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\usepackage{graphicx}

\title{PIR/corrector+Echidna Engineering Summary}
\author{Masayuki Akiyama (Astronomical Institute, Tohoku University)}
\date{
\begin{itemize}
\item Ver.4.0 : 2008/09/10 : Updated with 2008/09 engineering results (200805eng/200809eng.tex). 
\item Ver.3.0 : 2008/09/04 : Updated with 2008/08 engineering results (200805eng/200808eng.tex)
\item Ver.2.1 : 2008/08/11 : Bug fixed.
\item Ver.2.0 : 2008/08/10 : Updated with 2008/06 engineering results (200805eng/200806eng.tex)
\item Ver.1.0 : 2008/06/05 : Updated 2007/12 report with 2008/01,2008/05 engineering results (200805eng/200805eng.tex)
\item Ver.-1.1 : 2008/01/18 : Bug fixed, the FPI coordinate of the distortion center
is updated to be (2.4mm, 4.0mm) after applying the SKY and SPINE camera offset (positioner/ECH\_test/200801eng.tex).
Updated result is consistent with the offset between the CMM center and the
rotator axis.
The target was M38 (not M37). 
\item Ver.-1.0 : 2008/01/17 : Previous report circulated AAO/Kyoto/Subaru. \\
\end{itemize}
This is a summary of the Echidna engineering observations for the telesocpe
model construction.
}

\begin{document}
\maketitle

\section{Converting the SKY CCD coordinate to the FPI XY coordinate}

The relation between the SKY camera coordinate and the FPI XY 
coordinates is determined by taking stars with moving SKY camera
in FPI X and Y directions. The determined relation between 
CCD camera coordinate (local) and the FPI XY coordinate (global) is
described with 
\begin{eqnarray}
X({\rm FPI}) &=& A \times X({\rm CCD}) + B \times Y({\rm CCD}) + C + {\rm FPIx}, \nonumber \\
Y({\rm FPI}) &=& D \times X({\rm CCD}) + E \times Y({\rm CCD}) + F + {\rm FPIy} \nonumber \\
 &C&= {\rm X\ correction\ for\ the\ offset\ between\ the\ SKY\ and\ SPINE\ cameras} \nonumber \\
 &F&+ {\rm Y\ correction\ for\ the\ offset\ between\ the\ SKY\ and\ SPINE\ cameras} \nonumber \\
\end{eqnarray}
(We use the SKY coordinate definition measured on the FITS images, 
i.e. (0,0) at the bottom left of the images.)

\begin{table}[tb]
\begin{center}
\begin{tabular}{cccccccc}
\hline
Date   &    A       &     B    &  C   &    D     &     E    &  F   & Note              \\ \hline
200712 & $-8.103$   & $+0.677$ & $+$3445 & $-0.668$ & $-8.268$ & $+$2876 & 175.3deg rotation   \\
200712 &            &          & $+$3556 &          &          & $+$2821 &                     \\ \hline \hline
200703 & \multicolumn{4}{c}{Sky-camera reinstall}   &          &         &                     \\
200805 & $-8.082$   & $-0.251$ & $+$3556 & $+0.388$ & $-8.673$ & $+$2641 &                     \\
200805 &            &          & $+$3550 &          &          & $+$2639 &                     \\ \hline \hline
200807 & \multicolumn{4}{c}{Sky-camera reinstall}   &          &         &                     \\
200808 & $-8.041$   & $+0.203$ & $+3575$ & $-0.195$ & $-8.364$ & $+2610$ & {\bf Current value} \\ \hline \hline
\end{tabular}
\caption{
The measured sky-camera to FPI coordinate converstion factors.
}
\end{center}
\end{table}

The XY offset between SKY and SPINE cameras are determined by aligning a
guide fibre to a bright star. Two measurements were performed with the same
guide fiber to different stars. There results are shown as parameters $C$ and $F$
in Table 1. Using these values as the offset between the SKY and SPINE cameras 
in relation (1), the coordinates measured with the SKY camera
can be converted to the spine positions measured with the SPINE camera.
(In order to align a star to guide spine No.168, we need to move
telescope by $+878.48^{\prime\prime}$, $-243.09^{\prime\prime}$
(or $+901.76^{\prime\prime}$, $-258.28^{\prime\prime}$). 

The FPI XY directions and the celestial coordinates NE directions are 
aligned well. The relation is summarized in Figure~\ref{FPI_coordinate}.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\linewidth]{FPI_coordinate.eps}
\end{center}
\caption{
The relation between the sky camera CCD coordinate, 
the FPI XY coordinate, and the cerestial coordinate.
It should be noted that the SKY camera Y coordinate is
flipped on the data and the Echidna GUI window.
}
\label{FPI_coordinate}
\end{figure}

\section{Rotator axis on the FPI coordinate}

\begin{table}
\begin{center}
\begin{tabular}{cccc}
\hline
Date     &   EL     &                  & Note  \\ \hline
20071208 & $\sim$80 & ($-702$, $-727$) &       \\ \hline\hline
 \multicolumn{3}{c}{Echidna reinstall} & \\ 
20080125 & 85       & ($-826$, $-712$) &       \\
20080125 & 27       & ($-826$, $-712$) & No significant shift from El=85deg. \\ \hline\hline
 \multicolumn{3}{c}{Echidna reinstall} & \\
20080513 & $\sim$80 & ($-754$, $-567$) &      \\ \hline
20080617 & $\sim$80 & ($-739$, $-577$) & This value used for 200806 data \\ 
20080617 &          & ($-666$, $-597$) &                     \\ \hline\hline
 \multicolumn{3}{c}{Echidna reinstall} & \\
20080814 & $\sim$80 & ($-699$, $-645$) & \\ \hline\hline
 \multicolumn{3}{c}{FAM tilt adjusted} & \\
20080910 & $\sim$80 & ($-459$, $-531$) & {\bf Current value} \\ \hline
\end{tabular}
\caption{
The measured FPI coordinates of the rotator axis.
The shifts of the FPI coordinates
reflect the shift of the Echidna against the rotator axis.
}
\end{center}
\end{table}


The axis of the rotator is measured by taking stars with the sky-camera with rotating the
instrument. The measured CCD coordinate were converted to FPI coordinate
using the relation (1). On 20080125, the rotation axis was measured
at EL=85deg and at EL=27deg (At EL=27deg, the position of the FPI was 
keeped at ($0,0$) referring the FPI encoder values, otherwise the 
recorded star track is not on a circle due to the flexure of the FPI gantry. 
The movement of the FPI was $\pm200\mu$m at most at EL=27deg).
No shift was observed. The difference of 
the FPI coordinate of the pointing center between 20071208 and 20080125 
measurements is ($-124, +15$) (in $\mu$m), which represents the accuracy 
of the installation of Echidna unit to PIR enclosure (we remove Echidna  
20080110 and re-install 20080117, it should be noted that this
is not related to the accuracy of the PIR mounting accuracy).

\section{Measuring the movement of the center of the distortion pattern (1)}

We observed open cluster fields with tiling the SKY camera 
FoVs in vertical and horizontal strips on the FPI coordinate.
The data are taken with the instrument rotator with SYNC status, i.e.
the instrument is rotated against the corrector - primary mirror system.
The measured coordinates of the stars on the SKY camera
were at first converted to the FPI XY coordinate using the above relation.
Then the FPI XY catalog was cross matched with the RA-DEC
catalog of the stars in the FoVs. Using the (X, Y, RA, DEC) file, 
the distortion pattern of the field is determined by ccmap
command in IRAF. One measurement of the distortion pattern 
is shown in Figure~\ref{FPI_coo_distortion}.
The offset from the plate scale at the center of FoV is shown 
as the optical distortion pattern.
The distortions are determined by ccmap command in IRAF.
The measured distortion is consistent with the ray trace result
shown with green lines and dots. The FPI XY coordinate of 
the distortion center was
estimated to be ($-4.6$mm,$-1.9$mm) for this measurement. 

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.8\linewidth]{FPI_coo_distortion.eps}
\end{center}
\caption{
The measured distortion of the corrector lens in 20071224 with
rotator angle of $-97$deg.
Horizontal axis is the distance from the FPI (0,0) position in arcmin.
Vertical axis is the offset from the $0.01240$ (or $0.01237$) arcsec/$\mu$m
in arcsec. Left) in Y (or Dec) direction, MAP2-Y and MAP3 results are
shown in red and blue, respectively. Ray trace model is shown in green.
The model is offsetted by $-0.39$arcmin ($-1.9$mm) to match the observed distortion curve. 
Right) in X (or RA) direction, MAP2-X and MAP4 results are shown in 
red and blue, respectively. Ray trace model is shown in green. The model
is offsetted by $-0.95$ arcmin ($-4.6$mm). 
}
\label{FPI_coo_distortion}
\end{figure}

In the nights of 20080124 and 20080125, the same measurements are
done with changing the rotator angle. The results are summarized in 
Table 1 and Figure 3. The center of the circle of the distortion pattern
movement is consistent with the rotator axis described in Section 2.

\begin{table}
\begin{center}
\begin{tabular}{cccccccc}
\hline
 Date    & EL & Rotator &     CMM(mm)      & FAM  &  Distortion Cnt. ($\mu$m) & Scale ($\prime\prime$/mm) & \\ \hline
20071224 & 45 &  $-97$   &  ($+4.0, -4.0$) & $+0.60$ & ($-4600, -1940$) & (12.398, 12.401) & \\ \hline
20080124 & 30 &  $+0$    &  ($+3.7, -3.8$) & $+0.55$ & ($+ 980, -4630$) & (12.405, 12.409) & \\
         & 30 &  $-90$   &  ($+3.7, -3.8$) & $+0.55$ & ($-4540, -2720$) & (12.411, 12.401) & \\
         & 30 &  $+90$   &  ($+3.7, -3.8$) & $+0.55$ & ($+3180, + 970$) & (12.409, 12.401) & \\
         & 30 &  $+180$  &  ($+3.7, -3.8$) & $+0.55$ & ($-2740, +3260$) & (12.420, 12.416) & \\ \hline  
20080125 & 85 &  $+6$    &  ($+3.7, -3.8$) & $+0.60$ & ($+1680, -4260$) & (12.405, 12.405) & \\
         & 85 &  $-84$   &  ($+3.7, -3.8$) & $+0.60$ & ($-4670, -3720$) & (12.400, 12.403) & \\
         & 85 &  $-174$  &  ($+3.7, -3.8$) & $+0.60$ & ($-3350, +3130$) & (12.404, 12.402) & \\
         & 85 &  $+96$   &  ($+3.7, -3.8$) & $+0.60$ & ($+3000, +1650$) & (12.404, 12.408) & \\ \hline
20080513 &    &          &  ($+4.7, -4.2$) & $-5.00$ &                  &                  & \\ \hline
\end{tabular}
\caption{
Summary of the results of the distortion center measurements.
}
\end{center}
\end{table}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.5\linewidth]{plot_dist_cen_200801.eps}
\end{center}
\caption{
The measured positions of the distortion pattern center on the
FPI coordinate from 200801 engineering observations.
The numbers in the figure indicate the ROTATOR\_REAL\_ANGLE
during the measurements. The measured distortion centers locate
on a circle around the rotator center with radius of 4.444mm (shown
with the large circle). The asterisk indicate the best fit position
of the distortion center at ROTATOR\_REAL\_ANGLE=0.0.
(The direction of the ROTATOR\_REAL\_ANGLE is different from
the OFFSET\_ANGLE.)}
\label{plot_dist_cen_200801}
\end{figure}

\section{Measuring the center of the distortion pattern (2)}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.5\linewidth]{plot_dist_cen.eps}
\end{center}
\caption{
The measured positions of the distortion pattern center on the
FPI coordinate from 200805 (only two points marked with filled 
square with large open square)  and 200806 engineering observations.
200806 results shown with filled symbols are used for determine
the best fit circle. Black, blue, and red symbols represents
results with CMM at (4700,-4200), (3200, -4000), and (3700, -4500).
The asterisk indicate the best fit position
of the distortion center at ROTATOR\_REAL\_ANGLE=0.0.
The results of 200801 (shown in the previous figure) are shown 
with green symbols.
\label{plot_dist_cen}}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{cccc} \hline
         & CMM position  & Dist.Shift.Rot0 & Fitted Dist.Shift.Rot0 \\ \hline
20080125 & $(3700, -3800)^{*}$ & $(1230, -4652)$   & $(1807,-4540)$ \\ \hline\hline
\multicolumn{3}{c}{Echidna reinstall}          & \\ \hline
20080617 & $(4700, -4200)^{*}$ & $(2465, -5390)$   & $(2730,-5097)$ \\
20080617 & $(3200, -4000)    $ & $(1121, -4260)$   & $(1282,-4657)$ \\
20080617 & $(3700, -4500)    $ & $(1743, -4769)$   & $(1694,-5231)$ \\ \hline\hline
\multicolumn{3}{c}{Echidna reinstall}          & \\ \hline
20080815 & $(3700, -4500)    $ & $(1404, -5784)$   & $(1694,-5231)$ \\ 
20080815 & $(3700, -4200)^{*}$ & $(1658, -5384)$   & $(1743,-4935)$ \\ 
20080815 & $(4700, -4200)    $ & $(3392, -4970)$   & $(2730,-5097)$ \\ 
20080815 & $(3700, -3200)    $ & $(2196, -3868)$   & $(1905,-3948)$ \\
20080816 & $(3700, -4200)    $ & $(1776, -4518)$   & $(1743,-4935)$ \\
20080816 & $(3700, -3200)    $ & $(1864, -3680)$   & $(1904,-3948)$ \\
20080816 & $(2700, -4200)    $ & $(1141, -5115)$   & $(756,-4773)$  \\ \hline\hline
\multicolumn{3}{c}{FAM tilt adjusted} & \\ \hline
20080910 & $(3800, -4600)    $ & $(1445, -5941)$   & $(1777,-5346)$ \\ \hline
\end{tabular}
\end{center}
\caption{
The best-fit distortion pattern center at ROTATOR\_REAL\_ANGLE=0.0
compared with the CMM positions. Fitted Dist.Shift.Rot0 shows 
Dist.Shift.Rot0 determined with equation shown in Sect.4.
These Dist.Shift.Rot0 values correspond to telescope.model
parameters No.23 and No.24, {\bf but signs are inverted}. 
All values are in $\mu$m. Best CMM positions for each observation
are marked with asterisk.
\label{CMM0}}
\end{table}

The movement of the distortion pattern is further examined
with the sky-camera mapping. In order to reduce the error
caused by the tracking error etc, we only observed 10 stars
at the edge of the field of view with a few stars at the center
on the nights of 20080617. The results are summarized in 
Figure~\ref{plot_dist_cen}. As we expect, the radius of the
circle of the distortion pattern movement changes as we change
the position of the CMM. The radius of the circle and the
center of the distortion pattern at ROTATOR\_REAL\_ANGLE=0deg is determined
by fitting circle to the data points. The center of the circle
is fixed at the measured rotator axis. The resulting best fit
circles are shown in the figure. The residual of the fitting is
514$mu$m rms. for the $(4700,-4200)$ dataset taken on 20080617.

The center of the distortion pattern at ROTATOR\_REAL\_ANGLE=0.0degree is
summarized in Table~\ref{CMM0}. The conversions between the centers and
the CMM positions are determined assuming coordinate conversion with 
rotation. The conversion is,
\begin{eqnarray}
{\rm DISTCEN\_FPIX\_0} &=& A \times {\rm CMMX}(\mu{\rm m}) + B \times {\rm CMMY}(\mu{\rm m}) + C, \nonumber \\
{\rm DISTCEN\_FPIY\_0} &=& D \times {\rm CMMX}(\mu{\rm m}) + E \times {\rm CMMY}(\mu{\rm m}) + F  \nonumber \\
 &A&=  0.987  \nonumber \\
 &B&=  0.162  \nonumber \\
 &C&= -1228.6  \nonumber \\
 &D&= -0.162  \nonumber \\
 &E&=  0.987  \nonumber \\
 &F&= -191.3  \nonumber \\
\end{eqnarray}
During the fitting, scale is fixed to be 1.00 and X and Y orthogonalities are
assumed. The CMM coordinate is 9.3 degree rotated against
the FPI coordinate. The rotation is roughly consistent with the mechanical 
measurements of the CMM and FPI systems as shown in Appendex. 
The rms of the fitting is 346$\mu$m and 369$\mu$m in FPIX and FPIY coordinates.
The origin of the CMM is at $(-1228,-191)$ on the FPI coordinate. This
is roughly consistent with the measured offset between the CMM center and
the rotator center at $(-699,-645)$ ($\sim$1mm, $\sim$1mm).

Using the center of the distortion pattern as described above, we fit 
the sky-camera mapping results again in order to refine the distortion 
model parameters. The results are shown in Table~\ref{distpars}.
The distrotion model parameters are not consistent each other yet, the
measurements can be affected by telescope tracking error during measurements.
Further refinement need to be done.

\begin{table}
\begin{center}
{\small
\begin{tabular}{ccccrrrrrl} \hline
Date     & FieldID & PA(OFF)       & FAM  & Dist1($\theta$) & Dist2($\theta^3$) & Dist3($\theta^5$) & Dist4($\theta / \lambda$)     &  rms($^{\prime\prime}$) & Note \\ \hline
Design   & value   &               &      & 0.291950 & 0.0445   & 0.0364   & $-0.197$  &      & \\
Required & accuracy&               &      & 0.000032 & 0.000540 & 0.00842  &           &      & \\ \hline
20080514 & field5  & $0(0.52)$     &      & 0.290461 & 0.046430 & 0.001194 & $-0.197$F & 0.16 & \\
20080514 & field6  & $180(-118.1)$ &      & 0.290559 & 0.042003 & 0.049526 & $-0.197$F & 0.19 & \\
20080514 & field9  & $0(71.2)$     &      & 0.290434 & 0.051681 & 0.001150 & $-0.197$F & 0.16 & \\
20080617 & field50 & $90(108.1)$   &      & 0.290539 & 0.042683 & 0.049064 & $-0.197$F & 0.17 & \\ \hline
20080814 & field21 & $0(103)$      & 5.30 & 0.290528 & 0.042719 & 0.034522 & $-0.197$F & 0.20 & {\bf current value} \\  
20080814 & field30 & $0(-81)$      & 5.05 & 0.290529 & 0.043843 & 0.012562 & $-0.197$F & 0.17 & \\
20080814 & field31 & $0(-82)$      & 5.05 & 0.290495 & 0.046507 & $-0.017435$ & $-0.197$F & 0.17 & \\ 
20080814 & field32 & $180(97)$     & 5.05 & 0.290531 & 0.044360 & 0.032638 & $-0.197$F & 0.18 & \\ 
20080814 & field33 & $180(97)$     & 5.05 & 0.290436 & 0.048028 & $-0.025634$ & $-0.197$F & 0.18 & \\ 
20080814 & field50 & $0(-97)$      & 5.30 & 0.290629 & 0.038582 & 0.073852 & $-0.197$F & 0.22 & \\ 
20080814 & field54 & $0(92)$       & 5.25 & 0.290562 & 0.039828 & 0.080988 & $-0.197$F & 0.21 & \\ \hline
20080909 & field01 & $0(-95)$      & 4.25 & 0.290676 & 0.040246 & 0.054698 & $-0.197$F & 0.19 & \\ 
20080909 & field03 & $180(75)$     & 4.33 & 0.290585 & 0.043423 & 0.027744 & $-0.197$F & 0.19 & \\ 
20080909 & field09 & $0(-92)$      & 4.20 & 0.290665 & 0.041416 & 0.042362 & $-0.197$F & 0.21 & \\ \hline
\end{tabular}
}
\caption{
The distortion model parameters determined for each field. 
Distortion parameter change which will make $0.1^{\prime\prime}$ difference at
the edge of FoV shown as ``Required accuracy''. ``rms'' represents rms residual
after the fitting. Because at 20080909, the
FAM tilt was adjusted, the FAM value before and after the adjustment can have
systematic difference. 
\label{distpars}}
\end{center}
\end{table}
 
\section{Pointing center}

Currently, the rotator axis is refered as the pointing 
center of the telescope, but the rotator axis does not match
the origin of the FPI coordinate as described in Section 2.
The configuration of the fiber is caluculated from the
origin of the FPI coordinate, and the RA-DEC coordinate
of the origin need to be supplied to the fiber configuration 
software. The difference between the pointing center of the
telescope and the origin of the FPI coordinate is handled 
with the environmental variables, POINTING\_OFFSET\_RA and
POINTING\_OFFSET\_DEC. 

We evaluate the pointing errors with the current telescope 
pointing model (determined by PA) by checking RA-DEC coordinate 
of the rotator axis with the sky-camera mapping data.
The resuling offsets between the target coordinate and the
RA-DEC coordinate of the rotator axis is shown in Figure~\ref{plot_point_cen}.
There is a large offset between the data points taken before
and after meridian. Need to be checked..

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.5\linewidth]{plot_point_cen.eps}
\end{center}
\caption{
Measured pointing error for various field, the difference
of the RA-DEC coordinate of the rotation center from
the target RA-DEC coordinate. Data taken before meridian
are marked with open squares.
\label{plot_point_cen}}
\end{figure}

% \begin{table}
% \begin{center}
% \begin{tabular}{ccc}
% ID      & RAoffset              & DECoffset \\ \hline
% field5  & $-7.0^{\prime\prime}$ & $+16.5^{\prime\prime}$ \\
% field9  & $-4.5^{\prime\prime}$ & $+14.8^{\prime\prime}$ \\
% field10 & $-5.3^{\prime\prime}$ & $+16.6^{\prime\prime}$ \\
% field11 & $-3.0^{\prime\prime}$ & $+14.2^{\prime\prime}$ \\
% field12 & $-13.8^{\prime\prime}$ & $+7.0^{\prime\prime}$ \\ \hline
% average & $-5.0^{\prime\prime}$  & $+15.5^{\prime\prime}$ \\
%         & $-416$($\mu$m) & $-1292$($\mu$m) \\ \hline
% \end{tabular}
% \end{center}
% \caption{
% \end{table}

\begin{table}[tb]
\begin{center}
{\tiny
\begin{tabular}{lcl}
\hline
Parameter      & Value  & Note       \\ \hline
SKYC\_AKIYAMA\_A & $-8.041$ & see Sec.1. (20080814) \\
SKYC\_AKIYAMA\_B & $0.203$  & see Sec.1. (20080814) \\
SKYC\_AKIYAMA\_C & $3575.0$ & see Sec.1. (20080814) \\
SKYC\_AKIYAMA\_D & $-0.195$ & see Sec.1. (20080814) \\
SKYC\_AKIYAMA\_E & $-8.364$ & see Sec.1. (20080814) \\
SKYC\_AKIYAMA\_F & $2610.0$ & see Sec.1. (20080814) \\
SKYC\_CENTRE\_X\_OFFSET & 600 & Set to be stars can be catched at center. \\
SKYC\_CENTRE\_Y\_OFFSET &  70 & Set to be stars can be catched at center. \\
ROTATOR\_AXIS\_X & $459$ & see Sec.2 Tab.2. {\bf sign is inverted by definition} (20080909) \\
ROTATOR\_AXIS\_Y & $531$ & see Sec.2 Tab.2. {\bf sign is inverted by definition} (20080909) \\
POINTING\_OFFSET\_RA  & $-9.12$ & At PA=0, for FPI(0,0), this needs to be {\bf subtracted} from commanded RA \\
                      &         & Depending on ROTATOR\_AXIS\_X \\
POINTING\_OFFSET\_DEC & $+7.20$ & At PA=0, for FPI(0,0), this needs to be {\bf subtracted} from commanded DEC \\
                      &         & Depending on ROTATOR\_AXIS\_Y \\
telescope.model No.10 & 0.290528 & see Tab.5 need to be refined \\
telescope.model No.11 & 0.042719 & see Tab.5 need to be refined \\
telescope.model No.12 & 0.034522 & see Tab.5 need to be refined \\
telescope.model No.13 & $-0.197$ & need to be checked with tel.rastering. \\
telescope.model No.23 & $-1445$  & depends on CMM position, see Sec.4,Tab.5, {\bf sign is inverted from Tab.4,Fig.4 by definition} \\
telescope.model No.24 & $5941$   & depends on CMM position, see Sec.4,Tab.5, {\bf sign is inverted from Tab.4,Fig.4 by definition} \\
telescope.model No.25 & Not used & echTelMdl.cc revised such that No.23,24 refer to ROTATOR\_REAL\_ANGLE=0deg \\
telescope.model No.26 & Not used & echTelMdl.cc revised such that No.23,24 refer to ROTATOR\_REAL\_ANGLE=0deg \\
telescope.model No.27 & Not used & echTelMdl.cc revised such that No.23,24 refer to ROTATOR\_REAL\_ANGLE=0deg \\
telescope.model No.28 & Not used & echTelMdl.cc revised such that No.23,24 refer to ROTATOR\_REAL\_ANGLE=0deg \\ \hline
\end{tabular}
}
\caption{
Summary of the best estimate environmental variables for ICS and 
telescope model parameters related to the distortion pattern modeling. 
It should be noted that POINTING\_ERR does not relate to the pointing
error of the telescope, relate to the offset between the rotator
axis and the FPI origin. These values are described in 
\$ECONFIG/.zshrc and \$ECONFIG/telescope.model
}
\end{center}
\end{table}

\section{Guide fiber focus positions}

Memo: skycamera focus 4.7mm and AGSH focus 3.4mm.
Center 4.9mm then edge 5.1mm-5.15mm.
If sky-camera best focus at center is 5.15mm then, 
best focus for science fiber is 5.15mm, best focus
for sky-camera mapping is 5.30mm, roughly.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=1.0\linewidth]{dist_param.eps}
\end{center}
\caption{
\label{dist_param}}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\linewidth]{focus_scale.eps}
\end{center}
\caption{
The effect of the offfocus on the centroiding of the 
sky camera mapping. The position of four stars at the
edge of the FoV are measured relative to the position
of the star at the center. The movement of the relative
positions are plotted as a function of FAM 
focus position (mm). The best focus positon was around
FAM=4.4(mm) and the position at this focus was used as
the base position. Off focus by 0.2mm can affect the
measured position of the stars by 20$\mu$m at most.
\label{focus_scale}}
\end{figure}

\section{Appendix: calibration items summary}

{\bf IF WE RE-INSTALL SKY-CAMERA}, then we need to determine the
relation between SKY-camera coordinate and FPI coordinate
as described in Section 1 and Table1. SKYC\_ environmental 
variables need to be updated.

\noindent
{\bf IF WE RE-INSTALL ECHIDNA TO PIR}, the position of the rotator
axis on the FPI coordinate can be changed. We need to determine
the rotator center on the FPI coordinate as described in Section 2
and Table 2. ROTATOR\_AXIS\_X and ROTATOR\_AXIS\_Y, and
POINTING\_ERR\_RA and POINTING\_ERR\_DEC environmental
variables need to be updated. 

The relation between the CMM coordinate and the FPI 
coordinate can be changed (parameters shown in Section 4). 
The shift of the relation can be checked 
by measuring the center of the distortion pattern at 
ROTATOR\_REAL\_ANGLE=0.0. If the center is measured
at ROTATOR\_REAL\_ANGLE=0.0 additionally, the matching
between the rotator axis and the center of the circle of
the distortion pattern movement can be checked.

\noindent
{\bf IF WE RE-INSTALL PIR TO TELESCOPE}, the coma-free CMM position 
can be changed. No.23 and No.24 telescope.model parameters need
to be changed using the updated CMM position and the equation 
shown in Section 4. It should be noted that the equation may be
changed if we re-install Echidna to PIR.
(The radius of the circle of the distortion
pattern movement can be changed).

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\linewidth]{FPI_coordinate2.eps}
\end{center}
\caption{
The relation between the Echidna unit and the PIR/CMM unit at
rotator angle of 0 degree. CMM axis is 7.5 degree rotated 
against Echidna coordnate (measurements by M.Kimura)
}
\label{FPI_coordinate2}
\end{figure}

\end{document}












