Post-fabrication metrology and analysis of the LMT segmented secondary reflector.

David M. Gale(a)†, David R. Smith(b), Andrea León Huerta(a), Carlos Tzile Torres(a), Emilio

Hernández Rios(a), David Castro Santos(a), G. Valsecchi(c), R. Bianucci(c). (a) Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Luis Enrique Erro #1,

Tonantzintla, Puebla, CP72840 México. (b) MERLAB, PC 357 S. Candler St., Decatur, GA 30030, USA

(c) Media Lario S.r.l., Località Pascolo, 23842 Bosisio Parini (LC), Italy

ABSTRACT A new 2.7-meter segmented secondary reflector has been delivered to the Large Millimeter Telescope (LMT) for coupling to the recently completed 50-meter primary. The segmented reflector was designed and manufactured by Media Lario S.r.l. in Lombardy, Italy, using the same laminated Nickel panel technology employed by the LMT for the full 50-meter primary surface. Media Lario used their in-house coordinate measuring machine to adjust the surface during assembly, with the reflector panels facing upwards. As part of the Final Acceptance Review measurements of the surface were undertaken by LMT staff at the Media Lario factory, using both a laser tracker and photogrammetry. Measurements were also made of the electroforming mold for the central panel. The reflector was mounted on a rotating stand allowing surface measurements to be performed according with the respective gravitational load cases. Measurements at the Media Lario factory provided a useful reference for repeat data taken at the LMT site, since the reflector was shipped as a fully assembled unit, designed to require no further adjustment after leaving the factory. In this paper we present the surface measurements conducted during the review, and comparisons of the observed gravitational load deformations with those predicted by FEA. Although the latter were often at the level of measurement uncertainty, we were able to verify specific cases, as well as performing a sanity check on the manufacturer's design analysis. The measurements confirmed final surface error values leading to reflector acceptance by the project. An RMS surface error of the order of 25 microns over the entire reflector was recorded at 60 degrees elevation using photogrammetry data after adjusting to the best-fit parabola, showing compliance with the LMT specification. We also present final surface measurements taken at the LMT site prior to installation. Key words: segmented reflector, laser tracker, photogrammetry, laminated nickel panel, gravity deformations.

1. INTRODUCTION

The Large Millimeter Telescope is situated at the summit of the Sierra Negra volcano 200 km east of Mexico City, and is a joint venture between the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE) in Mexico, and the University of Massachusetts Amherst (UMASS) in the US. The telescope is located in the mountainous region to the East of the Mexican altiplano, at an altitude of 4,600 m (15,092 ft), and operates without any environmental enclosure. 2017 saw completion of the final construction phase of the LMT, with the installation of the full 50-meter primary reflector, following three years of operation as a 32-meter facility. A key activity during this period was the replacement of the existing 2.7-meter monolithic aluminium secondary mirror with a lighter segmented reflector, offering improved surface accuracy and better long-term maintenance capability through the acquisition of spare replacement surface panels. Although the existing monolithic mirror was improved within the central zone corresponding to the 32-m aperture in 20121, this component was not suitable for coupling to the full 50 m primary aperture.

† dgale@inaoep.mx phone +52 222 266 3100 x5307, 5317 www.lmtgtm.org

The segmented reflector was designed and manufactured by Media Lario S.r.l. in Lombardy, Italy, using the same laminated nickel panel technology employed by the LMT for the full 50-meter primary surface. Nine electroformed panels are supported on a steel back-structure by means of precision adjusters, see Figure 1, allowing limited surface shape adjustment during final assembly of the reflector. A finite element analysis (FEA) was carried out by the contractor during the design phase in order to characterize reflector performance for all major load cases. Details of the design, analysis and fabrication of the secondary reflector have been reported previously2.

Figure 1. General schematic and close-up view of the segmented subreflector, showing panel adjusters and back-structure.

2. IN-HOUSE METROLOGY DURING REFLECTOR INTEGRATION

Metrology for the panel molds and the individual panels was undertaken on Media Lario´s Poli TCX Coordinate Measuring Machine (CMM). The same instrument was used to align the panels to the back-structure during reflector integration2. The CMM has a Maximum Permissible Error (MPE) of 7 + 9·L µm over the measurement volume of 2.2 m x 3 m x 1 m, where L is the maximum measurement length in meters. The RMS measurement repeatability of the CMM was reported to be within 1 µm. The full surface was measured over 2,684 points, 540 uniformly distributed on the central panel and 268 with radially-dependent density on each of the eight lateral panels, see Figure 2. The measurement routine included correction for the radius of the measurement stylus. Since the diameter of the assembled subreflector exceeds the maximum CMM measurement volume in one direction, integration measurements were carried out in two steps, first on the central panel and 4 external segments on one side, then on the central panel and the remaining 4 external panels on the opposite side. The entire surface profile was reconstructed by stitching the two measurement sets, using the central segment as a common reference. Figure 2 shows the partially assembled reflector on the CMM table. It should be noted that the panels were integrated and aligned at room temperature and in a face-up orientation, while under operating conditions M2 will face downward, in a nominal 0°C environment. Therefore, during integration a compensation was made for temperature and gravitational load cases obtained from the design-phase finite element model. Applied corrections were for the 0 °C thermal load case and correcting Z gravity for face-up integration. For the thermal scaling calculation the CTE of Nickel (13·10-6 K-1) was used.

For each data point (Xi, Yi), the residual ∆Zi error between the measured coordinate Zi and the nominal value of the theoretical hyperboloid was calculated. Figure 3 shows the final surface residual map, together with P-V and RMS values of the surface error.

Figure 2. In-house reflector measurement: (L) distribution of probing points; (R) measurement of partially assembled reflector.

Figure 3. Final map of RMS surface error obtained on the Media Lario CMM. Higher errors can be seen at the panel boundaries.

3. METROLOGY PROCEDURE FOR THE FINAL ACCEPTANCE REVIEW The LMT project sent a metrology team to conduct laser tracker and photogrammetry measurements on the completed article at the final acceptance review, held at the Media Lario manufacturing facility in Lombardy, Italy, in May 2016. The objective was to carry out independent surface measurements, which, while expected to have lower accuracy than the CMM measurements, would provide due diligence that there were no obvious problems with the surface shape. Furthermore, the project request-for-quote specifically required that measurements be made for at least two orientations of the mirror, as a sanity check of gravity loading performance. Figure 4 shows the orientations for which measurements were obtained during the review. The client designed a rotating mount for the reflector as part of the requirement for provision of handling and installation tools, so this infrastructure was made sufficiently stiff for use as a mirror support during the metrology activity. Figure 4 illustrates the handling frame, a rectangular brace that is bolted kinematically to the mirror back-structure, and a detachable transport cart on which it rotates. All measurements were conducted at room temperature in a quiet, open area of the manufacturing hall. Temperature probes were attached to the reflector for continual monitoring of the daily thermal conditions in the work area. Prior to starting metrology activities the reflector surface was targeted for photogrammetry, see section 3.2 for details.

Figure 4. Top: reflector orientations considered in this paper. Bottom: mirror attached to rotating handling frame and transport cart.

3.1 Laser tracker measurementsThe LMT project hired an API Radian laser tracker locally from Microservice S.r.l., which was operated by project metrology staff. Due to the size and orientations of the reflector, Media Lario were asked to fabricate two custom tracker stands in advance. These allowed the tracker to be positioned in a stable manner both above and below the reflector depending on its orientation, see figure 5. A spherically mounted retroreflector (SMR) was scanned manually over the reflector surface with a sampling interval of 20 mm, and scan lines spaced roughly every 70 mm, giving a total of 4,900 raw data points. This data set was reduced by selecting data points with a sampling interval of 60 mm, resulting in 325

samples for the central panel, and 167 samples for each lateral panel. Figure 6 shows the scan trajectory for the full surface. Note that since the reflector had first been targeted for photogrammetry, all tracker scans had to be made carefully to avoid touching these targets. Each tracker measurement also included 4 fiducial points from SMR nests glued temporarily to the handling frame, taken before and after the surface scan to identify reflector orientation and monitor the stability of the setup, and the measurement of a reference invar scale bar.

Figure 5. Tracker location on purpose-built stands for face-up and face-down orientations of the secondary reflector.

Since the laser tracker measurements could be susceptible to increased noise as a result of local air turbulence, the overhead air conditioning system was turned off during and before periods of data collection. Screens were also located around the work area. Approximately 35 minutes were required to obtain each surface map.

Figure 6. Left: Typical surface scan path for tracker measurements. Right: Tracker SMR in contact with the reflector surface.

3.2 Photogrammetry measurementsThe LMT metrology team took a Geodetic Systems INCA3 photogrammetry camera to the Media Lario factory to provide a second metrology option. The reflector was targeted with 1,116 6mm reflective targets, 298 over the central panel, and 102 on each lateral panel. 20 coded targets on rectangular plates were distributed about the surface to assist the V-Stars bundling software with "stitching" of camera images. A 3-meter carbon scale bar for photogrammetry image scaling was placed to one side of the reflector, see figure 7.

Figure 7. Left: Photogrammetry sampling pattern. Right: measurement in progress. Note the carbon and invar scale bars lower left.

Camera locations were distributed around the surface at distances of between 1 and 3 m, and covering a range of angles with respect to the mirror axis. Coverage was 360° around this axis. The total number of images per map was of the order 360. Figure 8 shows camera station distribution for the reflector face-up, El -90°. Each photogrammetry map took approximately 20 minutes.

Figure 8. Photogrammetry camera stations for the face-up (-90°) orientation. Left: plan view, Right: side view.

4. TRACKER AND PHOTOGRAMMETRY DATA SETS 4.1 Data processing The secondary reflector is a hyperboloid of revolution with outer diameter 2.63 m, defined by Equation 1.

𝑍 𝑥, 𝑦 = − !!!!

1 + (!!!!!)!

∙ 1 + 𝑘 − 1 (1)

Where the design radius of curvature R=1768.63 mm and the conic constant K=-1.14269 at 0°C. In order to obtain the deviation of the surface respect to the ideal shape the data cloud is best-fitted to Equation (1) using a least-squares algorithm. The surface fit consists of transforming raw data in local coordinates by rotation and translation along the x, y, and z-axis, then fitting against the model to minimize the deviation between them. During this process the target used for the measurement is compensated at every point by subtracting the normal distance to the local surface, which is the SMR radius for laser tracker measurements, or the retro-reflective target thickness for photogrammetry. This ensures that the data point cloud lies on the surface and not above it. Since the mirror has high curvature, the mirror shape itself is used to constrain the fit, and in consequence no fiducial points are required to locate the surface, even though fiducial targets were located on the reflector support for orientation purposes. The fitting algorithm uses the Levenberg-Marquardt technique3 to solve the least-squares problem. This technique is a particular strategy for iteratively searching for the best-fit; it tries various values for the unknown coefficients, and for each attempt it computes chi-square searching. The best values of the coefficients are ones that minimize the chi-square value. The search process involves starting with an initial guess at the coefficients. After the data has been fitted, it is compared to the ideal geometry, obtaining residuals that represent the distance between the fitted and ideal surface. Finally the surface RMS is calculated, and where appropriate, any data outliers are detected and removed, and the RMS value adjusted accordingly. For the quantitative evaluation of reflector surface compliance with the design shape, the parameter of interest is the Root Mean Square (RMS) error, denoted by ERMS or simply "RMS":

𝐸!"# =(!!!!!)!

!!!!! (2)

Where 𝑧 represents the component of a measured surface coordinate in the direction of optical propagation and 𝑧 is the corresponding component of the theoretical or design surface. 𝐸!"# is calculated by computing the residual (𝑧! − 𝑧!) at each data point, after appropriate fitting of measured and theoretical point clouds. Data outliers occur periodically in the point clouds, and are usually the result of occasional contaminants trapped beneath the tracker SMR as it is scanned, or from damaged or poorly seated photogrammetry targets. The modified Z-score method in used to detect outliers, since it is a more robust statistical technique than the standard deviation The modified Z-score is defined as: 𝑀! = !.!"#$ (!!!! )

!"# (3)

Where 𝑥 is the mean and MAD denotes the median absolute deviation. With this indicator we determine how many MADs each data point lies from the mean. The cutoff value is 3.5. The MAD is defined as: 𝑀𝐴𝐷 = 𝑚𝑒𝑑𝑖𝑎𝑛 𝑌! − 𝑚𝑒𝑑𝑖𝑎𝑛 𝑌 (4)

The MAD is less affected by extremes in the tail because the data in the tails have less influence on the calculation of the median than they do on the mean. During the acceptance review all data was analysed independently by staff from the LMT metrology team and from MERLAB, allowing the continual cross-checking of results during the course of the meeting.

4.2 Data sets obtained during the acceptance review Measurements were conducted over a period of eight days, with days 1 and 2 used primarily for setting up, and training for API Radian tracker operation (the LMT metrology technicians have extensive experience with Faro and Leica trackers). A total of 39 surface maps were taken, together with two measurements of the steel mold used to fabricate the central panel as a reference test, see section 6. Initial metrology tests concentrated on the -90° orientation, (see Figure 7, right-hand image) since this places the reflector close to the floor and provides the most stable test setup. Early maps showed metrology artefacts such as SMR scanning deficiencies and non-optimised photogrammetry camera stations. These maps were discarded form the analysis, while the metrology process was fine-tuned for both techniques. Table 1 summarises the data sets considered to be sufficiently reliable for analysis. Three repeat measurements (column "N") were logged for each technique and for each reflector orientation. Table 1. Summary of reflector measurements showing computed RMS surface error and best-fit radius of curvature.

Orientation N Raw data Outliers removed Raw data Outliers removed Laser Tracker RMS/um SD RMS/um SD RoC/mm SD RoC/mm SD

-90 deg 3 28.3 4.4 21.7 0.3 1769.951 0.011 1769.968 0.011 90 deg 3 30.5 0.6 26.4 0.7 1770.336 0.021 1770.350 0.024 60 deg 3 34.0 2.7 28.5 1.1 1770.383 0.040 1770.403 0.040

0 deg 3 32.3 0.3 28.6 0.1 1770.180 0.016 1770.197 0.016 Photogrammetry RMS/um SD RMS/um SD RoC/mm SD RoC/mm SD

-90 deg 3 27.1 0.3 23.3 0.1 1770.061 0.023 1770.057 0.025 90 deg 3 27.9 0.3 23.2 0.2 1770.312 0.016 1770.341 0.022 60 deg 3 29.2 0.9 25.1 1.3 1770.192 0.009 1770.209 0.007

0 deg 3 30.1 0.2 26.4 0.1 1770.137 0.007 1770.153 0.007 For each data set the theoretical radius of curvature (RoC) was allowed to float in order to minimise the RMS surface error; small variations in RoC can be compensated later by a focus shift on the antenna. Table 1 shows final RoC and RMS error values for each data set, before and after removal of outliers. Raw tracker results give higher RMS error and larger standard deviation (SD) for all orientations, which we attribute to (i) variation of SMR scan path for each measurement, (ii) possible reflector motion resulting from SMR contact with the reflector, (iii) noise due to air currents. We note that the residual errors between repeat measurements for a single orientation are not much lower than the errors obtained from each individual measurement; this suggests that we are approaching the noise limit of the data for both techniques. Figure 9 shows residual maps for repeat measurements at 60° orientation.

Figure 9. Residual maps for repeat measurements using laser tracker and photogrammetry (60 degrees orientation).

5. MEASUREMENT RESULTS Subtraction of repeat surface maps taken at -90° orientation (the most stable reflector position) indicates a residual RMS error of around 10-12 µm for photogrammetry, and around 15-20 µm for laser tracker. These values may be considered to be the RMS contribution of each technique to the total reported measurement error. Given the higher repeatability of photogrammetry data, typically around one micron in the RMS value, it was considered as the preferred technique. Figure 10 shows averaged photogrammetry maps (three repeat measurements), for each reflector orientation. Note that the RoC value is optimised for minimum error in each case. Slight astigmatism is apparent for 0° and 60° orientations. If we consider a 10 µm error contribution for the photogrammetry technique, then the surface RMS error is of the order of 19.5 µm for face-up and face-down measurements, and does not exceed 23.5 µm for orientations with a gravitational component in the Y-direction. These results indicate compliance of the segmented secondary reflector with the upper limit of 26 µm specified by the LMT, and close approximation to the project goal of 19 µm under stable conditions.

Figure 10. Averaged surface error maps from photogrammetry data, for each orientation of the reflector.

5.1 Gravitational analysis Finite element analysis (FEA) performed by Media Lario predicts reflector surface deformation under specific load cases, including gravity and thermal loads2. While the tracker and photogrammetry measurements do not have the level of accuracy to allow quantitative verification of the FEA, the measurements can be used to check for any unexpected behaviour of the finished article arising from gravity loads. Here we consider the load cases for 1 g acceleration in the Z- and Y-directions, corresponding to the experimental load conditions facilitated by the rotating reflector mount. The FEA analysis predicts around 40 µm maximum edge deflection for 1 g in Z and Y, with overall RMS errors below 7 µm after fitting out piston, tip-tilt and RoC. See the left-hand images of Figure 11 and reference [2] for FEA results.

Figure 11. Surface error in the Z-direction corresponding to gravitational load cases. Top left: 1 g load in the Z direction, from FEA. Top right: Corresponding measured surface deformation according to equation (5). Bottom left: 1 g load in the Y direction, from FEA. Bottom right: Corresponding measured surface deformation according to equation (6).

From the experimental data we can attempt to look for the gravity deflections predicted by the FEA. A 1 g acceleration along the Z-axis can be obtained from face-up and face-down measurements as follows: 𝐺! = !"#$%&"!"!!"!!"#$%&"!"!!"

! . (5)

While 1 g acceleration along the Y-axis can be obtained from: 𝐺! = 𝐺! − (𝑀𝑒𝑎𝑠𝑢𝑟𝑒!"!!" + 𝑀𝑒𝑎𝑠𝑢𝑟𝑒!" !) (6)

The right-hand images of Figure 11 show surface error maps obtained from Equations (5) and (6). The observed deformations are of the order of the measurement uncertainty limit as expected. However we note that the magnitude of the RMS errors are similar for FEA and the measured surface form. The "measured" variation in RoC between -90° and +90°, determined by floating RoC for best fit, is around 280 µm (see the -90° and +90° images of Figure 10), while FEA predicts around 140 µm after inverting the mirror. Note also that the behaviour of the panels closely matches FEA for the Gz case, while the Gy case presents more obvious differences. The FEA indicates that a change in Radius of Curvature is the dominant effect; for surface measurements we systematically fit to the best RoC, hence avoiding the maps being dominated by a large focus term. 5.2 Thermal analysis The FEA analysis considered the thermal load case for ΔT=10°C, and predicted a change in RoC of 226 µm. The left-hand image of Figure 12 shows the FEA deformation plot. As a test on the real article we applied a 10°C temperature compensation (offset from room temperature) to the face-up data, and repeated the best-fit with floating RoC to obtain the new focus value. The right-hand image of Figure 12 shows the resulting residual map, with a focus term of 230 µm.

Figure 12. Left: FEA Thermal load ΔT=10°C (Media Lario). Right: Measurement residual map for a 10°C temperature change.

6. MEASUREMENT OF THE CENTRAL PANEL MOLD In order to further investigate photogrammetry accuracy, surface measurements were taken of the steel mold used to form the central reflector panel. Prior to the fabrication of this panel the mold was hand-polished to achieve a surface RMS error of 3.96 µm, as measured by the Media Lario CMM using a total of 1,247 probe points. Two photogrammetry measurements were taken using 314 targets, and differed from each other in the RMS error by 0.9 µm. Figure 13 shows a surface map from the CMM data, after reduction of the sampling density to 314 points, and a single photogrammetry map, obtained by allowing the RoC to float and after removing an astigmatic term. If we take the high-density CMM measurement to represent the true surface error for the mold, and consider the mean of the surface RMS error provided by the two photogrammetry maps, then the measurement error contribution introduced by the photogrammetry technique is 8.1 µm.

Figure 12. Left: CMM surface map of the central panel mold. Right: Photogrammetry measurement of the same object. Both data sets have 314 surface points.

7. CONCLUSIONS

Surface measurements of the LMT segmented secondary reflector were carried out at the manufacturer's facility in Italy prior to shipping the assembled unit to Mexico. The measurements were carried out using both a laser tracker and photogrammetry. By comparing the measurement results with data from the manufacturer's CMM, the error contribution due to metrology technique was determined to be of the order of 10-12 µm for photogrammetry and 15-20 µm for the laser tracker. Measurements of the reflector mounted at different orientations allowed the project to perform a sanity check on the surface deformations due to gravitational loads, predicted by Finite Element Analysis during the design phase. Although close to the noise limit, we were able to confirm the predicted deformations to within an order of magnitude for distinct cases. Most importantly, the measurements confirmed compliance of the new secondary reflector with the LMT surface error requirement. The setting procedure for the secondary surface panels had contemplated the possibility of bias-rigging the surface for a preferred working elevation angle, since the reflector was aligned in the face-up position on the factory CMM. The

measurements described in this paper indicated that bias-rigging was not feasible using our existing portable metrology equipment due to the inherent error of these systems. At the same time, the FEA analysis was not considered to be sufficiently reliable to risk bias rigging on the CMM. Photogrammetry maps tended to show some astigmatism at the extreme limit of sensitivity, which was not present in the CMM data. The reason for this is not known.

ACKNOWLEDGEMENTS

The authors would like to thank colleagues at INAOE and Media Lario Technologies for their help with logistics in preparation for the Final Acceptance Review, and for continual support at the Media Lario facility during the activities. The Large Millimeter Telescope project is grateful for the sustained support from the Mexican National Council for Science and Technology (CONACyT).

REFERENCES

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