Pasadena, California October 24-25, 2007 Contributors to the development effort: from IMTEC

1. TMT M1 Segment Support Assembly (SSA) Preliminary Design Review (PDR)Volume-2: SYSTEM LEVEL CALCULATIONS(See last slide for Revision History) Pasadena, California October 24-25, 2007 Contributors to the development effort: from IMTEC RJ Ponchione, Eric Ponslet, Shahriar Setoodeh, Vince Stephens, Alan Tubb, Eric Williams from the TMT Project George Angeli, Curt Baffes, Doug MacMynowski, Terry Mast, Jerry Nelson, Ben Platt, Lennon Rodgers, Mark Sirota, Gary Sanders, Larry Stepp, Kei Szeto TMT Confidential The Information herein contains Cost Estimates and Business Strategies Proprietary to the TMT Project and may be used by the recipient only for the purpose of performing a confidential internal review of the TMT Construction Proposal. Disclosure outside of the TMT Project and its External Advisory Panel is subject to the prior written approval of the TMT Project Manager. * Note: HYTEC, Inc. merged with IMTEC Inc. in March 2007.

2. Outline • Volume-2: System Level Calculations • M1 Segmentation • Segmentation Correction (for Variable Segment Geometry) • Budgets: • Installation & Alignment • Edge Gap • Actuator Stroke • Mass

3. System-Level Calculations SEGMENTATION(see Backup Slides for more detail)

4. y y y y y y y y y y y y y x x x x x x x x x x x x x Segmentation • M1 Array Segmentation • Six identical sectors of 82 unique segments • Unique hexagonal shape • Unique optical figure • Segments (PSAs) clock 60 deg between sectors A A A A B2 B1 C2 C1 PSA C-Sys. (XY) Actuator & AAP Locations (Dots)

5. Segmentation Scheme • Segmentation Overview: • Hexagons on a curved surface cannot be equal and regular (with regular gaps) • Segment outlines are determined by projecting a hexagonal array onto the optical surface, resulting in irregular hexagons (varied size and shape): • Constant gap segmentation • Segment size and shape variations are important for many reasons: • Affecting: Optical performance, Hex-correction, Size of mirror blanks, Length variation of Mirror Cell Top Chord members, AAP adjustment range… • By stretching the Base Pattern in-plane before projection, we can affect these resulting characteristics. Vertices in optical surface ZM1 project vertices and center onto optical surface, // ZM1 XM1 Center (scaled) (RM1) Scaled hex array Scaling rule Base pattern: regular hex array

6. Segmentation Scheme • Segmentation studies : • Radial scaling of base pattern: • 9 different objectives evaluated • “Rule 7” selected a = 0.165 • Minimize blank size • Other metrics also favorable when a = 0.165 (Only radial scaling studied, other parameters might be used for scaling, such as Azimuth) Where: a is the scaling parameter Rmax is the largest vertex radii (before scaling) R is the radial coordinate of a point in the base pattern to be scaled Rscaled is the scaled radius of the point in the scaled pattern. k is the paraxial radius of curvature of M1 All coordinates in the M1 system Scaling Rule1 • Objectives evaluated: • Minimize Irregularity • Minimize Variation of Segment Area • Minimize Variation of Circumscribed Dia. • Minimize Cell top bar length range • Minimize SSA aligner range • Minimize Edge angle scatter (diffraction) • Minimize diameter of largest circumscribed circle • Minimize Max Pivot Shifts to rebalance Whiffletree • Minimize Max figure residual after correction Note: 1. Per Mast and Nelson

7. Segmentation Scheme • Sensitivity of objectives: a = 0.165 Nominal cell member length = 1.24m 7: Max. Circumsc. Diam. – 1.21m 1: Max. Irregularity (mm RMS) 8: Max WT Pivot Shift (mm) Value of metric 5: Max In-Plane Alignment Range (mm) 6: STD Edge Angle Scatter (mrad) 9: Max Pivot Shift resid. (nm) 4: Range of Cell Bar Length (%) 2: Range of segment area (%) 3: Range Circum. Ø (%) Value of tuning parameter a

8. Segmentation Scheme • Implementation: • Scaling rule affects the following: • Mirror shape and size • Coordinates of mirror vertices • Definition of optical origin and PSA coordinate systems (unique for each seg.) • Mirror cell node locations • Mirror cell top-chord length & variation • Position of AAP Post relative to Fixed Frame hole • Segmentation Database contains the following parameters: • PSA origins and Coordinate Axes in M1 Coordinate system • Mirror vertices • expressed in PSA and M1 coordinate systems • Location of Segment Clocking Mark in PSA XY-plane (Arrow points to center of M1) • Coordinates of AAP mounting pads on mirror cell top chord • Best fit radius for AAP mounting hole in Fixed Frame • minimize adjustment range over 82 segments • Location of edge sensor positioning fiducials in PSA coordinate system • Segmentation Database under Revision Control at IMTEC • See backup slides for excerpt from Segmentation Database

9. System-Level Calculations SEGMENTATION CORRECTION(see Backup Slides for more detail)

10. Segmentation Correction Approach • Segment irregularity and size variations would degrade optical performance if not compensated for • Single support system design for all 82 types: • Adjusted for each segment geometry • Correction approach: • Rebalance each whiffletree: • Pivot-point shifts • Analysis of each type required • Drill holes for whiffletree pivots in custom locations for each segment type • Low cost, automated CNC operation • Balance masses would raise part count and add mass • Analysis of worst case corrections • Max pivot shift estimate: ~3.5 mm • Axial RMS error increases ~ 10% (~1nm)

11. Segmentation Correction Analysis • Calculate unit cases (1gz applied to distorted segment) • Size Variations (Grow and Shrink) • Clocking • De-center (X,Y) • Irregularity • Seven postulated cases (an approximate set, not orthogonal) • Unit effects isolated by subtracting 1g RMS (in quadrature) • Results show that pivot shifts are effective at compensating for segmentation (Next Slide) • Residual RMS is acceptable • Magnitude of pivot shifts practical • Note: This work was performed on the 1.2m segment • Results suggest the correction approach and have not been repeated on the 1.44m design.

12. Segmentation Correction Analysis • Evaluated 12 Cases Shown (for 1.2m segment) • Conclusion: Pivot shifts very effective correction method 10.2nm for 1.44m segment

13. Segmentation Correction Analysis • Hardware Implementation of Pivot Shifts Can shift Pivots Several mm in-plane

14. System-Level Calculations INSTALLATION & ALIGNMENT BUDGET

15. Installation & Alignment • Alignment & Registration • Estimate segment position errors • in-plane, clocking, piston and tip/tilt • Due to: • Registration - Clearance and Repeatability • PMA Assembly Errors - Tower to Optical Origin/Axes/Plane • Fixed Frame Alignment Errors - At Targets • Target to Fixed Frame Tower-Attachment Tolerances • Surveying Errors - Measurement Uncertainty (TMT Project Responsibility) • Position error estimates are based on RSS of various effects • Requirements are: • In-plane alignment: +/-0.200mm (0.400mm range) • Clocking alignment: +/-0.200mm at vertex (0.400mm range) • In-plane repeatability: +/-0.050mm (0.100mm range) • Clocking repeatability: +/-0.050mm at vertex (0.100mm range

16. Installation & Alignment • Alignment & Registration Inputs & Assumptions

17. Installation & Alignment • Alignment & Registration Inputs & Assumptions

18. Installation & Alignment • Alignment & Registration Results

19. Installation & Alignment • Optical impact of these positioning errors • using Terry Mast’s sensitivities • Conclusion: • Design is very close to meeting requirements • Need relaxation of clocking requirements for alignment and repeatability • In-plane repeatability: 0.100  0.125mm • Clocking repeatability requirement: 0.100mm  0.225mm at vertex • Alignment clocking: 0.400  0.450mm at vertex

20. System-Level Calculations GAP BUDGET

21. Gap Budget - Excluding Seismic • Nominal Segment-to-Segment Gap: 2.5 mm • Random Gap Reducing Effects: • PSA Manufacturing and Installation Tolerances 0.488 mm RSS Sum • Environmentally induced PSA motions 0.139 mm RSS Sum • Mirror cell deformations 0.486 mm RSS Sum RSS = 0.702 mm • Actuation Segment tip/tilt de-center: 0.754 mm • Adjacent segments with full differential tilt • Fault Condition – Controller or Human Error Linear Sum: 1.456 mm • Gap Margin = 2.500 mm – 1.456 mm = 1.044 mm • Note: Linear sum of all effects gives 2.479 mm gap change • within budget • See back-up slides for more details + Linear Sum

22. Gap Budget - Including Seismic • Assume 3.0g seismic with segment motions out of phase by 22.5 deg [0.39 factor * 3.0 * 0.203mm (1g deflection)] = 0.238 mm • Random Gap Reducing Effects: • PSA Manufacturing and Installation Tolerances 0.488 mm RSS Sum • Environmental PSA motions (w/seismic)0.275 mm RSS Sum • Mirror cell deformations 0.486 mm RSS Sum RSS = 0.744 mm • Actuation Segment tip/tilt de-center: 0.754 mm • Adjacent segments with full differential tilt • Fault Condition – Controller or Human Error Linear Sum: 1.495 mm • Gap Margin = 2.500 mm – 1.495 mm = 1.005 mm • Note: Linear sum of all effects gives 2.717 mm gap change • exceeds gap allowable (Segments may contact slightly during EQ.) • See back-up slides for more details + Linear Sum

23. Gap Budget • Gap Budget Summary: • Gap margin appears acceptable using RSS summation • Conservative linear summation shows little or no margin • No changes recommended

24. System-Level Calculations ACTUATOR STROKE BUDGET

25. Actuator Stroke Budget • Actuator Stroke Budget • PMA assembly errors are large terms, still being refined • Mirror Cell thermal distortion is significant TBD • 5mm actuator stroke seems sufficient

26. System-Level Calculations MASS BUDGET

27. Mass Estimate • Current design meets both fixed and moving mass limits: • Component sizing complete • Current CAD mass summary (no contingency included)

28. Acknowledgements Acknowledgements: The TMT Project gratefully acknowledges the support of the TMT partner institutions. They are the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology and the University of California. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Knowledge Development Fund, the Association of Universities for Research in Astronomy (AURA) and the U.S. National Science Foundation.

29. System-Level Calculations BACKUP SLIDES

30. System-Level Calculations Primary Mirror Segmentation: Detailed Discussion Of Segmentation Analysis Credit: Eric Ponslet, IMTEC

31. M1 Segmentation Problem • Define details of segmentation of M1 into hexagonal segments • M1 curvature and constant gaps leads to irregular hexagons • Infinite number of ways to define irregular hexagons on M1 surface • Limit choices by applying a radial scaling rule to an initial, regular hexagonal base pattern, in projection • Approach (3D) • start with regular hexagonal array in the XYM1 plane (“base pattern”) • use a scaling rule to distort array in plane • Current rule has one adjustable parameter • extrude (//ZM1) distorted array into optical surface • consider shape of resulting segments as projected into local frames • Implement gaps • Calculate various metrics

32. M1 Segmentation Problem • Tuning the scaling rule • Rule has one adjustable parameter • Parameter can be adjusted to achieve various goals • Tuning problem: • What are some useful goals to pursue? • What are the best adjustments of the parameter to achieve those goals? • Compromises…

33. Previous Work (1/3) • Original work by others • TMT.OPT.TEC.06.025.DRF01: Excel spreadsheet with calculated coordinates of 1.2m segmentation patterns for three scaling rules (Larry Stepp) • T. Mast and J. Nelson, “TMT Primary-Mirror Segment Shape,” TMT Report No. 58, TMT.OPT.TEC.04.001.REL01, November 2004. • L. Stepp, “Advantages and Disadvantages of Segment Geometries,” TMT.OPT.TEC.05.031.DRF01, December 6, 2005. • Initial presentations of HYTEC work • E. Ponslet, “Primary Mirror Segmentation: Issue with Rule #1,” TMT.OPT.PRE.05.087.REL01 (HPS-280001-0045), January 17, 2006 • E. Ponslet, “Primary Mirror Segmentation: Corrected Results,” TMT.OPT.PRE.06.004.REL02 (HPS-280001-0046A), January 30, 2006 • Detailed report • E. Ponslet, “TMT Primary Mirror Segmentation Studies,” TMT.OPT.TEC.06.005.REL01 (HTN-280001-0007), June 7, 2006 • Other relevant documents: • T. Mast, G. Angeli, and S. Roberts, “TMT Coordinate Systems,” TMT.SEN.TEC.05.016.DRF04, September 2005.

34. Previous Work (2/3) • Based on earlier scaling work by Larry Stepp & Jerry Nelson • Two scaling formulations • Three candidate goals for minimization (tuning of a) • Maximum irregularity of any segment • Range of segment area • Range of circumscribed diameter • Introduced concept of Best Fit Regular Hexagon (BFRH) • Least-square fit performed in local XY plane (XYSEG) • Minimizes RMS value of distances from vertices of segment to vertices of BFRH • Adjust radius of BFRH (1 parameter) • BFRH can be centered at OSEGor free to re-center (2 parameters) • BFRH can be aligned with XSEGor free to rotate (1 parameter) • Residual of LSQ fit is a measure of irregularity • Irregularity (and size variations) impacts performance (imperfect SuperHex correction)

35. Previous Work (3/3) • Showed that • Both scaling formulations are equally effective / equivalent • All 3 goals can be achieved by proper tuning, using a single formulation • Goal 1: maximum irregularity reduced by factor 11 (with BFRH rotation) • Goal 2: range of segment area reduced by factor 86 • Goal 3: range of circumscribed diameter reduced by factor 11 • Allowing rotation of BFRH results in large improvements • Only a factor for Goal 1 • Re-center of BFRH has negligible impact • Results in more complicated definition of XYZPSA • Abandoned to keep definition of center simple from value without scaling

36. d2 d3 Y radius clocking d1 d4 de-center X Best fit regular hexagon Segment outline in {XY}SSA plane d6 d5 Irregularity: Definition • irregularity  RMS distance between vertices of actual segment and vertices of LSQ best fit regular hexagon with arbitrary center, radius, and clocking angle • irregularity = RMS of residual of fit • general definition includes 4 variables: decenter (X & Y), radius, and clocking angle

37. Re-centering BFRH has Negligible Impact These results from Aplanatic Gregorian design with a=0.6m

38. Recent Changes and Additions (1/2) • Modified definition of segment center • Was: mean of XYM1 coordinates of 6 vertices (after scaling, before gaps) • Changed to: scaled location of centers in base pattern • New definition is closer to BFRH center > re-centering now even less useful • ~0.01mm difference in irregularity • New, more exact calculation of circumscribed circle • Was: centered at origin of local frame • Changed to: center is optimized to minimize diameter • Difference is small • Added representation of M1 cell • Cell nodes and Interface nodes • Added representation of SSA-Cell interface • 3 SSA support points per segment, at single location in local coordinate system • Represent SSA side of interface YPSA XPSA Free circle Centered circle

39. Recent Changes and Additions (2/2) • Added six additional Tuning Goals • Minimize range of bar lengths in top layer of cell • Minimize range of SSA alignment system • Minimize width of diffraction spikes from segment edges • Minimize largest circumscribed diameter (blank/boule size) • Minimize magnitude of WT pivot shifts • Minimize residual figure error from WT pivot shift correction • Repeated all calculations for Ritchey-Chrétien design and new segment size • Geometry and segmentation • K=60m, k=-1.00095 • a=0.716m, t=45mm, gap=2.4mm • 6*82=492 segments instead of 6*123=738 • WH pivot shift data not available for larger segments • Used sensitivities from a=0.6m – not directly applicable • Observations: • Optimal tuning almost identical • Conclusions unchanged • Can base decision on old baseline (a=0.6m, AG)

40. Segmentation Patterns New baseline: 6×82, 1.432m segments

41. Coordinate Systems • XYZM1 / RθZM1 • OM1 at apex of M1 optical surface • ZM1 along axis of symmetry of M1 optical surface, positive toward stars • XYZSEG • OSEG in M1 optical surface, at center of segment • ZSEG M1 optical surface • XSEG in RZM1 plane • XYZTEMP* (= XYZSSA in TMT.SEN.TEC.05.016.DRF04) • OTEMP = OSEG • ZTEMP= ZSEG • XTEMP // XZM1 plane • XYZPSA* • proposed as replacement for XYZSSA and XYZSEG in TMT.SEN.TEC.05.016.DRF04) • OPSA = OTEMP • ZPSA= ZTEMP • (XPSA,XTEMP) = rotation to BFRH • SSA uniquely located in XYZPSA • Coordinates of SSA features are invariant in XYZPSA • Suggest keeping only XYZM1 and XYZPSA as official systems • Possibly also XYSSEG if used by others *Not currently an official TMT coordinate system (TMT.SEN.TEC.05.016.DRF04, September 2005)

42. Defining “PSA” Reference Frame YTEMP XPSA XTEMP YPSA Rotation about ZTEMP, from XYZTEMP to XYZPSA BFRH BFRH Outline in XYTEMP Vertices in optical surface ZTEMP ZPSA XTEMP ZM1 extend vertices and center into optical surface // ZM1 XM1 center XYM1 = scaling rule × center Scaled hex array Scaling rule Base pattern: regular hex array

43. Defining/Positioning Hardware • Defining Segment outlines • Begin with circular blank from polishing • Engrave fiducials into optical surface of polished segment • Fiducials define location of XYZPSA reference frame • Cut segment outlines relative to fiducials • segment edges are straight lines in XYPSA plane (basic) • segment side faces // to ZPSA (basic) • Final-figure segments relative to XYZPSA • Optical prescription described in XYZPSA • Defining M1-Cell to SSA interface coordinates • All assembly tooling aligned to fiducials only • Physical outline or vertices are never used as datum • Coordinates of support points are identical in all segment types, when expressed in PSA frame • Converting those coordinates back to XYZM1 frame provides global coordinates of interface points

44. Cell Nodes and Cell-SSA Interface • Cell Nodes • At given distance Hcell behind 3 of 6 pre-gap vertices, along local normal to optical surface • Form irregular triangle whose geometry depends on scaling • Interface nodes • At 1/3 along length of cell members • SSA Supports • 3 points, representing nominal centers of AAP adjusters • At HSSA, RSSA from OPSA (same for all segments) • HSSA and RSSA optimized to minimize maximum distance to interface nodes YPSA ZPSA X,YPSA Segment Vertices (before gaps) ZM1 HSSA XPSA SSA Supports (3) RSSA XM1 Hcell Cell Nodes (3) Interface Nodes (3)

45. Cell Nodes and Cell-SSA Interface ZPSA Local Normal to optical surface XPSA YPSA Hcell HSSA Cell Nodes RSSA SSA Supports (“center” of adjusters) Interface Nodes

46. Cell Nodes and Cell-SSA Interface

47. MATLAB Segmentation Code • Produces regular Hex base pattern in XYM1 • given segment size (a), and ID and OD of M1 (cropping) • Scales the base pattern in radial direction • Scaling rule adjusts radial coordinate (only) of centers and vertices of base pattern • Adjustable parameter a (intensity of scaling) • Outermost vertex of array (at Rmax) is unchanged by definition of scaling rule • Extrudes segment centers and vertices into M1 optical surface • Defines segment-local coordinate systems • Z // normal to optical surface at segment center • Calculates coordinates of cell and interface nodes • Top layer nodes and interface nodes • Implements gaps • Calculates size and rotation angle of BFRH • Defines final local systems (XYZPSA) • Establishes coordinates of SSA support points • Calculates various metrics of resulting segmentation • Produces outputs files (ASCII) • segment vertex coordinates in M1 or PSA system • Cell node coordinates in M1 system • Creates various diagnostic plots • Distribution of metrics across array and corresponding statistical distributions • Segment outlines • 3D plots of array and cell

48. New Segmentation Goals (1/2) • Minimize segment irregularity • Minimize variation of segment area • Minimize variation in segment size • Minimize range of lengths of top members of cell • Avoid having to build different length members • Metric = Max(L)/Min(L)-1, in percent • Minimize required range of SSA alignment system • Minimize maximum in-plane (XYPSA) distance between SSA support points and interface nodes • Radial and depth location of SSA supports adjusted for best fit • Minimize width of diffraction spikes from segment edges • Based on edge angles projected on sky (?) • Minimize scatter of projected (into XYM1) angle of segment edges • Sort edges into 3 groups of angles (~0º, ~60º, ~120º) • Calculate standard deviation within each group (Std0, Std60,Std120) • Metric = RMS(Std’s) = √ 1/3 (Std02+Std602+Std1202) • This goal is optimized without scaling (a=0): Std0 = Std60 = Std120 = 0

49. New Segmentation Goals (2/2) • Minimize diameter of largest circumscribed circle • Minimize size of glass boules/segment blanks • Now using exact calculation of circumscribed circle (free center) • Minimize magnitude of WT pivot shifts (rough estimate) • Whiffletree pivot shifts are used to fine-tune axial support to actual segment shapes • Custom machining of pivot features for each segment type • Pivot shifts require “real estate” in the hub region of whiffletree components • Large shifts could be difficult to implement • Estimates based on study of pivot shifts for various modes of segment shape variations • Based on “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006) • Metric = maximum estimated shift at any WT joint, for any segment • Minimize residual figure error after pivot shift (rough estimate) • Pivot shift is very effective, but not perfect • Figure error after optimal pivot shift is slightly worse than nominal value • Estimates based on study of pivot shifts for various modes of segment shape variations • Based on “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006) • Metric = estimated value of √(RMScorrected2 – RMSnom2), where RMSnom and RMScorrected are the RMS values of the SSA-induced surface errors, for a nominal regular segment (for which the WT geometry was designed) and the actual segment, after correction via WT pivot shifts, respectively

50. Goals 8 and 9: Axial Support Pivot Shifts • BFRH is optimally clocked  no residual rotation • Correction for segment size (case 1) – using BFRH radius for size • Requires pivot shifts up to 0.49mm per mm of radial growth • Used nominal segment radius = mean(BFRH radii) (could have used midrange value instead) • Leaves residual figure error up to 0.425nm per mm of radial growth • Correction for Irregularity of segment (mean of cases 6 to 12) • Requires pivot shifts up to 0.636mm per mmRMS of irregularity • Leaves residual figure error up to 0.197nm per mmRMS of irregularity Mean = 0.197 nmRMS/mmRMS Mean = 0.636 mm/mmRMS Table from “Correction of Segmentation Effects by Shifting Whiffletree Pivot Locations” TMT.OPT.TEC.06.009.REL01 (Eric Williams, June 2006) Applicable to a=0.6m