2 Mathematical model for integrated sensor 
orientation 
For sensor calibration a common model of all three groups of 
observations (image coordinates of tie and ground control 
points (GCP), pre-processed data for the projection centre 
coordinates, pre-processed data for the angles of exterior 
orientation) needs to be set up. This model is the same as 
used in integrated sensor orientation, where all available 
information is processed simultaneously in order to obtain the 
highest accuracy. In this section, the model is outlined in 
some degree of detail. We finally turn to the description of 
the GPS network solution. 
2.1 Model for image coordinates 
The image coordinates are connected to the object space 
coordinate system via the well-known collinearity equations. 
The collinearity equations are based on an orthonormal 
coordinate system. Obviously, the national coordinate 
systems (e.g. UTM) do not correspond to this requirement. In 
traditional photogrammetry the effects of non-orthogonality 
are sometimes compensated by an earth curvature correction 
applied to the image coordinates. For a combined adjustment 
with image coordinates and direct observations of position 
and attitude this approach is not sufficient, because also the 
attitude observations need to be corrected, and different scale 
factors must be introduced for planimetry and height (see e.g. 
Jacobsen 2002; Ressl 2002). A more straightforward way is 
to transform all object space information into an orthonormal 
system a priori, e.g. a local tangential system. In the 
following we use such a local tangential system. Where 
necessary, this system is denoted by the symbol m. 
The system of non-linear observation equations reads 
; NX -kX0) 1 (Y=Yo)}+ 7, (2 - 40) 
Xy mx lf tL TEES 
n.CX — No)rr.O — Yo) + 1. (7 — Zp) 
f AO Er Yo tr (Zo) 
  
+ dx! 
(1) 
  
Yu = ph pit ; A 
X - Xo) en -Yo)rr.(Z-Zo) 
xX’, Y’, Vy, Vy image coordinates and related residuals 
XYZ object space coordinates for tie points and 
GCP 
Xo,Yo,Zo object space coordinates of projection centre 
i elements of rotation matrix R"; (0,0,K) 
between the image coordinate system and the 
object space coordinate system 
Kar Yo image coordinates of the principal point 
dx’,dy” corrections of the image coordinates 
f calibrated focal length 
In order to introduce stochastic properties for the coordinates 
of the GCP, they are introduced as unknowns into the 
adjustment, and direct observations for the control point 
coordinates are set up within the model. 
2.2 Position equations 
The second part of the integrated sensor orientation model 
deals with the pre-processed observations for the position of 
the projection centre. These observations are often given in 
some Earth-fixed reference system, e.g. WGS 84, and need to 
be transformed into the local tangential system prior to using 
them in integrated sensor orientation. The model is extended 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
by bias and drift parameters to describe linear systematic 
effects of position, as is sometimes also done in GPS- 
supported aerial triangulation (Jacobsen 1991; Ackermann 
1994). An additional parameter dt takes care of a possible 
time synchronisation offset between the instant of image 
exposure and the GPS/IMU time. 
rs Xo 
Y mue ute zy de dry 
Zn IMI y z J 
(2) 
Ce (b_Xo (d. Xe 
R' (t dt)(e.p,k)) dy" | - | b Yo| + (1+dt-to) d. Yo 
de pA v5. Zo C d Zo 
X, Y,Zicpsamuy Object space coordinates of IMU centre of 
Vx.Vy,VziGps/iuu) mass and related residuals 
t observation time of GPS/IMU 
dt synchronisation offset between GPS/IMU 
time and instant of camera exposure 
to reference epoch for drift computation 
Xo,Yo,Zo object space coordinates of projection 
centre, time dependent 
R'i(c, q,K) rotation matrix between the image 
coordinate system and the object space 
coordinate system, time dependent 
dx dy dz S MU ro components of offset vector 
between IMU centre and. projection centre, 
expressed in image coordinate system 
b XoYoZo bias in position (one parameter per strip, or 
one for the whole block) 
d XoYoZo drift in position (one parameter per strip, or 
one for the whole block) 
2.3 Attitude equations 
The third part of the model describes the pre-processed 
attitude observations. In principle, these observations 
describe the rotation between the IMU coordinate system (so 
called body system 5) and an inertial coordinate system. In 
strap down navigation, the inertial system is replaced by a 
local level system, the so called navigation system 7 (see e. g. 
Bäumker, Heimes 2002). The x-axis of the navigation system 
points northwards, the z-axis downwards along the local 
plumb line, the y-axis completes the right-handed system. 
Besides other corrections, the transformation from inertial to 
navigation system requires the knowledge of Earth rotation 
and gravity parameters, and is usually integrated into 
GPS/IMU pre-processing. 
Thus, the pre-processed attitude observations describe the 
rotation of ^ around n. In aerial applications, the body system 
is fixed to the aircraft. » and the related rotation angles roll, 
pitch, and yaw are defined according to the aviation norm 
ARINC 705 (ARINC 2001). 
Since the IMU measurements refer to the local plumb line, 
the navigation system is not an Earth-fixed system, but 
moves together with the IMU. A connection to the 
photogrammetric rotations is given via the Earth fixed system 
e. The instantaneous position of the aircraft is expressed in 
   
     
   
    
  
   
   
  
  
  
  
  
  
  
   
   
    
  
  
  
  
   
  
  
   
  
  
    
  
   
    
   
    
  
  
  
  
  
  
   
  
   
  
  
  
   
  
  
  
  
  
    
   
   
    
   
  
  
  
  
  
   
  
  
  
  
   
   
  
  
   
  
   
  
  
  
   
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