SDSS 2.5-m telescope secondary measurements
Sloan Digital Sky Survey Telescope Technical Note
19980903
Walter
Siegmund, Larry Carey, Russell Owen and Patrick Waddell
Contents
Introduction
The SDSS 2.5-m secondary mirror is supported in a manner similar
to the Apache Point Observatory 3.5-m telescope secondary mirror.
Separate systems support the axial and transverse components of the
secondary mirror weight vector. The axial support is provided at 9
points by three whiffletrees, each of which is attached to the back
of the mirror at three points. An axial actuator is present in the
link connecting each whiffletree to the secondary mounting frame
(Figure 1). These enable focus and tilt adjustments of the secondary
mirror. Each axial actuator consists of a 200 step/revolution motor
coupled to a 80:1 harmonic drive reducer (Harmonic Drive
Technologies, Peabody, MA, HKC-20-080-2). The motor is driven by a
Galil motor controller with a resolution of 50 microsteps per motor
step. The reducer drives a 635 µm pitch (40 threads/inch) screw
(Universal Thread Grinding, Faifield, CT). Thus, each microstep
corresponds to 0.7938 nm and each revolution of the motor shaft
(10,000 microsteps) corresponds to 7.938 µm.
The transverse support is applied at the center of the mirror by a
lever that is attached to the secondary mounting frame by a gimbal
and that extends into the secondary to its center of gravity. A
linear bearing couples the lever to the secondary to allow focussing.
Two actuators that act along the diagonals (parallel to the secondary
support vanes) drive the end of the lever opposite the secondary.
Each of these actuators consists of a 200 step/revolution motor that
drives a nut threaded on a 317.5 µm pitch (80 threads/inch)
screw. The motor is driven by a Galil motor controller with a
resolution of 50 microsteps per motor step. The lever supports are
separated by 133.7 mm and the load is 35.3 mm from the gimbal with
the mirror at the center of its focus range. Consequently, the lever
provides a nominal 3.787:1 reduction ratio that varies with focus.
Each motor microstep corresponds to a nominal lateral displacement of
the mirror of 8.38 nm.
Figure 1: Axial actuator. Each axial
actuator consists of a 200 step/revolution motor coupled to a 80:1
harmonic drive reducer. The reducer drives a 635 µm pitch (40
threads/inch) scew. The cad
drawing in .dxf format is not quite current.
Measurements
Two Mitutoyo 519-332 cartridge head probe electronic indicators
were used. The indicators are half bridge type and specified to be
linear to 0.5%. Mitutoyo 519-404a Mu-checkers were used to drive the
indicators and produce analog signals. These signals were digitized
and logged by a 12-bit A/D card (National Instruments PCI-1200) using
LabView 3.1 running on a Macintosh. This system provided graphical
display of the data as they were acquired, a feature that improved
the quality of the data and reduced the time required to collect
it.
To measure the performance of an axial actuator, an indicator was
mounted as close as possible to the actuator (Figure 1). The rear
surface of the whiffletree was indicated. Transparent plastic
self-adhesive tape was used to electrically isolate the gauge head
from the whiffletree to reduce electronic interference (the indicator
was electrically isolated from the magnetic base by a hot-melt
adhesive joint used to mount the indicator).
Figure 2: A Mitutoyo electronic
gauge head is supported by a magnetic base just below the shiny
actuator assembly on the right. It indicates the axial motion of
the aluminum whiffletree that is mounted on the back of the
secondary mirror. The third arm of the whiffletree is not visible
because it is behind the black tubular steel secondary mounting
frame. Click on this image to see a larger one.
Figure 3: Small move (ABC03). The
mirror was translated axially two cycles with an amplitude of 60
µm. The motor controllers were instructed to move at a
constant rate in each segment except for a brief acceleration
phase at the ends. Data for the B and C actuators are shown offset
vertically for clarity. The four motion segments shown are
numbered 01 through 04 in the following discussion.
Figure 4: Actuator difference. The
difference in displacement of the two actuators of Figure 3.
Electronic indicator scale differences and the linear drift of the
offset in time were removed. The displacement difference is 112 nm
RMS or 79 nm RMS per actuator. Significant hysteresis occurs where
the actuators reverse.
Figure 5: Actuator non-linearity.
The residual error after removal of the best fit straight line to
the first and third segments of Figure 3. These data were fit to y
= a0 + a1*cos(2*pi*x/7.938µm+ø1) +
a2*cos(2*(2*pi*x/7.938µm+ø2)).
Data were taken August 18-21, 1998. Long moves of 1.0 mm, typical
of an initial focus motion at the beginning of the night, and short
moves of 60 µm, typical of a refocus motion during the night,
were examined. For both moves, the three actuators were commanded to
move two cycles of a triangle wave. For a long move, the amplitude
was 1,260,000 microsteps (1.000 mm) and the rate was 50000
microsteps/sec (40 µm/sec). Acceleration times at direction
reversals were about 100 msec. For a short move, the amplitude was
75600 microsteps (60.0 µm) and the rate was 2520 microsteps/sec
(2.0 µm/sec). Acceleration times at direction reversals were
about 5 msec.
The data set ABC03 was a short move with the telescope near
0° elevation (Figure 3). Displacement differences of the
actuators cause tilts of the secondary mirror. The difference of the
displacements of actuators B and C (Figure 4) is 112 nm RMS or 79 nm
RMS per actuator. The corresponding 2D image motion is 41 mas RMS.
(To calculate 2D image motion, multiply by root 3 to account for the
three actuators, divide by 0.483 m to convert to 2D mirror tilt and
multiply by twice the secondary/focal surface separation divided by
the final focal length or 0.709.) Electronic indicator scale
differences, offset, and linear drift of offset in time were removed.
Putative hysteresis effects are small but apparent near direction
reversals, particularly at 40 and 100 seconds.
To study nonlinearities, a straight line was fit to each segment
of the triangle wave. The residual nonlinearities were between 46 and
58 nm RMS (Figure 5). The nonlinearity with harmonics through second
order in motor shaft rotation (shown) removed is between 40 and 44 nm
RMS. The fit coefficients are given in Table 1. The residual errors
in the reverse direction were larger. For the C actuator, the second
harmonic was larger than the first.
After moving the telescope to 21.2° elevation, the secondary
was moved two cycles with 1.0 mm amplitude (ABC11). This long move
exhibited nonlinearities with a range of 8 µm and a differential
motion between two actuators of 9.2 µm. The latter number
corresponds to an image motion of 2.4 arc seconds.
The subsequent short move of 60 µm (ABC12) exhibited
nonlinearities and hysteresis as well (Figure 6). The total range of
2.1 µm corresponds to an image motion of 0.55 arc seconds. The
500 to 700 nm of direction reversal hysteresis corresponds to about
130 to 180 mas of image motion. Within a motion segment, the
nonlinearities are comparable to those of Figure 5, except for the
first segment.
After these anomalies were noted, the rest of the data taken at
0° were examined. In a long move (ABC07) about 10 µm of
differential motion occurred. In the subsequent short move (ABC08)
after a 0.56 mm move to near the center of the indicator range, 1.8
µm of differential hysteresis was associated with the first
reversal of motion. This is equivalent to 470 mas of image motion.
Hysteresis at subsequent direction reversals was 500 to 1000 nm.
Table 1: Least squares fit
parameters for the residual motion errors of Figure 5. The data
were fit to y = a0 + a1*cos(2*pi*x/7.938µm+ø1) +
a2*cos(2*(2*pi*x/7.938µm+ø2)).
|
a1 (µm)
|
ø1 (deg)
|
a2 (µm)
|
ø2 (deg)
|
B01
|
31
|
-156
|
28
|
12
|
B03
|
39
|
-155
|
27
|
9
|
C01
|
53
|
-26
|
29
|
35
|
C03
|
38
|
-34
|
29
|
33
|
Figure 6: Actuator difference. At an
elevation of 21.2°, large differences appeared in the first
cycle. These decreased in subsequent cycles. However, about 500 to
700 nm of hysteresis was associated with direction reversal.
Data obtained April 19, 1998, allowed repeatability of an actuator
to be examined. At that time, the mirror was being tested face down
in the SDSS support building, i.e., corresponding to a telescope
elevation of 90°. The indicator was registered to the back of
pad B3 that is attached to the back of the secondary mirror with
adhesive (visible at the bottom of Figure 2). As describe above, the
three axial actuators were commanded to move two cycles of a triangle
wave (Figure 7). However, in this case, each segment of motion
consisted of 200 discrete 0.198 µm steps for a total range of
39.7 µm. Since the steps were apparent in the data, it was
possible to fold and stack the four motion segments so that each step
had the same ordinate (Figure 8). The displacement differences for
each step that are apparent in the Figure are due to hysteresis.
Subtracted from the data from each motion segment is the mean of
segments 02, 03 and 04 (Figure 9). Segments 01, 02, 03, and 04 have
standard deviations of 405, 99, 88, and 61 nm, respectively. These
correspond to a range of 32 to 210 mas RMS of 2D image motion.
Figure 7: Actuator
non-repeatability. With the secondary mirror off the telescope and
facing down,the three axial actuators were moved 39.7 µm in
200 discrete 0.198 µm steps (individual steps not visible).
This was followed by the reverse motion. Two cycles were measured.
The four motion segments are numbered sequentially 01 through 04.
The starting point was 4,000,000.
Figure 8: Actuator
non-repeatability. This detail of Figure 7 shows the 0.198 µm
steps that comprise the 39.7 µm actuator displacement.
Measurements of the four motion segments have been folded and
overlaid so that measurements of the same commanded displacement
appear at the same ordinate. The vertical offsets indicate
non-repeatability of the actuator.
Figure 9: Actuator
non-repeatability. The mean of segments 02, 03, and 04 is
subtracted from the four motion segments of Figure 8. Segments 01,
02, 03, and 04 have standard deviations of 405, 99, 88, and 61 nm,
respectively.
To investigate the feasibility of closed loop control of the
actuators, data obtained April 19, 1998 were examined. The mirror
was measured face down in the laboratory. Four data sets (NP 10 A
50, NP 10 A 50 V2, NP 10 A 100 and NP 10 A 100 V2) were aligned in
time and plotted (Figure 10). The starting point for each motion
was 4,000,000. The actuators commanded to move in steps of either
one or two full steps (50 or 100 microsteps) of 39.7 nm each at
intervals of 0.409 seconds. Ten negative steps (toward the primary
mirror) were followed by ten positive steps. This was repeated
once.
Upon reversal of direction, the first step or two is small or
missing and is followed by a larger amplitude step or two. This
suggests that stick-slip is present as well as hysteresis. A
linear control system with unity gain would measure the position
error and command the nearest integer number of steps to correct
that error. This will be stable if the behavior of Figure 10 is
typical of the other actuators and at other elevations of the
telescope. It would be unstable if a step command resulted in a
motion larger than 1.5 steps, upon direction reversal, but such
behavior is not present in the data examined. If closed loop
control is feasible, it will be limited only by the step
quantization error of about 10 nm RMS and the noise and systematic
errors of the position transducer. Dynamic effects at 0.409
second/step are small. This suggests that small-signal response
time can less than 1 second.
Figure 10: Actuator hysteresis. Four
separate data sets are plotted. They are aligned in time but
displaced vertically for clarity even though the starting point
was the same in each case (4,000,000)
Conclusions
The existing focus actuators operate open loop. Their behavior
upon direction reversal limits their performance to roughly 300 to
500 nm RMS or 70 to 130 mas RMS (2D) on the sky. At this level, only
the least stringent tracking error specification of 200 mas RMS (2D)
is satisfied and the tracking error is likely to be dominated by the
focus actuators.
With suitable position feedback transducers, the existing
actuators should provide performance approaching 20 nm RMS, if our
measurements are typical. Performance at this level would satisfy the
most stringent specification for tracking error, i.e., 50 mas RMS
(2D). The error allocated to each secondary actuator is 40 nm RMS
which corresponds to a contribution of 10 mas RMS to the tracking
error. A specification for the small-signal response time of the
focusing system has not been adopted, but is likely to be about 1
second.
Measurements of the secondary axial actuators indicate the
following.
- During long focus moves, secondary tilts equivalent to image
motion of 2.4 arc seconds occur at 0° and 21.2°
elevation. No long moves were made at 90°.
- Immediately after long focus moves, e.g., those appropriate
for initial focus, short focus moves, e.g., 50 to 60 µm moves
appropriate for adjusting the focus during imaging, exhibit
direction reversal hysteresis. The hysteresis decreases in
subsequent cycles. The magnitude of the differential hysteresis is
equivalent to image motion of 470 mas, but quickly decreases to
130 to 260 mas and becomes negligible in some cases.
- Over short focus moves and in the same direction, secondary
tilts equivalent to 2D image motions of 30 to 40 mas RMS occur.
The residual error at the frequency of the first and second
harmonics of motor shaft rotation is a significant
contributor.
- Closed loop control of the existing actuators may be feasible.
Such control should result in performance limited by the step
quantization error of roughly 10 nm RMS and the noise and
systematic errors of the position transducer providing feedback.
Glass scale transducers such as the Heidenhain
CT 25 appear to have adequate performance, e.g., an accuracy
of +/-30 nm or an estimated error of 15 nm RMS. The unit cost of
the CT 25 is under $7000, but less expensive transducers may be
found. The small signal response time can be 1 second without
encountering dynamic effects.
Date created: 9/03/98
Last modified: 11/5/98
Copyright © 1998, Walter A. Siegmund
Walter A. Siegmund
siegmund@astro.washington.edu