[Lucia, Enzo, Eleonora]
As a first step to perform the PI horizontal LVDT diagonalization we have updated the sensing matrix LVDT2EUL, as reported by Enzo in klog #6073
Here are reported some more details about the convention used and the matrix computation:
- x and y are the interferometer global coordinates.
- The HR surface faces +y and identifies the longitudinal direction L.
- Transverse direction T is 90° anticlockwise from L ( - x in the IFO coordinates).
- Positive Yaw motion (Y) is anticlockwise.
- Leg 1 of the IP is along the +x direction and angles are measured anticlockwise from it.
- The horizontal LVDs are numbered from H0 to H2 anticlockwise, staring form x axis and the positive direction of their axis is also taken anticlockwise.
- According to the CAD (and visual inspection) they are separated by 120° and the first one is place at an angle alpha = 22.5° from the x axis.
- NOTE that in the initial sensing matrix this angle was taken to be 30° since it was measured considering the center of the LVDT+COIL unit. While it is more accurate to consider the position of the only LVDT (see first attaches picture of entry #6073)
According to the convention above, the projection of any horizontal motion of the Preisolator (with components L, T, Y) on each horizontal LVDT is
H_n = L* cos (alpha + 2n*pi/3) + T *sin (alpha+ 2n*pi/3) + R_h*Y
where n goes from 0 to 2 and R_h is the distance from the LVDT to the PI center.
Therefore, the matrix S connecting EUL to LVDT as [H0 H1 H2] = S* [L T, Y] is
cos (alpha) sin(alpha) R_h
cos (alpha + 2pi/3) sin(alpha+ 2pi/3) R_h
cos (alpha + 4pi/3) sin(alpha+ 4pi/3) R_h
and the sensing matrix LVDT2EUL to be put in the model will be its inverse: S^-1
Finally, the sensing matrix LVDT2EUL we obtain is
0.6159 -0.5289 -0.0870
0.2551 0.4058 -0.6610
0.5612 0.5612 0.5612
(taking alpha = 22.5° and R_h = 0.5940)
Note that it is consistent with what Mark computed in its document T1807663-v1, if we take alpha = 30° instead of 22.5°.
0.6159 -0.5289 -0.0870
0.2551 0.4058 -0.6610
0.5612 0.5612 0.5612