The Habitable Exoplanet Observatory (HabEx) mission has unique optical performance requirements which drive the mirror design process beyond the traditional criteria. While mass and stiffness are still important, the response to inertia loading (expressed in terms of Zernike coefficients) to omni-directional excitation dominates the effort. While a Zerodur mirror is the current baseline, as mass budgets change, a ULE design is being studied as a potential alternative. This trade study looked at over 264 design variations using the Arnold Mirror Modeler and ANSYS(c) to investigate the influence of various design elements, including: substrate thickness, core cell size, hexapod geometry and local reinforcement. Design 'goodness' was evaluated based on the mirror's inertial deformation response to omni-directional input. This response was calculated via RSSing Zernike polynomial responses to (XYZ) accelerations

Arnold, William R.

Stahl, H. Phillip

NASA Technical Reports Server

  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
Influence of core and hexapod geometry, and local reinforcement on 
the performance of ultra-lightweight ULE mirror  
William R. Arnolda, H. Philip Stahla, 
aNASA Marshall Space Flight Center, Redstone Arsenal, Huntsville, AL, USA 35812 
ABSTRACT  
The Habitable Exoplanet Observatory (HabEx) mission has unique optical performance requirements which drive the 
mirror design process beyond the traditional criteria. While mass and stiffness are still important, the response to inertia 
loading (expressed in terms of Zernike coefficients) to omni-directional excitation dominates the effort. While a Zerodur 
mirror is the current baseline, as mass budgets change, a ULE design is being studied as a potential alternative.  This 
trade study looked at over 264 design variations using the Arnold Mirror Modeler and ANSYS© to investigate the 
influence of various design elements, including: substrate thickness, core cell size, hexapod geometry and local 
reinforcement.  Design ‘goodness’ was evaluated based on the mirror’s inertial deformation response to omni-directional 
input.  This response was calculated via RSSing Zernike polynomial responses to (XYZ) accelerations. 
Keywords: HabEx, primary mirror design, trade studies, ULE alternatives 
 
1. INTRODUCTION  
The HabEx Architecture-A telescope requires a 4-meter off-axis circular-aperture monolithic primary mirror.  An open-
back Zerodur mirror is the baseline, with a closed-back ULE mirror as an alternative.  This paper is one in a series of 
design studies supporting the project. Previous papers1,2 covered the Zerodur option extensively and limited ULE cases 
as well as discussing the overall scope of the HabEx work at MSFC.  This paper expands the ULE design parameters 
with an emphasis on the impact of suspension system geometry and local reinforcement.  The HabEx mission and the 
unique requirements of the coronagraph control the output format of this study.  For HabEx, inertial deformation 
(response to inertial loads) is an important performance parameter both as it relates to manufacturing for diffraction 
limited performance and to ultra-stable on-orbit wavefront performance required by coronagraph.  
The ranges of all the parameters used in this study are set by published industrial capabilities.  The actual mirror 
manufacturer may or may not be the raw material supplier.   A total of 264 separate models were created and run in 
period of two weeks using the AMM (Arnold Mirror Modeler) and ANSYS.  
1.1 Purpose of HabEx 
The basic goals of the HabEx Program are explained in Figure 1. 
Figure 1:  Basic goals of HabEx program 
https://ntrs.nasa.gov/search.jsp?R=20180006819 2019-08-31T18:54:15+00:00Z
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
1.2 Baseline HabEx Architecture A Concept 
The HabEx Science and Technology Definition Team (STDT) chose the following parameters for Architecture-A 
(Figure 2).  The telescope would have a 4-meter aperture, with a 72-meter external Starshade occulter flying in 
formation with the satellite.  Four instruments would be included on the satellite; a coronagraph instrument for 
Exoplanet Imaging, a Starshade Instrument for Exoplanet Imaging, a UV-NIR Imaging Multi-object Slit Spectrograph 
for general observatory science and a High-Resolution UV Spectrograph for general observatory science. 
 
 
 
 
 
 
 
 
 
 
 
 
 
1.3 HabEx Baseline Telescope Design 
This paper concentrates on the primary mirror for the telescope shown in Figure 3.  While the baseline mirror is an open-
back Zerodur mirror, other considerations such as mass allocations may require alternative designs.  This trade study was 
conducted to get a better understanding of the driving design parameters for a ULE mirror to meet the HabEx criteria. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The 4-meter telescope design is off-axis to minimize diffraction, and the station keeping is done using Micro-thrusters, 
instead of the traditional reaction wheels.  The main instrument for planet finding is the advanced coronagraph.  Most of 
the design (mirror performance) requirements are driven by this coronagraph.  
Figure 2:  Concept for Architecture A 
Figure 3:  Current Baseline Concept for Telescope portion of satellite 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
1.4 HabEx WFE Stability Specification 
The HabEx telescope has a Zernike Polynomial based WFE (wave front error) budget (Figure 4) divided between LOS 
(line-of-sight) jitter, PM (primary mirror) deformations due to inertial loading, from station keeping, and PM thermal 
deformations.  The LOS jitter is a system level issue (includes secondary mirror and metering structure motions as well a 
PM motion) and is not addressed in this study.  The thermal deformations also depend upon system inputs and are not 
addressed in this study.  The inertial allocation is used to compare trade study results; but remember, the purpose of the 
study is to understand which parameters influenced WFE, not to find a single point design which meets the specification. 
 
RSS Allocation 100% 1% 60% 80% 10%
VVC-6 Allowable LOS Inertial Thermal Reserve
K N M Aberration [pm rms] [pm rms] [pm rms] [pm rms] [pm rms]
TOTAL RMS 416 4 250 333 41
2 1 1 Tilt 0 0 0 0
3 2 0 Power (Defocus) 250 2.5 150 200 24.75
4 2 2 Pri Astigmatism 200 2 120 160 19.8
5 3 1 Pri Coma 175 1.75 105 140 17.325
6 4 0 Pri Spherical 200 2 120 160 19.8
7 3 3 Pri Trefoil 2.6 0.026 1.56 2.08 0.2574
8 4 2 Sec Astigmatism 0.35 0.0035 0.21 0.28 0.03465
9 5 1 Sec Coma 0.35 0.0035 0.21 0.28 0.03465
10 6 0 Sec Spherical 0.35 0.0035 0.21 0.28 0.03465
11 4 4 Pri Tetrafoil 0.35 0.0035 0.21 0.28 0.03465
12 5 3 Sec Trefoil 0.35 0.0035 0.21 0.28 0.03465
13 6 2 Ter Astigmatism 0.1 0.001 0.06 0.08 0.0099
14 7 1 Ter Coma 0.1 0.001 0.06 0.08 0.0099
15 8 0 Ter Spherical 0.1 0.001 0.06 0.08 0.0099
16 5 5 Pri Pentafoil 0.35 0.0035 0.21 0.28 0.03465
17 6 4 Sec Tetrafoil 0.1 0.001 0.06 0.08 0.0099
18 7 3 Ter Trefoil 0.1 0.001 0.06 0.08 0.0099
19 8 2 Qua Astigmatism 0.1 0.001 0.06 0.08 0.0099
20 9 1 Qua Coma 0.1 0.001 0.06 0.08 0.0099
21 10 0 Qua Spherical 0.1 0.001 0.06 0.08 0.0099
22 12 0 Qin Spherical 0.1 0.001 0.06 0.08 0.0099
Order
 
Figure 4:  JPL Initial Tolerance Allocation for Primary Mirror 
2. MIRROR TRADE STUDY 
The purpose of the trade study was to determine the mirror and suspension system design parameters which influence 
the mirror’s on-orbit performance.  The study varied mirror design parameters of core thickness, core cell size, cell wall 
thickness and facesheet thickness (Figure 5).  For most of the study, the reinforcement thicknesses, pad diameter and pad 
perimeter diameters were held constant. A sub-study compared the influence of no reinforcement (web thickening) and 
smaller pad size.  The sub-study used the same mesh as the reinforced designs.  The study also traded support system 
variations.  The hexapod (kinematic) support has three primary variable:  number of attachment pads, location of 
attachment pads and hexapod strut angles relative to the ground plane.  This study compared the performance of three 
attachment pads on mirror with two legs each versus a six pad configuration with one leg on each pad.  The pads were 
attached to the mirror at 100%, 85% and 70% of the mirror’s radius (Figure 6).  Hexapod stiffness determines the ratio 
of horizontal modes (X and Y) WFE relative to optical direction (Z) WFE.  Two angles are significant in hexapod 
stiffness.  The horizontal angle controls the rotation about Z and together with attachment diameter controls ratio of tilts 
X and Y to loading in a given direction.  As one of the input HabEx design goals is a balanced response in all three 
directions, the horizontal angle was fixed (maximized the rotation about Z torsional modes) and the hexapod elevation 
angle (or vertical angle) was varied.  The method of varying this parameter in the AMM4,5 was to increase the hexapod 
height variable while keeping the upper and lower diameters and plan-view angles the same.  (To understand how the 
modeler works, see the references). 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
 
 
2.1 Scope of Trade Study  
The AMM (Arnold Mirror Modeler) software was used to generate separate FEM models of each design point.  A 
separate archive file was generated for each point to permit rerunning or expanding the search about a specific design 
point.   The AMM generates an ANSYS APDL which inputs the model, runs 1g static acceleration loadcases in X,Y and 
Z directions, post processes these results for mean displacements and RMS surface error for each case, then calculates 
and plots the first 10 supported modes and the first 10 free-free (mirror only) modes.  Each run generates a summary file, 
and displacement files (of the optical surface nodes only) for input into a Zernike decomposition program.  Over 264 
cases were created, run and post-processed (in this instance, run thru the Zernike Decomp program) in a two weeks span.  
The longest time was transferring the results in EXCEL for graphs and tables. 
Figure 6:  3 Point Hexapod Geometries 
Figure 5:  Core Depth and Hexapod Angles 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
2.2 Three Points Hexapod Results 
The results are presented in two formats, graphical (Figure 7) and tabular (Figure 8).  The thicker core depths produce 
higher mounted frequencies but at higher mass.  Larger core cells reduce the mass with negligible frequency change.  
The higher mount diameters increase frequencies.  Mount vertical angle has negligible effect on first mode frequencies.  
 
 
Figure 8:  Displacement results for 3 Points Hexapods 
Figure 7:  3 Points Hexapod Mass versus Frequency 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
2.3 Six Points Hexapod Results  
The Six Points Hexapods (Figure 9) had similar trends (Figures 10 and 11), core thickness increase mass and stiffness 
(as measured by first modal frequency) and cell size decreases mass with small stiffness effects.  But, the six-point 
mount results in greater frequency spread with mount geometry changes.  Increases in hexapod height (i.e. vertical 
mount leg angle) decreases first modal frequency. 
 
 Figure 10:  Six Points Hexapod mass versus Frequency 
Figure 9:  Six Points Hexapod Geometries 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
 
3. INERTIAL DEFORMATION 
To our knowledge, inertial deformation may be a new performance metric and is defined as the mirror deformation 
produced by an acceleration.  For example, gravity sag is the deformation produced when the mirror (on its mount) is 
exposed to a 1-G acceleration.  On-orbit, inertial deformation is the response of the mirror (reacting against its mount) 
when it is exposed to accelerations from sources such slews, solar wind, reaction wheel noise, etc.  And, can be modeled 
simply as gravity sag deformation scaled by the ratio of accelerations.  In the case of HabEx, point stability will be 
maintained via micro-thrusters with a specified noise of 0.1 micro-Newton (~0.01 micro-G).  This error is then 
decomposed into Zernike polynomials and compared to the specified error budget. 
3.1 Analysis Process  
Due to numerical accuracy inherent in finite element modeling and types of models and elements used, all analyses were 
performed in meters at one gravity (9.802 m/s^2), the displacements were then scaled to the appropriate levels.  The 
load-cases were unit 1 g accelerations in the three axes (X, Y and Z) (Figure 12).  These displacements were output into 
individual files and then input into a purpose-built Zernike Decomposition program which processed each load-case.  
Zernikes were scaled to the appropriate units then RSSed (square root of the sum of squares) together (Figure 13) to be 
compared to the tolerance table.  The dedicated program created a results file for insertion into EXCEL tables reported in 
Figures 14, 15 and 21. 
 
Figure 11:  Displacement results for Six Points Hexapods 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
 
Figure 12:  Typical Static Load-Case Results 
 
Figure 13:  RSS of X,Y,Z Zernike Polynomials 
 
 
 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
3.2 Core Depth Trade Study 
This portion of the study looked at one cell size and the 100% mount diameter.  The main result was that none of the 
variations met the WFE error budget specification in all terms.  But, the thicker and stiffer the mirror, the better.  
 
Figure 14:  Processed results of Core Depth Study 
 
3.3 Mount Diameter Trade Study 
This portion looked at the 400mm Core Depth and 290mm Cell Size and varied the mount diameters for 3 and 6 point 
cases.  Different mount pad and radial distance locations result is different Zernike polynomial WFE distributions. 
 Figure15:  Mount Diameter Trade Study 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
3.4 Trade Study on Back Profiles 
This study investigated the influence of the back profile.  Based upon earlier open-backed results we were expecting an 
effect.  But, for the closed back design, the back-sheet seems to have negated the benefit of a deeper outer zone.  
 
 
 
Figure 17:  Mass versus Frequency for Back Profiles. 
 
 
 
 
 
Figure 16:   Basic Parameters looked at in Back Profile Study. 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
 
 
3.5 Local Reinforcement Study 
Local reinforcement usually refers to localized increases in core web thicknesses.  With ULE mirrors in particular, 
waterjet cutting of the core makes this relatively easy to do.  The front and back sheets are usually uniform thickness.  
There is a mass penalty for this feature, but launch loads and local stresses under pads become a serious issue as the 
mirrors become lighter and thinner sections.  Most of the cases used a fixed reinforcement thickness pattern and pad and 
perimeter diameters.  To see if there was any significant benefit to this local reinforcement, a series of runs were made 
using the same mesh as reinforced, but keeping all core webs uniform (minimal) thickness (Figure 19).  While the 
previous open back design study indicated that local reinforcement provided a significant benefit, the closed back design 
does not indicate the same advantage - back plate seems to have eliminated much of the need for local reinforcement. 
 
Figure 19:  Local Reinforcement Study 
Figure 18:  Results of Back Surface Profile Study 
 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
3.6 Hexapod Leg Stiffness Study 
All the previous runs used a common strut stiffness value.  In order to see if that value was influencing the results, a 
short parametric study was done using just one geometry case from each mount diameter and 3 and 6 point 
configurations.  The strut stiffness was reduced by two orders of magnitude for each step.  The result (Figure 20) showed 
that once a certain threshold was reached, the actual leg stiffness did not influence the results. 
3.7 Facesheet Thickness Study 
Finally, a study of the effect of the facesheet on the mid-frequency Zernikes was done.  In the second row of Figure 21, 
the first number is the thickness of the front-sheet and the second number is the thickness of the back-sheet. Ignoring 
quilting, no significant improvement is provided by increases up to 13mm for either the front only or both plates. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 21:  Facesheet Thickness Study Results 
Figure 20:  Hexapod Leg Stiffness Results 
  
 
 
SPIE Proceedings 10743, Optical Modeling and Performance Predictions, 2018. 
4. CONCLUSIONS 
The Habitable Exoplanet Observatory (HabEx) mission requires a telescope whose optical wavefront has ultra-stable 
mid-spatial frequency error to enable coronagraphy of exo-Earth class planets.  While mass and stiffness are still 
important design parameters, for coronagraphy, the more important metric is inertial WFE, i.e. the response of the mirror 
to on-orbit acceleration noise.  While an open-back Zerodur mirror is the current baseline, this paper summarizes a series 
of trade studies for a potential alternative closed-back ULE design.  The study looked at over 264 design variations using 
the Arnold Mirror Modeler and ANSYS© to investigate the influence of various design elements.  As expected, the 
stiffer the mirror, the better its total rms inertial deformation.  But, the distribution of that deformation (i.e. its 
decomposition into Zernike polynomials) varies with mount configuration (3 vs 6 pads, radial pad location, and strut 
angle).  While previous studies indicated that back shape and local reinforcement were beneficial for open-back mirrors, 
their benefit was smaller for the closed-back mirrors in this trade study.  One conclusion is that the back-sheet eliminates 
the need for local reinforcement and edge support. 
 
    REFERENCES 
[1] Stahl, H. Phillip, “Overview and performance predictions of the baseline 4-meter telescope concept design for 
habitable-zone exoplanet Observatory”,  
[2] Arnold, W.R., and Stahl, H. Phillip, "Structural design of a 4-meter off-axis space telescope for the Habitable-zone 
Exoplanet Direct Imaging Mission", SPIE 10398-7 (2017).  
[3] Arnold, W.R., "AMTD design process," NASA/ SPIE Mirror Technology Days, Albuquerque, NM (2014). 
[4] Arnold, W.R., Etal, “Next Generation lightweight mirror modeling software”, SPIE Opto-mechanical Engineering 
2013, San Diego, CA  SPIE 8836-15 (2013).  
[5] Arnold, W.R., Etal, “Integration of mirror design with suspension system using NASA’s new mirror modeling       
software”, SPIE Opto-mechanical Engineering 2013, San Diego, CA  SPIE 8836-17 (2013). 
[6] Arnold, W.R., Etal, “Next Generation lightweight mirror modeling software”, NASA Mirror Tech Days, Redondo 
Beach, CA (2013).  
 


Influence of Core and Hexapod Geometry, and Local Reinforcement on the Performance of Ultra-Lightweight ULE Mirror

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