# Intro:

The telescope pitch and roll was fit to a 2-d function of azimuth and zenith angle. Data from the ao9 theodolite survey and the tilt sensors mounted in the dome were used for the fit. This page describes how the data was taken, what  corrections were made to the data, the results of the fit, and how far off the fit is.

• theodolite and corrected tilt sensor data for the theodolite positions.
• Tilt sensor corrected data and the 2d fits.
•  fits to the individual azimuth swings for the tilt sensor corrected data.
• #### Quick links to sections:

The procedure used.
The AO9 survey.
Converting tilt sensor pitch/roll to theodolite pitch/roll.
Fitting a 2d function to the corrected tilt sensor data.
Fitting the azimuth swings to constant+1az+3az terms.
Summary.

# The procedure used: (..top)

• Level the platform using the tilt sensors. Spin the azimuth 360 degrees and then adjust the tiedowns to remove any 1 azimuth terms.
• Tilt sensor run on 04aug01 using the tilt sensors mounted on the rotary floor in the dome. Did azimuth swings at za=(2,4,6...,18,19.5) deg and za strips from 2 to 19.5 deg za with the azimuth=(62.87,122.87,182.87,242.87, and 302.87)
• AO9 survey using a theodolite on 08aug01 measuring two azimuth swings (za=10,19.6) and two za strips (az=242.87 and 302.87).
• Use the laser ranging data to verify that the platform average height did not move much during the above measurements.
• Fit the pitch and roll from the theodolite to f(az,za)=c0+c1*(za-10)+c2*(za-10)^2+c3*(za-10)^2 +  1az+2az+3az cos and sin terms. The 1 az term will be needed below to correct the tilt sensor data.
• Convert the tilt sensor data from voltage to degrees using the sensors transfer function.
• Measure and then correct the tilt sensor data for the angle they made with the rotary floor (by spinning the floor 180 degrees and subtracting the average).
• Smooth the tilt sensor data to 1 degree in azimuth and .05 degrees in za.
• Move the tilt sensor data to the theodolite reference.
• Fit a 2d function to the tilt sensor data. Use many terms so we can follow as many of the wiggles as possible (for possible use in future pointing corrections).
• Check how good the 2d fit using the fit residuals from the tilt sensor data and the theodolite data.
• Refit the azimuth swings with a simpler fit (c0 +c1*az +sin(az-phase)+sin(3az-phase) to look at some of the physical motions of the structure.
• # The AO9 survey:  (..top)

The AO9 survey  is described here. Ideally we would have run the tilt sensors while the survey was being done, but
someone forgot to hit return on the computer (me...). This gave us the pitch, roll, and focus at a limited number of points (60). By looking at the 1 azimuth term you can see that the center of rotation of the platform is .2 inches east and 2.2 inches north of AO9. This data takes a long time to acquire (9 pm to 3:30 am for 60 points). The tilt sensor data is easy to acquire but has unknown offsets. We use the AO9 data to solve for the tilt sensor offsets. The focus information is only available from the theodolite (although you can make some assumptions about how the focus should change with the pitch).

# The tilt sensor data:

The tilt sensor data was taken on 04aug01 at night. The azimuth swings were done  at .2 degrees/second while the za strips used .1 degrees/second. Sampling was at 5 hz. The data was smoothed to 1 degree in azimuth and .05 degrees in za (fwhm).

#### Converting tilt sensor pitch/roll to theodolite pitch/roll.  (..top)

There are offsets in the tilt sensor data because the sensors are not perfectly aligned in the uphill direction. Another problem is that the center of rotation of the azimuth arm is not centered directly above AO9. This will cause a 1 azimuth term in the data. To move the tilt sensor to the theodolite pitch roll:
• For each of the 60 theodolite measurements, all tilt sensor points within 3 degrees az encoder and .05 degrees za encoder of theodolite point were averaged.  3 and .05 were chosen by trying all az distance < 5 deg and all za dist < 1  and using the pair that minimized the residuals. The az swing at 19.6 degrees was note used since it was not within .05 degrees of the tilt sensor swing (at 19.5).
• The theodolite 1az term was removed from the theodolite data and the 1az term was removed from the tilt sensor data.
• A linear fit to za enc. was done to the differences: dif(i)= A0 + A1*za*(theod(i)-tilt(i)) for pitch and roll separately.
• The corrected tilt sensor data was then: tsCor(az,za)=ts(az,za) + ((A0)+A1*za) + theod1Azterm(az)
•     The figures show the theodolite and corrected tilt sensor data for the theodolite positions.
1. Fig 1 has the pitch,roll versus za. The solid line is the tilt sensors after correction while the dash line is the theodolite. The rms is for all of the points used.
2. Fig 2 plots the pitch, roll versus az (az swing at 10 degrees).
The residual rms is .008 deg for pitch and .009 degrees for za.

#### Fitting a 2d function to the corrected tilt sensor data:  (..top)

A function P(az,za) and R(az,za) was fit to all of the corrected tilt sensor data. It will be used to correct the pitch,roll using the tie downs. The 2d function used had:
• za terms: let x=(za-10)/4.  then fit a 13th order polynomial in x.
• az terms: fit sin,cos terms in (1az) (3az)
• za*az terms: fit terms in za*(cos(3az) ,cos(4az),cos(6az)) and the corresponding sin terms.

• The fit ends up having 24 terms. There are 5450 data points. The plots show the Tilt sensor corrected data and the 2d fits.
1. Fig 1 plots the tilt sensor corrected data (pitch top, roll bottom) for the 5 za strips (color coded by azimuth position). The smooth lines are the fit.
2. Fig 2 has the residuals (data - fit) for the za strips. The rms is for all of the data (za strips and az swings). The residuals are large for the pitch around 10 degrees for the strips at az=62 and az=182. These azimuths are at the minimums of the 3az term (see fig 3). They also were in the downhill za direction: az (62,182,302) down and az:(122,242) up. 302 (downhill) does not show the same large residual.  There is also a large excursion at 17 degrees for the strip at az=242.87.
3. Fig 3 plots the pitch and roll data for the 10 azimuth swings. The smooth line is the fit.
4. Fig 4 shows the residuals (data-fit) for the azimuth swings. You do not see any large jumps in the pitch for  the azimuth swing at 10 degrees za. The roll residuals have larger residuals when doing the azimuth swings.
5. Fig 5 are the theodolite points - 2dfit.. The residuals get large at the edges of the tilt data. There is no tilt sensor data for 19.6 or 1.09 za so the fit residuals are very large.
For all of the tilt sensor data the residuals are .005 deg in pitch and .0038 deg in roll. The residuals are large when the telescope is moving in the direction of the axis (za strips , pitch) (az swings, roll). This is probably from the shaking and irregularities of the track. Looking at the az swings you can see that there are some shimming bumps on the rail (of order .05 degrees after smoothing to 1 deg az). When we do the theodolite za strips it is important that we don't position the azimuth on one of these bumps. If we did, then that bump would be weighted extra in the za fit terms. The same goes for the azimuth swings and any za bumps.

#### Fitting the azimuth swings to constant+1az+3az terms.  (..top)

The 2d fit is good to reproduce the pitch and roll but covariance between the fit terms is not zero so it is hard to see the structural response in an individual term. A simpler function of az was fit to the individual azimuth swings. F(az)=C0 +C1*az+C2*sin(az-C3) + C4*sin(3az-C5). The A1*az term is essentially a term linear in time since azimuth increased or decreased linearly. The figures show the fits to the individual azimuth swings for the tilt sensor corrected data.
1. Fig 1 is the pitch - (constant + linearTime) term for each swing over plotted. The 1 an 3 azimuth terms stick out.
2. Fig 2 plots the roll for each azimuth swing. The dashed lines are the individual fits. The fit values are listed below.
3. Fig 3 has the pitch residuals for each swing and the average at the bottom.
4. Fig 4 plots the roll residuals for the individual swings and the average residual at the bottom. There is a strong 6az terms for az 0-180 and then it decreases for az 180-360. There also appears to be a 3az*sin(za) term for az 0-180 (see fig 2 for the za to color mapping).
5. Fig 5 plots the pitch coefficients versus za. The top plot has  plots (c0- (a+b*za)) versus za.  With a-.0258 and b-.0037 degP/degza. Moving down the page has sin(az-ph) amplitude, sin(az-ph) phase, sin(3az-ph) amplitude, and sin(3az-ph) phase. Except for the constant term, there is not much za dependence. The constant term fit of a+b*za gives a=.0258. The 2-d fit had a constant term of .011. The is not surprising since the many za terms in the 2d fit can modify the y intercept.
6. Fig 6 is the same plots for Roll. The fit to the constant term give a y intercept of -.06 while the 2d fit had -.1 . The 3az roll amplitude drops by .007 degrees from 10 deg za out to za of 19.5.

# Summary:  (..top)

A table of results is listed below,Ph is the phase angle for the maximum value. (??) --> that the amplitude is so small that the phase angle should not be relied on. Amplitudes are in degrees.

 1az pitch Amplitude phase (deg) 1az roll Amplitude phase (deg) 3az pitch Amplitude phase (deg) 3az roll Amplitude phase (deg) 1. tiltSensor .003 73?? .002 7?? .022 11 .049 74 2. Theod. at AO9 .021 197 .025 273 .017 9 .036 65 2b. theod. focus error .64Inch 179 .30in 21 3. TheodEnc-PlatEnc   from az swings   Gives platform offset   dx=.2 east,2.2north(in) .024(zadif) 6 (186) .024,(azdif) 277 4. pointings errors:za,gcAz .027(zaerr) 5 .027,(azErr) 274 .004 62 .007 65 5.Pitch,roll or dome Using center of reflector. .028 163 .035 248 .017 9 .036 65 5b. reflector based focus error .91 inch 156 .28 22 6.ReflectorEnc-platformEnc platform is: dx:1.14 west,2.89 north(in) .033(zadif) 339 .033,(azdif) 250
• 1. Shows that the platform was level relative to the tilt sensors when the tilt sensor data was taken.
• 2. Has a 1az term for the thedolite at ao9. The pitch and roll phase of the 1az term differ by 80 rather than 90 degrees.
• 1,2. The 3az amplitude for the tilt sensors are 1.3 times larger then the amplitudes measured by the theodolite. The pitch phase is close but the roll phase is off by 9*3 degrees az.
• 3. The theodolite - platform encoders have a 1 az term difference do to the horizontal offsets. The amplitudes agree with the 1 az terms of the thedolite pitch and roll. The phase for the roll/azdiff is the same. The phase for the pitch/zadiff is off by 191 degrees. To figure out the correct direction: look along  az=6deg from AO9 (where the theodZa-encZa is a max). Suppose the encoder za is 10 deg. Then the theod za would be 10.024 deg za. The pitch of the dome is set by the rails and the encoder za so it is 10deg. It should be 10.024 degrees. This shows that the maximum za difference is the minimum pitch so the phase should differ by 180 degrees. It differs by 191 so it's close.
• 4. Model 13 great circle pointing errors for the 1az term agree in phase and amplitude with the 1 az term from the theodolite-encoder readings. The 3 az terms do not. This may be because the pointing model includes a sinza*cos(3az) term.
• 5. This is the 1az terms you get for the offset of the platform relative to the reflector. It includes the offset of the platform from A09 and the offset of the reflector from AO9.

•

A 1 azimuth term in pitch and roll can be caused by tilting the platform. A horizontal translation of the platform can create a 1 az term in pitch but should not affect the roll. This is because the elevation rails are curved in the pitch direction but are flat in the roll direction (spinning in azimuth or za does not change the roll angle of the dome).
We have an inconsistancy.  The pitch 1az term is explained by a horizontal offset, but there is an identical roll error. If both of these were caused by a platform tilt, then the tilt sensors should have measured it. The tilt sensor data was taken on 04aug01 while the survey was on 09aug01. Maybe the platform moved. Even if this is so, the difference in encoder values shows that there is  a horizontal offset that should give a pitch error close to what we measure. So where is the 1az roll error coming from ??

Some other values....
 tilt sensor Floor offsets: pitch:-.141,roll:.672 degrees. hor offset platform from AO9 .2 inches east, 2.2 inches north Theod 1az term: pitch: ampl:.021 deg, peak@az=196.64 Roll  : ampl:.025 deg, peak@az=272.56 tiltSens 1az term: pitch: ampl:.002 deg, peak(not measured). roll   : ampl:.002 deg, peak(not measured). convert tiltSens to Theod: tscorpitch=9.78186-.99575*za + tsrawPitch+theodPitch1Az tscorRoll  =  .3524   -.02124*za + tsrawRoll + theodRoll1Az Residuals (Theod-tiltsenCor) 47 pnts pitch:.008, roll:.009 degrees 2d-fit smoothing: 1 degree az, .05 degrees za tiltsensor input data: 5450 points.  sample at 5hz za strips 2-19.5 at az:(62.87+60*n) n=0..4  at .01deg/sec,(down/up) azwings at za=2,4,6..18,19.5                             at .2 deg/sec residuals(tiltSenCor - 2d fit) pitch: .0052 degrees roll   : .0038 degrees. residuals(theod - 2dfit) pitch: .0054 degrees roll   : .0084 degrees.

The main coefficints to the 2d fit are:

 term pitch roll constant 0.011280838 -0.10458930 Asin(az-ph) A:0.02141,ph:106.57 A:0.02479,ph:181.027 Asin(3az-ph) A:0.02168,ph:304.570 A:0.04892,ph:133.993

To evaluate the 2d fit in idl:
@phil
@tsinit