
MEASUREMENTS OF STRUCTURAL DEFORMATIONS OF LARGE REFLECTOR ANTENNAS
Michael J. Brenner
Engineering Metrology Services
Tucson, Arizona USA
ABSTRACT
Optical surveying techniques with theodolites have been utilized for many years for static measurements of reflector antennas. This paper reports on updated optical surveying systems used to measure the accuracies and structural deformations of reflector antennas. Deformations of large Cassegrain tracking antennas during elevation rotation and a fixed, billboardstyle compact range reflector over time are discussed.
A simple surveying method is shown for the integrated measurement of Cassegrain antennas (both primary and secondary reflectors) from near the primary vertex. Other topics covered include accurate prediction of interpolated gravity deformations of rotating reflectors based on a small measurement sample, and a method for taking differences between measurements.
The use of EDM (Electronic Distance Measurement) theodolites as well as angleonly devices is described, along with software which manages both the measurement and datareduction systems.
Keywords: Reflector Structures, Reflector Metrology, Errors, Compact
Ranges
1.0 INTRODUCTION
Most large, multipanelled reflectors are aligned, at least initially, using some variant on the theodolitetape method [1][2][3], in which a vertexmounted theodolite is used to measure angles, and a radial arm of some sort measures target distance. This paper describes extensions of this general technique to the measurement of structural deformations of reflectors, including the position of the Cassegrain subreflector. The use of integrated EDM (Electronic Distance Measuring) instruments is discussed.
These techniques were applied during three recent field installations,
including a 45foot diameter Cassegrain tracking antenna used for millimeterwavelength
radio astronomy, a 31 x 22 foot billboardstyle compact range reflector,
and an 86foot diameter centimeterwavelength tracking radar antenna.
2.0 CHANGES OVER TIME
A large, serratededge compact range reflector surface
was measured using multiple triangulating theodolites on two occasions
separated by several weeks. During that time, the temperature in the anechoic
chamber changed substantially, and there was possible subsidence of the
reflector's foundation. Figure 1 shows the difference in the surface between
the two measurements.
Surface deformations normal to the reflector surface based on separate measurements are determined by taking the vector dot product of the displacement vector with the surfacenormal unit vector. This in effects projects the displacement into the surfacenormal direction. Symbolically this can be written as 

Figure 1 Compact Range Reflector Deformations 
d_{n} = D dot product with n 
d_{n} = surfacenormal displacement
D = displacement vector n = parabola normal unit vector 
3.0 CASSEGRAIN OPTICS MEASUREMENT
Cassegrain secondary reflectors (subreflectors) are often constructed with a central targeting mirror normal to the axis of rotation with etched crosshairs, and 2 or more perimeter targets at known locations. A vertexmounted theodolite is commonly used to align the subreflector to its theoretical location with respect to the primary. Such a targeting system with a theodolite can also be used to determine the position of the subreflector. Figure 2 shows a schematic layout of the theodolite and the subreflector with a central mirror target and one perimeter target. One additional perimeter target can provide information in the orthogonal axis. More targets provide redundant information, which can be used for error checking and statistical evaluations.
Geometric evaluation of Figure 2 shows that given R_targ,
Z_targ and Z_offset (target coordinates) from the subreflector manufacturer,
R_th, Z_th, tilt_th (theodolite position) calculated from field measurements
of the primary (a standard 3 degreeoffreedom parabola bestfit as described
below will provide these data), th1 and th2 theodolite measurements of
subreflector targets, the subreflector position R_subr and Z_subr can be
calculated. The subreflector tilt angle tilt_subr is based on the theodolite
angle measurement from autocollimation (or autoreflection) on the subreflector
mirror.
Unlike the use of a theodolite to measure the primary reflector targets from the vertex region, subreflector measurements require the use of nearzenith look angles, which can compromise accuracy. This accuracy can be critical due to the sensitivity of the subreflector focal distance calculation to small changes in angle. Therefore, it is important that the inherent instrument errors be removed during the measurements. The best way to do so is to take both face1 and face2 theodolite measurements. By proper combination of measured angles, the theodolite vertical index error and axis inorthogonality can be removed. 

Figure 2 Secondary Reflector Position Measurement 
Having measured the primary reflector surface targets in the usual manner
[1][2][3], the subreflector position information can be combined to provide
information on the full Cassegrain optics [4][5]. Such an analysis provides
information on where the subreflector may be moved to peak the gain (i.e.
minimize the RMS of the path length errors [6]), the rf beam pointing position
before and after subreflector adjustment, and the adjustments at the primary
reflector screw points which will produce the most accurate paraboloid
of the desired focal length.
4.0 MEASUREMENT OF GRAVITY SAG
With minor adjustments to the theodolite and a comfortable working platform for the operator, primary surface and subreflector position measurements as described above can be performed at any elevation angle for a tracking antenna. For antenna structures which behave in a linear elastic fashion, measurements at 3 separate elevation angles provides sufficient information to determine the relative positions of all targets at any other angle. (A common example of nonlinear structural deformation is the bending of bolted flanges.) The process requires the intermediate calculation of the faceup (zenith pointing) and faceside (horizon pointing) gravity deformations of the reflector, and their subsequent vector superposition. The best results will be achieved if the elevation angles at which measuree ments are mad are well separated, such as 0, 45 and 90 degrees.
The gravity deformation of a linear elastic structure at elevation angle can be derived by vector superposition of the faceup and faceside gravity vectors as
d = d_{s} cosq + d_{u} sinq 
where
d_{s} = faceside gravity deformation
d_{u} = faceup gravity deformation 
The faceup and faceside deformations described here are of the type derived from a computer structural analysis in which gravity is "turned on" from a particular direction. As such, they cannot be directly measured in the real world.
For a linear elastic structure which rotates from angle q_{1} to q_{2} the relative deformation, or motion, will be d_{2}  d_{1} as shown above. For 3 measurements, two independent linear equations can be written for d_{2}  d_{1} and d_{3}  d_{1}. Simple matrix algebra shows that
d_{u }=
d_{s }= 
sinq_{3 }sinq_{2}cosq_{3
}cosq_{2}^{1}
sinq_{3 }sinq_{1}cosq_{3 }cosq_{1} 
d_{3}  d_{2}
d_{3}  d_{1} 
Now that d_{u} and d_{s} are known, for any angle q, the predicted alignment error at a target is the gravity deformation travelling from a measured reference angle ( q_{2}) to angle superposed onto the actual measured alignment error at the reference angle. This can be expressed as
d(q) = d_{s} (cosq  cosq_{2}) + d_{u} (sinq  sinq_{2}) 
For targets on the primary reflector, the surface errors ë can be determined using the vector dot product method described above.
These concepts were recently applied to a 45foot diameter
millimeterwavelength radio telescope. For that project, upon completion
of the primary surface alignment effort, measurements were taken at 0,
45 and 90 degrees elevation. The measurements included the subreflector
position as was the primary reflector surface at 480 target locations.
Figure 3 shows the calculated RMS surface accuracy for 5 cases.
The primary reflector surface measurement data was analyzed using 3 and 6 DegreeOfFreedom (DOF) least squares parabola bestfit routines. The 3DOF solution calculates the required translation along the boresight axis (Dz), as well as rotations about the X and Y axes (ax , ay), while the 6DOF solution additionally solves for the bestfit translations of the X and Y axes ( D x, D y) and the bestfit focal length (F). Additionally, a full Cassegrain optics analysis was performed [4][5], both for the case of the subreflector in the asmeasured position and for the calculated bestfit subreflector position. As expected, the 6DOF primary analysis gives identical results to the full optics analysis with subreflector motions allowed. The only difference between them is that the full optics analysis accounts for path length effects of the feed misposition, which in this case was zero. 


Figure 3 shows that the reflector was wellaligned at the selected optimal
elevation angle of 45 degrees, and that the residual (after bestfitting)
RMS gravity deformations of the primary are greater than 7 mils. In this
case (and many others in the author's experience) the measured gravity
deformations are greater than those predicted by analysis by substantial
factor. The figure also shows that the subreflector is misaligned to optimize
performance at 75 degrees elevation instead of 45 degrees.
5.0 USE OF ELECTRONIC DISTANCE MEASUREMENT
In some cases, it is necessary to measure targets on a
primary reflector surface where the radial target locations are not known,
such as the upgrade an older antenna. For these applications, the use of
a precision tape is not practical. Fortunately, in recent years commercially
available coaxial (with the sighting telescope) Electronic Distance Measuring
(EDM) theodolites have become sufficiently accurate to be of use in the
measurement of some of the most accurate large reflector antennas. The
RMS distance accuracy of wellcalibrated instruments is about 10 mils (0.010
inch) over the ranges of interest here. A typical vertexmounted theodolite
views perimeter targets on a large parabolic primary reflector offset by
no more than 17 degrees from the surface tangent. The RMS measurement error
normal to the surface due to the EDM is therefore approximately 10 mils*sin(17
deg) = 3 mils, which is good enough for most applications.
Recently, an EDM theodolite was used to measure 1200 targets on the primary reflector of an 86foot diameter tracking radar. Measurements were made at 4 elevation angles. The surface deviations derived from the measurements were combined in 4 ways, thus providing 4 separate calculations of the faceup and faceside gravity deformations. The differences among these results gives an indication of the degree to which the assumption of linear elastic behavior is valid, as well as a check on the measurement accuracy. The three most reliable data sets were combined as described above to determine the faceup (90 deg) and faceside (0 deg) gravity deformations, as shown in Figure 4. Two features are apparent in the figure. First, as expected, in the faceup orientation the reflector open up, increasing the focal length. while in the faceside measurement the opposite is true. Second, a ring structure is apparent, which indicates the action of the reflector backstructure. 
Gravity Deformations at 0 Degrees Elevation

Figure 4 FaceUp and FaceSide Gravity Deformations 
6.0 COMPUTERS AND SOFTWARE
Clearly, the techniques described above require the collection and analysis
of large amounts of data. Custom software has been developed for field
use which downloads theodolite angles and distance measurements to a laptop
computer via a serial connection. In practice, the operator aims at a target
and clicks a trigger switch to measure a target location. The software
provides information needed to take all of the necessary measurements,
and performs all of the necessary field calculations. It provides surface
maps as shown above, RMS accuracies based on various degrees of bestfitting,
as well as the present location and adjustment requirements to the subreflector
position which will optimize antenna performance.
7.0 CONCLUSIONS
These techniques have proved useful both in the installation of new equipment and the upgrading of existing facilities. They can be used with finite element structural analysis tools to qualify new designs or to upgrade existing facilities.
While the full optics analysis of Cassegrain antenna systems was described,
these general techniques would apply as well to prime focus or Gregorian
systems.
REFERENCES
[1] M. J. Brenner, A. Joel Ellder, M. S. Zarghamee, "Upgrade of a Large MillimeterWavelength Radio Telescope for Improved Performance at 115 GHz", Proc of the IEEE, vol. 82, no 5, pp. 734741, May 1994.
[2] M. J. Kesteven, B. F Parsons, D. E. Yabsley, "Antenna Reflector Metrology: The Australia Telescope Experience", IEEE Transactions on Antennas and Propagation, vol. 36, no.10, pp.1481 1484, October 1988.
[3] A. Greve, "Reflector Surface Measurements of the IRAM 30m Radio Telescope", International Journal of Infrared and Millimeter Waves, vol. 7, no. 1, 1966.
[4] M. S. Zarghamee, J. A. Antebi, "Surface Accuracy of Cassegrain Antennas,", IEEE Transactions on Antennas and Propagation, vol. AP33, no. 8, pp. 828837, August 1985.
[5] M. S. Zarghamee, "Peak Gain of a Cassegrain Antenna with Secondary Position Adjustment," IEEE Transactions on Antennas and Propagation, vol. AP30, no. 6, pp. 12281233, November 1982.
[6] J. Ruze, "Antenna Tolerance Theory  A Review", Proc of the IEEE,
vol. 54, pp. 633640, 1966.
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