Construction of a feedback-controlled positioning system for
the Eöt-Wash LISA small force measurement
Colin Connolly, Stanford University
advisor: Jens Gundlach
UW Research Experiences for Undergraduates Program in Physics
We constructed a positioning system for the attractor plate of the Eöt-Wash LISA small force
measurement feedback-controlled via software using position input from an infrared laser
position sensor. The system was used to drive a simulated attractor plate in sinusoidal motion
with average precision of one micron. The systematic influence of hysteresis in the piston
pump was the limiting factor; an improved pump should improve the performance of the
system by nearly an order of magnitude.
must be eliminated. The 2- to 4-mm spacing
INTRODUCTION
Direct evidence for gravitational waves
between the proof masses and spacecraft
(GW) has yet to be found. Observations of the
housing potentially allows for small forces
PSR 1913 + 16 binary pulsar have indirectly
between the metal surfaces. Forces due to
indicated their existence, but current
patch effect potentials are one class of small
observatories still lack the sensitivity
forces that has not been studied in much detail.
necessary to observe GW events on laboratory
Macroscopic metal surfaces are a mosaic of
timescales. The Large Interferometer Space
patches of different exposed crystal planes
Antenna (LISA) is a joint collaboration
with different surface dipole moments. A
between NASA and the European Space
difference in the work function of two closely-
Agency (ESA) that will achieve sensitivities
spaced metals causes a small voltage between
great enough to observe GW events such as
them. The forces due to this effect are
black hole mergers and potentially map the
expected to be small, but may be large enough
GW background left over from the creation of
to require greater spacing between the proof
the universe.
masses and spacecraft for the LISA
The LISA satellites will form vertices
observatory.
of an equilateral triangle in space with laser
Measurement of these small forces, as
beams traveling between them. Each satellite
well as others due to outgassing, thermal
will serve as an interferometer. The laser
gradients or other effects, is being made with a
beams will reflect from cubical metal proof
torsion step pendulum by the Eöt-Wash group
masses within each spacecraft. For LISA to
at the University of Washington. The
achieve its target sensitivity of a 10-23
pendulum will be driven by a closely-spaced
minimum fractional length change, any force
attractor plate which will move sinusoidally
coupling of the proof masses to the spacecraft
over a range of about 1 to 2 mm. To avoid
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electrostatic coupling the attractor plate will be
drives the positioning system, which directly
moved with an oil bellows fed by a tube from
influences the sensor’s measurement. User
outside the vacuum chamber.
control on the software side defines the
This paper details the construction of a
equilibrium point.
feedback-controlled positioning system for the
The position sensor consists of an
attractor plate component of the Eöt-Wash
infrared laser beam reflecting from a moveable
small force measurement for LISA.
mirror onto a split photodiode. The laser beam
is generated by a 5-mW infrared
semiconductor laser. The light passes through
EXPERIMENTAL METHOD
The experimental apparatus for the
a lens and collimator to produce a parallel
feedback-controlled positioning system can be
beam 1.1 mm in diameter. The beam strikes
summarized as three interdependent systems:
the mirror with an incident angle of 45º and
first, the infrared laser position sensor; second,
reflects to reach the photodiode orthogonally.
the motor-driven positioning system; and third,
As the mirror moves along the normal, the
the software feedback loop. The sensor feeds
beam translates across the photodiode surface.
position information to the feedback loop,
The dynamic range of the position sensor is
where the information is analyzed and an
approximately 1 mm. The fraction, θ, of the
output voltage is produced. This voltage
differential photocurrent between the two sides
FIG 1: Experimental setup: the motor and pump assembly at right drives the bellows at center to move the steel
block at left. The position of the block is measured by the position sensor at left.
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of the photodiodes to the total photocurrent is
notch limits the P term’s effect for small errors
roughly linear in mirror displacement over that
while preserving it for large errors. With this
range.
notch, the P constant can be set very high so
The positioning system compares its
that it supplies a strong restoring force to keep
input voltage to a voltage divider output via a
the mirror near its target position, yet it will
high-gain differential amplifier and a push-pull
not cause the output to become unstable near
transistor driver to power the motor. The
the target. Likewise, the I term will only
voltage divider is set by a geared multi-turn
change significantly once the P term has
potentiometer driven by a gear on the motor
reduced the error to a manageable level,
shaft. The turning of the motor drives a
reducing unwanted overshoot.
hydraulic piston which draws or pushes oil
A hysteresis was observed in the
through a tube to a copper bellows in contact
piston’s performance. When the piston
with the mirror mount. Since the piston
changed direction, there was a “dead zone” in
location defines the potentiometer voltage,
which the motor would spin without oil
choosing an input voltage will cause the motor
flowing, presumably due to the o-ring in the
to spin in the appropriate direction until the
piston assembly rolling over. To overcome
voltage difference is eliminated and the piston
this effect, the feedback loop coupled to every
has reached the desired location.
large direction change a short (approximately
The choice of input voltage is made by
700-ms) burst of maximum voltage to preserve
a modified PID feedback loop. The loop
linearity.
defines the current error as the difference
between the current position measurement and
RESULTS AND ANALYSIS
the target position. The sum of two terms, a P
Two periods of sinusoidal plate
term proportional to this error and an I term
motion are displayed in Figure 2. The lower
proportional to the integral of the error over
peaks follow a sine wave closely with only a
time, defines the output voltage to the
small effect of the piston hysteresis still visible.
positioning system. The usual D term
The peaks for positive θ, however, show a
proportional to the change in error is omitted
sharp step and are followed by recovery
because phase delays causes it to increase the
oscillations. The step is caused by the voltage
noise in the system. An additional Gaussian
burst, since the “dead zone” is not truly “dead”
envelope limits the range of the I term to very
when going in the negative θ direction. When
small errors and a corresponding Gaussian
the motor spins quickly for the burst period
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there is a small oil movement that jogs the
then there would be no large deviation from
measurement off the target. There is then a
this noise level.
“dead zone” in the other direction as the
Figure 5 is a Fourier transform of the
feedback loop attempts to compensate. This
periodic motion. The 300-s period component
unwanted effect is comparable in magnitude to
is clearly visible, as is a small second
the motor motion without applying a burst.
harmonic near 6.7 mHz and a multitude of
Thus the burst was only effective for half of
resonances with periods of less than 10 s.
the peaks.
These high-frequency peaks are natural
While the voltage burst was not able to
recovery frequencies of the positioning system
eliminate the effects of the piston hysteresis,
and would be much smaller if the system
the waveform is still very reproducible. The
maintained consistent sub-micron accuracy,
blue points in Figure 3 are a superimposition
thus avoiding the need to recover. The size of
of all 50 periods of the data set. The red points
the second harmonic should also be reduced
are the residuals of each period from a spline
by smoothing the top peak. All these
of a single period (the period chosen for the
contributions are at least two orders of
spline was the one that minimized the average
magnitude weaker than the 300-s resonance,
residuals). The horizontal lines are the 1-σ
however, especially near the primary
average deviation. Clearly the points of
resonance itself.
greatest nonreproducibility occur at the upper
peaks at the onset of and during the recovery
SUMMARY
from the voltage burst.
The positioning system was observed
The red curve in Figure 4 shows twice
to reproducibly maintain the mirror on average
the average standard deviation of a set of one
within a micron of its target, although
point from each period for each point in time
deviations of 5 to 8 microns were observed
(e.g. a set of all the first points, a set of all the
from the common waveform. The limiting
second points, etc.). The plate position
factor is the hysteresis of the piston in the
deviated from the target within these bounds
positioning system. It is reasonable to
for 95% of the data collected. The quiet
estimate a factor of 5 to 10 improvement with
region before the top peak is very calm, the
the elimination of the hysteresis alone. The
position staying within half a micron of the
infrared laser position sensor is sensitive to at
target very consistently. It is then reasonable
least this precision.
to expect that if the hysteresis was removed
The hysteresis could be eliminated by
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the use of a peristaltic pump, which could also
potentially increase the resolution of the
positioning system, as well; or by the use of a
second bellows instead of a piston.
ACKNOWLEDGEMENTS
I am very grateful for the guidance and
patience of Jens Gundlach. I also very much
appreciate the advice, assistance and expertise
of Stephan Sclamminger and Eric Swanson.
REFERENCES
C. C. Speake, Class. Quantum Grav. 13, A291
(1996)
L. Carbone et al., Achieving geodetic motion
for LISA test masses: ground testing
results, (2003)
J.B. Camp et al., J. Appl. Phys. 71 783 (1992)
J.B. Camp et al., J. Appl. Phys. 69 7126 (1991)
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FIG 2: two periods of attractor plate motion.
FIG 3: 50 periods of attractor plate motion (blue, printed for phase information); residuals from cubic spline of one
period (red).
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FIG. 4: Two-σ deviation of averaged data from cubic spline. The attractor plate motion was within these bounds
95% of the time.
FIG. 5: Fourier transform of attractor plate motion
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