The distribution of the detected position difference between top and bottom. Detected position is calculated with nominal VDC position. It is easily seen that relative shift for x and z directions are about 1 and 4 mm, respectively.
The distribution of the difference of the direction cos between top and bottom. Systematic difference exist for z, which may originate from layer position difference from the nominal one.
The procedure of the positionning is as follows:
1. Study the rotation of VDCbottom. Then, apply the inverse-rotation to the vector obtained by VDCbottom.
2. Remove the remaining shift of VDCbottom. Then, track vector from VDC bottom can be converted into E549 global coordinate expression.
The calculated distribution of Eular angle theta, which represents the rotation with respect to y axis. The calculation is exactly done assuming phi and psi being 0.
The correlation between z component of down direction vector and down-up difference of x component. In the linear approximation of the Eular rotation matrix, the slope gives the rotation angle theta.
The profile of the previous two-dimensional correlation. The vertical axis is enlarged. Then, rotation angle is about 5.4 mmrad.
The correlation and profile between x component of up direction vector and up-down difference of z component. In the linear approximation of the Eular rotation matrix, the slope also gives the rotation angle theta, and it is deduced to be about 4.9 mmrad.
The correlation and profile between y component of up direction vector and up-down difference of x component. In the linear approximation of the Eular rotation matrix, the slope gives the rotation angle phi+psi, and it is deduced to be about 3.1 mmrad.
Even within the linear approximation, theta and psi+phi rotation couples. We eliminated the theta rotation first, then serched for the best inverse psi+phi rotation, and found the best combination of theta (rotation angle around y axis) and pshi+phi (z). The result of search is found here as a ASCI file.Theta and psi+phi rotation were found to be 5.65 and 7.10 mrad, respectively.
The systematic offset of direction cos x and z AFTER the elimination of rotation.
The correlation and profile between x and z position (at y=0) up-down difference vs the fraction of x(z)/y components of detected direction cos. Note that this is done AFTER the elimination of rotation. This indicates the presence of x,y,z offset from the nominal ones. x,z offsets appear as the vertical-sections of the correlation, while the y the slope. Now, we conclude that x, y, and z offsets for VDCdown are 0.7, -0.7, and 4.0 mm, respectively.
Now, VDC bottom rotation and shift with respect to VDC top are completely known.