24 bit encoder resolution and accuracy
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Phil Moore and myself visited the BEI encoder factory in 
Little Rock last week. The primary reason was to check 
progress on the three encoders which they are currently 
making (they have made some progress and produced a fairly
realistic looking schedule for the remainder of the work). 
However, we also  got a chance to have some technical 
discussions. This note is just to briefly describe some 
information which they may choose to make available about 
encoder accuracy, which may in the future be useful on
either JCMT or UKIRT.

BEI divide the encoder error into three components: 
quantising error, wide-angle error and narrow-angle error.

The 24-bit encoders have a resolution of 0.08" (360 deg/2**24). 
For these encoders, the quantising error (rms = lsb/root(12)) 
is negligible.

The wide-angle error is the result of code disk eccentricity, 
ellipticity, etc. These are periodic over one revolution with 
various higher order harmonics. 

The "narrow-angle errors" are those which arise during the 
interpolation within one cycle of the fine code track. 
For our encoders, this is 360/2**16, or 20". 
These are caused by gap change and pattern shading variations.

Adding the above three sources of error in quadrature,
BEI arrive at the figure of 0.28" rms for the accuracy of 
the 24-bit encoders (nowhere have I seen a figure quoted for 
the peak-peak error).  
For the UKIRT encoders, BEI checked the performance with an 
ultradex table, and a Moller-Wedel autocollimator. They showed 
us some very crude measurements, giving an rms of 0.1 arcsec, 
and peak-peak values of around 0.5 arcsec.

However, BEI now have a 29-bit encoder, and an automatic 
measuring system, which allows them to check the accuracy of 
every transition! Using this, they showed us two curves for high 
accuracy encoder (I can't remember the model, or whether it was 
22 or 24 bit; Phil may).

i) a plot of maximum and minimum error found within each 
successive 1/1024 of a revolution. This formed a band with a 
separation of around 0.2 arcseconds, a smoothly varying mean 
dominated by low-order harmonics, and a peak-peak of around 
0.7 arcseconds.

ii) a plot of the narrow-angle error measured for every bit over 
one 20" cycle. This showed a number of cycles of error, with a 
peak-peak of around 0.2 arcsec. It is this component, sampled at 
it's maximum and minimum value, which produced the input to the 
first plot.

The two BEI staff members who were giving us the tour argued 
between themselves as to whether (a) either form of error was 
constant with time, and (b) whether the narrow-cycle error was 
constant with fine-track position. We also had a long discussion 
as we whether we would get copies of the complete calibration 
data. Apparently it is not policy to release this (or at least 
not in the detail we would like), however, they have 
unofficially agreed to make the data available, and I hope to 
pursue this further.

I suppose the bottom line on this is simply that BEI can 
characterise the encoders once they are manufactured, and can 
then guarantee that they have a resolution of 0.08", and 
an rms error of better than 0.28". However, depending upon our 
ability to get the data, and it's stability with time and 
position, we may also be able to get calibration curves for 
each encoder which would allow us to:

a) calibrate out the wide-angle error, potentially removing 
   up to about 0.7 arcseconds of peak-peak variations,

b) (less likely) calibrate out the narrow-cycle error, removing  
   another 0.2 arcseconds or so of peak-peak variation.

If both of these were stable, it would seem that in theory 
at least we could use the calibration data to correct the 
encoder output so that the maximum error was of the order of 
0.1 arcsecond or less.

All of this is rather speculative at the moment, and it may 
in fact be impractical or irrelevant, but I thought people 
might like to know that at least the possibility exists.


Richard P.  22 March 1995