of the
equation. Traditionally, this is done in stationary mode. In that
case, the initial velocity is zero, the initial position is obtained
from GPS, and the initial orientation matrix is determined by an
alignment procedure that makes use of accelerometer leveling
and gyro compassing. The alignment is usually done in two
steps. In the coarse alignment simplified formulas are used to
obtain pitch and roll from accelerometer measurements and
azimuth from gyro measurements, within an accuracy of a few
degrees. In the fine alignment small-angle error formulas are
used in a filtering scheme to obtain more accurate estimates of
; ; € ;
the parameters. The rotation matrix Ry can then be obtained
from the estimated pitch, roll, and azimuth or an equivalent
parameterization, as for instance quaternions. This is the
standard alignment procedure if sensor measurements from a
navigation-grade inertial system are available. For medium and
low accuracy systems, this method cannot be applied. Because
: : : b
of the unfavorable signal-to-noise ratio for Q;, the gyro
compassing procedure will not converge. Thus, stationary
alignment cannot be used with MEMS-based or other low-
accuracy IMUs.
The alternative is in-motion alignment. This method has mainly
been used in airborne applications, specifically for the in-air
alignment of inertial systems. It is obviously dependent on
using additional sensors, which in this case are GPS receiver
outputs, 1.e. position and/or velocity in the e-frame. In-motion
alignment makes use of the fact that very accurate position
information is available at a high data rate. It is therefore
possible to determine accurate local-level velocities. By
combining these velocities with the ones obtained from the
i ; e ;
strapdown IMU, the rotation matrix Rj can be determined.
When implementing this approach for medium or low accuracy
IMUs two difficulties have to be addressed. The first one is that
cither no azimuth information at all is available, or that it is
derived from GPS velocities and is rather inaccurate. Thus, the
standard small-angle error models cannot be used any more
because the azimuth error can be large. By reformulating the
velocity error equations, one can arrive at a set of equations that
converges quickly for azimuth, even if the initial errors are
large. Scherzinger (1994) has given a thorough discussion of
the problem and its solutions, based on earlier work by Benson
(1975). A land vehicle application, using a MEMS-IMU
integrated with GPS, is given in Shin and El-Sheimy (2004).
Convergence is fast in this case, taking only about 50 s. This
bodes well for airborne applications where, due to the higher
velocities, a better signal-to-noise ratio should further improve
the results.
The second difficulty is more fundamental. It has to do with the
way in which the non-linearity in equation system (2) and in the
corresponding GPS measurements are handled. The standard
approach is to expand the errors in position, velocity, and
orientation into a Taylor series and to truncate the series after
the linear term. The error equations obtained in this way are
then cast into state vector form. By adding the linearized GPS
measurements to the model and by representing the state
variable distribution by a Gaussian random variable, the
