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ations
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| the
ystem
ed in
geographic coordinates (A; ¢;) taken from the GPS/IMU
positions. A rotation can then be set up from n to e, another
one to the local tangential system m centred at the geographic
coordinates (A,, ¢,) and used in the collinearity equations.
Additional rotations need to be introduced to take into
account the different axes direction of the navigation and the
local tangential system on the one hand, and the body and the
image coordinate system on the other hand. Finally, the
misalignment between the image coordinate system and the
body system must be included. Since the misalignment is
considered as constant, it must be expressed in the body
system rather than in the image coordinate system. As in the
case of the position equations the system is completed by bias
and drift parameters.
The non-linear observation equations for attitude are given by
An
roll Vroll
ll
pitch |. (t) Vpitch
yaw b
= NS n a
T [REOS 0)- R^ (o: 09): Ri (50,2)
0 er
Vyaw
R(t +d) (dc) Rl (m0, 0)- RD (Aroll, Apitch, Ayaw) ]
( b roll Y! d roli Y!
cb pitch | +(t+dt—to) d_ pitch
b yaw b d yaw b
(roll, pitch, yaw)", elements of rotation matrix between 5
YrolbVpitdu aw and n and related residuals
T function to extract a vector of rotation
angles out of the rotation matrix
œ,p,K angles of exterior orientation, time
dependent
Ais O; geographic coordinates of IMU at the
instant of exposure
Ado geographic coordinates of origin of local
tangential system
Aroll, Apitch, Ayaw angles of boresight misalignment
Re rotation matrix between Earth-fixed and
navigation system
Rw rotation matrix between navigation system
at the origin of the local tangential system
and Earth-fixed frame
RU rotation matrix between local tangential
system and fixed navigation system
R"™, rotation matrix between image coordinate
system and local tangential system
Rod rotation matrix between body system
altered by misalignment and image
coordinate system
RU rotation matrix of boresight misalignment
b roll, pitch, yaw bias in attitude (one parameter per strip or
one or the whole block)
d roll, pitch, yaw drift in attitude (one parameter per strip or
one or the whole block)
t, dt, t, see above
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
2.4 Least-squares adjustment
Equations (1) to (3) form the mathematical model of
integrated sensor orientation, which is also used to determine
the system calibration parameters necessary for direct sensor
orientation, as mentioned above. Different sets of calibration
parameters can be chosen. Besides the boresight
misalignment, the bias parameters of equation (2) are often
selected as calibration parameters. Depending on the
available data and the intended use of the calibration results,
also the interior orientation parameters of the camera, and the
synchronisation offset dt can be considered as unknowns.
Of course, it must be ensured that the selected parameters can
be computed in a numerically stable way from the provided
information. For instance, assuming a flat test field a
correction to the calibrated focal length and a bias parameter
in Z can only be determined if imagery from at least two
different flying heights is available, and a correction to the
principal point in flight direction together with a bias in flight
direction requires at least two strips flown in opposite
directions. Also, the synchronisation offset dt is highly
correlated with the correction for the principal point in flight
direction, a separation needs again two different flying
heights. Finally, in order to interpolate the corrected exterior
orientation parameters at the instant of image exposure after
having estimated dt at the end of each adjustment iteration,
high resolution information for position and attitude must be
available.
The unknowns of the approach are computed in a least-
squares adjustment, based on the principle
v'Pv -> minimum (4)
As usual, P is the weight matrix of the observations. It should
be noted that often no stochastic information is available for
the GPS/IMU measurements after the Kalman filtering if
commercial software is used for GPS/IMU pre-processing.
2.5 GPS network solution
We now address the GPS network solution. It is well known,
that the observation quality in differential GPS depends on
the length of the baseline (Seeber 2003). In other applications
GPS network solutions, i.e. solutions involving more than
one reference station, have found major interest and are
increasingly also used in practical applications. In
photogrammetry, such possibilities have not yet been
exploited.
Based on the described formulae, we can create a network
solution in two different ways: first, we can consider each
available reference station individually and set up equations
(2) and (3) for each image as often as there are reference
stations. An alternative is to combine the observations of the
different reference stations during GPS/IMU pre-processing,
and to subsequently set up equations (2) and (3) only once
per image. The first possibility has the advantage of
simplicity, since pre-processing can be done in the
conventional way. The second possibility has the advantage
of being able to analyse in detail the different high frequency
GPS/IMU observations, and to eliminate any error at an early
stage. Also, different sets of calibration parameters can be
taken into account for the different reference stations. Since
at present we do not have the possibility to do GPS/IMU pre-
processing, we have decided to use this first possibility for
our studies.
