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Fig. 4 Placements of reference bar on test volume; camera setup 
e : targeted points. 
4.1. Calibration of the relative orientation and scale for 
distance measurements 
When the system is set up, a moveable (hand-held) reference 
bar equipped with 3 LEDs is placed in 16 predetermined 
positions and orientations, enclosing the test measurement 
volume of 1x1x1 m° (see fig. 4). The distances between the 
target points (LEDs) are precalibrated with an accuracy of 
better than 5.0 microns (2 x sigma level). For each reference 
bar position, the observed sensor coordinates of the targets 
are recorded for each camera. Sensor coordinate observations 
are subsequently put into the setup adjustment, which is a 
free network bundle adjustment only constraining the given 
distances on the reference bar. 
A simple accuracy check follows the setup adjustment. For 
each reference bar position of the setup measurements, the 3D 
coordinates of the target points on the reference bar are 
calculated by intersection, employing the orientation 
parameters found in the setup adjustment. The distances 
between the intersected target points are compared with their 
nominal values. Based on the found differences (errors), a 
distance measurement accuracy can be estimated as RMSqE. 
For the test measurement volume (1x1x1 m3), this distance 
measurement accuracy is found to be better than 0.10 mm (2 
X sigma level). 
4.2 Volvo/Renalult test 
In April 1991 and March 1992, extensive accuracy 
evaluations were carried out by Volvo Car Corporation. The 
test procedure conforms to the relevant parts of the German 
VDI-VDE norm, (see GMA, 1986) for accuracy evaluation of 
CMM's. In the tests, a dual-camera system was set up to give a 
measurement volume of 1x1x1 m3 with a camera 
configuration approximately as showed in fig. 4. The most 
important results were (given as U95 i.e. 2 x sigma level): 
- Repeatability of Light Pen measurements better than: 
In the depth direction, Z : 0.08 mm 
In the X or Y direction : 0.02 mm 
- Diagonal bar length measurement. Distances along all 4 
spatial diagonals are compared to nominal values using a 
high precision step block gauge (see GMA, 1986). The Light 
Pen is used for the length measurements. Total bar length is 
1000 mm: 
Uncertainty of measurement (GMA, 1986) : 0.12 mm 
The repeatability numbers are mean values for the total 
measurement volume. Uncertainty of measurement is defined 
in (GMA, 1986), and includes both statistical spread of single 
measurements at both ends of the bar, as well as systematic 
errors. Early experiments with a modified Light Pen having 
more optimal geometry, indicates even better repeatability. 
The standard deviations on estimated distances that is found in 
the simulation studies, can not be directly compared to the 
experimental U95 uncertainty of measurements. The accuracy 
characteristics of MNS is described in more detail in 
(Pettersen, 1992). 
APPENDIX. Constrained relative orientation 
We will give the formulas for bundle adjustment assuming 
that the cameras of the stereo vision system retain their 
relative orientation when taking several sets of pictures. 
The collinearity conditions are (see e.g. Freyer, 1989): 
X-X Fdx mper ote re erroe 
(X-X9)5 v (Y Yo) a32* (Z- Zo) 833 
A(1) 
(X-X9) a2, (Y -Yo) a22* (Z- Z9) 325 
ya year SE EN 
(X-Xo)azı+(Y-Yo)az2+(Z-Zo)az3 
C, X Yo: principle distance and principle image point 
X,Y,Z: object point 
Xo; Yo; Yo; exposure station 
X,y: image point 
aj element of a rotation matrix R 
dx = -x'b,+y'b,+x"dr/r+p,(r2+2x"2)+2 p2 x'y" 
dy = y'b,+x"b,+y'dr/r+p,( r2+2y"?)+2 p,x"y” 
dr/rs f((r?-r?)f,(r*-r* f3(ré- r, 6) 
12 = x'24y" 
X - X-X 
Y. Yrs 
zero-radius 
b,,b,: affinity and lack of orthogonality 
f,,f,, f;: radial distortion 
decentering