Sloan Digital Sky Survey Telescope Technical Note 19980903
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.
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 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
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
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)
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)
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.
Date created: 9/03/98 Last modified: 11/5/98 Copyright © 1998, Walter A. Siegmund Walter A. Siegmund
