Extremely Large Segmented Mirrors: Dynamics, …scmero.ulb.ac.be/Publications/Thesis/R_Bastaits_PhD_Thesis10.pdfExtremely Large Segmented Mirrors: Dynamics, Control and Scale Eﬁects

Universite Libre de Bruxelles

F a c u l t e d e s S c i e n c e s A p p l i q u e e s

Extremely Large Segmented Mirrors:Dynamics, Control and Scale Effects

Renaud Bastaits

First vibration mode (l/2)

Flat mirror

l/2

Thesis submitted in candidature for thedegree of Doctor in Engineering Sciences June 2010

Active Structures LaboratoryDepartment of Mechanical Engineering and Robotics

ii

Jury

Supervisor : Prof. Andre Preumont (ULB)

Members :

Prof. Claude Jamar (AMOS, Liege)

Dr Martin Dimmler (ESO, Germany)

Dr Yvan Stockman (CSL, Liege)

Prof. Olivier Verlinden (UPMons)

Prof. Frank Dubois (ULB)

Dr Arnaud Deraemaeker (ULB)

iii

Remerciements

Les competences scientifiques et pedagogiques du Professeur Andre Preumont,son dynamisme et sa personnalite ont constitue le ciment de ce travail. Je luisuis tres reconnaissant de m’avoir accepte comme eleve. Son contact restera pourmoi source d’enrichissement personnel sous bien des egards, au-dela des seuls as-pects scientifiques.

Je tiens egalement a remercier tout particulierement Christophe Collette parl’intermediaire de qui j’ai commence mes travaux au sein du Laboratoire desStructures Actives. Ses conseils, sa bonne humeur a toute epreuve et son soutienm’ont aide a trouver mes marques et a avancer.

De meme, je souhaite remercier Goncalo Rodrigues avec qui j’ai travaille en etroitecollaboration depuis ma reorientation dans le domaine des telescopes. Partagerles sujets de recherche, le meme bureau, ainsi que la plupart des repas a ete sourced’une stimulation constante.

Le Laboratoire des Structures Actives a constitue pour moi un environnementstimulant; par leurs qualites humaines et scientifiques, les personnes que j’y aicotoyees m’ont aide a avancer dans une atmosphere d’emulation toujours souri-ante.

Je remercie egalement le Prof. Frank Dubois, et les Dr Yvan Stockman etStephane Roose pour m’avoir apporte a de nombreuses reprises leur aide precieusepour progresser face a des questions d’optique.

Je remercie egalement le Fonds National de la Recherche Scientifique pour le sou-tien financier qu’il m’a apporte, via la bourse FRIA FC76554.

Enfin, je suis heureux de partager le fruit de ce travail avec mes proches qui,chacun a leur maniere, apportent ce qui fait le sel de ma vie.

v

Abstract

All future Extremely Large Telescopes (ELTs) will be segmented. However, astheir size grows, they become increasingly sensitive to external disturbances, suchas gravity, wind and temperature gradients and to internal vibration sources.Maintaining their optical quality will rely more and more on active control means.

This thesis studies active optics of segmented primary mirrors, which aims atstabilizing the shape and ensuring the continuity of the surface formed by thesegments in the face of external disturbances.

The modelling and the control strategy for active optics of segmented mirrors areexamined. The model has a moderate size due to the separation of the quasi-staticbehavior of the mirror (primary response) from the dynamic response (secondary,or residual response). The control strategy considers explicitly the primary re-sponse of the telescope through a singular value controller. The control-structureinteraction is addressed with the general robustness theory of multivariable feed-back systems, where the secondary response is considered as uncertainty.

Scaling laws allowing the extrapolation of the results obtained with existing 10mtelescopes to future ELTs and even future larger telescopes are addressed and themost relevant parameters are highlighted. The study is illustrated with a set ofexamples of increasing sizes, up to 200 segments. This numerical study confirmsthat scaling laws, originally developed with simple analytical models, can be usedin confidence in the preliminary design of large segmented telescopes.

vii

Contents

1 Introduction 11.1 Evolution of telescopes: From 1600 to 1980 . . . . . . . . . . . . . 11.2 Modern-days telescopes . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 A context of multiple technological breakthroughs . . . . . 31.2.2 The advent of active optics for monolithic mirrors . . . . . 51.2.3 Segmented mirrors . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Long baseline interferometry . . . . . . . . . . . . . . . . . 91.2.5 Space telescopes . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Future Extremely Large Telescopes . . . . . . . . . . . . . . . . . . 131.4 Scale effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Basics of Telescope Optics 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Quantifying the wavefront error . . . . . . . . . . . . . . . . 23

2.3 Common optical configurations of optical telescopes . . . . . . . . 262.3.1 Newtonian telescopes . . . . . . . . . . . . . . . . . . . . . 262.3.2 Two-mirror telescopes . . . . . . . . . . . . . . . . . . . . . 272.3.3 Telescopes with 3 or more mirrors . . . . . . . . . . . . . . 28

2.4 Wavefront error due to deviations from the design . . . . . . . . . 282.4.1 Shape of optical elements . . . . . . . . . . . . . . . . . . . 292.4.2 Relative position of optical elements . . . . . . . . . . . . . 292.4.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.4 Design trade-offs . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Diffraction-limited imaging . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ix

x CONTENTS

2.5.3 Diffraction from obscurations . . . . . . . . . . . . . . . . . 342.5.4 Diffraction-limited versus aberrated . . . . . . . . . . . . . 35

2.6 Telescopes with a segmented primary mirror . . . . . . . . . . . . . 352.6.1 Conditions for optimal performances . . . . . . . . . . . . . 352.6.2 Design trade-offs . . . . . . . . . . . . . . . . . . . . . . . . 372.6.3 Diffraction in segmented telescopes . . . . . . . . . . . . . . 38

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Active control of telescopes 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 External disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Layers of active control in modern telescopes . . . . . . . . . . . . 50

3.3.1 Pointing and tracking . . . . . . . . . . . . . . . . . . . . . 513.3.2 Active optics . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.3 Adaptive optics . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Active optics of the Keck telescope . . . . . . . . . . . . . . . . . . 583.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Dynamics and control 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Quasi-static approach . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . 734.3.2 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.3 Static response . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.4 Dynamic response in modal coordinates . . . . . . . . . . . 79

4.4 Control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4.1 Dual loop controller . . . . . . . . . . . . . . . . . . . . . . 824.4.2 Extended Jacobian SVD controller . . . . . . . . . . . . . . 83

4.5 Loop shaping of the SVD controller . . . . . . . . . . . . . . . . . . 844.6 Control-structure interaction . . . . . . . . . . . . . . . . . . . . . 87

4.6.1 Multiplicative uncertainty . . . . . . . . . . . . . . . . . . . 884.6.2 Additive uncertainty . . . . . . . . . . . . . . . . . . . . . . 88

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Scale effects 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

CONTENTS xi

5.2 Static deflection under gravity . . . . . . . . . . . . . . . . . . . . . 975.3 First resonance frequency . . . . . . . . . . . . . . . . . . . . . . . 985.4 Control bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.5 Control-structure interaction . . . . . . . . . . . . . . . . . . . . . 1015.6 Wind response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.7 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . 1065.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Structural response of large truss-supported segmented reflec-tors 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 Wind model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2.3 Random response . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Results in open-loop . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Controlled response . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.5 Effect of damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.6 Effect of mean wind velocity . . . . . . . . . . . . . . . . . . . . . . 1276.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Conclusions 1337.1 Original aspects of the work . . . . . . . . . . . . . . . . . . . . . . 1337.2 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3 Future perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A Definitions of optical design parameters 137A.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

B Primary aberrations 141B.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C Shack-Hartmann sensors 145

D Small-gain theorem 149D.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 149D.2 Stability robustness tests . . . . . . . . . . . . . . . . . . . . . . . . 149

D.2.1 Additive uncertainty . . . . . . . . . . . . . . . . . . . . . . 150D.2.2 Multiplicative uncertainty . . . . . . . . . . . . . . . . . . . 150

D.3 Residual dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xii CONTENTS

E Mode shapes of segmented mirrors with supports 1-3 153

F Wind response of Set 3 157

Chapter 1

Introduction

1.1 Evolution of telescopes: From 1600 to 1980

Astronomy distinguishes itself from other scientific disciplines by the fact that itsdevelopments are mainly based on observations, as most experiments are unprac-tical or impossible by nature. Throughout the ages, astronomers have relentlesslydeveloped instruments to exploit the full potential of the sky that their nakedeyes were not able to catch. Amongst the most notable inventions, it led to thedevelopment of elaborate calendars and early positioning systems, based on thepatterns formed by the astral objects.

The invention of the first refracting telescope at the end of the XVIth century,and its improvement and use by Galileo to observe the sky, revealed the poten-tial of such instruments to push back the limits of the observation of objectsin the skies, by focusing more light than what the naked eye is capable of, andby magnifying the image. The technological and mathematical developments toimprove the refracting telescope led Isaac Newton to the construction of the firstreflecting telescope around 1670, based on the use of a parabolic mirror insteadof a lens as the light collector. The reflecting telescope exhibits some advantageswith respect to the refracting one, in particular the fact that they are exempt ofchromatic aberrations, as reflection laws do not depend on the wavelength of thelight, while refraction laws do.

The brightness of the faintest objet that a given telescope can observe is limitedby the effective area of its primary mirror (M1) (Enard et al., 1996). Furthermore,the diameter of M1, D, also affects the resolution and contrast characteristics ofthe images formed by the telescope in ideal conditions (see section 2.5). Conse-quently, improving the performance of the telescopes has called for a constant

1

2 1 Introduction

increase of D along history. Fig.1.1 shows the evolution of the aperture diameterof optical and infrared telescopes throughout the years. Until the beginning of theXXth century, reflecting and refracting telescopes were competing against eachother, benefiting from respective technological developments leading to innova-tive designs. For apertures larger than 1m, the reflecting telescopes have provedthe most efficient.

1600 1700 1800 1900 2000

Galileo

Apert

ure

dia

mete

r [m

]

0.1

1

10

100

Newton

VLT,Gemini,Subaru

Keck

TMT,JELT

E-ELT

OWL?

Year

Monolithic reflecting

Refractive

Segmented reflecting

Hubble

JWST

NTTActive optics

GTC

2100

Figure 1.1: Telescope aperture diameter in time [adapted from (Bely, 2003), p.2].

The role of the telescope structure is to maintain the optical performances ofthe telescope during observations, by preserving the shape and alignment of theelements in the optical train. As those optical elements were built larger andthicker, and consequently heavier, so were their supporting structures. But theirsensitivity to the effects of changing gravity and temperature grew accordingly.

1.2 Modern-days telescopes 3

Innovative solutions were developed, commonly referred to as passive, combin-ing, e.g., new design approaches for the sub-structures, the generalized use ofkinematic mountings and an optimized choice of materials.

1.2 Modern-days telescopes

Not only should the next generations of telescopes collect more light. To be reallyeffective, they should also ensure that the collected light is always focused on thesmallest area, otherwise faint stars and slight details of extended objects are lostin a blurry luminous background. Consequently, the challenge is to build largertelescopes with an improved optical accuracy, maintained along time in spite ofexternal disturbances, such as gravity, wind and thermal gradients.

1.2.1 A context of multiple technological breakthroughs

Before 1980, the mounts of the largest telescopes were of the equatorial type(Fig.1.2.a), in which rotations around the polar axis (parallel to the Earth’s ro-tation axis) and around the declination axis (perpendicular to the polar axis)allow the initial pointing towards an object. The tracking was then simply per-formed by rotating the telescope around its polar axis, at a constant speed, tocompensate for the Earth’s rotation. This simple principle could be performed inopen-loop by the use of clock mechanisms. However, while elegant in its princi-ple, the structural constraints induced by that configuration revealed unpracticalfor their implementation in ever-growing telescopes, mostly because of their in-trinsic heaviness [(Bely, 2003), p.234], and cannot correct errors due to externaldisturbances.

The orientation of telescopes of the altitude-azimuth (alt-az ) type is based on avertical (azimuth) axis and on a horizontal (altitude or elevation) axis. Tracking isintrinsically more difficult in this case, because it requires the axes to be rotatedat variable speeds depending non-linearly on the orientation of the telescope.However, computer control has almost cancelled that drawback. Moreover, theirstructures are more compact, much simpler and lighter than those of equivalentequatorial telescopes (cfr Fig.1.3), implying so significant cost savings that alt-azmounts have become the standard 1 . Finally, as the orientation of the altitudeand azimuth axes do not change with respect to the orientation of the gravityfield, the implementation of feedforward corrections (based on lookup tables) inactive optics is easier and more efficient (Enard et al., 1996).

1Other particular configurations are used in some projects such as the SALT and HET (seesection 1.2.3), but they are out of scope for this thesis, as they are not envisioned for any futureELT.

4 1 Introduction

Altitudeaxis

Azimuthaxis

Earth

Declinationaxis

Polaraxis

Earth

’s rota

tion a

xis

a) b)

Figure 1.2: (a) Equatorial mount - (b) Altitude-Azimuth mount.

The construction of telescopes with primary mirrors of diameters in the range of3.5m showed the practical limits of passive techniques with respect to the severeoptical tolerances required to attain the best performances, calling for complexperiodical readjustments (on a timescale of weeks) (Wilson, 2003). The goal ofactive optics is to automate that optical maintenance procedure, during observa-tions, on much shorter time scales (from a few tens of seconds to a few minutes).Consequently, active optics both increases the optical performance and lengthensthe timescales over which they can be maintained, making telescopes more effi-cient.

The implementation of active optics in modern telescopes has had a considerableimpact on their overall design. First, it permitted the use of thinner mirrors(meniscus and segmented mirrors), while passive mirrors were relying solely ontheir thickness to minimize the sag under gravity. This consequently alleviatedthe requirements on the overall structure, and therefore the overall cost of thetelescope. It also allowed a significant relaxation of the requirements on the lowspatial frequency quality of the meniscus mirrors made active, letting the man-ufacturer focus on mid- and high spatial frequencies that also are of practicalimportance (Noethe, 2009). Fig.1.3 shows the concurrent effects of the resort toactive optics and alt-az mounts on the mass of telescopes.

In parallel, adaptive optics has permitted the correction of the optical aberrationsinduced by the continuous local changes in the index of refraction of the atmo-


M1 diameter [m]

Tota

l m

ass [to

ns]

VLT Keck

D2.5

0 5 10

102

103

101 Equatorial

Alt-az

1

Figure 1.3: Total mass versus M1 diameter [(Bely, 2003), p.235].

sphere, giving access to unprecedented optical performances (cfr section 3.3.3).Finally, the development of Charged-Coupled Devices (CCD) cameras and theiruse in replacement of photographic plates has allowed a much higher efficiencyin the use of the collected light. It also permitted the development of wavefrontsensors exhibiting the required performances for their implementation in activeand adaptive optics control loops.

1.2.2 The advent of active optics for monolithic mirrors

Active optics was first implemented in the New Technology Telescope (NTT), a3.5m telescope completed by the European Southern Observatory (ESO) in 1989(Wilson et al., 1987); the implementation has two aspects, as can be seen fromFig. 1.4. The shape of the primary mirror (M1) is controlled by actuators pushingagainst its back, while the alignment of the secondary mirror (M2) with respectto M1 is maintained through the control of its rigid-body degrees of freedom.An optical sensor, located downwards M2, measures the aberrations induced inthe output wavefront and transmits the information to a controller. The lat-ter determines the changes in shape and alignment that are responsible of thoseaberrations, and calculates the signals to apply to the actuators to compensatefor them and thus obtain the best images.

Compared to similar passive primary mirrors of that time, NTT M1 was twicethinner (Noethe, 2009). This represented a major improvement as the require-ments on the structural design were substantially softened, and as the decreaseof the thermal inertia of the mirror also has a direct impact on the image qualitythrough the phenomenon of mirror seeing (see section 3.2). Fig.1.5 comparesthe images produced by the NTT to those produced by other state-or-the-art

6 1 Introduction

M1 shape actuators

M1

M2

Defocus

Scienceinstrument

Wavefrontsensor

Controller

Decenter

a) b)

Figure 1.4: Active optics at the New Technology Telescope (NTT) - (a) Funda-mental principles [adapted from (Wilson et al., 1987)] - (b) Back of the primarymirror: Each square corresponds to the cell of an actuator (ESO, 2010).

Figure 1.5: (a) ESO 1m Schmidt; (b) ESO 3.6m (passive); (c) ESO 3.5m NTT(raw image); (d) ESO 3.5m NTT (after post-processing) (Wilson, 2003).

telescopes in 19892.

The successful results of the technology developed for the NTT served as thebasis for the design of the Unit Telescopes of ESO’s Very Large Telescope (VLT)(Fig.1.6): Four active telescopes with a M1 of 8.2m (completed successively be-tween 1998 and 2001). It is worth noting that the thickness of VLT M1 (0.17m)is actually smaller than that of NTT M1 (0.24m), in order to fully exploit its

2The primary mirror of NTT suffered from spherical aberration resulting from an error inpolishing. Fortunately, the active optics system of NTT was able to correct it, a fact which,although it was consuming 80% of the control authority, can be seen as its first success.


a) b)

Figure 1.6: Very Large Telescope: (a) 8.2m Unit Telescope after completion - (b)Detail of the back-structure and actuators of its primary mirror (ESO, 2010).

potential in terms of light weight, thermal inertia and control authority 3.

The success of NTT gave rise to two projects very similar to VLT: The Subarutelescope in Hawaii with a primary mirror of 8.2m diameter that was completedin 1999 by Japan (Iye et al, 2004) and the Gemini observatory, consisting of two8.1m telescopes at different sites in Hawaii and Chile completed in 2000 by aninternational consortium (USA, UK, Canada, Chile, Brazil, Argentina, and Aus-tralia) (Mountain et al., 1994).

An other approach to manufacturing lightweight mirrors was developed in par-allel, based on the mechanical properties of honeycomb-like structures, allowingto reduce the mass without affecting significantly the stiffness, thus minimizingthe deformation under gravity. This can be achieved either by direct casting(Angel and Hill, 1982) or by machining the exceeding material. It has been usedto produce mirrors such as the 8.4m mirrors of the Large Binocular Telescope(University of Arizona, 2010) and the 6.5m mirrors of the two Magellan Tele-scopes (AURA, 2010). However, mirrors of those dimensions still require activecorrections to be used at their full potential [see (Noethe, 2009) e.g.].

3This fact comes from a requirement to the design of NTT that it could be used for astro-nomical observations even if the active optics fails (Noethe, 2009), while the images producedby the VLT are not usable without active optics (Wilson, 2003)

8 1 Introduction

1.2.3 Segmented mirrors

The idea of segmentation consists in replacing a monolithic mirror by an assem-bly of contiguous segments, constituting the tessellation of an optical surface,supported by a single mechanical structure. Segments in the 1- to 2-m-diameterrange can be designed to exhibit individual deformation under gravity lower thanoptical tolerances, while still providing a mass per surface unit much lower thanthat of equivalent monolithic mirrors. However, active control is required tomaintain the overall shape and continuity of the surface formed by the segmentsdue to the deformations of the supporting structure. This is particularly criticalif an optical or near infrared (IR) telescope is to be used close to its diffractionlimit (see section 2.6).

The most sophisticated form of segmentation has been first implemented success-fully in the optical/near IR Keck I & II telescopes, that saw first light respectivelyin 1993 and 1996 (Fig. 1.7.a). Their respective primary mirrors are made of 36hexagonal 1.8m-diameter segments, for an effective aperture of approximately10m. Fig. 1.7.b shows a picture of the primary mirror of the Keck telescopes:Each segment is equipped with a set of sensors that measure the relative normaldisplacements between two adjacent segments and with 3 actuators that correcttheir positions (piston and tilts).

Figure 1.7: Keck I & II 10m telescopes: Left, the telescopes inside their enclo-sures; right, front view of the segmented M1 (Keck Observatory, 2010).

Again, the success of Keck gave rise to other projects. Inaugurated in 2009,the Gran Telescopio Canarias (GTC - Spain) is based on a design very similarto that of Keck, with a slightly larger segmented primary mirror (10.4m) madeup of 36 segments (Alvarez and Rodriguez-Espinosa, 2004). The Hobby-EberlyTelescope (HET - USA) (University of Texas, 2008) and the Southern African


Large Telescope (SALT - South Africa, Germany, Poland, USA, UK and NewZealand)(Blanco et al., 2003) also both use rectangular segmented primary mir-rors of 11×9.8 meters, made up of 91 hexagonal segments and were completedrespectively in 1997 and 20054.

1.2.4 Long baseline interferometry

Fig.1.8 shows the basic principles of long baseline interferometry; it consists oftwo or more independent telescopes separated by a distance called the baseline,B, that point at the same object. Instead of being driven to their respectiveinstruments, the image they produce are combined in a single beam illuminatinga camera. Because of the wave nature of light, instead of an image, the combi-nation produces interference fringes containing information about the image thatcan be accessed through post-processing. However, as shown by the figure, thewavefront enters the optical train of each telescope with a certain delay time.Obtaining the best fringes requires that delay to be reduced to a portion of thewavelength; this is done through so-called delay lines. Once phased, the aper-tures composing the interferometer can be seen as elements of a single collectingoptical surface (Enard et al., 1996).

B

dela

y

delayline

beamscombination

baseline

D

q

Figure 1.8: General principles of interferometry [adapted from (ESO, 2010)]

The benefit is that the resolution of such an interferometer is proportional to(B sin θ)−1 instead of D−1. Consequently, for a given total collecting area, an in-

4Their very specific design and their use without cophasing, mainly for spectroscopy, aimedat lower construction costs, making them difficult to compare to Keck or the GTC.

10 1 Introduction

terferometer can have a resolution several orders of magnitude higher than thatof a single telescope5. Moreover, the use of interferometry at its full potentialrequires that active and adaptive optics should be very efficient to ensure a goodphasing of the beams. Finally, the post-processing of the fringes requires exten-sive computer power and an important observing time. Therefore, in the nearfuture it will most likely be complementary to conventional observing techniquesinvolving telescopes with large apertures.

Interferometric techniques are implemented in modern optical and infrared tele-scopes. The Keck Observatory uses the 85m baseline between Keck I and II(Colavita et al., 2004). In the VLT-Interferometer, up to three of the eight tele-scopes can be combined: The four 8.2m Unit telescopes have fixed locations whilethe four 1.8m Auxiliary telescopes can adopt different configurations to modifythe length and the orientation of the baseline (Glindemann et al, 2004).

One could also mention other particular projects such as the Large BinocularTelescope (University of Arizona, 2010) or the Giant Magellan Telescope (Johnset al., 2004) that are based on respectively two and seven 8.4m lightweight pri-mary mirrors assembled on a single back-structure (Fig.1.9). The goal is to beable to operate them either with or without interferometry mode, in which theoptical trains must be phased to attain the best resolution permitted by theiroptical designs.

Figure 1.9: Giant Magellan Telescope project (AURA, 2010).

5But the sensitivity remains a function of the sum of the areas of the mirrors composing theinterferometer.


1.2.5 Space telescopes

The atmosphere sets strong boundaries to ground-based astronomy. First, itis transparent only to a small portion of the electromagnetic spectrum, namelythe visible and the near-infrared and it blocks or absorbs the rest (ultraviolet,gamma- and X-rays,. . . ). Furthermore, the quality of the wavefront emitted bycelestial objects is continuously degraded by turbulence in the successive layersof the atmosphere.

Those reasons led to the launch of space telescopes programs from early 1980.The Hubble Space Telescope (HST), launched in 1990 , is probably one of themost emblematic projects. The HST is depicted in Fig.1.10, its primary mirroris a 2.4m diameter monolithic lightweight mirror; it produces diffraction-limitedimages in the ultraviolet, visible and near-IR and is also used for spectrometry.Its initial results were poor due to an error in the fabrication of its M1: Theinstallation of optical elements to compensate for that error required the launchof a dedicated space mission three years later6. Thanks to this correction, thetelescope was able to reach its full potential and the data it produced led tocountless scientific publications.

Figure 1.10: Hubble Space Telescope (NASA, 2010a).

6The error was quite similar to that affecting the M1 of the NTT but the active devices inHST could not correct it.

12 1 Introduction

Following the same trend as ground-based telescopes, space telescopes with largerprimary mirrors are planned for the future. Fig.1.11 depicts the James WebbSpace Telescope (JWST), to be launched in 2014. Its 6.5m primary mirror willconsist of 18 hexagonal segments. During the launch, JWST is folded configura-tion in order to comply to the limited available volume in the cap; once in orbit,its active structure deploys itself and then maintains its optical configuration, toprovide diffraction-limited imaging in the IR (Gardner et al, 2006).

Figure 1.11: James Webb Space Telescope: Left, folded configuration of theJWST, during the launch - Right, after deployment, once in orbit (NASA, 2010b).

However, space telescopes suffer from a limited lifetime and few or no possibilitiesof maintenance, from long development times and from costs that are far abovethose of ground-based systems. The advent of adaptive optics allows ground-based telescopes to compensate for most of the optical aberrations induced byatmospheric turbulence. On the other hand, space telescopes can observe wave-lengths unattainable by ground-based ones, their operation does not depend onthe weather, they do not suffer from luminous backgrounds,. . .

1.3 Future Extremely Large Telescopes 13

1.3 Future Extremely Large Telescopes

Monolithic mirrors larger than the current 8m generation are difficult to produce,and would set severe constraints on the design of their support structures, tomaintain their shape and alignment to severe optical tolerances. As a result,segmentation seems the only promising solution to reach diameters of 20m andbeyond (Strom et al., 2003), to form the class of the so-called Extremely LargeTelescopes (ELTs) on which the remainder of this text will focus.

Figure 1.12: Thirty Meter Telescope project (TMT, 2010)

Several projects of future ELTs have been proposed since the end of the XXth

century. Three different telescopes were investigated in North America: The Cal-ifornia Extremely Large Telescope (involving many of the persons that workedon the Keck telescopes) (California Institute of Technology, 2002), the GiantSegmented Mirror Telescope (National Optical Astronomy Observatory, 2002),and the Very Large Optical Telescope (Roberts et al, 2003), with segmented pri-mary mirrors of resp. 30m and 20m for the latter. In 2003, those projects wereabandoned; their respective consortia joined their efforts into a new commonproject, namely the Thirty Meter Telescope (TMT), depicted in Fig.1.12. Its30m segmented primary mirror will be tesselated with approximately 500 hexag-onal segments (TMT Obs. Corp., 2007). Its construction officially started in2009; TMT is expected to see first light around 2018.

14 1 Introduction

In Europe, ESO studied the concept of the Overwhelmingly Large TelescopeOWL (ESO, 2004), a very ambitious project combining six reflectors, amongstwhich both the primary and the secondary mirrors would be segmented, the firstbeing a 100m spherical reflector made up of more than 3000 segments, and thesecond a 20m reflector made up of more than 200 segments. In parallel, a consor-tium led by the Lund Observatory in Sweden proposed a concept for the Euro50(Lund Observatory, 2003), a telescope with a 50m primary mirror composed of618 segments.

Eventually, some aspects of OWL were judged too risky, especially with respectto its high projected cost. A new project was developed, involving both ESOand the team working on the Euro50 (that was abandoned too): The EuropeanExtremely Large Telescope (E-ELT), with a segmented 42m primary mirror tes-selated by approximately 1000 hexagonal segments, that is depicted in Fig.1.13.As a compromise between ambition and timeliness, certain high-risk items ofOWL were avoided, such as the spherical M1 and the segmentation of M2; it isscheduled to see first light in 2017 (Gilmozzi and Spyromilio, 2008).

Japan has also started the conceptual study of a 30m telescope with a segmentedprimary mirror, called the Japan Extremely Large Telescope (JELT), that shouldbe made up of approximately 1080 segments (Iye et al, 2004).

Figure 1.13: European Extremely Large Telescope (ESO, 2010)

1.4 Scale effects 15

1.4 Scale effects

The implementation of different layers of active control have allowed telescopesto reach an unprecedented higher level of optical performances. In particular,active optics has allowed a much more efficient use of the telescope structureand has made segmented optics possible in the visible and in the near infrared.The success of Keck is the promise to attain a significantly larger size of primarymirrors in the near future.

Fig.1.14 compares the M1 of some of the most celebrated telescopes, the existingones (HST, VLT and Keck) and the future ones due to be built within the nextdecade (JWST, TMT and E-ELT), that will all be segmented. Note that thesize of the earth-based telescopes is one order of magnitude larger than that ofspace telescopes. The gap between the largest existing segmented telescope inuse today (Keck) and the future ones is large and appears even larger in Table1.1, that compares some key aspects of Keck and E-ELT.

VLT - 8.4 m

JWST - 6.5 m

TMT - 30 m

E-ELT - 42 m

Keck - 11m

HST - 2.6 m

Space Telescopes Ground-based Telescopes

Figure 1.14: Primary mirrors of current and future optical and infrared telescopes.

16 1 Introduction

Keck E-ELTM1 diameter: D 10 m 42 mSegment size 1.8 m 1.4 mCollecting Area 76 m2 1250 m2

# Segments: N 36 984# Actuators 108 2952# Edge Sensors 168 5604fsegment (+ Whiffle Tree) 25 Hz ∼ 60 Hzf1 (M1) ∼ 10 Hz ∼ 2.5 Hzf2 (M2) ∼ 5 Hz ∼ 1-2 HzAdaptive Optics (# d.o.f.) ∼ 350 ∼ 8000Tube and mount mass ∼ 110 t ∼ 2000 t

Table 1.1: Keck vs. E-ELT

Moreover, as the size of the telescopes increases, they become increasingly sensi-tive to external disturbances such as thermal gradients, gravity and wind, and tointernal disturbances from support equipments such as pumps, cryocoolers, fans,etc. These disturbances can deteriorate significantly the image quality. As aresult, the shape stability of ELTs relies more and more on active control means:The control system involves larger loop gains, and therefore a larger bandwidth.At the same time, the natural frequency of future ELTs is expected to be sub-stantially lower than any operating telescopes. Those conditions, combined tothe very low inherent damping of welded steel structures, increase the risk ofcontrol-structure interaction. Therefore, one can reasonably wonder if the pastexperience with Keck is sufficient to warrant a sound design and optimum oper-ation of the future ELTs, and this point alone deserves a careful attention.

1.5 Outline 17

1.5 Outline

This text is concerned with the extrapolation of the active optics of current 10-meter class telescopes (Keck, GTC, VLT) to the next generation of 30m to 40mELTs, and future, even larger ones. It studies how the various factors affectingthe structural response and the control-structure interaction are influenced bythe size of the telescope.

Chapter 2 presents the basics of telescope optics. It is focused on the optome-chanical parameters that affect the optical quality.

Chapter 3 describes the various layers of control of large telescopes, with an em-phasis on the active optics of the Keck telescopes.

The first part of chapter 4 is devoted to the numerical modelling of active opticsin large segmented mirrors. The second part studies the problem of control-structure interaction in future ELTs. A parametric study is conducted, based onthe numerical model developed previously.

Chapter 5 is concerned with the extrapolation of active optics of current tele-scopes to the future ELTs. Scaling laws are proposed to evaluate the optome-chanical performances of a telescope without resorting to complicated analysis.

Chapter 6 is dedicated to the comparison of those scaling laws with numericalparametric studies involving representative models based on the approach de-scribed in the first part of chapter 4.

1.6 References

Alvarez, P. and Rodriguez-Espinosa, J. M. The GTC project: in the midst ofintegration. In Oschmann, J. M., editor, Ground-based Telescopes - SPIE 5489,pages 583–591, 2004.

Angel, J. R. P. and Hill, J. M. Manufacture of large glass honeycomb mirrors. InBurbidge, G. and Barr, L. D., editors, International Conference on AdvancedTechnology Optical Telescopes - SPIE 332, pages 298–306, 1982.

AURA. Giant Magellan Telescope Observatory website, 2010. URLhttp://www.gmto.org/.

18 References

Bely, P. Y. The Design and Construction of Large Optical Telescopes. Springer,2003.

Blanco, D. R., Pentland, G., Winrow, E. G., Rebeske, K., Swiegers, J., andMeiring, K. G. SALT mirror mount: a high performance, low cost mount forsegmented mirrors. In Angel, J., R. P. and Gilmozzi, R., editors, Future GiantTelescopes - SPIE 4840, pages 527–532, 2003.

California Institute of Technology. California Extremely Large Telescope : con-ceptual design for a thirty-meter telescope. Technical report, 2002. URLhttp://celt.ucolick.org/reports/greenbook.pdf.

Colavita, M. M., Wizinowich, P. L., and Akeson, R. L. Keck Interferometer statusand plans. In Traub, W. A., editor, New Frontiers in Stellar Interferometry -SPIE 5491, October 2004.

Enard, D., Marechal, A., and Espiard, J. Progress in ground-based optical tele-scopes. Reports on Progress in Physics, 59:601–656, 1996.

ESO. European Southern Observatory website, 2010. URL http://www.eso.org.

ESO. OWL Concept Design Report - Phase A design report. European SOuthernObservatory, 2004.

Gardner et al. The James Webb Space Telescope. Space Science Reviews, 123(4):485–606, April 2006.

Gilmozzi, R. and Spyromilio, J. The 42m European ELT: status. In Stepp, L. M.and Gilmozzi, R., editors, Ground-based and Airborne Telescopes II - SPIE7012, 2008.

Glindemann et al. VLTI technical advances: present and future. In Traub, W. A.,editor, New Frontiers in Stellar Interferometry - SPIE 5491, 2004.

Iye et al. Current Performance and On-Going Improvements of the 8.2 m SubaruTelescope. Publications of the Astronomical Society of Japan, 56(2):381–397,April 2004.

Johns, M., Angel, J. R. P., Shectman, S., Bernstein, R., Fabricant, D. G., Mc-Carthy, P., and Phillips, M. Status of the Giant Magellan Telescope (GMT)project. In Oschmann, J. M., editor, Ground-based Telescopes - SPIE 5489,pages 441–453, 2004.

Keck Observatory. Keck Observatory website, 2010. URLhttp://www.keckobservatory.org/.

References 19

Lund Observatory. Euro50 - A 50m Adaptive Optics Telescope. Andersen, T.,Ardeberg, A. and Owner-Petersen, M., 2003.

Mountain, C. M., Kurz, R., and Oschmann, J. Gemini 8-m telescopes project.In M., S. L., editor, Advanced Technology Optical Telescopes V - SPIE 2199,pages 41–55, June 1994.

NASA. The Hubble Space Telescope website, 2010a. URLhttp://hubblesite.org/.

NASA. The James Webb Space Telescope website, 2010b. URLhttp://www.jwst.nasa.gov/index.html/.

National Optical Astronomy Observatory. The Giant Segmented Mirror Tele-scope Book, 2002. URL http://www.gsmt.noao.edu/book/.

Noethe, L. History of mirror casting, figuring, segmentation and active optics.Experimental Astronomy, 26(1-3):1–18, August 2009.

Roberts et al. Canadian very large optical telescope technical studies. In Angel,J., R. P. and Gilmozzi, R., editors, Future Giant Telescopes - SPIE 4840, pages104–115, January 2003.

Strom, S. E., Stepp, L., and Brooke, G. Giant Segmented Mirror Telescope: apoint design based on science drivers. In Angel, J., R. P. and Gilmozzi, R.,editors, Future Giant Telescopes - SPIE 4840, pages 116–128, 2003.

TMT. The Thirty Meter Telescope website, 2010. URL http://www.tmt.org/.

TMT Obs. Corp. Thirty Meter Telescope - Construction Proposal, 2007. URLhttp://www.tmt.org/docs/OAD-CCR21.pdf.

University of Arizona. Lare Binocular Telescope Observatory website, 2010. URLhttp://medusa.as.arizona.edu/lbto/.

University of Texas. The Hobby Eberly Telescope website, 2008. URLhttp://www.as.utexas.edu/mcdonald/het/het.html.

Wilson, R. N. The History and Development of the ESO Active Optics System.The Messenger, 113:2–9, September 2003.

Wilson, R. N., Franza, F., and Noethe, L. Active Optics I. A system for optimizingthe optical quality and reducing the costs of large telescopes. Journal of ModernOptics, 34(4):485–509, 1987.

20 References

Chapter 2

Basics of Telescope Optics

2.1 Introduction

M1

M2

...8 focalsurfaceo

bje

ct

wavefr

ont

Figure 2.1: Principles of imaging with a telescope.

A telescope is an instrument designed to image objects located at large distancesfrom the observer. Those objects can be either point-like (e.g. stars) or extendedobjects (e.g. nearby planets) that can be seen as an ensemble of points. Thespherical wavefront emitted by such point sources located at the infinite can beconsidered as plane at the level of the telescope (Fig.2.1). The role of the tele-scope aperture, namely its primary mirror (M1), is to collect the light energy;as the radiated energy is distributed over the area of the wavefront, the largerthe aperture, the more energy collected and the fainter the objects that can beobserved by that telescope.

The focusing of the light is performed by the optical elements composing theoptical path of the telescope, including M1. The designs vary greatly dependingon the use and on the cost of the telescope, and can include from 1 to 6 mirrors,for the most complex design published in the literature (OWL). The quality ofa telescope can be summarized by its ability to focus the energy emitted by a

21

22 2 Basics of Telescope Optics

a) b) c)

Figure 2.2: (a) Point object; (b) Image spot subject to diffraction; c) Image spotsubject to aberrations (and diffraction).

point object into the smallest area possible on the focal surface. Diffraction aswell as deviations from the initial design (aberrations) cause a spreading of theenergy away from the nominal focus, setting physical and practical boundariesto the performances of the telescope in terms of resolution and contrast (Fig.2.2).

This chapter summarizes the basic concepts of optics that govern the optome-chanical performances of a telescope.

2.2 Aberrations

2.2.1 Definition

a)image plane

b)image plane

Figure 2.3: (a) Rays emerging from a spherical wavefront converge towards asingle point in the image plane; (b) Rays emerging from an aberrated wavefronthit the image plane over an extended area, spreading the light energy [adaptedfrom (Geary, 2002), p.79].

On a strictly geometric point of view, a perfect telescope (like in Fig.2.1) shouldfocus the light of a distant (dimensionless) point source into a (dimensionless)image point on the focal surface, to establish a point-to-point correspondence be-tween an object and its image (Fig.2.3.a). In other words, it should transform anincoming diverging spherical wavefront into a spherical wavefront converging to-wards a point on the focal surface [(Schroeder, 2000), p.45]. However, deviations

2.2 Aberrations 23

from the initial design in the shape or in the position of the optical elements,or the observation of off-axis objects will induce distortions of the output wave-front, causing the light energy to be spread on the image surface as depicted inFig.2.3.b. Those deviations are called aberrations.

Accordingly, the images generated by aberrated wavefronts result from a super-position of light spots of finite size rather than points. It creates a blur in theimage, the amplitude and shape of which is roughly determined by those of theaberrations present. Historically, a classification of five primary aberrations hasbeen established, namely spherical aberration, coma, astigmatism, field curva-ture and distortion1. They were classified according to analytical developmentsmade by Seidel and they correspond to the optical signatures of some of the mosttypical deviations from the initial design. Combined with the derivation of theiranalytical expressions, they were the basis of the measurements of optical qualitysince the XIXth century. A deeper discussion is provided in appendix B.

2.2.2 Quantifying the wavefront error

The wavefront error with respect to its reference sphere can be expressed as afunction of space coordinates W (r). Its root mean square (RMS) value, com-puted over its whole surface provides an effective indicator of the quality of awavefront (or of the surface of a mirror)2. It is mostly expressed either as anabsolute measurement (in units of microns e.g.) or as a relative measurement, afraction of the wavelength of observation, λ. Conventionally, a system is consid-ered as nearly perfect if the RMS wavefront error of the output beam is less thanλ/14 (cfr section 2.5).

In many applications, it is not required to know point by point the shape of theerror. By extending the principles behind the use of the primary aberrations, itcan be more convenient and efficient to express the wavefront error as a linearcombination of a set of orthogonal functions defined over the whole aperture.One of the most common analytic representation uses the Zernike polynomials,in the form

W (r, θ) =n∑

i=1

aiZi(r, θ), (2.1)

where W (r, θ) and Zi(r, θ) are respectively the wavefront and the ith Zernike

1By nature, reflecting telescopes are not affected by chromatic aberrations, the reader canrefer e.g. to [(Walker, 1998), p.138] for more information.

2Other indicators, such as the peak-to-valley wavefront error (P-V) can be misleading as theygive no information about the the area over which the error is occurring.


polynomial expressed in polar coordinates. The coefficient ai results from theprojection of W on Zi, and may be computed either by direct integration overthe unit circle or by least square fitting. The Zernike polynomials have elegantanalytical expressions, the formulation of which can be automated easily [see e.g.(Malacara, 1992), p.464]3. They are given up to number 11 in Table 2.1 anddepicted in Fig.2.4.

Polynomial Denomination

1 Piston√4r cos θ Tilt√4r sin θ Tilt√3

(2r2 − 1

)Defocus√

6(r2 sin 2θ

)Astigmatism√

6(r2 cos 2θ

)Astigmatism√

8(3r3 − 2r

)sin θ Coma√

8(3r3 − 2r

)cos θ Coma√

8r3 sin 3θ Trifoil√8r3 cos 3θ Trifoil√5

(6r4 − 6r2 + 1

)Spherical aberration

Table 2.1: Zernike polynomials [convention from (Zemax Corp., 2005)].

The piston and the tilt terms correspond respectively to a constant and to alinear phase shift all over the wavefront; the latter only change the location ofthe focus on the image surface. Accordingly, none of those terms have an impacton the image quality. Defocus corresponds to a change of the overall radius ofcurvature of the wavefront, changing the position of the focus either upstreamor downstream the initial image surface. The shapes of astigmatism, coma andspherical aberrations as Zernike polynomials are close (but not identical) to thoseof the corresponding primary aberrations.

The Zernike polynomials are usually classified with respect to their radial andazimuthal orders: The higher the orders of the polynomial, the higher its spatialfrequency and, usually, the lower its amplitude in the wavefront error [this is re-ferred to as the principle of St Venant in (Wilson et al., 1987)]. In general, mostof the wavefront errors due to misalignment, mechanical and thermal distortions

3Care should be taken, however, that the ordering and the normalizing of the Zernike poly-nomials currently admit no single standard.

2.2 Aberrations 25

Tilt

Astigmatism Astigmatism

Trefoil Coma Trefoil

Tetrafoil Tetrafoil

Radia

l ord

er

Piston

Defocus

Spherical Aberration

Azimuthal Order

0

Tilt

Coma

1 2 3 4-1-2-3-4

0

1

2

3

4

Figure 2.4: Zernike polynomials ranked according to their azimuthal and radialorders.

and misfiguring can be described by combining the first 20 polynomials. On thecontrary, they are not best fitted for the description of errors at very high spatialfrequencies, such as surface roughness of mirrors, point defects, or the highestfrequencies of air turbulence, that would require an unpractically high number ofterms.

The Zernike polynomials have a zero mean and are orthogonal over the unitcircle. The mean square error of the total aberration is the weighted sum of themean square errors of each Zernike term4 [(Schroeder, 2000), p.264]. With thenormalization used in Table 2.1, the total RMS error of the wavefront is simply

4The weights depend on the normalization.


RMS =

√√√√n∑

i=4

a2i . (2.2)

The sum of Eq.(2.2) does not include piston (i = 1) and tilt (i = 2, 3) as they donot affect the image quality.

The analytical expressions above are defined on unobstructed circular pupils.Other polynomials have been proposed for other pupil shapes on which the wave-front error is analyzed: Circular aperture with a central circular obstruction(Mahajan, 1981), hexagonal or rectangular aperture (Mahajan and Dai, 2007),etc.

2.3 Common optical configurations of optical telescopes

The notions of f-number (f/#) and field of view (FOV), that are used extensivelyin the following, are defined in appendix A.

2.3.1 Newtonian telescopes

F1

F2

M1 M1

a) b)

Figure 2.5: Newtonian telescope: (a) Configuration giving access to the primefocus F1 - (b) A folding mirror gives an easier access to focus F2.

According to the fundamental property of conics, the simplest telescope could bebuilt with a single parabolic reflector, taking advantage of the fact that one ofits foci is at infinity. This design was implemented by Isaac Newton in his firsttelescope, commonly referred to as Newtonian telescope (Fig.2.5). However, asillustrated in appendix B, wavefront errors may arise as well from errors in theshape of the mirror, or by observing off-axis objects. Table 2.2 synthesizes thedependence of the primary aberrations with respect to the field angle θ, and thef/# of the overall telescope. Those relations set severe constraints on the designand use of such a telescope, and limit its implementation to telescopes with rather

2.3 Common optical configurations of optical telescopes 27

small diameters of M1, large f/# and small FOV. The most limiting aberrationin this case is coma. Finally, the focus of such telescopes is difficult to access,and would complicate the design of the supporting structure for large mirrors.

Spherical (f/#)−3

Coma θ(f/#)−2

Astigmatism θ2(f/#)−1

Table 2.2: Scaling laws of primary aberrations affecting a Newtonian telescope[(Bely, 2003), p.111].

2.3.2 Two-mirror telescopes

M1

M2

M1

M2

a) b)

Figure 2.6: (a) Cassegrain telescope - (b) Gregorian telescope.

The limitations of the Newtonian telescope can be (at least partially) overcomeby increasing the complexity of the optical design, consisting in a secondary conicmirror, with one of its foci collocated with that of the paraboloidal M1. Thereare two important classes of two-mirror telescopes differing in the shape of thesecondary mirror: The Cassegrain uses a convex hyperboloid and the Gregoriana concave ellipsoid (Fig.2.6).

However, the relations of Table 2.2 still apply to the Cassegrain and Gregoriantelescopes because of their paraboloidal M1. The dominant off-axis aberration isstill coma. The difference lies in the overall f/# of the telescope, that are largerthan that of a Newtonian telescope with the same diameter and tube length, thusallowing for larger fields. However, the need for still larger fields has called fora more efficient use of the geometrical parameters of the conics, based on twoconsiderations. First, the requirement for sphericity only applies to the outputwavefront. Second, it is possible, by a proper choice of geometrical constants ofdownstream mirrors, to compensate fully or partially for the aberrations induced


by upstream mirrors5.

This led to design variations of the classical Gregorian and Cassegrain. In thosevariations, the paraboloid constituting M1 is replaced respectively by an ellipsoidand a hyperboloid. It can be shown from the equations [see e.g. (Schroeder, 2000),p.115] that the departure of M1 from a paraboloid causes the reflected wavefrontto be different from a sphere, but that it can be corrected by a proper choiceof M2 as equations show that there is an infinite number of combinations of theconic constants of M1 and M2 that ensure the correction of spherical aberrationof the output wavefront. Amongst them, some particular combinations allowto compensate for off-axis aberrations as well, of which coma is the prevalentone. Designs that compensate for both coma and spherical aberrations are calledaplanatic and the aplanatic Cassegrain is better known as the Ritchey-Chretien;the field of such telescopes is larger than that of their classical versions, and islimited by astigmatism according to Table 2.2.

2.3.3 Telescopes with 3 or more mirrors

The principles of the generalized Schwarzschild theorem have been put in practiceboth in analytic studies of theoretical designs, and in actual projects of futureELTs. The use of 3 or more mirrors allows for the compensation for aberrationssuch as the off-axis astigmatism and the distortion and field curvature on theimage plane. It also opens the way to using a segmented spherical M1 thatwould exhibit significant advantages in terms of the manufacturing, testing andmaintenance of the segments, balanced by the need for at least two correctormirrors to compensate for the significant on-axis aberrations induced by suchfast primaries6. Those designs also permit a better integration of beam steeringand deformable mirrors, respectively for image motion and wavefront correction.

2.4 Wavefront error due to deviations from the design

During observations, a telescope is subjected to external disturbances that canmodify the shape of its mirrors and their relative positions in the optical train.This section presents basic relations on the sensitivity of the wavefront withrespect to those deviations in the case of a two-mirror telescope (they are equally

5This is a general principle stated in the generalized Schwarzschild theorem: ”For any geom-etry with reasonable separations between the optical elements, it is possible to correct n primaryaberrations with n powered elements.” [(Bely, 2003), p.123]. In this context, the term ”pow-ered elements” refers to conics; surfaces of higher degrees could compensate for more than 1aberration but are difficult to produce and test.

6A mirror is said to be fast if it has a small f/#, cfr appendix A

2.4 Wavefront error due to deviations from the design 29

valid for a Cassegrain, a Gregorian and their aplanatic variations and can beextended to telescopes with more mirrors).

2.4.1 Shape of optical elements

To a first approximation, after reflection on an aberrated mirror, the error af-fecting the wavefront is twice that of the mirrors; it can be expressed in termsof Zernike coefficients, according to Eq.(2.3). Therefore, the optical tolerances,when referring to a mirror, are twice as severe as when referring to a reflectedwavefront. For curved mirrors, simulations show that the value of the coefficientis slightly smaller than 2, and that the difference grows with the amplitude of theinput aberrations and when the mirror is faster (smaller f/#).

ai, output wavefront = 2.ai, mirror misfigure (2.3)

2.4.2 Relative position of optical elements

M1

M2

d

l

optical axis

a

Figure 2.7: Despace d, tilt α and decenter l.

If the mirrors are all made up of surfaces of revolution, the influence of their rel-ative positions is essentially governed by three relative parameters (for each pairof mirrors) that describe the deviations with respect to the initial design; theyare defined on Fig.2.7, namely (axial) despace, (lateral) decenter and tilt. Table2.3 summarizes the scaling laws of the primary aberrations induced when suchdeviations are present [(Schroeder, 2000), p.132]. A dependence in θ indicatesthe variation of the induced aberration with the field angle. Those aberrationsmust be added to those induced when observing off-axis objects (cfr Table 2.2),either due to the observation of extended objects or due to errors in pointing.

In addition to the aberrations, the position of the image on the focal surface isshifted of an amount proportional to l and to α. A general conclusion regarding


Aberration Despace d [m] Decenter l [m] Tilt α [rad]

Sp d (f/#)−3 / /C θd (f/#)−2 l (f/#)−2 α (f/#)−2

A / l (f/#)−1 /

Table 2.3: Scaling laws of primary aberrations affecting a two-mirror telescopeunder the effect of deviations in the relative position of the mirrors. The spher-ical aberration (Sp), coma (C) and astigmatism (A) refer to the correspondingprimary aberrations.

structural aspects that can be drawn from Table 2.3 is that a telescope with afaster M1 (smaller f/#) is more sensitive to position errors of any kind.

2.4.3 Linearity

The output wavefront error in terms of primary aberrations can be computed byadding those induced along the optical train of a telescope [(Schroeder, 2000),p.93]. The analytical expressions relating the low-order Zernike polynomials tothe primary aberrations are developed in (Wyant and Creath, 1992). Althoughthose relations are non-linear, simulations show that, over a quite extended regimeof aberrations, the wavefront errors induced by the various elements can be addedin terms of their Zernike coefficients without generating significant deviations withrespect to the actual output wavefront error (Angeli and Gregory, 2004; Noethe,2002; Whorton and Angeli, 2003). Therefore, linear optomechanical models canapproximate the Zernike coefficients of the output wavefront by

ai, output = ai, input +∑

shape

ai +∑

alignment

ai +∑

off−axis

ai . (2.4)

2.4.4 Design trade-offs

The trends in the design of telescopes can be summarized roughly by three re-quirements: A wide unaberrated field of view, a good resolution and a good lightgathering power. The trade-offs consist of balancing between contradictory ad-vantages from optical and structural points of view.

The combination of good resolution and light gathering power call for large andfast primary mirrors. The evolution of the f/# of M1 in time is illustratedin Fig.2.8. A small f/# of M1 brings several other advantages: The structureand the enclosure are comparably smaller, having a significant impact on their

2.4 Wavefront error due to deviations from the design 31

Keck I & II

E-ELT, TMT

GTC

VLT

1950Year

1970 1990 20100

1

2

3

4

5

Gemini,Subaru

NTT

Prim

ary

mirro

r f/#

Figure 2.8: Evolution in time of the f-number of the primary mirrors of opticaland infrared telescopes [adapted from (Bely, 2003), p.136].

total costs. The compactness of the structure also ensures a comparably higherstiffness, a lower overall mass (and thus less thermal inertia) and a smaller M27,leading to higher eigen frequencies. On the other hand, as shown in section 2.3, asmaller f/# induces tighter tolerances on the alignment of the optical elements,mirror shapes that are more difficult to produce and test and is responsible forhigher off-axis aberrations [cfr Table 2.3 and(Strom et al., 2003)].

Telescopes in the range 2-10m largely rely on two-mirror configurations, amongstwhich the Cassegrain type (mostly in its Ritchey-Chretien variation to improvethe FOV) is the most common: For a given f/# , a Cassegrain is more compactand has a smaller secondary. Those advantages overcome their slightly worseoff-axis aberrations and difficulty to test convex M2.

More elaborate designs are considered for some future ELT projects: In thethree-mirror design of JELT (Nariai and Iye, 2005) and the five-mirror design ofE-ELT (ELT Telescope Design Working Group, 2006; Spyromilio et al., 2008),the motivations are essentially to extend the usable field of view. In the six-mirror concept of OWL, it is also constrained by the envisioned spherical M1.

7A smaller secondary offers advantages in terms of mass, thermal inertia, optical testing,obscuration of M1 (more light gathering power), area exposed to the wind and diffraction (seesection 2.5.3).


However, mirrors are the most critical elements with respect to optical quality.Moreover they, and their supporting structures, represent massive elements, theposition and shape of which must be maintained with tight tolerances. Therefore,the smaller and the less numerous they are, the simpler and more effective thestructure. It is worth noting that the designers of the 30m TMT have chosena Ritchey-Chretien configuration (TMT Obs. Corp., 2007), building on theirsuccessful experience with Keck.

2.5 Diffraction-limited imaging

2.5.1 Definitions

Because of the wave nature of light, even a perfect (unaberrated) optical sys-tem will not image a point source as a true point, but rather as a bright coresurrounded by a halo. This spreading of the light energy is called diffraction.Light is diffracted at the edges of any opaque body that it crosses on the pathbetween the object and the image plane: Diaphragms, mirrors, lenses, structuralelements,. . . Those edges modify the interferences of the light waves as they travelthrough space, which in turn spreads the light energy in deterministic patternsdefined by the shape of the opaque body.

An optical system is said to be diffraction-limited when the aberrations are suffi-ciently small so that the size of the image point is only limited by the diffraction.It is a lower physical boundary to the size of the image spot that a perfect imag-ing system can produce. Therefore, it sets a limit under which the aberrationshave little impact on the image quality: Roughly speaking, an imaging systemcan be considered as perfect if the area over which the rays hit the image planeis encompassed by the central bright spot produced by diffraction.

2.5.2 Imaging

The image of a point object formed by an imaging system is called its PointSpread Function (PSF). The PSF takes the diffraction and aberrations into ac-count. The image formation consists in a convolution of each point of the objectby the PSF. Therefore, the narrower the PSF, the sharper the image. The PSFof a diffraction-limited imaging system with an unobstructed circular aperture,Fig.2.9, is called the Airy disk (e.g. the on-axis image formed by a parabolicmirror). It consists in a bright core surrounded by concentric rings. The centralspot contains approximately 84% of the total light energy on the image surface;its diameter is proportional to λ, the wavelength of the light, and to the f/# ofthe system.

2.5 Diffraction-limited imaging 33

d1 d2d1= 2.44 f/#l

d2= 4.48 f/#l

(EE = 84%)

(EE = 91%)

image planeintensity profile

Figure 2.9: Airy disk, diffraction pattern produced by a perfect imaging systemwith a circular pupil (EE refers to the percentage of encircled energy) [adaptedfrom (Born and Wolf, 1997), p.416 and (Walker, 1998), p.51].

The resolution of a telescope, δθ, is the minimum angular separation between twopoint objects (of the same brightness) to appear as two separate images. Becauseof diffraction, there is an overlap between the energies associated with each imagespot; therefore, if the images are too close to each other, they might be seen asa single spot by the detector. By convention, the limit of resolution is expressedto a first order in terms of the equivalent Airy disk produced by a telescope, andcorresponds to

δθ = 1.22λ/D. (2.5)

This corresponds to a situation where the peak of one Airy disk falls in the firstdark ring of the other Airy Disk8. Therefore, increasing D improves the resolu-tion (to the limit of a constant level of aberrations RMS).

An other important practical aspect of that definition is that λ actually definesthe order of magnitude of the optical precision required to consider an opticalsystem as perfect. A diffraction-limited image requires that the phase differencesover the wavefront be inferior to a fraction of λ. Conventionally, a system isconsidered as diffraction limited if the RMS wavefront error of the output beamis less than λ/149. Therefore, it is much easier to build large and fast radio-telescopes that operate at significantly longer wavelengths (λ > 0.5m).

8This convention is known as the Rayleigh criterion. There are other criteria, amongst whicha less conservative definition that simply uses δθ = λ/D.

9Therefore, a mirror is considered as perfect if its RMS surface error is inferior to λ/28.Practically, the requirements change in function of the spatial frequencies of the defects. Thedisturbances and means of compensation that we consider in this work are concerned only withlow spatial frequencies, i.e. larger than approx. D/10.


PSF(peak)

PSF(peak)unaberrated

aberrated

Figure 2.10: Comparison of the PSF of a diffraction-limited (unaberrated) tele-scope (continuous line) with that of the same telescope undergoing aberrations(dashed line). The lateral shift of the peak is due to tilt and causes image motion.

The ratio between the peak intensity of a telescope undergoing aberrations tothat of the same telescope working as diffraction-limited is called the Strehl ratio,S (cfr Fig.2.10). Eq.(2.6) is an approximation of S in terms of the RMS outputwavefront error (denoted by ϕ and expressed in meters)10 (Wyant and Creath,1992). Coupling that approximation to the condition above leads to defining atelescope as diffraction-limited if S ≥ 0.8, corresponding to a wavefront RMSerror less than λ/14. While often used, this criterion can be misleading as it isdefined for one wavelength and it gives no information on the shape of the halo.

S =PSF(peak)aberrated

PSF(peak)unaberrated' e−(2πϕ/λ)2 (2.6)

2.5.3 Diffraction from obscurations

An obscuration is any opaque object located on the path of the wavefront, suchas M2 and its support structure in telescopes with two (or more) mirrors, asshown in Fig.2.11. Their presence tends to increase the spreading of the energyaway from the central core, thus enforcing the halo while modifying its globalshape at the same time. Moreover, they cause a global decrease of the totalenergy in the image plane, proportional to their relative area with respect to thatof M1. The effects of the obscurations can be incorporated in S, according toS∗ = S.(1−Aobs/AM1), where AM1 and Aobs are respectively the area of M1 andthe total area of the obscurations, projected on the incoming (plane) wavefront.

10This approximation is valid for S ≥ 0.1

2.6 Telescopes with a segmented primary mirror 35

a) b)PSF intensity profile

M1

M2

M1

no obscuration

obscuration by M2energyloss

Figure 2.11: (a) The obscuration induced by M2 keeps the rotational symmetrybut causes some of the energy to be transferred from the central core to itssurrounding rings as shown by the dashed curve (normalized to unit); the dottedline is normalized so as to take into account the energy loss due to the obscuration.- (b) A thin rectangular obscuration produces a perpendicular spike in the PSF(independently on the location of the obscuration with respect to the center ofthe aperture) [based on (Born and Wolf, 1997), p.416 and (Harvey and Ftaclas,1995)].

2.5.4 Diffraction-limited versus aberrated

The effects of diffraction are important only when the aberrations are negligible,or when the diameter of the spot calculated by the geometric approach is of thesame order of magnitude as that of the equivalent Airy disk of the telescope(2.44λf/#). For the latter case, the so-called extended Nijboer-Zernike method(Braat et al., 2008) provides simplified analytical methods of calculation of thePSF, derived from the exact approach [see (Schroeder, 2000), p.258]. In caseswhere aberrations are big enough to make the geometric spot significantly largerthan the equivalent Airy disk, the spreading of the energy is such that the effectsof diffraction become negligible [(Schroeder, 2000), p.271].

2.6 Telescopes with a segmented primary mirror

2.6.1 Conditions for optimal performances

Fig.2.12.a illustrates the three fundamental requirements for a densely segmentedmirror to be optically equivalent to their parent monolithic mirror: Coaligningconsists of stacking the images produced by each of the N segments, cofocusing


a) b)Coaligning Cofocusing Cophasing

unphased

cophased

parent mirror

N

d D l/D

l/d

Figure 2.12: Focalization, alignment and cophasing [(Bely, 2003), p.334].

by ensuring that they have the same size in their common image plane (i.e. thatthey are focused on the same plane) and cophasing ensures the continuity be-tween the edges of the segments11. Diffraction-limited imaging at wavelengths≥ λ requires a wavefront error less than λ/14. Fig.2.12.b shows the significantincrease of resolution resulting from the phasing. An unphased mirror has thesame resolution as one of its segments (the total energy being the sum of thatreflected by N segments). When phased, the resolution corresponds to that ofa monolithic mirror with the same D and f/#, approximately

√N better than

that of a single segment. Therefore, phasing in future ELTs is critical to meetthe best optical performances of the telescope.

Segments are usually fabricated so that their surfaces match that of the parentmonolithic mirror. As M1 of current and next segmented telescopes are aspheric,so are their segments. Furthermore, their asphericity grows as they are locatedfurther away from the vertex of M1. Therefore, technologies have been developedduring the construction of the Keck-I telescope to fabricate off-axis segments effi-ciently and at an affordable cost (Nelson et al., 1980). The supporting structurealso ensures a good positioning and orientation of the segments with respect totheir parent shape. However, external disturbances such as the wind-, gravity-and the thermal-induced deformations cause deviations in the shape of the struc-ture, resulting in changes in the position and orientation of the segments, but alsoin their shape. Therefore, passive means as well as offline and active correctionsare required to ensure the verification of the three conditions during operation ofthe telescope.

11Cophasing might seem redundant with cofocusing at a first glance. But, in a diffraction-limited telescope, the depth of focus due to diffraction [see e.g. (Bely, 2003), p.121] might belong enough to induce a significant phase delay between some of the N segments.


2.6.2 Design trade-offs

Figure 2.13: Tessellation geometries: Left, honeycomb; right, keystone.

Two geometries, shown in Fig.2.13 are commonly envisioned to achieve a densetessellation12. The honeycomb-type has proved the most interesting for largetelescopes since Keck (Strom et al., 2003): Hexagonal segments are easier to fab-ricate and to polish, their supports and the location of their edge sensors andposition actuators are the same for all the segments and their position control isbetter understood. On the other hand, the number of surfaces to polish and testis slightly bigger with hexagons (N/6 due to the six-fold symmetry) than withthe keystone (1 shape per ring).

The segment size is also subjected to a complex trade-off (Padin, 2003; Stromet al., 2003). The deformations under gravity call for smaller segments because oftheir evolution as d4/h2 (d and h are respectively the diameter and the thicknessof a segment), which also induces a better thermal behavior and increases theeigen frequencies of the structure (by diminishing its overall mass). Smaller seg-ments are also easier and cheaper to handle, fabricate, polish and test. This alsomakes them less sensitive to deviations in their figuring or in their positioning.However, the number of position actuators and edge sensors grows with N , whichhas a significant impact on the overall cost, makes them more difficult to con-trol (more error propagation due to sensor noise, higher probability of failures,increased computational requirements and higher requirements on the opticalsensors used for calibration, . . . ) and increases the complexity of the supportingstructure. Currently, there is a consensus to consider the 1-2 meter-diameterrange as optimal.

12Fig.2.13 represents the projection of the pattern on the mirror surface. As tessellating acurved surface with regular polygons is not possible, their edges exhibit a slight curvature tominimize the gaps between the segments (Nelson and Sheinis, 2006).


2.6.3 Diffraction in segmented telescopes

Some instruments envisioned to equip future ELTs need imaging capabilities withrequirements on resolution and contrast that are so high that a detailed study ofthe diffraction halo is mandatory [analytical formulations are derived in (Floreset al., 2003; Yaitskova et al., 2003b; Zeiders and Montgomery, 1998) e.g.]. Further-more, the sources of potential degradation of those performances are numerous,as can be seen on Fig.2.14(left), both due to segmentation and to obscurations bystructural members used to stiffen the telescope. One can recognize the character-istic halo corresponding to a hexagonal segment of M1 (Fig.2.14-right), periodicpatterns of peaks due to the periodicity of both segmentations of M1 and M2,and bright lines induced by the beams and cables constituting the support of M2.

Figure 2.14: Left, simulation of the diffraction figure of OWL (ESO, 2004); right,diffraction halo produced by a segment of OWL M1 taken alone.


A non-exhaustive list of sources of image degradation in a cophased and diffraction-limited ELT can be established (Chanan and Troy, 1999; Harvey and Ftaclas,1995; ESO, 2004; Sacek, 2010; Troy and Chanan, 2003; Yaitskova et al., 2003a,b;Yaitskova, 2007, 2008; Zeiders and Montgomery, 1998):

- segmentation of M1 (both due to the gaps and to the shape of the edges ofthe whole aperture when it is not circularized);

- obscurations by the secondary mirror and its support (beams and cables);

- variations of reflectivity between the segments;

- errors in the size of the segments and in their positioning (tangentially tothe parent mirror shape);

- fillet radius at the edges of the segments (often referred to as turned edges);

- piston and tip/tilt errors of the segments;

- shape aberrations of the segments;

Table 2.4 summarizes the effect of these sources on the final Strehl ratio, S∗, inthe form of simple analytical coefficients, κj . By extension of section 2.5.3, S∗

can be computed as

S∗ = S .∏

j

κj (2.7)

where S is the Strehl ratio due to aberrations as defined in Eq.(2.6), taking intoaccount both the aberrations in the segments shape, and the aberrations in theglobal shape of the mirror.

40 2 Basics of Telescope OpticsSo

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2.7 Conclusions 41

2.7 Conclusions

This chapter has reviewed the main design parameters affecting the optical qual-ity of a telescope. The main aberrations have also been discussed, as well as thequantification of the wavefront error. A telescope can be considered as diffraction-limited if the Strehl ratio is such that S ≥ 0.8.

As the telescopes become larger, it is increasingly difficult to have perfectly stablereflecting surfaces and the image quality becomes more sensitive to thermal gra-dients and time-varying gravity loads. In modern large telescopes, the wavefronterrors induced by those disturbances are compensated by various layers of activecontrol which are analyzed in the next chapter.

2.8 References

Angeli, G. Z. and Gregory, B. Linear optical model for a large ground-based tele-scope. In Kahan, M. A., editor, Optical Modeling and Performance Predictions- SPIE 5178, pages 64–73, January 2004.


Born, M. and Wolf, E. Principles of optics. Pergamon Press, 1997.

Braat, J. J., Dirksen, P., and Janssen, J. E. Detailed description of the ENZapproach. T.U. Delft, 2008.

Chanan, G. and Troy, M. Strehl Ratio and Modulation Transfer Function forSegmented Mirror Telescopes as Functions of Segment Phase Error. AppliedOptics, 38(31):6642–6647, November 1999.

ELT Telescope Design Working Group. E-ELT Telescope Design. ESO, 2006.


Flores, J. L., Paez, G., and Strojnik, M. Optimal aperture configuration for asegmented and partially diluted extremely large telescope. Journal of ModernOptics, 50(5):729–742, May 2003.

Geary, J. M. Introduction to Lens Design - With Practical ZEMAX Examples.Willmann-Bell, 2002.

42 References

Harvey, J. E. and Ftaclas, C. Diffraction effects of telescope secondary mirrorspiders on various image-quality criteria. Applied Optics, 34(28):6337–6349,1995.

Mahajan, V. N. Zernike annular polynomials for imaging systems with annularpupils. Journal of the Optical Society of America, 71:75–85, 1981.

Mahajan, V. N. and Dai, G. M. Orthonormal polynomials in wavefront analysis:analytical solution. Journal of the Optical Society of America A, 24(9):2994–3016, September 2007.

Malacara, D. Optical Shop Testing. Wiley, 1992.

Nariai, K. and Iye, M. Three-Mirror Anastigmat Telescope with an UnvignettedFlat Focal Plane. Publications of the Astronomical Society of Japan, 57:391–397, April 2005.

Nelson, J. and Sheinis, A. Modern Optical Design. In Foy, R. and Foy, F. C.,editors, Optics in Astrophysics, pages 37–48. Springer, 2006.

Nelson, J. E., Gabor, G., Hunt, L. K., Lubliner, J., and Mast, T. S. Stressedmirror polishing. 2: Fabrication of an off-axis section of a paraboloid. AppliedOptics, 19(14):2341–2352, 1980.

Noethe, L. Active Optics in Modern, Large Optical Telescopes. Progress inOptics, 43:1–70, 2002.

Padin, S. Design Considerations for a Highly Segmented Mirror. Applied Optics,42(16):3305–3312, 2003.

Sacek, V. Telescope Optics, 2010. URL http://www.telescope-optics.net/.

Schroeder, D. Astronomical Optics. Academic Press, 2000.

Spyromilio, J., Comeron, F., D’Odorico, S., Kissler-Patig, M., and Gilmozzi, R.Progress on the European Extremely Large Telescope. The Messenger, 133:2–8, September 2008.



References 43

Troy, M. and Chanan, G. A. Diffraction effects from giant segmented mirror tele-scopes. In Angel, J., R. P. and Gilmozzi, R., editors, Future Giant Telescopes- SPIE 4840, pages 81–92, January 2003.

Walker, B. H. Optical Engineering Fundamentals. SPIE Press, 1998.

Whorton, M. S. and Angeli, G. Z. Modern control for the secondary mirror ofa Giant Segmented Mirror Telescope. In Angel, J. R. P. and Gilmozzi, R.,editors, Future Giant Telescopes - SPIE 4840, volume 4840, pages 140–150,January 2003.


Wyant, J. C. and Creath, K. Basic Wavefront Aberration Theory for OpticalMetrology. In Shannon, R. R. and Wyant, J. C., editors, Applied Optics andOptical Engineering, Volume XI, volume 11, 1992.

Yaitskova, N. Influence of irregular gaps between primary mirror segments ontelescope image quality. Journal of the Optical Society of America A, 24(9),September 2007.

Yaitskova, N. Diffraction halo by a segmented telescope. In Stepp, L. M. andGilmozzi, R., editors, Ground-based and Airborne Telescopes II - SPIE 7012,2008.

Yaitskova, N., Dohlen, K., and Dierickx, P. Diffraction in OWL: Effects of seg-mentation and segments edge misfigure. In Angel, J., R. P. and Gilmozzi, R.,editors, Future Giant Telescopes - SPIE 4840, pages 171–182, January 2003a.

Yaitskova, N., Dohlen, K., and Dierickx, P. Analytical study of diffraction effectsin extremely large segmented telescopes. Journal of the Optical Society ofAmerica A, 20(8):1563–1575, August 2003b.

Zeiders, G. W. and Montgomery, E. E. Diffraction effects with segmented aper-tures. In Bely, P. Y. and Breckinridge, J. P., editors, Space Telescopes andInstruments V - SPIE 3356, 1998.

Zemax Corp. Zemax - Optical Design Program - User’s Guide. Zemax Corp.,2005.

44 References

Chapter 3

Active control of telescopes

3.1 Introduction

Telescopes up to the 4-meter range were essentially passive in the sense thatthe unavoidable deviations from their initial optical design were only correctedonce in a while through complex offline maintenance procedures. Because of thesimultaneous increase in their size and in the requirements on their optical per-formances, maintaining optical tolerances over long time periods by relying solelyon the intrinsic stiffness of the structural and optical elements or on elaboratestructural designs (Serrurier truss, athermal design,. . . ) have proved increasinglyinsufficient to enable diffraction-limited imaging in 2-meter telescopes (or larger)(Noethe, 2002).

The implementation of active control in telescopes aims at automating the wholemaintenance procedure, making it faster and more efficient by minimizing humanintervention. Furthermore, active correction can be performed during observa-tions; this makes optical quality of active telescopes almost constant in time,allowing them to operate close to their best performances whenever the meteo-rological conditions are good. Moreover, active control has led to considerablerelaxations on the (passive) structural design. This, in turn, alleviates the de-mand in terms of amplitude to the active control, allowing it to focus on issuesof precision, to achieve the best optical quality possible (Enard et al., 1996).

45

46 3 Active control of telescopes

3.2 External disturbances

Errors in the output wavefront may arise from several different sources; we chooseto classify them according to their main characteristics. Table 3.1 summarizestheir corresponding spatial and temporal bandwidths and typical amplitudes.

• As explained in the previous chapter, the design and manufacture are in-creasingly critical as telescopes grow larger. The wavefront errors theyinduce are mostly constant in time and hence should be minimized as theyconsume a permanent portion of the control authority. Wavefront errorsfrom the design and from the structural elements mainly involve lower spa-tial frequencies, typically tilt, defocus, astigmatism, coma and sphericalaberration1. Manufacturing errors in mirrors involve a large range of spa-tial frequencies, that are of equal optical importance; active control allowsa relaxation on the manufacturing tolerances for the lowest spatial frequen-cies, letting the manufacturer focus on eliminating the higher ones. Agingmostly modifies those constant errors on very long timescales.

• Operation and maintenance induce sporadic deviations in the positioningand aligning of instruments and optical elements. Therefore, they mostlyinvolve the same lower spatial frequencies as design and manufacture errors,with slow variations in time, up to the order of days or weeks.

• Gravity induces deformations in the structure, consisting mainly in bend-ing and compressing the M2 support structure, as well as in the mirrorsthemselves, also dominated by low spatial frequencies [in the case of VLT,most of the deformation is covered by the first 16 Zernike polynomials (Wil-son, 2003)]. However, the timescale of the gravity loads is shorter since thetracking of stars during observations follows the rotation of the Earth.

• Temperature changes are the cause of two different types of effects. The firstone is mainly related to dilatation of both the structure and the mirrors,producing errors very similar to those produced by gravity both in termsof amplitudes and spatial2 and temporal3 bandwidths.

The second one is related to the thermal inertia of the mirrors, that delaysthe temperature changes of the mirrors with respect to those of the air andacts like a lowpass filter. The resulting difference in temperature gener-ates natural convection at the surface of the mirror, hence distorting the

1For the remaining of this text, if unspecified, aberrations must be understood as Zernikepolynomials rather than primary aberrations.

2The contribution from the structure is dominated by defocus; that of the mirrors dependson the geometry of the structure and on the elevation angle.

3The temperature decrease rate is of ∼ 0.5− 1oC/hour at the best astronomical sites.

3.2 External disturbances 47

wavefront by modifying locally the index of refraction: This phenomenonis called mirror seeing and, as suggested by Fig.3.1, is very detrimentalto the image quality. Additionally, any source of heat inside the domemight induce similar distortions to the wavefront (referred to as dome see-ing). The latter can be alleviated by using liquid cooling in those sourcesof heat. Correcting mirror seeing requires natural ventilation to performa cooling by convection and to wash away the thermal eddies: For thispurpose, windscreens allow the external air to flow through the enclosure.Dome and mirror seeing cover broad spatial and temporal frequencies; theiramplitudes tend to decrease when the frequencies increase.

0 2-2

1

2

DT [°C]

Rela

tive s

pot siz

e [/]

Figure 3.1: Influence of ∆T , the difference between the temperature of M1 andthat of the air, on the relative spot size [adapted from (Enard et al., 1996)].

• Similarly to dome and mirror seeing, atmospheric seeing results from localvariations of the index of refraction of the air in the highest layers of theatmosphere4. Atmospheric seeing is compensated for by Adaptive optics.

• Wind carries energy covering broad spatial and temporal frequencies (Wil-son et al., 1987). The wind loads produce substantial aberrations whichare partly compensated by the active optics. The enclosure design aimsat reducing the wind gusts while keeping a good natural ventilation5. Thewind response of a telescope will be analyzed in detail in section 5.6.

4One could also mention site seeing, resulting from heat exchanges between the ground andthe consecutive layers of air above it. The effects of seeing in general are very similar on thespatial aspect, small differences arise only from a slight shift between their respective temporalbandwidths.

5At the best astronomical sites, the wind speeds are in a typical range of 15− 20m/s; insidethe enclosure, the wind speed is maintained to 2− 4m/s.


• The operation of modern telescopes and instruments requires electro-mecha-nical devices such as pumps, cryocoolers, fans, . . . These act as internalsources of vibrations that are transmitted to the structure of the telescopeand may be responsible of severe degradation of the image quality by ex-citing some of its resonances. Vibrations appear very detrimental to theoperation of long-baseline interferometry.

To illustrate how a tiny vibration can deteriorate the image quality, Fig.3.2 showsan hypothetical free flying truss structure supporting a segmented mirror of 2mdiameter (space telescope); the figure shows the first mode (which is essentiallyastigmatism) and the PSF corresponding to the largest circular aperture insidethe mirror, when it is flat and when the vibration amplitude is λ/2 (peak tovalley).

First vibration mode (l/2)

Flat mirror

l/2

Figure 3.2: First vibration mode of a free flying truss supporting a flat segmentedmirror. The diameter of the segments is d=2m. The first mode is essentiallyastigmatism. The figure also shows the PSF corresponding to the largest circularaperture inside the mirror, when it is flat and when the vibration amplitude isλ/2 (peak to valley).

3.2 External disturbances 49

SourceBandwidth Typical

[Hz] amplitude

Design, manufacture, operation,[0 , 10−6

]À 100λ

maintenance and aging (fixed→ weeks)

Gravity (pointing and tracking)[10−4 , 10−3

]À 100λ

and thermal deformations (hours → minutes)

Wind-induced deformations[10−2 , 10

] À 100λ

Site, dome and mirror seeing[10−4 , 10

]< 20λ

Atmospheric seeing[10−2 , 102

]< 20λ

Tracking errors (vibrations)[1 , 102

]

Table 3.1: Main sources of image degradation in telescopes with the correspond-ing bandwidths and typical amplitudes [adapted from (Wilson et al., 1987; Wil-son, 2003)]. In general, structural deformations due to gravity and temperatureare slower processes involving larger amplitudes (up to a few millimeters) andlower spatial frequencies, as they rely on the elasticity of massive solids. Onthe contrary, seeing is related to the physics of air, which involves shorter timeconstants and smaller amplitudes (up to a few microns), with a very extendedspatial frequency range.


3.3 Layers of active control in modern telescopes

The preceding errors can be subdivided into 4 categories: Those affecting thepointing of the whole telescope, those affecting the relative position/orientationbetween M1 and M2, those affecting the shape of M1 and those affecting theoptical properties of the air. The fact that only the errors affecting the finalwavefront matter, and their quite good natural separation in terms of spatial andtemporal bandwidths, schematized in Fig.3.3 has called for a similar separationof the active control problem into 3 main layers:

- Pointing/tracking controls the main axes of rotation of the telescope.

- Active optics, compensates both for the deformations of M1 and for the errorin relative position/orientation between M1 and M2. Typical aberrationamplitudes up to a few millimeters are expected for future ELTs.

- Adaptive optics, compensates for the errors induced by seeing in general.Typical aberration amplitudes up to a few microns are expected for futureELTs.

Main axes

M1Shape

M2 RigidBody

Adaptive Optics

Bandwidth [Hz]

0.01 0.1 1 10 100

Spatial fr

equency 100

3

2

Figure 3.3: Temporal and spatial frequency distribution of the various controllayers of a large telescope [adapted from (Angeli et al., 2003)]. Active opticswill correct aberrations amplitudes up to a few millimeters in future ELTs, whileadaptive optics will correct aberrations up to a few microns.

The first two categories mostly aim at correcting the deviations in a quite directfashion (the actuation opposes physically the actual deviation) while the correc-tion of seeing can only be an indirect process (an optical compensation). The

3.3 Layers of active control in modern telescopes 51

aspects of thermal control (cooling) that directly affect heat transfers are beyondthe scope of this text; one could summarize the correction of temperature-inducedaberrations by stating that active optics performs the correction of thermoelasticdistortions (both in the mirror and in the structure), while adaptive optics com-pensates for the seeing errors.

Some overlap exists between those somewhat arbitrary separations. It is the av-erage amplitude of the disturbances and their temporal bandwidth that tend todefine the most effective means of correction between those that have an author-ity on the same type of error. For instance, the correction of image motion by alightweight fine steering mirror allows more precision and a much higher band-width than attempting to correct it with the main axes of a telescope weightinghundreds of tons, but the amplitudes it corrects are limited by its position insidethe optical train. Similarly, as M2 of future ELTs grow larger (and heavier), itis impossible to tilt them at the high temporal rate required by the correction ofimage motion induced by atmospheric seeing. Furthermore, the amplitude of thetilt that M2 can compensate for is also limited to avoid inducing significant coma(Guisard et al., 2000). Therefore, those corrections are complementary, and canbe performed in parallel.

3.3.1 Pointing and tracking

Pointing designates the orientation of the telescope along a predefined line ofsight by rotating it around its main axes. Tracking consists in compensatingfor the earth’s rotation and external disturbances to maintain the object in theline of sight. Their distinction is thus mainly based on their respective tem-poral bandwidths. Pointing and tracking accuracy are intrinsically limited bymanufacturing errors, by the precision of the encoders, by the atmospheric re-fraction and turbulence and by the structural deformations induced by externaldisturbances, amongst which wind may be particularly severe. The systematicparts of these error sources can be accounted for by a feedforward compensationbased on lookup tables. However, the residual error is usually significant andrequires a closed-loop compensation called guiding. It is based on the monitoringof the position of a star in the vicinity of the object to be observed. It mustbe close enough to be in the field of the telescope, and bright enough to allowmeasurements with a sufficient bandwidth6. A centroid algorithm, based on theimage captured by a dedicated sensor, compares the position of the guide starwith respect to the desired reference and calculates the required corrections. Asstated previously, the slower components of those corrections, involving largeramplitudes, are controlled by the main axes of rotation of the telescope.

6Light detectors must be feeded with enough photons to provide accurate measurements withgood signal-to-noise ratios. Therefore, the brighter the object, the faster the measurement.


3.3.2 Active optics

The NTT project

The New Technology Telescope (NTT), that saw first light in 1989, was the firsttelescope to be implemented with active optics7. The NTT is a Ritchey-ChretienAlt-Az telescope with a 0.24m-thick monolithic meniscus M1 of D = 3.5m andf/# = 2.2. Its locked-rotor frequency is ∼ 10 Hz, while the first resonance in-volving a deformation of M1 is at ∼ 115 Hz (Enard et al., 1996; Noethe, 2002)8.Some key aspects of the principles of active optics as implemented in NTT arerepresented in Fig.3.4.

M1

M2

Defocus

Scienceinstrument

Shack-Hartmann25x25 µ-lenses

Coma

Astigmatism,Spherical aberration,

Trefoil, Tetrafoil,

CCDcamera

Controller

( = 0.03Hz)fc

Controller

( ~ 0.1Hz)fc

Alt-az axes

Tilt

Figure 3.4: Principal elements of the tracking and active optics control loops inthe New Technology Telescope

Active optics at NTT is designed to compensate for 7 (primary) aberrations.They are measured on a bright reference star in the field of the telescope by aShack-Hartmann wavefront sensor (see appendix C) with 25 × 25 microlenses.Because the reference star is usually a few arcminutes away from the object, theyare not affected identically by atmospheric seeing; therefore, the effects of seeingmust be filtered out by averaging the wavefront measurements over a sufficientlylong time, 30s in this case, in order to measure only the aberrations induced by

7Up to now, no adaptive optics correction is implemented in NTT, therefore it is a seeing-limited telescope.

8Those values change slightly with the elevation angle.


the telescope itself. This constitutes the limit to the bandwidth of the activeoptics in NTT (Wilson, 2003).

Moreover, it is difficult to distinguish if some low spatial frequency aberrationsare due to deviations in shape or in alignment of the mirrors. The assumptionof linearity (cfr section 2.4.3) allows correction in NTT at a system level: Defo-cus and coma are corrected respectively by axial (despace) and lateral (decenter)rigid-body motions of M2 (3 DOF), while tilt (image motion) is compensated forby the alt-az axes of rotation (2 DOF) and the remaining aberrations (sphericalaberration, astigmatism, trefoil and tetrafoil) are corrected by the shape controlof M1 [(Wilson, 1999), p.286].

The primary mirror is supported by a set of 78 axial force actuators, 3 of whichbeing used as fixed points. A spring inside each actuator cell allows the man-ual correction of constant shape errors. The primary mirror is also supportedlaterally by a set of 24 astatic force actuators. They consist in counterweights at-tached to levers that apply forces that change continuously (and passively) withthe elevation angle of the telescope [(Wilson, 1999), p.268]. Shape control ofM1 is based on the measurements made by the wavefront sensor to compensatefor the gravitationally- and thermally-induced shape errors based on a modalapproach(Wilson et al., 1987). The feedback compensation is constructed asfollows:

1. The Jacobian J relating the optical aberration w to the control forces f isconstructed column by column by applying successively a unit force to eachactuator and measuring the resulting aberrations, leading to

w = Jf (3.1)

J is a rectangular matrix with more columns than lines.

2. The Moore-Penrose pseudo-inverse is computed

J+ = JT(JJT

)−1(3.2)

Each column of J+ represents the actuator force vector compensating thecorresponding optical aberration of unit amplitude. The force vector (ofminimum quadratic norm) compensating a given aberration w0 is obtainedaccording to

f0 = J+w0 (3.3)


The shape control is entirely based on that feedback loop, it is not supplementedby a feedforward (open-loop) stage based on lookup tables. The relative error isof the order of 0.1% RMS on the generation of all aberrations but the sphericalaberration term, which exhibits an error of ∼ 2% (Noethe, 2002)9.

Segmented mirrors

Figure 3.5: M1 of the future E-ELT telescope consisting of 984 segments, eachof them equipped with 6 edge sensors and 3 two-stage position actuators. Thesegments are connected to the actuators by a whiffletree (Dimmler et al., 2008).

The implementation of the shape control of M1 depends closely on the type ofmirror envisioned. Segmented mirrors such as the 10m M1 of the Keck telescopesuse a local, indirect metrology based on the relative displacements between thesegments measured by a set of edge sensors (see Fig.3.5). The controller acts onthe position of the segments to maintain the shape of M1 and its continuity, thusallowing the conditions for the optimal use of a segmented mirror to be met. Onthe contrary to wavefront sensors, the edge sensors do not bring bandwidth lim-itations in the system. In practice, the bandwidth of the control loop will ratherbe bounded by the avoidance of control-structure interaction (see section 4.6).However, edge sensors suffer from a drift in time that requires periodical offlinecalibrations with the use of adapted wavefront sensors. A deeper description isgiven in section 3.4 dedicated to the Keck telescopes.

9This is due to both facts that, at constant spatial order, rotationally symmetric shapes aremore difficult to generate and that NTT actuators are unable to pull on the mirror. In practice,gravity acts as the pulling force, but its equivalent amplitude is limited by the portion of themirror on which it acts.


Secondary mirrors

Reflecting surface

Housing

Hexapod(despace, tilts)

Longitudinalactuators(decenter)

Axialactuators(shape)

Figure 3.6: Actuators controlling the position and shape of TMT M2 [adaptedfrom (TMT Obs. Corp., 2007; Cho, 2008)].

The control of M2 in future ELTs will also be more complex than in NTT and willinvolve a flexible mirror. The rigid-body motion of M2 will be actively controlledthrough five degrees of freedom (despace, tipt/tilt and lateral decenters; clock-ing is maintained passively, as it is irrelevant to optical quality for rotationallysymmetric telescopes). M2 of TMT and E-ELT will be respectively 3.1m and6m, thus requiring correction to maintain their shape Fig.3.6 shows the variousactuators used to control M2 of TMT.

Some aberrations result from deviations that are highly predictable and thatsuffer from negligible hysteresis or friction (gravity). In those cases, an open-loop(feedforward) correction strategy can remove large correction terms that wouldotherwise overload the feedback loops; in that approach, during observations,a set of sensors (rotational encoders or temperature gauges e.g.) measure theinput parameters to determine the adequate correction signals from a lookuptable (established through simulations or by measurements in the telescope).Feedback control provides a precise correction of the residual error. To be fullyefficient, open-loop correction needs periodical offline calibration of its sensorsand actuators, and the latter should be able to maintain their commands overlong time periods of up to several hours.


3.3.3 Adaptive optics

Object

AtmosphericTurbulence

Distortedwavefront

Planewavefront

8

Deformablemirror

Controller

Wavefrontsensor

Instrument

a)

b)

AO OFF AO ON

Figure 3.7: (a) General principles of adaptive optics (AO) [adapted from (Hickson,2008)];(b) Images with and without adaptive optics (resp. AO on and AO off).

Turbulence occurring along the course of the wavefront in the atmosphere modi-fies locally the refraction index of the air, with random spatial and temporal struc-tures, resulting in corresponding distortions of the wavefront [see e.g. (Tyson,2000), p.33]. The amplitude of the aberrations induced are somewhat smallerthan those induced by other external disturbances. For a long time, the impos-sibility to build and maintain telescopes with a high optical quality reduced therelative impact of turbulence although it is sufficient to spread the energy in theimage plane in spots 10 to 100 times larger than the diffraction limit. Recently,as active optics has allowed telescopes to work closer to their diffraction limit,the effects of seeing have become of practical importance.

Astronomers have been aware of the problem of atmospheric seeing since the lateXVIIth century, and for long, the only mitigation solution was to build telescopesat the top of mountains. More recently, methods for measuring the quality of aportion of sky in terms of seeing have been used to improve the choice of a site.The availability of efficient wavefront sensors and fast computers, in parallel toextensive developments in the field of deformable mirrors have led to the recentdevelopment of adaptive optics.

The general principles of adaptive optics are depicted in Fig.3.7.a: Distortions af-fect an initially plane wavefront as it crosses the atmosphere, and are transmitted


along the optical path of the telescope, causing a spreading of the energy in thefocal plane (Fig.3.7.b). When the adaptive optics is turned on, a portion of the in-coming light energy is deviated through a beam-splitter located upward the imageplane, towards a wavefront sensor (typically a Shack-Hartmann) which measuresthe wavefront aberrations. This information is processed by a controller thatdetermines the commands to apply to a deformable mirror in the optical train ofthe telescope, in order to compensate for the incoming aberrations by reflection,so as to obtain a nearly spherical wavefront upward the focal plane (Hardy, 1994).

10 501

D [m]10

0

102

104

Visible,= 0.5

m

l

m

Near Infra

red,= 2.2

m

l

m

1010.1

l m[ m]

#DOF Bandwidth[Hz]

100

102

104

a) b)

Figure 3.8: (a) The number of degrees of freedom of the adaptive optics systemwith D required to obtain S = 0.5 - (b) Evolution of the control bandwidth withλ (Rodrigues, 2010).

Many parameters influence the design of an adaptive optics system. Figure 3.8.ashows the evolution of the number of actuators required to achieve S = 0.5 as afunction of the diameter of the primary mirror, for various wavelengths. Figure3.8.b shows the evolution of the AO control bandwidth with the wavelength.

Adaptive optics is complementary to active optics in achieving near diffraction-limited imaging. It compensates for aberrations induced by seeing of any typeover an extended temporal bandwidth. Correction at high speeds requires wave-front sensors with a good quantum efficiency, bright reference objects [whethernatural or artificial guide stars, see e.g. (Hardy, 1998), p.70] and lightweightactive mirrors of two types: a deformable mirror that corrects wavefront aberra-tions and a fine steering mirror that corrects image motion at high frequencies.

The recent work dedicated to the field of adaptive optics at the Active StructuresLaboratory is reported in Goncalo Rodrigues’ PhD thesis (Rodrigues, 2010).


3.4 Active optics of the Keck telescope

Figure 3.9: Keck elevation structure [adapted from (Bely, 2003), p.227,232,233]

This text being devoted to the control of large segmented mirrors, the activeoptics of the Keck telescope deserves a special attention.

The Keck telescope10, Fig.3.9, is a Ritchey-Chretien Alt-Az telescope with a M1of D = 10m and f/# = 1.75 tesselated with 36 hexagonal segments, each witha diameter of d = 1.8m (corner to corner), separated by a gap of 3mm (Aubrunet al., 1987; Noethe, 2002). Table 3.2 summarizes the masses of some of thesubstructures. The locked-rotor frequency is ∼ 5 Hz, while the first eigen modeinvolving a deformation of M1 is at f1 =∼ 10 Hz (Medwadowski, 1989)11.

Each segment is supported by three 12-point whiffletrees represented in Fig.3.10.a-b; this relatively homogeneous support, combined with their mass and the ratioof their diameter over their thickness (75mm) guarantees that they can be consid-ered as rigid bodies. In practice, however, low-order spatial deformations, suchas manufacturing errors and the residual gravity and thermal distortions, are cor-rected by applying forces in the 36 supports of the whiffletrees (Noethe, 2002).

10There are two telescopes at the Keck observatory, as they can be considered as identical,they are referred to without distinction as the Keck telescope in the remainder of this text.

11Those values change slightly with the elevation angle.

3.4 Active optics of the Keck telescope 59

Element mass [kg]Segment 400M1 (incl. active optics system) 21000M1 support structure 20500M2 (incl. active optics system) 3000M2 support structure 17000Elevation ring 25500Elevation structure 110000Total moving mass 220000

Table 3.2: Mass of selected substructures composing Keck (Medwadowski, 1989).

Figure 3.10: (a) Support system for the segments of Keck - (b) Footprint of thesupport points of the whiffletrees, the hollow circles show the location of theposition actuators - (c) Detail of the lateral support [adapted from (Bely, 2003),pp.220,223].

Each segment possesses 6 rigid-body degrees of freedom. Amongst them, onlythree need an active control, namely piston and tip/tilt (Mast and Nelson, 1982).They are controlled by three position actuators that connect the whiffletrees tothe support structure and have a range of 1.1mm (Fig.3.10.a). The other threedegrees of freedom (lateral decenters and clocking about the segment axis) arecontrolled passively because they are less sensitive to external disturbances andthe optical performance is less sensitive to their deviations. The lateral supportdetailed in Fig.3.10.c consists in a diaphragm that provides a sufficient stiffnessto constrain passively the desired degrees of freedom while opposing negligible


150

200

100

50

0

Eig

en fre

quency [H

z]

150 20010050

Mode number [/]

Segmentpiston

Segmenttilts

Figure 3.11: Eigen frequencies of Keck [adapted from (Aubrun et al., 1987)]; thedrift in the frequencies of the local piston and tilt modes of the segments is dueto the elastic coupling with the structure.

Conducting surfaces

gap = 3mmsegment

reflecting side

55mm

4mm

Sensor bodySensorpaddle

a)b)

Edgesensor

Positionactuator

Figure 3.12: (a) Capacitive edge sensors used in Keck - (b) Layout of the 168edge sensors and 108 position actuators with respect to the 36 segments (Jaredet al, 1990).

axial resistance to piston and tip/tilt imposed by the position actuators. Thestiffness provided by their support induces local modes of the segments at ∼ 25Hz (piston)12 and ∼ 40 Hz (tilts)(Aubrun et al., 1988). The eigen frequencies ofKeck are given in Fig.3.11.

The continuity of M1 during observations is guaranteed by capacitive edge sensors(Fig.3.12). There is a total of 168 edge sensors, 2 per inter-segment gap13, locatedclose to the corners of the segments to maximize their sensitivity. Due to their

12This mode turned out to be critical for control-structure interaction.13Segmented mirrors made up of M regular hexagonal rings, without the central segment,

have N = 3M(M + 1) segments, 6(M + 1)(3M + 2) edge sensors and 3N position actuators.


interlocking design, they are sensitive to the dihedral angle between two segmentsas well as to their relative piston errors.

The sag under gravity of M1 is ∼ 0.6mm P-V, with a change of ∼ 170µm RMSbetween the pointing at zenith and at the horizon, mainly as a defocus mode(Cohen et al., 1994); during tracking, the deformation changes up to a rate of∼ 30nm/s RMS. The range of temperature variations is approx. 16oC, causingrelative changes of M1 of up to 0.1mm P-V, for a rate of ∼ 3µm/h RMS (Cohenet al., 1994). The despace under gravity between M1 and M2 is maximum atzenith, with ∼ 35µm and varies continuously up to ∼ −10µm at horizon14; thedecenter varies between ∼ 30µm and ∼ 90µm (Medwadowski, 1989).

During observations, active optics is performed at two levels (Jared et al, 1990):Feedforward (open-loop) control compensates for most of the deviations in theshape of M1 induced by gravity and temperature changes. Closed-loop controlmaintains the continuity of M1 in face of the residual of those disturbances (itstabilizes a pre-established continuity). A deeper description of the loops, thedelays they encounter and the filters they use are provided in (Jared et al, 1990;Cohen et al., 1994).

The lookup tables for the open-loop control, the re-alignment of M2 and there-phasing of the segments are performed offline, once a month, using a dedi-cated camera [a modified Shack-Hartmann, described in (Chanan et al., 1988)],that samples the wavefront from a guide star downwards the optical train. Itsmeasurements must be averaged over 30s to cancel atmospheric turbulence. Itoperates in four modes (Chanan et al., 1994):

1. The passive tilt mode that measures the tilt errors of the segments throughone aperture per segment;

2. The fine screen mode that measures both the errors due to tilts of thesegments and misalignments of M2 through 13 apertures per segment;

3. The ultra fine screen mode that measures the figure errors of one segmentwith 217 apertures;

4. The segment phasing mode that measures the relative height between theedges of adjacent segments; M1 is sampled by 84 apertures straddling theinter-segment gaps.

Cofocusing (Fig.2.12) is performed segment by segment, using the third mode ofthe camera to measure the low order aberrations (residual errors from manufac-turing, mostly); their correction is performed by the warping harness, through a

14The minus sign indicates a growing distance.


least-squares calculation based on its influence matrix (Noethe, 2002).

Using mode 1 of the camera, errors in the position and alignment of M2 appearas tilt errors in the segments; therefore, M2 must be aligned before attemptingthe coaligning of the segments. This is done with the second mode of the camera,in which global defocus and coma due to M2 cannot be distinguished from thoseinduced by the segments; consequently the rigid-body corrections to apply to M2are determined by cancelling the average value of defocus and coma over the 36segments (Witebsky et al., 1990).

Coaligning the segments is then performed either with mode 1 or mode 2 byadjustment of the position actuators in closed-loop. This step is repeated atmultiple zenith angles and temperatures to establish the feedforward lookup table(Witebsky et al., 1990).

Cophasing is performed with mode 4. Each aperture produces a diffraction figureresulting from the interferences between the parts of the wavefront issuing fromeach side of the gap that is sampled. When the phase error is an integer multiple ofπ, there is only one luminous core, other phase errors split this core into two peaksthe relative energy of which are proportional to the phase error as illustrated inFig.3.13. The broadband algorithm inspects the diffraction patterns produced bythe wavelengths in the range 700-900nm, giving a capture range of 1µm and anaccuracy of 30nm (Chanan et al., 1998). The narrowband algorithm inspects thepatterns produced by a wavelengths around ∼850nm, giving a capture range of∼ 120nm and an accuracy of 6nm (Chanan et al., 2000a). The command signalsapplied to the actuators are computed through least squares with a zero-mean,based on the influence matrix of the 36 piston motions of the segments.

A second technique called Phased Discontinuity Sensing (PDS) has also beenimplemented in Keck (Chanan et al., 1999). It is based on the comparison ofdefocalized images of a star by the telescope. Amongst its advantages with respectto the techniques described above, it only requires an infrared detector of thesuitable size, while the broadband and narrowband algorithms rely on a dedicatedcamera that could prove very costly and difficult to build for ELTs (Chananet al., 2000b). As for the segment phasing mode described previously, the PDSalternates between two algorithms that offer a trade-off between capture rangeand accuracy (Chanan and Pinto, 2004). Currently, those techniques are beingstudied and compared to other ones (based on a pyramid sensor and on a Mach-Zender interferometer) in the view of their possible integration in future ELTs[see (Gonte et al, 2008, 2009) e.g.].

Fig.3.14 illustrates the theoretical PSFs obtained at different wavelengths, for dif-


l/8 l/4 3 /8l l/20a) b)

Figure 3.13: (a) Sequence of PSFs produced by one aperture in function of therelative piston error between the edges - (b) Each of the 78 apertures used inthe segment phasing mode have a diameter of 12cm (when projected on M1) andstraddle one inter-segment edge; the missing apertures correspond to areas thatare obscured by the optical train [adapted from (Chanan et al., 1998)].

ferent RMS piston errors under representative seeing conditions (without adaptiveoptics). It shows that, at optical wavelengths, phasing is irrelevant as errors in-duced by atmospheric seeing dominate. As the wavelength grows larger, so doesthe coherence length of the atmospheric seeing [see e.g. (Hardy, 1998), p.103];as the latter reaches the size of one segment, phasing errors begin to limit imagequality and become more critical at the longest wavelengths, where the coher-ence length is larger than the diameter of the telescope. The generalization ofhigh-level adaptive optics systems extends considerably the range of wavelengthsover which phasing is of critical importance.

Several error sources can affect the active control. First, the position actuatorsand edge sensors have a limited resolution of approximately 4nm and are proneto drift in time, slowly degrading their performances between two phasings andthey are both subject to a noise level of approximately 5nm RMS (Cohen et al.,1994). However, their potential for image degradation is lowered by the factthat the residual errors affecting the output wavefront tend to be dominated bysegment misfigures at low-order spatial modes, the discontinuities at the edges ofthe segments being somewhat smaller (Troy et al., 1998)15. Finally, small randomerrors also appear during phasing, due to seeing (Padin, 2002).

The high degree of redundancy and the symmetrical repartition of the edge sen-sors make the system substantially less sensitive to systematic errors in the edge

15In practice, typical contributions to the total RMS error are as follows: 24nm RMS of tip-tilt, 23nm RMS of phasing errors and 59 nm RMS of segment figure errors, for a total of 76nmRMS. The error is dominated by figure errors in the segments (Troy et al., 1998).


0.5 1 2 5 10

0.5

0.2

0.1

0R

MS

Pis

ton E

rror

[m

]m

8

l m[ m]

Figure 3.14: Theoretical PSF for the Keck telescopes assuming representativeseeing conditions, for a variety of wavelengths and piston errors of the seg-ments.(Chanan et al., 1998).

sensors than to noise. Moreover, it also allows fault detection and makes thesystem only slightly sensitive to the elimination of defective sensors. Actuatorfailures are more critical; in practice, three actuators are locked to define a ref-erence orientation with respect to the structure; as any actuator can be usedfor that purpose, up to three defective ones can be assigned that role, but anysupplementary defective actuator will induce image degradation to occur (Mastand Nelson, 1982).

The shape control is based on the quasi-static relation between the readings ofthe edge sensors, y, and the commands to the position actuators, a, according toEq.(3.4) (Cohen et al., 1994):

y = Ja, (3.4)

where J is the Jacobian of the edge sensors, totally defined by the geometry ofthe sensors and actuators and their position with respect to each segment. It canbe determined analytically [complete formulations are provided in (Chanan et al.,2004)] or experimentally to take into account the errors in the manufacturing, inthe assembly and in the gains of the sensors and actuators. The equation govern-ing the control consists in determining the optimum values of a that minimizesthe quadratic norm of the output error ‖y‖2

a = J+y, (3.5)

3.5 Conclusions 65

where J+ =(JT J

)−1JT is the pseudo-inverse of the (rectangular) Jacobian

matrix (see section 4.2). In practice, J+ is determined offline and stored in a fileto optimize the online calculation time. The cophasing strategy will be discussedfurther in the next chapter.

In practice, the bandwidth of the closed-loop control is limited to 0.5 Hz, toavoid control-structure interaction with the modes at ∼25 Hz involving coupledresonances of the segments on their whiffletrees (piston). Therefore, in operation,the image quality suffers from small blurs due to the local piston and tip/tiltmodes of the segments (25-40 Hz) that are excited by external disturbances andare not taken into account by the control (Aubrun et al., 1987). To avoid thatproblem, the whiffletrees in future ELTs will be designed in such a way thatthe first segment resonance is increased to ∼60 Hz. However, the problem ofcontrol-structure interaction is more than ever of critical concern, as discussedlater.

3.5 Conclusions

The optical quality of modern large telescopes is based on different layers of activecontrol that compensate for the aberrations induced by external disturbances.They can be classified in function of their spatial and temporal bandwidth:

- Pointing/tracking controls the orientation of the telescope through its mainaxes of rotation. It compensates for slowly varying image motion with largeamplitudes.

- Active optics acts in two ways: The control of the shape of the primarymirror and of the rigid-body movements of the secondary mirror. It com-pensates for slowly varying deformations induced by gravity or temperaturechanges in M1 and in the telescope structure involving low spatial frequen-cies with large amplitudes (up to a few millimeters).

- Adaptive optics compensates for rapidly varying errors induced by seeingin general. The aberrations cover a broad range of spatial frequencies withamplitudes up to a few microns.

The shape control of segmented primary mirrors is based on edge sensors to mea-sure the relative position of the segments and position actuators to compensatefor the deviations induced by the external disturbances. In Keck, the bandwidthof the controller is limited by the control-structure interaction due to the firstcoupled mode involving the vibration of the segments on their supports. Control-structure interaction is likely to be even more critical for future ELTs, becauseof the increased bandwidth of all control systems, and the lower resonance fre-quencies of the structure. This is the subject of the next chapter.

66 References

3.6 References

Angeli, G. Z., Cho, M. K., and Whorton, M. S. Active optics and control ar-chitecture for a giant segmented mirror telescope. In Angel, J., R. P. andGilmozzi, R., editors, Future Giant Telescopes - SPIE 4840, pages 129–139,January 2003.

Aubrun, J. N., Lorell, K. R., Mast, T. S., and Nelson, J. E. Dynamic Analysisof the Actively Controlled Segmented Mirror of the W.M. Keck Ten-MeterTelescope. IEEE Control Systems Magazine, pages 3–10, December 1987.

Aubrun, J. N., Lorell, K. R., Havas, T. W., and Henninger, W. C. PerformanceAnalysis of the Segment Alignment Control System for the Ten-Meter Tele-scope. Automatica, 24(4):437–453, July 1988.


Chanan, G. and Pinto, A. Efficient Method for the Reduction of Large Piston Er-rors in Segmented-Mirror Telescopes. Applied Optics, 43(16):3279–3286, June2004.

Chanan, G., Troy, M., Dekens, F., Michaels, S., Nelson, J., Mast, T., and Kirk-man, D. Phasing the Mirror Segments of the Keck Telescopes: The BroadbandPhasing Algorithm. Applied Optics, 37(1):140–155, January 1998.

Chanan, G., Troy, M., and Sirko, E. Phase Discontinuity Sensing: A Methodfor Phasing Segmented Mirrors in the Infrared. Applied Optics, 38(4):704–713,February 1999.

Chanan, G., Ohara, C., and Troy, M. Phasing the Mirror Segments of the KeckTelescopes II: The Narrow-band Phasing Algorithm. Applied Optics, 39(25):4706–4714, September 2000a.

Chanan, G., MacMartin, D. G., Nelson, J., and Mast, T. Control and alignment ofsegmented-mirror telescopes: matrices, modes, and error propagation. AppliedOptics, 43(6):1223–1232, February 2004.

Chanan, G. A., Mast, T. S., and Nelson, J. E. Keck Telescope Primary MirrorSegments: Initial Alignment and Active Control. In Ulrich, M. H., editor, ESOConference on Very Large Telescopes and their Instrumentation, volume 30,pages 421–428, 1988.

Chanan, G. A., Nelson, J. E., Mast, T. S., Wizinowich, P. L., and Schaefer,B. A. W.M. Keck Telescope phasing camera system. In Crawford, D. L. and

References 67

Craine, E. R., editors, Instrumentation in Astronomy VIII - SPIE 2198, pages1139–1150, June 1994.

Chanan, G. A., Troy, M., and Ohara, C. M. Phasing the primary mirror segmentsof the Keck telescopes: a comparison of different techniques. In Dierickx, P.,editor, Optical Design, Materials, Fabrication, and Maintenance - SPIE 4003,pages 188–202, July 2000b.

Cho, M. K. Performance prediction of the TMT secondary mirror support system.In Atad-Ettedgui, E. and Lemke, D., editors, Advanced Optical and MechanicalTechnologies in Telescopes and Instrumentation - SPIE 7018, July 2008.

Cohen, R. W., Mast, T. S., and Nelson, J. E. Performance of the W.M. Kecktelescope active mirror control system. In Stepp, L. M., editor, AdvancedTechnology Optical Telescopes V - SPIE 2199, volume 2199, pages 105–116,June 1994.

Dimmler, M., Erm, T., Bauvir, B., Sedghi, B., Bonnet, H., Muller, M., andWallander, A. E-ELT primary mirror control system. In Stepp, L. M. andGilmozzi, R., editors, Ground-based and Airborne Telescopes II - SPIE 7012,2008.


Gonte et al. Active Phasing Experiment: preliminary results and prospects. InStepp, L. M. and Gilmozzi, R., editors, Ground-based and Airborne TelescopesII - SPIE 7012, 2008.

Gonte et al. On-sky Testing of the Active Phasing Experiment. The Messenger,136:25–31, June 2009.

Guisard, S., Noethe, L., and Spyromilio, J. Performance of active optics atthe VLT. In Dierickx, P., editor, Optical Design, Materials, Fabrication, andMaintenance - SPIE 4003, pages 154–164, July 2000.

Hardy, J. W. Adaptive Optics. Scientific American, 270:40–45, June 1994.

Hardy, J. W. Adaptive Optics for Astronomical Telescopes. Oxford UniversityPress, 1998.

Hickson, P. Fundamentals of Atmospheric Adaptive Optics. Unpublished seminarsupport, 2008.

68 References

Jared et al. W. M. Keck Telescope segmented primary mirror active controlsystem. In Barr, L. D., editor, Advanced Technology Optical Telescopes IV -SPIE 1236, July 1990.

Mast, T. S. and Nelson, J. E. Figure control for a fully segmented telescopemirror. Applied Optics, 21(14):2631–2641, July 1982.

Medwadowski, S. J. Structure of the Keck Telescope - an overview. Astrophysicsand Space Science, 160:33–43, October 1989.


Padin, S. Wind-induced deformations in a segmented mirror. Applied Optics, 41(13):2381–2389, May 2002.

Rodrigues, G. Adaptive Optics with Segmented Deformable Bimorph Mirrors.PhD thesis, Active Structures Laboratory - Universite Libre de Bruxelles,February 2010.


Troy, M., Chanan, G. A., Sirko, E., and Leffert, E. Residual misalignmentsof the Keck Telescope primary mirror segments: classification of modes andimplications for adaptive optics. In Stepp, L. M., editor, Advanced TechnologyOptical/IR Telescopes VI - SPIE 3352, pages 307–317, August 1998.

Tyson, R. K. Introduction to Adaptive Optics. SPIE Press, 2000.

Wilson, R. N. Reflecting telescope optics I and II. Springer-Verlag, 1999.



Witebsky, C., Minor, R. H., Veklerov, E., and Jared, R. C. Alignment andcalibration of the W.M. Keck Telescope segmented primary mirror. In Barr,L. D., editor, Advanced Technology Optical Telescopes IV - SPIE 1236, July1990.

Chapter 4

Dynamics and control

4.1 Introduction

Modelling for control and controlling large complex active structures such astelescopes poses several challenges:

(1) Finite element techniques classically used in structural modelling tend to use avery large number of degrees of freedom that reflect the complex geometry of thestructural components. The size of the control model (used in the control designand to evaluate the control-structure interaction) must be drastically reducedwhile preserving the main features of the system, statically and dynamically.

(2) The control model must give access to all actuator inputs, sensor outputs,and optical performance metrics, and it is essential to preserve the kinematicrelationship between the position actuators and the quasi-static position of thesegmented mirror (primary response).

(3) From a control viewpoint, the number of inputs and outputs are unusuallylarge (2952 position actuators and 5604 edge sensors for the future E-ELT).

The objectives of this chapter are threefold:

(1) To develop a representative numerical model of a large segmented mirror andits supporting truss that can be used for control design and robustness evaluation;this model should have a minimum complexity to allow extensive parametricstudies. At the same time, it should provide enough insight of the dynamicbehavior to be able to predict control-structure interaction.

(2) To implement a control strategy based on the primary response of the mirrorand two sensor arrays, the edge sensors measuring the relative displacementsof adjacent mirrors and a Shack-Hartmann sensor measuring the normal to thesegments.

69

70 4 Dynamics and control

(3) To combine the structural dynamics and the control to compute the stabilityrobustness margins, closed-loop random response under wind loads and evaluatethe optical performance of the system.

This chapter follows closely (Bastaits et al., 2009).

4.2 Quasi-static approach

H s( )

Edge Sensorsd

Supportingtruss

m

ka Faca

y1

y2

Normal tosegment +

n Atmosphericturbulence

Normal to wavefront(Shack-Hartmann)

if

aSegment

PositionActuators

(gravity, wind)

Figure 4.1: Active optics control flow for large segmented mirrors. Only the axiald.o.f. at both ends of the actuators are kept after a Craig-Bampton reduction isperformed.

The shape control of the primary mirror can be represented schematically as inFig.4.1, based on position actuators of displacement vector a and edge sensorsoutput y1. Each position actuator is represented by a force Fa acting on a springka that is taken as the stiffness of the whiffletree; the force is related to the un-constrained displacement by a = Fa/ka. In this way, the local flexibility of thewhiffle tree is well represented in the system, which is very important if one wantsto simulate the effect of the whiffletree in the control-structure interaction (thisturned out to be essential in Keck). The position actuators rest on a support-ing truss carrying the whole mirror. The disturbances d applied to the systemcome from thermal gradients, changing gravity vector with the elevation of thetelescope, and wind.

Currently, similarly to Keck (Jared et al, 1990; Cohen et al., 1994), the controlstrategy envisaged by ESO for co-phasing the segments (Dimmler et al., 2008;

4.2 Quasi-static approach 71

Dimmler, 2008) assumes that the supporting truss is rigid and that the naturalfrequency of the whiffle tree (connecting the actuators to the segments) is wellabove the bandwidth of the control system (which is realistic). In this case, thebehavior of the segmented mirror is assumed quasi-static and the relationshipbetween the actuator displacements a and the edge sensors output y1 is simply

y1 = Jea (4.1)

where Je is the Jacobian of the edge sensors of the segmented mirror. In thisstudy, it has been assumed that the edge sensors are sensitive to the relativedisplacements normal to the segments only, but the approach is general providedEq.(4.1) holds. The pseudoinverse of the Jacobian J+

e is best obtained by SingularValue Decomposition (SVD):

Je = UΣVT (4.2)

where the column of U are the orthonormalized edge sensor modes, the column ofV are the orthonormalized actuator modes, and Σ is a rectangular matrix whichcontains the singular values σi on its diagonal. The control system works ac-cording to Fig.4.2, called SVD controller. The diagonal matrix Σ−1 contains theinverse of the singular values, σ−1

i , on its diagonal; it provides an equal authorityon all singular value modes. Only the modes with non-zero singular values areconsidered in this control block. The set of filters H(s) provide adequate distur-bance rejection and stability margins (we will come back to this). H(s) may bea scalar function if the same loop shaping is applied to all SVD modes; in thiscase, all the loops have essentially the same control gain.

Primarymirror

Positionactuators

Edgesensors

a y

d

-

+

V H )(s S-1 U T

J U= S V T

Figure 4.2: Block diagram of the cophasing control system (SVD controller). Theblock Σ−1 is limited to the non-zero singular values. (Chanan et al., 2004).


0 25

10-15

10-5

100

Index

s

Coma

Trifoil

Astigmatism

Piston

Tilt

Defocus

…

50 75

Figure 4.3: Singular values of the Jacobian Je of the edge sensors, ranked byincreasing values for the segmented mirror of Fig.4.4. Piston, tilt and defocuscannot be controlled from the edge sensors.

Fig.4.3 shows a typical distribution of the singular values of Je, for the segmentedmirror shown in Fig.4.4. The singular values are ranked by increasing values; onesees that the lowest singular values correspond to the lowest optical modes, piston,tilts and defocus, which are unobservable from the edge sensors used in this study,and the modes with the lowest singular values next to them are astigmatism,trefoil and coma. In this study we assume that, in addition to the edge sensors,the control system also includes another set of sensors, y2, measuring the tiltangles of the segments (normal sensor in Fig.4.1). Once again, if the supportingtruss is rigid, there is a linear relationship between the actuator displacementsand the normal to the segments

y2 = Jna (4.3)

where Jn is the Jacobian of the normal to the segments; it is block-diagonal,because the two components of the normal to a segment depend only on thethree displacement actuators under that segment. A practical way to measurethe normal to the segments is to use a Shack-Hartmann sensor (one micro lens persegment); however, the Shack-Hartmann array measures the normal to the wave-front which, in addition to the normal to the segment, includes the atmosphericturbulence which appears as noise (Fig.4.1). The bandwidth of the active opticsbeing much lower than the frequency content of the atmospheric turbulence, ncan be significantly attenuated by low-pass filtering.

4.3 Structural dynamics 73

4.3 Structural dynamics

The dynamics of the mirror consists of global modes involving the supportingtruss and the segments, and local modes involving the segments dominantly. Theglobal modes are critical for the control-structure interaction, especially for largemirrors; the segments are designed in such a way that their local modes havefrequencies far above the critical frequency range1, but their quasi-static response(to the actuator as well as to gravity and wind disturbances) must be dealt withaccurately. In order to handle large optical configurations, it is important toreduce the model as much as possible, without losing the features mentionedbefore. The starting point is a standard finite element (FE) model

(M11 M12

M21 M22

)(x1

x2

)+

(K11 K12

K21 K22

)(x1

x2

)=

(f10

)(4.4)

where the degrees of freedom (d.o.f.) have been partitioned in boundary d.o.f.,x1, including all the loaded ones, especially the axial d.o.f. at both ends of theactuators, at the connection with the truss and the segments (they are highlightedin Fig.4.1), and the internal d.o.f., x2 which are not loaded and may be eliminatedby appropriate reduction.

4.3.1 Model reduction

A Craig-Bampton reduction [(Geradin and Rixen, 1994), p.343; (Craig and Bamp-ton, 1968)] is conducted in two steps: First a Guyan reduction (Guyan, 1965)assumes a static relationship between the internal and the boundary d.o.f.

(K11 K12

K21 K22

)(x1

x2

)=

(f10

)(4.5)

leading to x2 = −K−122 K21x1. In a second step, this solution is enriched by a set

of fixed boundary modes, solution of

M22x2 + K22x2 = 0 (4.6)

[obtained by setting x1 = 0 in Eq.(4.4)]; they constitute the column of the matrixΨ2 and are normalized according to ΨT

2 M22Ψ2 = I. Note that these modes arein a plane orthogonal to the actuators. The number of fixed boundary modesincluded in the reduction will be discussed shortly. Overall, the coordinate trans-formation reads

1As mentioned before, Keck experienced control-structure interaction with one of its seg-ments, with a local resonance at 25 Hz. Later designs use stiffer whiffletrees.


(x1

x2

)=

(I 0

−K−122 K21 Ψ2

)(x1

α

)= T

(x1

α

)(4.7)

where α is the vector of modal amplitudes of the fixed boundary modes. Usingthe transformation matrix T, the mass matrix M and the stiffness matrix Kof the reduced system are readily obtained according to the classical formulaeexpressing the conservation of the kinetic and strain energy, respectively:

M = TTMT K = TTKT (4.8)

leading to the final equation in reduced coordinates:

(M11 M12

M21 I

)(x1

α

)+

(K11 00 Ω2

)(x1

α

)=

(SaFa + d

0

)(4.9)

where the external force f1 has been split into the actuator force SaFa (Sa is thematrix describing the topology of the actuators and Fa = kaa the control force)and the external disturbance d. In this equation:

• The stiffness matrix is block diagonal, with K11 = K11−K12K−122 K21 being

the Guyan stiffness matrix and Ω2 being a diagonal matrix with entriesequal to the square of the natural frequencies of the fixed boundary modes.

• M11 is the Guyan mass matrix, given by

M11 = M11 −M12K−122 K21 −K21K−1

22 M21 + K12K−122 M22K−1

22 K21

and the off-diagonal blocks are given by

M21 = MT12 = ΨT

2

(M12 −M22K−1

22 K21

)

• Equation (4.9) assumes that the fixed boundary modes are not loaded byexternal disturbance d (wind, gravity, etc) and the control force Fa.

To be complete, a damping matrix should be included; it will be included whenwe transform into modal coordinates. For a segmented mirror with N segments,the size of x1 is typically ∼ 6N (the vertical components at both ends of the3N position actuators.), down from > 24N , depending on the details of theoriginal model shown in Eq.(4.4). Because of the way they have been selected,the reduced coordinates x1 describe fully the rigid body motion of the segmentsand the sensor output can be expressed by

y1 = Sy1x1 (4.10)

y2 = Sy2x1 (4.11)

where the matrices Sy1 and Sy2 describe the topology of the sensor arrays.


4.3.2 Modal analysis

xy

z

A

B

Figure 4.4: Segmented mirror consisting of 91 segments supported by a truss.The figure shows the support conditions assumed in this chapter, and the first 3modes.

In order to illustrate our analysis, consider the segmented mirror of Fig.4.4, con-sisting of 91 segments supported by a truss. The mass is assumed to be distributedequally between the segments, 50 kg each, and the truss, 4550 kg. The actuatorstiffness ka has been selected in order that the first piston mode of the segmentsis 100 Hz. The truss stiffness has been chosen so that the first global mode isf1 = 20 Hz. The modal damping will be assumed uniformly ξi = 0.01 unlessotherwise specified.

Fig.4.5 shows the eigen frequency distribution of the full FE model; the first 20modes or so are global modes; their mode shapes are a combination of opticalaberration modes of low order. Next follow the local modes of the segments (tiltnear 75 Hz and piston near 100 Hz). Fig.4.6 shows the natural frequencies ob-tained after the Craig-Bampton reduction described above, for various numbersof fixed boundary modes. They are solution of Eq.(4.9) after setting the righthand side to 0. One sees that a small number of fixed boundary modes (three inthis case) improves the agreement of the eigenfrequency distribution. The modeshapes are displayed in Fig.4.7; the frequencies listed at the top of the figure arethe natural frequencies of the model; those on the left side are the frequencies ofthe fixed boundary modes; their relationship with the frequencies of the missingmodes in the reduced model is interesting.


50 100 150 200 250 3000

50

100

150

200

Mode index

Eig

en fre

quency [H

z]

globalmodes

PistonTilt

Figure 4.5: Eigen frequency distribution of the full FE model.

For a large segmented mirror, the model obtained after the Craig-Bampton re-duction may be still quite big (∼ 6N , down from > 24N , where N is the numberof segments) for coupling it directly to the control; before going into state-spacecoordinates, it can be further reduced by transforming into modal coordinatesand truncating the high-frequency modes that are of no concern in the control-structure analysis. However, the modal truncation must be performed in such away that the primary response of the structure (i.e., the kinematic relationshipbetween the position actuators and the rigid-body motion of the segments) ispreserved, as discussed below.


50 100 150 200 2500

50

100

150

200

250

Mode index

Eig

en fre

quency [H

z]

0 10 2010

30

50

70

Guyan

C-B 1 mode

C-B 2 modes

C-B 3 modes

Full FE model

300

Figure 4.6: Eigen frequency distribution of reduced models (Guyan and Craig-Bampton) for an increasing number of fixed boundary modes.


Fig

ure

4.7:

Eig

enm

odes

ofth

efu

llm

odel

and

vari

ous

redu

ced

mod

els

(Cra

ig-B

ampt

on)

wit

han

incr

easi

ngnu

mbe

rof

fixed

boun

dary

mod

es(o

fre

spec

tive

natu

ralfr

eque

ncie

sf 1

=27

Hz,

f 2=

43H

zan

df 3

=55

.1H

z).


4.3.3 Static response

From Eq.(4.9), the static response of the system is

x1 = K−111 [SaFa + d] (4.12)

y1 = Sy1K−111 [SaFa + d] = Sy1K

−111 Sakaa + Sy1K

−111 d (4.13)

Comparing this equation with Eq.(4.1), one gets

Je = Sy1K−111 Saka (4.14)

Similarly, using the second output equation, one gets

y2 = Sy2K−111 [SaFa + d] = Sy2K

−111 Sakaa + Sy2K

−111 d (4.15)

and

Jn = Sy2K−111 Saka (4.16)

Both Je and Jn can be obtained geometrically without resorting to the FE model.

4.3.4 Dynamic response in modal coordinates

Assume that the eigenvalue problem has been solved for the reduced system andthat the eigen modes have frequencies ωi and have been normalized to a unitmodal mass. Let φi be the partition of the eigen modes corresponding to theboundary d.o.f. x1. Because the control force and the disturbance are onlyapplied to those d.o.f., the equation governing the dynamic response of mode i is

zi + 2ξiωizi + ω2i zi = φT

i SaFa + φTi d (i = 1, . . . ,m) (4.17)

However, only the m lowest frequency modes, within or close to the bandwidthof the disturbance respond dynamically; the higher ones (i > m) respond in aquasi-static manner and may be regarded as a singular perturbation. The staticresponse of the previous section includes all modes, and if a flexible mode isaccounted for dynamically, one must remove its contribution from the flexibilitymatrix; this is obtained by subtracting from K−1

11 the contribution Fm of the mmodes which respond dynamically according to Eq.(4.17):

Fm =m∑

i=1

φiφTi

ω2i

(4.18)


Thus, the flexibility matrix of the high frequency modes is K−111 −Fm (depending

on m) [(Preumont, 2002), p.22] and the overall dynamic response at the d.o.f. x1

reads

x1 = Φmz + [K−111 − Fm][SaFa + d] (4.19)

where the components of z are solutions of Eq.(4.17) (dynamic response), Φm =(φ1, . . . , φm) is the matrix of mode shapes at the boundary d.o.f., and the secondterm is the quasi-static response (singular perturbation) of all modes beyond m.From Eq.(4.10), the edge sensor output y1 is

y1 = Sy1Φmz + [Je − Sy1FmSaka]a + Sy1 [K−111 − Fm]d (4.20)

where Eq.(4.14) has been used. Similarly, from Eq.(4.11), the normal sensoroutput y2 reads

y2 = Sy2Φmz + [Jn − Sy2FmSaka]a + Sy2 [K−111 − Fm]d (4.21)

after using Eq.(4.16). In these equations, Sy1Φm and Sy2Φm are the modal com-ponents, for the first m modes of the edge sensor output and the normal sensoroutput, respectively. Equations (4.20) and (4.21) are reduced to (4.13) and (4.15)if none of the modes respond dynamically (m = 0), which confirms that the re-duced model in modal coordinates does convey the full kinematic relationshipbetween the position actuators and the two sets of sensors (primary response).Fig.4.8 shows the complete input-output relationship between the external forcesacting on the boundary d.o.f. and the measurements of the sensors, combiningthe static and dynamic parts of the structural response.

Fm

iiii

iifT

s2+2x w s ii

2+w

...

...

Fm

+

Flexible modes

i = 1,…,m

a+

++

K-1

-

ka SaFa

d

x1

z

Sy1

Sy2

y1

y2

y

11

Figure 4.8: Input-output relationship between the external forces acting on theboundary d.o.f. and the measurements of the sensors.

4.4 Control strategy 81

The matrices describing the structural response can be reorganized as shownin Fig.4.9 to decompose the input-output relationship in terms of the primaryresponse G0(s) = J and the residual response GR(s); the latter is not taken intoaccount in the controller structure and is regarded as an uncertainty.

GR

GO

+

+

a y

kaSa

Fm

iiii

iifT

s2+2x w s ii

2+w

...

...

Fm

+

-

Sy1

Sy2

y1

y2

Flexible modesi = 1,…,m

(Residual response)

aJ =

JeJn( (

(Primary response)

Jacobian

y1

y2

+ +

y

Figure 4.9: Input-output relationship of the segmented mirror. The nominalplant G0(s) = J accounts for the quasi-static response (primary response) andthe dynamic deviation GR(s) is regarded as an additive uncertainty (residualresponse).

4.4 Control strategy

Given the size and complexity of the system, its variability with temperature andthe elevation angle of the telescope, and the large amount of real-time calcula-tions, it seems reasonable to base the control on the quasi-static behavior of thesystem (primary response) and address the dynamic amplification as a perturba-tion to the nominal system. The baseline co-phasing strategy currently envisagedby ESO is the SVD controller of Fig.4.2. The edge sensor output y1 is projectedinto the sensor modes, UTy1, then scaled according to the inverse of the singularvalues σ−1

i , shaped in the frequency domain according to the diagonal controllerH1(s) (possibly variable with the order of the mode), and then applied to theactuator modes through the matrix V:

a1 = VH1(s)Σ−1UTy1 (4.22)

This approach reflects the fact that, for each mode, the loop gain must be adjustedaccording to the singular value. If the same scalar controller is used for all loops,they will have the same gain. However, the sensor noise will be amplified onthose modes with the lowest singular values [which are the lowest optical modes,


Fig.4.3 (Angeli et al., 2004)]. Although (Chanan et al., 2004) concludes thatthe wavefront information is not useful for realistic edge sensor noise, this studyassumes a second set of sensors measuring the normal to the segments (tilt), y2;two controller structures coupling this information are examined below.

4.4.1 Dual loop controller

The idea consists of using the SVD controller based on the edge sensors forall the modes of the Jacobian Je with significant singular values, and to add asecond loop based on the normal sensors y2 to control a selected set of low orderZernike modes. This is done as follows: Let A = (a1, . . . ,ai, . . .) be the matrix,the column of which are the actuator displacements for the Zernike modes ofincreasing order. They are assumed to be normalized according to

ATWA = I (4.23)

(the matrix W depends on the geometry of the actuator network). Let N =(n1, . . . ,ni, . . .) be the output of normal sensors for the same low order Zernikemodes (with the same normalization). The control input is constructed as follows:The sensor output y2 is first projected in the space of the selected Zernike modes,NTy2; next, the proper frequency shaping with the diagonal controller H2(s)is applied (possibly variable with the order of the mode), and the correctionamplitude of the various Zernike modes is then applied to the segment actuators:

a2 = AH2(s)NTy2 (4.24)

This part of the control includes all the low order optical modes of the system.The problem with this dual loop approach is that the two set of modes (singularvalue decomposition of the Jacobian of the edge sensors on the one hand, andZernike modes on the other) are not orthogonal and that the control input a1

based on the edge sensors may excite the low order Zernike modes and vice versa.In fact, the control input a1 can be made orthogonal to the Zernike modes bypassing it through a spatial filter

I−AATW (4.25)

This filter will remove any component of the control which belongs to the col-umn space of A: (I −AATW)A = 0 while any vector orthogonal to A will beunaltered, since, if ATWB = 0 , (I−AATW)B = B. Finally,

a1 = [I−AATW]VH1(s)Σ−1UTy1 (4.26)

and the control input is a = a1 + a2.

4.4 Control strategy 83

Generated in this way, the control input of the co-phasing (edge sensor) controllerwill not excite the optical modes controlled by the Zernike modes controller. Thecophasing loop, however, may be excited by control input a2, although weakly,if only those optical modes corresponding to the lowest singular values in Je arecontrolled. Fig.4.10 shows the block diagram of the dual loop control strategypreviously described.

Normal to segmentsSegmentposition actuators

Edge sensors

Edge sensors controller

Zernike modes controller

Spatialfilter

d

a

2y

1y

2n

2a

1a

Primarymirror

M1-

+

+V S

-1UT

H ( )s1I AA W-

T

NT

H ( )s2A

+

+

Figure 4.10: Block diagram of the dual loop control strategy for the Active Opticsof M1. The edge sensors controller consists of a SVD controller acting on all SVDmodes with significant singular values. The Zernike modes controller acts on thelow-order aberrations which are unobservable from the edge sensors.

Note that the piston mode of the mirror is not observable, neither by the edgesensors, nor by the normal sensors. In practice, in the model, the central seg-ment is removed and replaced by edge sensors that are rigidly attached to thesupporting truss; in terms of singular values, this is equivalent to locking threeactuators to define a reference plane as is performed in Keck.

4.4.2 Extended Jacobian SVD controller

An alternative way to solve the problem of the poorly observable modes associatedwith the Jacobian Je of the edge sensor consists of building an extended Jacobiancoupling the two sets of sensors (with appropriate scaling):

y =(

y1

y2

)=

(Je

Jn

)a = Ja (4.27)


The singular values of the Jacobian of the edge sensor Je and the extendedJacobian J are compared in Fig.4.11. One sees that, except for the global pistonmode which is not observable (see above), the singular values of the extendedJacobian are quite well conditioned and ready for the implementation of Fig.4.2.After conducting a SVD of the extended Jacobian J = UΣVT , the controllerreads

a = VH(s)Σ−1UTy = K(s)y (4.28)

with as many modes as necessary. This control strategy is followed in the sequelof this chapter.

50 250

Coma

Trifoil

Astigmatism

Piston

Tilt

Defocus

0 25 275

Trifoil, Coma

Trifoil, Defocus

Piston

Tilt

Astigmatism

edge sensors

edge + normal

Index

100

10-2

10-15

Figure 4.11: Comparison of the singular values of the Jacobian Je of the edge sen-sors with the extended Jacobian J (edge sensor measurements and displacementactuators are expressed in meters, tilt angles in radians).

4.5 Loop shaping of the SVD controller

Referring to Fig.4.12, the governing equations are

y = Gd(s)d + G(s)a (4.29)a = −K(s)y (4.30)

or y = (I + GK)−1 Gd(s)d (4.31)

where Gd(s) is the transfer matrix between the disturbance d and the sensoroutput y, G(s) is the transfer matrix between the actuator input a and thesensor output and K(s) is the feedback controller. The nominal controller is

4.5 Loop shaping of the SVD controller 85

K

+

-

SVD controller

aPositionactuators

Disturbancesd

G y Edge sensors &Normal sensors

Gd

Figure 4.12: Block diagram of the control system.

designed for the primary response of the system alone, which consists of theconstant Jacobian matrix:

G(s) = G0(s) = J = UΣVT (4.32)

The SVD controller has the form (4.28) and, if one assumes the same loop shapefor all singular value modes, H(s) = Ih(s), the controller

K(s) = h(s)VΣ−1UT (4.33)

essentially inverts the Jacobian of the mirror, leading to KG0(s)(s) being diago-nal, with all non-zero singular values being equal to

σ (KG0) = |h (jω) | (4.34)

Thus, the loop shaping can be done as a SISO controller, according to classicaltechniques [(Franklin et al., 1986), p.129] or frequency shaping using the Bodeintegrals [(Lurie and Enright, 2000), p.74]. The control objective is to maximizethe loop gain in the frequency band where the disturbance has a significant en-ergy content while keeping the decay slow enough near crossover to achieve agood phase margin. An integral component is necessary to eliminate the staticerror in the mirror shape and a large gain at the earth rotation frequency tocompensate for the gravity deformations.

Fig.4.13 shows the Bode plots and the Nichols chart of the controller used inthis study; the compensator h(s) consists of an integrator, a lag filter, followedby a lead and a second order Butterworth filter. The crossover is fc = 0.25 Hzand the attenuation at the earth rotation frequency is 125 dB. The robustnessmargins are clearly visible on the Nichols chart [the exclusion zone around thecritical point (−1800, 0 dB) corresponds to a phase margin PM = ±450 and again margin GM = ±10 dB].


10

-410

-210

010

2

-1000

100

[Hz]

[dB

]

-250

-200

-150

-100

[°]

[dB

]

[°]

45° -1

80°

f

10 d

B0 d

B

PM

= 7

0°

GM

= 2

5dB

125dB

10

-510

-310

-110

1

f=

0.2

5H

zc

Eart

h r

ota

tion

(1.1

6 1

0H

z)

-5 10

-410

-210

010

2

[Hz]

10

-510

-310

-110

1

Fig

ure

4.13

:C

ompe

nsat

orh(s

)co

mm

onto

alllo

ops

ofth

eSV

Dco

ntro

ller.

Lef

t:N

icho

lsch

art

[the

excl

usio

nzo

near

ound

the

crit

ical

poin

t(−

1800

,0

dB)

corr

espo

nds

to(P

M=±4

50,G

M=±1

0dB

)].R

ight

:B

ode

plot

ssh

owin

gth

ega

inof

125

dBat

the

eart

hro

tati

onfr

eque

ncy

and

acu

t-off

freq

uenc

yof

f c=

0.25

Hz.

The

Nic

hols

char

tis

inva

rian

tw

ith

resp

ect

toa

shift

ofth

eB

ode

ampl

itud

ean

dph

ase

diag

ram

sal

ong

the

freq

uenc

yax

is.

4.6 Control-structure interaction 87

4.6 Control-structure interaction

From Fig.4.2, if the mirror would respond in a quasi-static kinematic manner, thecontroller transfer matrix would essentially invert that of the mirror. However,because the response of the mirror includes a dynamic contribution at the fre-quency of the lowest structural modes and above, the system behaves accordingto Fig.4.9 and the robustness with respect to control-structure interaction mustbe examined with care (Aubrun et al., 1987, 1988). The structure of the controlsystem is that of Fig.4.14.a, where the primary response G0(s) corresponds tothe quasi-static response described earlier and the residual response GR(s) is thedeviation resulting from the dynamic amplification of the flexible modes; K(s) isthe controller.

Primary

K

+ +

- +

Residual

Control

a)

G G K= O

G KG= O

L G G= O R-1

L G K= R

OG

RG

+ +

- +

c)

G

L

+

-

b)

GI L+

a y

Figure 4.14: Block diagram of the control system. (a) Mirror represented by itsprimary and residual dynamics. (b) multiplicative uncertainty, and (c) additiveuncertainty.

The control-structure interaction may be addressed with the general robustnesstheory of multivariable feedback systems (Doyle and Stein, 1981; Maciejowski,1989; Kosut et al., 1983), with the residual response being considered as uncer-tainty, either multiplicative or additive.


4.6.1 Multiplicative uncertainty

For a multiplicative uncertainty, the standard structure of Fig.4.14.b applies withG(s) = K(s)G0(s) and L = G−1

0 GR. The general theory of MIMO systemsshows that a sufficient condition for stability is that2

σ[L(jω)] < σ[I + G−1(jω)] , ∀ ω > 0 (4.35)

(σ and σ stand respectively for the maximum and the minimum singular value),which is transformed here into

σ[G−10 GR(jω)] < σ[I + (KG0)−1(jω)] , ∀ ω > 0 (4.36)

This test is quite meaningful and is illustrated in Fig.4.15.a. The left hand side isindependent of the controller; it starts from 0 at low frequency where the resid-ual dynamics is negligible and increases gradually when the frequency approachesthe flexible modes of the mirror structure, which are not included in the nominalmodel G0; the amplitude is maximum at the resonance frequencies where it isonly limited by the structural damping. The right hand side starts from unity atlow frequency where | KG0 |À 1 (KG0 controls the performance of the controlsystem) and grows larger than 1 outside the bandwidth of the control systemwhere the system rolls off (| KG0 |¿ 1).

The critical point A corresponds to the closest distance between these curves.The vertical distance between A and the upper curve should not be smaller thanthe requested gain margin GM . When the natural frequency of the structurechanges from f1 to f∗1 , point A moves horizontally according to the ratio f∗1 /f1

(increasing the frequency will move A to the right). Similarly, changing the damp-ing ratio from ξ1 to ξ∗1 will change the amplitude according to ξ1/ξ∗1 (increasingthe damping will decrease the amplitude of A).

4.6.2 Additive uncertainty

For an additive uncertainty, the standard structure of Fig.4.14.c applies withG(s) = G0(s)K(s) and L(s) = GR(s)K(s); a sufficient condition for stability isthat

σ[L(jω)] < σ[I + G(jω)] , ∀ ω > 0 (4.37)

which is translated into

σ[GRK(jω)] < σ[I + G0K(jω)] , ∀ ω > 0 (4.38)2The inverse of rectangular matrices should be understood in the sense of pseudo-inverse.


A typical result for this robustness test is depicted in Fig.4.15.b; in this case,both terms depend on the controller. The robustness conditions in Eq.(4.36) and(4.38) come from small-gain theorem (see appendix D); being sufficient condi-tions, they are both conservative and one may be more conservative than theother.

10

1

0.1

0.01

GM

s [ + ( ) ]I KG0-1

A

f1

Structuraluncertainty

s [ ]G G0 R

-1

10

1

1

0.1

0.01 100.1 100

[Hz]

0.01

s [ + ( ) ]I KG0

f1

GM

A

s [ ]G KR

Structuraluncertainty

a)

b)

Figure 4.15: (a) Multiplicative uncertainty: σ[I +(KG0)−1] refers to the nominalsystem used in the controller design. σ[G−1

0 GR] is an upper bound to the relativemagnitude of the residual dynamics - (b) Additive uncertainty: σ[I +G0K] refersto the nominal system used in the controller design. σ[GRK] is an upper boundto the effect of the controller on the residual dynamics. In both tests, the criticalpoint A corresponds to the closest distance between the nominal system and thestructural uncertainty. The vertical distance between A and the upper curve isa lower bound to the gain stability margin.


4.7 Discussion

10-1

100

101

102

-20

0

20

40

60

[Hz]

[dB

]

GM=45dB

A

ALimit of stability

f1 = 20 Hz

f1 = 2.2 Hz-20

0

20

40

60

[dB

]

a)

b)

Figure 4.16: Robustness test based on multiplicative uncertainty: (a) Stiff sup-porting truss (f1 = 20 Hz); the gain margin is GM = 45 dB - (b) Soft supportingtruss at the stability limit.

To illustrate the foregoing discussion, consider again the segmented mirror ofFig.4.4. Although this example does not correspond to a particular existing tele-scope, it is representative of the current generation of large telescopes (Keck,GTC); the parameters have been chosen so that f1 = 20 Hz and ξi = 0.01uniformly. The SVD controller is that of Fig.4.13 with a crossover frequencyfc = 0.25 Hz (we do not distinguish between the bandwidth and the crossoverfrequency).

4.7 Discussion 91

GM=36dB

A

A

Limit of stability

[dB

]

-120

-80

-40

0

f1 = 20 Hz

f1 = 3.7 Hz

10-1

100

101

102

[Hz]

[dB

]

-120

-80

-40

0

a)

b)

Figure 4.17: Robustness test based on additive uncertainty: (a) Stiff supportingtruss (f1 = 20 Hz); the gain margin is GM = 36 dB - (b) Soft supporting trussat the stability limit.

Figure 4.16.a shows the robustness test (4.36); with a ratio f1/fc ' 80 the systemexhibits a substantial gain margin of 45 dB with respect to control structureinteraction. Figure 4.16.b shows the stability limit when A touches the uppercurve; f1 = 2.2 Hz in this case. The same procedure is followed for the robustnesstest (4.38) shown in Fig.4.17.a, for f1 = 20 Hz. The system exhibits a lower butstill substantial gain margin of 36 dB with respect to control structure interaction.Figure 4.17.b shows the stability limit when A touches the upper curve, for f1 =3.7 Hz. Therefore, the test based on multiplicative uncertainty appears to be lessconservative than that based on additive uncertainty and will be the only testconsidered in the remainder of this text.


Figure 4.18 shows the evolution of the gain margin with the frequency ratio f1/fc

for various values of the damping ratio ξ in the case of the multiplicative uncer-tainty. This curve is obtained by gradually softening the Young’s modulus of thesupporting structure. If the first structural mode of the supporting structure hasa damping ratio of 1%, a gain margin GM = 10 requires a frequency separationf1/fc significantly larger than one decade.

101

102

-20

0

20

40

60

80

GM

[dB

]

UNSTABLE

GM=10dB

0.005

0.05

0.02

0.01

x

f /f1 c

Figure 4.18: Robustness test based on multiplicative uncertainty: Evolution ofthe gain margin with the frequency ratio f1/fc for various values of the dampingratio ξi.

4.8 Conclusions 93

4.8 Conclusions

The numerical modelling of large segmented mirrors has been addressed. Thismodel gives a full account of the quasi-static behavior of the segmented mir-ror and has minimum complexity to account for control-structure interaction.Two control strategies have been discussed, both based on a quasi-static behav-ior of the mirror. The control-structure interaction has been addressed througha frequency domain robustness test for MIMO systems; this test separates thedynamic response (residual) from the quasi-static response (primary) of the mir-ror. The evolution of the robustness test with the dominant frequency f1 of thesupporting truss and the damping ratio ξ has been analyzed. The study hasbeen illustrated with an example involving 91 segments, 270 inputs (excludingthe central segment) and 654 outputs (edge sensors and normal sensors), and aSVD controller; the procedure is very fast and allows extensive parametric stud-ies. An important conclusion of this study is that, for a structural damping ratioof ξ = 1%, the frequency gap between the crossover frequency of the controller,fc, and the critical flexible mode, f1, must be at least one decade. This conditionmay be more and more difficult to fulfill as the size of the telescope grows as wediscuss in the following chapter.

4.9 References

Angeli, G. Z., Upton, R. S., Segurson, A., and Ellerbroek, B. L. Active opticschallenges of a thirty-meter segmented mirror telescopy. In Ardeberg, A. L.and Andersen, T., editors, Second Backaskog Workshop on Extremely LargeTelescopes - SPIE 5382, volume 5382, pages 337–345, July 2004.



Bastaits, R., Rodrigues, G., Mokrani, B., and Preumont, A. Active optics oflarge segmented mirrors: Dynamics and control. AIAA Journal of Guidance,Control and Dynamics, 32(6):1795–1803, Nov.-Dec. 2009.


94 References


Craig, R. R. and Bampton, M. C. C. Coupling of substructures for dynamicanalyses. AIAA Journal, 6(7):1313–1319, July 1968.

Dimmler, M. E-ELT Programme: M1 Control Strategies. Number 1. ESO, Un-published report, E-TRE-ESO-449-0211, April 2008.


Doyle, J. C. and Stein, G. Multivariate Feedback Design: Concepts for a Mod-ern/Classical Synthesis. In IEEE Trans. on Automatic Control - Vol. AC-26,pages 4–17, February 1981.

Franklin, G. F., Powell, J. D., and Emani-Naemi, A. Feedback Control of DynamicSystems. Addison-Wesley, 1986.

Geradin, M. and Rixen, D. Mechanical Vibrations. Wiley, 1994.

Guyan, R. J. Reduction of stiffness and mass matrices. AIAA Journal, 3(2):380,February 1965.


Kosut, R. L., Salzwedel, H., and Emami-Naeini, A. Robust Control of FlexibleSpacecraft. AIAA J. of Guidance, 6(2):104–111, March-April 1983.

Lurie, B. J. and Enright, P. J. Classical Feedback Control. Marcel Dekker, 2000.

Maciejowski, J. M. Multivariable Feedback Design, pages 75–136. Addison-Wesley,1989.

Preumont, A. Vibration Control of Active Structures, An Introduction, 2nd edi-tion, pages 321–346. Kluwer, 2002.

Chapter 5

Scale effects

5.1 Introduction

As illustrated in Fig.1.14, the extremely large telescopes planned for the nearfuture are almost one order of magnitude larger than the largest existing ones,and there is no doubt that even larger ones will be planned in a non distantfuture (the OWL project, now cancelled, was based on a primary mirror of 100mdiameter). On the other hand, from the discussion of the foregoing chapters, itis clear that the optomechanical design of large telescopes becomes increasinglydifficult as their size increases, and that they require a control system of increasedcomplexity. The gap is so big between the existing telescopes and those underdesign that it is legitimate to worry about their feasibility; can the aberrationscaused by the structural response be actively removed? Or is there a limit inD/λ over which it becomes impossible?

In an attempt to answer these questions, a study has been conducted to establishthe scale effect and the extrapolation rules concerning the structural response(static deformation, natural frequency and wind response), the control band-width and the control-structure interaction as the size of the telescopes increases.These scaling rules are intended for preliminary design purposes and parametricanalysis. This study was conducted in two steps: In a first step, using very sim-ple models or existing scaling laws in other field of structural dynamics, simpleextrapolation rules were developed; this study is reported in this chapter. In asecond step, an extensive numerical study has been conducted with models ofincreasing size to validate the scaling laws; this study is presented in the nextchapter.

95

96 5 Scale effects

Wind response

The response of a large telescope to wind gusts can be approached in several ways:(i) random vibration, (ii) small scale wind tunnel tests, (iii) computational fluiddynamics (CFD) (numerical wind tunnel) or (iv) full scale measurements. Therandom vibration analysis as classically applied in wind engineering is delicateand requires guessing a few parameters such as the tributary areas, the drag co-efficients or the spatial correlation which have a huge impact on the quantitativeestimate of the random response. Small scale tests are good for evaluating theflow inside the enclosure but are inadequate for estimating the residual vibrationof an actual telescope. Currently, CFD does not have the capability to handle theturbulent response of huge and complex structures, while full scale measurementsare obviously limited to existing telescopes.

The understanding of the wind response of large telescopes has made significantprogress over the past few years, including its interaction with the wave fronterror (Padin, 2002). The wind response increases at least quadratically with theexternal wind speed (MacMynowski et al., 2006a), which calls for a careful designof the enclosure. The understanding of the flow inside the enclosure has made sig-nificant progress by combining measurements in existing telescopes, wind-tunneltests and CFD analyses (MacMynowski et al., 2006b). The wind induced imagedegradation comes primarily from the surface deformation of the primary mirrorM1 and the motion of the secondary mirror M2 (MacMynowski and Andersen,2010). Besides, depending on the design of the structure supporting M2, theremay be a significant structural coupling between M2 and the deformations of M1(Padin and Davison, 2004; MacMynowski et al., 2004, 2006b).

As the RMS wave front error of the primary mirror scales as the diameter D(Padin and Davison, 2004), one expects the wind response to become increas-ingly important as the telescope size increases. However, quantitative resultspublished in the literature so far are somewhat surprising and contradictory; forexample, in a preliminary study of the OWL telescope (Quattri et al., 2003)(which is in open air and has a primary mirror of diameter 100 m), the RMSwind response is estimated below 20 µm, with only 1 µm above 1 Hz (which iswell in the range of adaptive optics). On the contrary, another study of the GiantMagellan Telescope (Kan and Eggers, 2006) (which has an equivalent aperture of21.5m and is placed in an enclosure) stresses that a wind screen is very importantand, using recorded pressures on Gemini South (Cho et al., 2003), concludes thatthe RMS focus error is in the range of 250 µm, enough to be of concern. Bothstudies are based on random vibration and classical wind engineering; they arejust an illustration of the difficulty to obtain realistic figures.

5.2 Static deflection under gravity 97

A different approach is developed in this chapter: Using fairly simple models, an-alytical scaling laws for the main dynamic characteristics of the primary mirrorand its active control system have been developed. These scaling laws show thetrends, identify the important design parameters, and are expected to allow ex-trapolation from an existing telescope to a larger one with a similar architecture.

This chapter is based on (Preumont et al., 2009).

5.2 Static deflection under gravity

g

M

K

g

=D =M

K

g

w12

g

Figure 5.1: The static deflection under gravity scales according to ∆ ∝ f−21 .

A spring mass system subjected to gravity (Fig.5.1) undergoes a deflection∆ = Mg = K = g/ω2

1. More generally, for given boundary conditions, thegravity-induced deflection of a truss structure scales according to

∆ ∝ f−21 (5.1)

where f1 is the lowest natural frequency of the structure. Referring to Table 1.1,this means that the primary mirror of E-ELT will undergo gravity disturbances16 times larger than Keck; the control gains will have to be increased accordingly.

98 5 Scale effects

5.3 First resonance frequency

hD

Figure 5.2: Geometry of the truss supported reflector.

The foregoing section has shown the central role played by the first natural fre-quency in the static deflection. It is interesting to investigate the scaling lawfor the first natural frequency of a truss supported segmented reflector. Trusssupported reflectors have received a special attention in space applications. Ac-cording to (Lake et al., 2002, 2006; Wu and Lake, 1994), the first natural frequencyof a free flying truss follows

f1 ∼ 0.852D

(h

D

)√η.

E

%(5.2)

where η is the structural mass fraction

η =Truss Mass

Truss Mass + Reflector Mass(5.3)

(using lighter reflectors increases η). In Eq.(5.2), h and D are respectively thethickness and the diameter of the supporting truss (Fig.5.2); E is the Youngmodulus and % the material density of the truss. The coefficient 0.852 refers tothe free-free boundary conditions, but this result also applies to other boundaryconditions with another coefficient. For a given family of telescopes, the ratioh/D is fixed and f1 ∝ D−1 (this fits the data of Table 1.1). Eq.(5.2) showsclearly the advantage of building the supporting truss with a material of highspecific modulus E/%.

Table 5.1 compares the mechanical properties of traditional structural materialsfor telescope structures (steel and aluminum) with carbon fiber reinforced com-posites (CFRP) [(Agarwal, 1990), p.8]. Observe that the latter have a specific

5.4 Control bandwidth 99

modulus 4 times larger than either steel or aluminum, which doubles the naturalfrequency f1 if everything else is equal. As a side effect, the outstanding thermalstability of CFRP is worth noting; the thermal expansion coefficient α of CFRPgiven in Table 5.1 is the minimum value; it can be tailored to a large extent.Let us now examine the impact of the foregoing discussion on the active opticscontrol system.

E % E/% α(GPa) (g/cm3) (10−6 oC−1)

Steel 210 7.8 27 12Al 70 2.7 26 23CFRP 180/230 1.5/1.6 120/140 -0.2/0.1

Table 5.1: Mechanical and thermal expansion properties of Steel, Aluminium(Al) and carbon fiber reinforced plastics (CFRP).

5.4 Control bandwidth

In Keck, the active optics controllers do not include a real-time dynamic modelof the support structure; the control system assumes that the support truss isrigid and that the input-output relationship is fully described by the geometryof the system (cfr section 4.2). A similar assumption was done throughout allchapter 4. In this section, we examine the impact of the scaling laws for the staticdeflection and the natural frequency f1 on the control bandwidth of the gravitycompensation.

Figure 5.3 shows the open-loop transfer function |GH| of an hypothetical con-troller for gravity compensation; the decay rate has been assumed of -20 dB/decadein the vicinity of crossover, consistent with (Aubrun et al., 1987, 1988)1. Thelower curve has been drawn with a bandwidth equal to that of VLT, fc = 0.03Hz (ESO, 1998); the corresponding amplitude at the earth rotation frequency(1.16 10−5 Hz) is 68 dB.

The disturbance rejection is governed by the sensitivity function S(f) [e.g.(Franklinet al., 1986), p.91]:

y

d= S(f) =

11 + GH

(5.4)

which means that 68 dB reduces a static deformation of 110 µm to 40 nm.

1This gives a phase margin of 90o.

100 5 Scale effects

10-4

10-3

10-2

10-1

100

10110

0

102

104

106

68 dB

24 dB

1

Earth rotation

| |HG

20 dB/decade

f = 0.03 Hz

(VLT)c f = 0.46 Hz

(ELT)c

[Hz]

f = 2.5 Hz(ELT)

1f = 10 Hz(VLT)

H G y

d

-++

-5(1.16 10 Hz)

Figure 5.3: Comparison of the control bandwidth of the gravity compensation ofVLT and ELT for an integral controller (a = 1).

From the previous section and the data from Table 1.1, the gravity deformationsof a telescope of the ELT class are likely to be 16 times larger than those ofa current 10-meter class telescope such as VLT or Keck2. Thus, achieving thesame accuracy on the controlled shape will require that the disturbance rejec-tion, and therefore |GH| be 16 times larger (+24 dB); with the same decay rate,this corresponds to the upper curve in Fig.5.3, and increases the bandwidth fromfc = 0.03 Hz to fc = 0.46 Hz. Note that, at the same time, following our scalinglaws, the first natural frequency of the support structure has been reduced fromf1 = 10 Hz for VLT to f1 ' 2.5 Hz for E-ELT.

More sophisticated controllers like those discussed in the previous chapter will beable to achieve a steeper decay rate than s−1; if the average roll-off rate in thevicinity of the earth rotation frequency follows s−a, the loop gain |GH(f)| at thefrequency f is related to the bandwidth by

|GH(f)| = (fc/f)a

Thus, if the loop gain at the earth rotation frequency is to be scaled accordingto f−2

1 , this means that the controller bandwidth must be scaled according to

fc ∝ f−2/a1 (5.5)

2Even if a feedforward control is used, it is reasonable to assume that the same fraction ofthe deflection will be corrected by the feedback loop.


5.5 Control-structure interactionControl-structure interaction is well known to people dealing with flexible struc-tures; it occurs when the control bandwidth fc and the frequencies of the flexiblemodes fi become close to each other; it depends strongly on the frequency ratiofi/fc and also on the modal gain (measuring the capacity of a mode to be excitedby the control) and the damping ratio ξi. In Keck, the critical mode for control-structure interaction turned out to be a local mode of the segment and whiffle treenear 25 Hz (Aubrun et al., 1988). The segments of ELT have been designed to bestiffer (∼60 Hz, cfr Table 1.1) to alleviate this problem, but the joint effect of theincrease of the bandwidth and the reduction of the natural frequencies deservesa careful attention; according to Fig.5.3, the ratio f1/fc is reduced from about300 for VLT to about 5 for ELT. With a more general controller, the frequencyratio fi/fc scales according to

f1

fc∝ f

1+2/a1 (5.6)

where a is the average decay rate (expressed in number of poles) near the earthrotation frequency (0 < a < 2).

H s( )

d

kw

Fa

cw

m

m

ks cs

s

x

xs

Figure 5.4: Position control of a two-mass system. kw refers to the whiffle tree andks to the support structure. Model used to study the control-structure interactionby reducing ks.

The control-structure interaction can be illustrated with the two-mass model ofFig.5.4 (which is a simplified version of Fig.4.1); m is the mass of a segmentand the subscripts w and s refer to the whiffle tree and the support structure,respectively; the damping ratio is assumed the same for all modes. The controlobjective is to keep the segment position x fixed in spite of the low frequencydisturbance d applied to it, and the controller is (integral plus low-pass filter):

H(s) =g

s

11 + τs

(5.7)

102 5 Scale effects

20 dB/decade

|| [d

B]

HG

10-4

10-2 100

f [Hz]10

2

0

50

-50

-100

-150

100

92 dB

f = 0.43 Hzc

GM = 43 dB

4 Hz

1f

GM = 3 dB

Earth rotation-5

(1.16 10 Hz)

Figure 5.5: Position control of a two-mass system. Open-loop transfer functionfor a rigid support (full line) and when the first resonance is reduced to 4 Hz(dashed line).

Figure 5.5 shows the open-loop transfer function for the system of Fig.5.4 for arigid support, and when the first resonance is reduced to f1 = 4 Hz. Figure 5.6shows the evolution of the gain margin GM as a function of the frequency ratiof1/fc for various values of the damping ratio ξ; the figure has been obtained byreducing gradually the stiffness ks.

UNSTABLE

101

102

-20

0

20

40

60

GM

[dB

]

GM=10dB

x0.05

103

f /f1 c

0.010.02

Figure 5.6: Position control of a two-mass system. Evolution of the gain marginwith the frequency ratio f1/fc, for various values of the structural damping ξ.

5.6 Wind response 103

5.6 Wind response

In this section, we only consider the scale effect of the along-wind response ofa bluff body subjected to turbulent wind gusts according to the random vibra-tion approach classically used in wind engineering (Davenport, 1961; Scanlanand Simiu, 1996; Blevins, 1990). We assume that the turbulent wind force isdistributed according to Davenport’s spectrum [(Davenport, 1961), the spectrumis fully developed in section 6.2.2]; its power spectral density Φ(f) has the shapeof Fig.5.7.a, which is such that the flexible modes of the structures are in the tailof the distribution, where it behaves according to

Φ(f) ∝ f−5/3 (5.8)

(other power laws are met in the literature: f−2 or f−7/3, but this is only ofminor importance for our discussion). The structural response consists of a largequasi-static contribution and a dynamic (resonant) response. Due to the decayingshape of (5.8), the resonant response is generally dominated by the first mode,Fig.5.7.b. One can also observe that the quasi-static response will be attenu-ated by the active optics controller while the resonant response, which is beyondthe control bandwidth is amplified by the sensitivity, Fig.5.7.c. The amplifica-tion at resonance of the controlled response corresponds to a reduced dampingξ∗ = ξ/S(f1) where S(f1) is the sensitivity at the resonance frequency f1.

The cumulative RMS response is defined as [∫ f0 Φ(ν)dν]1/2. It represents the part

of the RMS response coming from the frequency range [0, f [. Comparing the cu-mulative RMS response with and without control, one sees that, with control,the resonant part becomes the largest contributor to the RMS response, Fig.5.7.d.

The random response of a structure to a point force with random amplitude ofpower spectral density (PSD) Φ0(f) can be evaluated by

Φ(f) = Φ0(f)|H(f)|2 (5.9)

where H(f) is the frequency response function (FRF) between the excitation andthe response. If the structural response is dominated by a single mode, the FRFis essentially that of a single degree of freedom oscillator of mass m equal to themass of the mode, frequency f1 and damping ξ. When the resonant responsedominates, the mean square (MS) value can be approximated by the so-calledwhite noise approximation, where Φ0(f) ' Φ0(f1). [e.g. see (Crandall and Mark,1963), p.76 or (Preumont, 1994), p. 80]; this leads to

104 5 Scale effects

10-1

2

Disp. response PSD

1

a)

b)

c)

d)

Wind pressure PSD

Sensitivity

Cumulative disp. RMS

without control

with control

without control

with control

f = 0.43 Hzc

1

100

10

10-5

100

0

resonantresponse

10-3

10-2

-4

10-2

10-3

10-1

100

101

f [Hz]

FF/

00m

ax

FF/

max

ss/

max

Figure 5.7: (a) Davenport’s turbulent wind spectrum, normalized to its maxi-mum. (b) Structural response to turbulent wind, with and without control (nor-malized to its maximum). (c) Sensitivity function; the first resonance belongsto the frequency range with a significant amplification. (d) Cumulative RMSresponse [

∫ f0 Φ(ν)dν]1/2 (normalized to its maximum); the resonant response is

amplified by the control.

5.6 Wind response 105

σ2 =∫ ∞

0Φ(f)df ∝ Φ0(f1)

m2ξf31

∝ 1

m2ξf14/31

(5.10)

where Eq.(5.8) has been used.

The foregoing result assumes a point force excitation, representative of a globalmode being excited by the pressure field on one segment. If one looks at the globalresponse, one has to consider the correlation between the excitations acting atdifferent points on the telescope; according to Davenport’s model, the spatialcoherence follows

C(∆z, f) = exp

(−fc|∆z|u10

)(5.11)

where c is a correlation constant, c ' 7, u10 is the mean wind reference velocity,f is the frequency and ∆z is the vertical separation between two points. Thecorrelation length is therefore δ ∼ u10/fc. Setting u10 '10 m/s as an order ofmagnitude, and f = f1 ' 2.5 Hz, one sees that the correlation length is typicallylower than the size of one segment. One can therefore assume that the forcesacting on the various segments are statistically independent, leading to a globalresponse

σ2 =N

m2ξf14/31

(5.12)

where N is the number of segments. Using the data given earlier, we know thatthe natural frequency scales as f1 ∝ D−1, N scales as N ∝ D2 and the massof the telescope tube and mount scales between D2 and D3 [(Bely, 2003), p.79].Assuming m ∝ D5/2 as an average, one gets

σ2 = D−3f−14/31 ξ−1 (5.13)

Besides, from Eq.(5.2), for a given aspect ratio h/D and a given specific stiffnessE/%, the natural frequency scales as f1 ∝ D−1, giving

σ2 = D5/3ξ−1 (5.14)

This result is based on the general theory of wind engineering and random vi-bration and on the scaling laws discussed previously for telescopes of the samefamily; it assumes that the wind response is dominated by the resonance of thefirst mode; it is very crude and does not take into account specific features ofthe telescope and the skill of the wind engineers in the design of the telescopeenclosure, but it probably gives orders of magnitude. Under that viewpoint, itshould be noted that the dependency with respect to D is consistent with (Padinand Davison, 2004), and that with respect to the damping is consistent with (Kanand Eggers, 2006).

106 5 Scale effects

5.7 Summary and conclusion

This study has been devoted to the extrapolation of the active optics of current10-meter class telescopes (Keck, VLT) to the next generation of 30m to 40mExtremely Large Telescopes (ELT), and possibly future ones. The various aspectsof the structural response and the control-structure interaction affected by thesize of the telescope have been considered and orders of magnitude have beenestimated. Based on crude analysis, the following results have been obtained:The static deflection under gravity scales as

∆ ∝ f−21 (5.15)

The first resonance frequency f1 varies according to

f1 ∝ D−1

√E

%(5.16)

where D is the diameter of M1 and E/% is the specific modulus of the structuralmaterial.

The control bandwidth fc of the gravity compensation system follows

fc ∝ f−2/a1 (5.17)

where a is the average decay rate (number of poles) near the earth rotation fre-quency (0 < a < 2; a = 1 for integral control).

The RMS response to wind gust scales as

σ ∝ D−3/2 f−7/31 ξ−1/2 (5.18)

or (for given truss geometry and material properties)

σ ∝ D5/6 ξ−1/2 (5.19)

which is roughly proportional to D. It must be kept in mind, however, thatbecause the resonant response occurs in the frequency range with the highestsensitivity, a reduction of ξ due to control-structure interaction must be consid-ered.

5.8 References 107

The parameters governing the control-structure interaction are the damping ξand the frequency ratio f1/fc which scales according to

f1

fc∝ f

1+2/a1 (5.20)

Once the diameter D has been fixed, there are only 3 free parameters3: Thespecific modulus of the structural material, E/%, the structural damping ξ andthe average decay rate a in the vicinity of the earth rotation frequency, whichdepends on the control system design. The use of CFRP instead of metallicstructures has a potential for doubling f1; various options for damping activelytruss structures with active struts or cable networks are discussed in (Preumont,2002). In the following chapter, the above scaling laws will be evaluated with acomprehensive set of models of increasing size.

5.8 References

Agarwal, B. D. Analysis and Performance of Fiber Composites, 2nd Ed. Wiley,1990.




Blevins, R. D. Flow-Induced Vibration, 2nd edition, pages 272–291. Krieger,1990.

Cho, M. K., Stepp, L. M., Angeli, G. Z., and Smith, D. R. Wind loading of largetelescopes. In Oschmann, J. M. and M., S. L., editors, Large Ground-basedTelescopes - SPIE 4837, pages 352–367, February 2003.

Crandall, S. H. and Mark, W. D. Random Vibration in Mechanical Systems.Academic Press, 1963.

3For a family of telescopes, the ratio h/D may be regarded as fixed.

108 References

Davenport, A. G. The application of statistical concepts to the wind loading ofstructures. In Proceedings of the Institution of Civil Engineers - Vol.19, pages449–471, August 1961.

ESO. The VLT White Book. European Southern Observatory, 1998.


Kan, F. W. and Eggers, D. W. Wind vibration analyses of Giant MagellanTelescope. In Cullum, M. J. and Angeli, G. Z., editors, Modeling, SystemsEngineering, and Project Management for Astronomy II - SPIE 6271, July2006.

Lake, M. S., Levine, M. B., and Peterson, L. D. Rationale for Defining StructuralRequirements for Large Space Telescopes. Journal of Spacecraft and Rockets,39(5):674–681, September 2002.

Lake, M. S., Peterson, L. D., and Mikulas, M. M. Space Structures on the Backof an Envelope: John Hedgepeth’s Design Approach. Journal of Spacecraft andRockets, 43(6):1174–1183, November 2006.

MacMynowski, D. and Andersen, T. Wind buffeting of large telescopes. AppliedOptics, 49(4):625–636, February 2010.

MacMynowski, D., Vogiatzis, K., Angeli, G. Z., Fitzsimmons, J., and Nelson,J. Wind loads on ground-based telescopes. Applied Optics, 45(30):7912–7923,October 2006a.

MacMynowski, D. G., Angeli, G. Z., Vogiatzis, K., Fitzsimmons, J., and Padin,S. Parametric modeling and control of telescope wind-induced vibration. InCraig, S. C. and Cullum, M. J., editors, Modeling and Systems Engineering forAstronomy - SPIE 5497, pages 266–277, September 2004.

MacMynowski, D. G., Blaurock, C., Angeli, G. Z., and Vogiatzis, K. Modelingwind-buffeting of the Thirty Meter Telescope. In Cullum, M. J. and Angeli,G. Z., editors, Modeling, Systems Engineering, and Project Management forAstronomy II - SPIE 6271, July 2006b.


Padin, S. and Davison, W. Model of image degradation due to wind buffeting onan extremely large telescope. Applied Optics, 43(3):592–600, January 2004.

References 109

Preumont, A. Random Vibration and Spectral Analysis. Kluwer, 1994.


Preumont, A., Bastaits, R., and Rodrigues, G. Scale effects in active optics oflarge segmented primary mirrors. Mechatronics, 19(8):1286–1293, December2009.

Quattri, M., Koch, F., Noethe, L., Bonnet, A. C., and Noelting, S. OWL windloading characterization: a preliminary study. In Angel, J., R. P. and Gilmozzi,R., editors, Future Giant Telescopes - SPIE 4840, pages 459–470, January 2003.

Scanlan, R. H. and Simiu, E. Wind Effects on Structures, page 135. Wiley, 1996.

Wu, K. C. and Lake, M. S. Natural frequency of uniform and optimized tetrahe-dral truss platforms. NASA Technical Paper, (3461):30, November 1994.

110 References

Chapter 6

Structural response of largetruss-supported segmentedreflectors

6.1 Introduction

The purpose of this chapter is to validate the scaling laws of chapter 5 throughextensive numerical simulations performed on models of increasing size. Thedifferent models and the controller are constructed as described in chapter 4.(Bastaits and Preumont, 2010)

6.2 Methodology

6.2.1 Structure

The mirror is assumed flat; two truss geometries have been considered, respec-tively with one and two layers (Fig.6.1), and three support conditions, as indi-cated in Fig.6.2 (the actual supporting conditions of M1 in a telescope are morecomplicated, but it will be enough to show that the results are, to a large extent,independent of the supporting conditions). The static deflection under gravityis computed with the mirror in zenithal position (pointing vertically) while thewind response is estimated with the mirror pointing horizontally. In one set ofcalculations, the size of the segments is taken identical to those of Keck (1.8 m)and the size of M1 is changed by changing the number of rings, from M = 2(corresponding to 18 segments) to M = 8 (N = 216 segments) - the Keck geom-etry corresponds to M = 3, N = 36 segments. In a second set of calculations,the configuration with M = 5 has been considered and the size of the mirror has

111

112 6 Structural response of large truss-supported segmented reflectors

Dhh

a) b)

Figure 6.1: Segmented mirror supported by a truss structure with (a) one layerand (b) two layers. h/D is the aspect ratio.

xz

y

Elevation axis

#1 #2 #3

Figure 6.2: Support conditions considered in this study.

been changed from D = 17m to 42.5m by increasing the size of the segmentsuniformly by a constant factor γ between 0.5 and 2.5 (Fig.F.1). The segmentsare considered as rigid and are attached to the supporting truss at 3 points withelastic connections such that the local modes of the segments are above 75 Hz.A structural damping of ξi = 0.01 is assumed uniformly, except when stated ex-plicitly. The various models used in this analysis are constructed as follows:

Reference model: The reference model is taken as a uniform truss (bars ofequal length) with M = 3 rings and N = 36 segments of 1.8 m, representative ofthe Keck telescope. The mass of one segment is ms = 400 kg; the total mass ofM1 is 40 t. The stiffness of the bars is adjusted to obtain a first natural frequencyf1 = 10 Hz. The truss geometry leads to an aspect ratio of h/D = 0.12 whichwill be kept the same in all the analysis. The structural mass ratio is η = 0.64.

6.2 Methodology 113

Set 1: A first set of models has been generated by varying the number of ringsfrom M = 2 to M = 8 while keeping the same size and weight for the segmentsand changing the length of the bars out of plane to keep the same aspect ratioh/D. For all models, the specific stiffness of the truss material, E/%, is keptconstant and the cross section is selected in order to achieve approximately thesame value of the structural mass ratio (0.64 ≤ η ≤ 0.71)1. For this set, the threesupport conditions depicted in Fig.6.2 have been analyzed. Figure 6.3 comparesthe mode shapes of a primary mirror with 8 rings (216 segments) for the 3 sup-port conditions; the first four modes of primary mirrors with 2 to 8 rings for the3 support conditions are depicted in appendix E. One sees that the frequencies fi

and the mode shapes vary substantially from one support condition to another;thus, considering all these cases allows to explore the robustness of the scalinglaws with respect to the support conditions.

Set 2: When the number of rings becomes large, for M = 5 and above, someof the bars tend to have an excessive length and a truss structure involving twolayers becomes more realistic (Fig.6.1.b). This situation has been investigatedwith a set of models starting from M = 5; all the models have the same specificstiffness E/% and the bar cross section is selected to keep the same structuralmass ratio as for the single layer truss. Only one support condition (#2) hasbeen analyzed with this set.

Set 3: A third set of models, all with M = 5 rings was generated, starting fromthe model with 5 rings of set 1; the geometry has been scaled by a constantfactor γ between 0.5 and 2.5 (this leads to a diameter D from 17m to 42.5m).The areal density of the segments is kept the same as well as the structural massratio η. The purpose of this set of models is to explore the behavior of extremelylarge mirrors with a numerical model of reasonable size (90 segments). Becauseof the way these models are constructed, the scaling law of the wind response,Eq.(5.18), does not apply. A new scaling law is constructed in Appendix F.

1In a truss structure, changing the cross section of the bars changes the stiffness and themass by the same amount and leaves the natural frequency unchanged.


3.2

Hz

5.3

Hz

5.3

Hz

7.2

Hz

3.5

Hz

3.7

Hz

8.4

Hz

9.6

Hz

Support

#1

Support

#2

Support

#3

f=

2.9

Hz

1f

= 3

.5 H

z2

f=

3.7

Hz

3f

= 5

.2 H

z4

Fig

ure

6.3:

Mod

es1

to4

ofa

prim

ary

mir

ror

wit

h6

ring

s(2

16se

gmen

ts)

for

the

3su

ppor

tco

ndit

ions

.

6.2 Methodology 115

6.2.2 Wind model

0 10

10

20

30

40

Wind speed [m/s]

Heig

ht [m

]

Exponentialwind profile

M 2=

M 3=

M 4=

M 5=

M 6=

M 7=

M = 8AS

3

0

Figure 6.4: Wind profile and segmented mirrors of increasing size considered inthis study.

As explained in section 5.6, we follow a classical wind engineering approach (Dav-enport, 1961; Scanlan and Simiu, 1996; Blevins, 1990); the primary mirror is as-sumed to be a bluff body subjected to turbulent wind gust. The turbulent windforce is assumed distributed according to Davenport’s spectrum:

ΦFiFj (ω) = 4νiνjΦζiζj (ω) (6.1)

where ζi = u′i/ui is the reduced turbulent velocity and νi is the drag force asso-ciated with the mean wind ui at node i:

νi =12%AiCDu2

i (6.2)

3 nodes are associated with every segment, at their support points on the truss;


each is given a tributary area Ai corresponding to one third of the area of onesegment, As/3 (Fig.6.4). The cross power spectral density of the non-dimensionalturbulent velocity field for two points distant of ∆z is that proposed in (Daven-port, 1961):

Φζiζj (∆z;ω) =2κu2

10

uiuj |ω|(600ω

πu10)2

[1 + (600ωπu10

)2]4/3exp(− |ω|

2πu10C|∆z|) (6.3)

where κ is a constant depending on the roughness of the ground at the site(κ = 0.03 is used in all the calculations) and C is a correlation constant (C ' 7).Our calculations have been performed with a reference velocity2 of u10 = 10 m/s;for frequencies of 2.5 Hz and above, the correlation length is typically lower thanthe size of one segment, so that it has been assumed that the wind excitation isfully correlated within one segment and uncorrelated between different segments.Comparisons conducted with a power distribution of the mean wind, ui ∝ zα

with α = 0.16, and a constant profile (Fig.6.4) led to very small differences and itwas decided to perform the comparison assuming a constant mean wind profile.

6.2.3 Random response

The power spectral density (PSD) matrix of the structural response to the sta-tionary random excitation (6.1) is given by (e.g.(Preumont, 1994), p.108)

ΦX(ω) = G(ω)ΦF (ω)G(ω)∗ (6.4)

where G(ω) is the frequency response matrix between the excitation forces and thenodal responses and ∗ stands for the conjugate transpose. The diagonal elementsof ΦX(ω) are the PSD of the 3N nodal responses, each of them corresponding toone third of a segment, so that the PSD of the spatial average is

ΦX(ω) =1

3NTrace[ΦX(ω)] (6.5)

and the RMS deformation of the mirror (temporal and spatial average) is

σx = [∫ +∞

−∞ΦX(ω)dω]1/2 (6.6)

Because of the low-pass nature of the wind excitation, only 4 flexible modes ofthe supporting truss have been included in the wind response analysis; however,the full static deformation is accounted for in the model (section 4.3), whichincludes the differential motion of adjacent segments, responsible for high spatialfrequency distortion in the wavefront.

2The influence of the reference velocity is discussed in section 6.6.

6.3 Results in open-loop 117

6.3 Results in open-loop

2 4 6 8 10 200

1

2

f1

[Hz]

#1#2#3

Support

Set

1 2 3f1

D2

fD2

RR

Figure 6.5: Reduced Static deflection in the center [∆f21 /∆Rf2

R] (normalized tothe reference model for D = 11 m) versus the first natural frequency f1 for thevarious sets of models and the various support conditions.

Tables 2 and 3 synthesize the numerical results obtained in this study; the sup-port conditions refer to Fig.6.2; f1 is the first natural frequency, ∆ the staticdeflection at the center when the mirror points vertically and σx is the RMS(spatial and temporal average) wind response of the mirror according to Eq.(6.6).κ = (ηE/%)1/2 (in m/s) is the parameter appearing in Eq.(5.16). The referencevalues DR, fR, ∆R and σR are taken from the reference configuration (D = 11m),for the appropriate boundary conditions.

Figure 6.5 shows the reduced static deflection ∆f21 /∆Rf2

R (normalized to the ref-erence model for D = 11m with the same support conditions) as a function ofthe natural frequency f1. One sees that all the reduced values are very close to1, which confirms the scaling law Eq.(5.15).

Similarly, Fig.6.6 shows the reduced first natural frequency f1D/κ, normalized tothat of the reference structure (D = 11 m) with the same boundary conditions, asa function of the diameter D of the primary mirror, for the various sets of modelsand the various support conditions. Again, the reduced data are reasonably closeto 1, which supports the scaling law (5.16), regardless of the support conditions.

Finally, Fig.6.7 shows the reduced RMS response σ to the wind excitation, nor-malized to the reference structure, as a function of the diameter D. For themodels belonging to set 1 and set 2, the reduced RMS response is defined asσ = σxD3/2 f

7/31 ξ1/2, consistently with Eq.(5.18), and the reference structure is


M N D Support f1 ∆ σx κ[m] [Hz] [mm] [µm] [m/s]

3 36 11 #1 10 2.2 7.34 1819#2 10 3 7.29 1789Ref.#3 10 2.3 8.44 2247

Set 1

2 18 7.8 #1 13.2 0.9 6.28 1833#2 13.7 1.5 5.91 1803#3 12.9 1.2 7.04 2265

4 60 14 #1 6.7 4.3 9.76 1842#2 7.6 5.4 9.42 1811#3 7.8 3.9 10.51 2275

5 90 17.1 #1 5.4 7.7 12.46 1862#2 5.9 9 12.28 1832#3 6.2 6.2 13.27 2301

6 126 20.3 #1 4.2 12.3 16.21 1882#2 4.7 14.1 15.88 1851#3 5 9.3 16.64 2324

7 168 23.4 #1 3.5 19.2 20.25 1899#2 3.8 21.1 20.28 1868#3 4.2 13.3 20.62 2346

8 216 26.5 #1 2.9 28 25.7 1915#2 3.2 30.5 25.54 1884#3 3.5 18.5 25.44 2366

Set 2

5 90 17.1 #2 5.5 9.7 14.17 18326 126 20.3 #2 4.6 13.9 16.1 18517 168 23.4 #2 4 19.2 18.5 18688 216 26.5 #2 3.4 25.8 21.26 1884

Table 6.1: Synthesis of the numerical results obtained in open-loop with themodels belonging to set 1 and set 2; the support conditions refer to Fig.6.2; f1

is the first natural frequency, ∆ is the static deflection at the center when themirror points vertically and σx is the RMS wind response of the mirror (temporaland spatial average) for u10 = 10 m/s. κ = (ηE/%)1/2 (in m/s) is the structuralparameter appearing in Eq.(5.16).

6.3 Results in open-loop 119

8 10 20 30 40

D [m]

#1#2#3

Support

Set

1 2 3f1D

fRDR

k

kR

0

1

2

50

Figure 6.6: Reduced first natural frequency f1D/κ (normalized to that of thereference structure for D = 11 m) as a function of the diameter D of the primarymirror, for the various sets of models and the various support conditions.

8 10 20 30 40

D [m]0

1

2

#1#2#3

Support

Set

1 2 3

sR

s

50

Figure 6.7: Reduced RMS response σ to the wind excitation, normalized to thereference structure, as a function of the diameter D.

that of D = 11 m. For the models belonging to set 3, because the size of thesegments is proportional to the diameter, the reduced RMS response is definedas σ = σxf

7/31 ξ1/2, consistently with Eq.(F.2) developed in Appendix F. In this

case, the normalization is done with respect to the response of the mirror with adiameter of D = 17.1 m. For the RMS wind response also, the numerical resultssupport reasonably the scaling formulae, Eq.(5.18) and (5.19), in spite of the factthat these formulae are based on a simplified model which ignores the quasi-staticresponse of the mirror.


6.4 Controlled response

Next, the active compensation of gravity (active optics) is applied to all models ofset 3. The control strategy (cfr chapter 4) uses a Singular Value Decomposition(SVD) controller (Chanan et al., 2004), which relies on a SVD of the Jacobianbetween the segment actuators and the edge sensors (Fig.4.2); the sensor signalsare projected into the sensor modes, filtered, and then applied to the actuatormodes after proper scaling, in order to achieve equal gain for all loops. This strat-egy neglects the flexible modes of the mirror; the filter used in this study is thatshown in Fig.4.13; the bandwidth is adjusted by moving the Bode plots alongthe frequency axis until the appropriate gain is achieved at the earth rotationfrequency (fe = 1.16 10−5 Hz); this leaves the Nichols chart (and therefore thestability margins) unchanged. The gain is adjusted in such a way that the staticdeflection under gravity ∆ (listed in Table 6.2) is reduced to a closed-loop valueof ∆∗ = 30 nm (uniform in all cases); the loop gain at fe necessary to achieve this,and the corresponding bandwidth fc (we do not distinguish the control bandwidthand the crossover frequency) are given in Table 3.

The controller design is based on the Jacobian of the mirror, which assumes apurely quasi-static relationship between the actuator command and the sensorresponse, y = Ja. As the controller bandwidth increases and the natural fre-quency of the structure decreases, control-structure interaction becomes possibleand the robustness with respect to spillover instability must be addressed3. Wefollow the approach developed in section 4.6: The residual dynamics (i.e. theflexible modes of the supporting truss) are considered as a multiplicative uncer-tainty and a sufficient condition for stability is obtained from a singular valueinequality [Eq.(4.36)]. The minimum distance in this inequality is used as GainMargin (GM).

3The spillover is the phenomenon whereby the closed-loop poles of the residual dynamics(not included in the controller model) are slightly perturbed by the controller and, because ofthe small stability margin associated with lightly damped systems, some may become unstable.

6.4 Controlled response 121M

ND

Sup.

f 1∆

σx

κG

ain

f cG

M∆∗

σ∗ x

[m]

[Hz]

[mm

][µ

m]

[m/s

][d

B]

[Hz]

[dB

][n

m]

[µm

]5

908.

6#

110

.91.

93.

2118

6296

0.00

912

130

3.16

#2

11.8

2.2

3.09

1832

970.

0111

430

3.01

#3

12.4

1.5

3.27

2301

940.

008

123

303.

28

590

17.1

#1

5.4

7.7

12.4

618

6210

80.

032

7230

10.3

7#

25.

99

12.2

818

3211

00.

037

6430

10.3

2#

36.

26.

213

.26

2301

106

0.02

675

3011

.75

590

25.7

#1

3.6

17.4

28.8

918

6211

50.

071

4330

21.7

8#

23.

920

.228

.918

3211

70.

082

3530

22.7

5#

34.

113

.931

.52

2301

113

0.05

645

3026

.17

590

34.3

#1

2.7

30.9

53.0

118

6212

00.

124

2530

38.3

#2

2.9

3653

.65

1832

122

0.14

517

3041

.2#

33.

124

.658

.63

2301

118

0.09

926

3047

.16

590

42.9

#1

2.2

48.3

85.3

718

6212

40.

194

1130

61.2

1#

22.

456

.287

.19

1832

126

0.22

55

3068

.79

#3

2.5

38.5

95.3

323

0112

20.

155

1330

76.0

6

Tab

le6.

2:Sy

nthe

sisof

the

num

eric

alre

sult

sob

tain

edw

ith

the

mod

elsbe

long

ing

tose

t3

inop

en-loo

pan

dcl

osed

-loo

pfo

ru

10

=10

m/s

.T

hefir

stco

lum

nsha

veth

esa

me

mea

ning

asin

Tab

le2;

∆∗

isth

est

atic

defle

ctio

nw

ith

cont

rol,

f cis

the

band

wid

thof

the

cont

rols

yste

m,σ

∗ xis

the

RM

S(s

pati

alan

dte

mpo

rala

vera

ge)

win

dre

spon

sew

ith

cont

rol

and

GM

isth

ega

inm

argi

nw

ith

resp

ect

tosp

illov

er.


This part of the study has been conducted with the models of set 3 only, whichallow to explore the full range of diameter with a model of only 90 segments. Thenumerical results are synthesized in the last columns of Table 3. In the vicinityof the earth rotation frequency fe, the loop gain behaves closely to f−1 (integralcontrol, see the amplitude plot in Fig.4.13), which means that a = 1 must beused in Eq.(5.17).

f12

R

fc

1fc

( )

( )f2

2 4 6 8 10 200

1

2

f1

[Hz]

#1#2#3

Set 3 - support

Figure 6.8: Reduced bandwidth of the gravity compensation controller, normal-ized to the reference for D = 17.1 m.

Figure 6.8 shows the reduced bandwidth (fcf21 )/(fcf

21 )R as a function of the nat-

ural frequency f1, for the various models investigated; the reference mirror is thatwith D = 17.1 m. The results are close to 1, which supports the scaling law ofEq.(5.17), with a = 1 in this case.

Figure 6.9 shows the Gain Margin GM obtained by the robustness test mentionedbefore assuming a modal damping ξ = 0.01 uniformly for all modes. From the wayit is constructed, this test guarantees against the spillover of the flexible modes;it is a sufficient condition, which means that violating it does not necessarilymean that a spillover instability will occur, but it is possible. GM is plottedas a function of the frequency ratio f1/fc between the first natural frequencyand the control bandwidth; the continuous line on this figure is that obtained inFig.4.18 for the same value of ξ = 0.01. Once again, one observes that, for thisdamping value, protecting against spillover requires a frequency ratio of at leastone decade.

6.4 Controlled response 123

101

102

103

-20

0

20

60

100

GM [dB]

GM=10dB

UNSTABLE /f1fc

#1#2#3

Set 3 - support

Figure 6.9: Gain Margin GM obtained from the stability robustness test againstspillover for ξ = 0.01, as a function of the frequency ratio f1/fc between the firstnatural frequency and the control bandwidth. The continuous line correspondsto the results of Fig. 4.18.

sR

s

#1#2#3

Support

Set 3 open-loop

closed-loop

8 10 20 30 40

D [m]0

1

2

50

Figure 6.10: Reduced RMS response σ to the wind excitation, normalized to thereference structure, as a function of the diameter D.

Figure 6.10 shows the reduced RMS response σ to the wind excitation, normalizedto the reference structure as a function of the diameter D; as explained before,for the models of set 3, the reduced RMS response is defined by σ = σxf

7/31 ξ1/2,

consistently with Eq.(F.2), and the normalization is done with respect to the re-sponse of the mirror with a diameter of D = 17m. We note that the discrepancy(with respect to 1) tends to be larger in closed-loop than in open-loop, which isdue to the fact that the control bandwidth increases with the mirror size. This isillustrated in Fig.6.11 which compares the wind response, with and without con-trol, of two mirrors of respectively 8.6m and 42.9m diameter. The figure shows the


0

20

40

60

80

10-3

10-2

10-1

100

101

10-14

10-12

10-10

0

1

2

3

10-14

10-12

10-10

10-8

f [Hz]

open-loopclosed-loop

sRMS

[µm]

XF ( )f [m /Hz]2

XF ( )f [m /Hz]2

sRMS

[µm] open-loopclosed-loop

g x= 2.5 - = 0.01

g x= 0.5 - = 0.01

Figure 6.11: Wind response, with and without control, of two mirrors of respec-tively 8.6m (above) and 42.9m (below). The figure shows the PSD of the spatialaverage of the mirror displacements, ΦX , and the cumulative RMS response,σRMS(f) = [

∫∞f Φ(ν)dν]1/2.

6.5 Effect of damping 125

PSD plots as well as the cumulative RMS response σRMS(f) = [∫∞f Φ(ν)dν]1/2.

Note that this definition is different from that used in section 5.6. One observesthat, for mirrors of small size (D = 8.6m), the wind response is still dominated byits quasi-static component; which is only slightly reduced by the gravity compen-sation controller. For the large mirror (D = 42.9m), the dynamic component issignificantly larger, because of the lower natural frequencies, and the quasi-staticcomponent is substantially attenuated by the control; the wind response of verylarge mirrors is clearly dominated by the resonant response.

6.5 Effect of damping

In order to assess the effect of damping, the analysis of the models belongingto set 3 has been repeated for 3 values of the modal damping, ξi = 0.005, 0.01and 0.02. The numerical results are synthesized in Table 6.3; for each diameter,the same controller is used for the three damping values. One observes that thegain margin with respect to spillover is reduced by 6 dB when the damping isreduced by 2; this is exactly what one would expect from the way the robustnesstest is constructed, because the amplitude of the peaks of the residual dynamics(Fig.4.15) varies as ξ−1.

On the other hand, Fig.6.12 shows the evolution of the RMS wind response (whenthe gravity compensation system is in operation) with the diameter, for the threedamping values; two sets of curves are displayed, the solid lines correspond to σ∗,the full RMS response and the dashed lines correspond to the RMS response above1 Hz (i.e. the resonant response), σ∗[1,∞]. For mirrors of small diameters, the twosets of curves are significantly different, because the quasi-static response makesup a large fraction of the global response (see upper Fig.6.11). The differencereduces gradually as the size of the mirror increases and the wind response oflarge mirrors is dominated by the resonant part (see lower Fig.6.11). All the casesanalyzed have the same values of h/D and κ = (ηE/%)1/2, so that, according toEq.(5.16), f1 ∝ D−1; substituting in Eq.(F.2), one finds σ ∝ D7/3ξ−1/2. Theresults of Fig.6.12 confirm that the resonant response scales according to ξ−1/2

and D7/3.


D f1 fc ξ GM σ σ∗ σ∗[1,∞]

[m] [Hz] [Hz] [dB] [µm] [µm] [µm]8.6 12.4 0.008 0.005 117 3.72 3.73 2.48

0.01 123 3.27 3.28 1.720.02 129 3.05 3.06 1.26

17.1 6.2 0.026 0.005 69 16.6 15.4 13.10.01 75 13.3 11.7 8.520.02 81 11.8 10.1 6.00

25.7 4.1 0.056 0.005 39 40.8 36.7 33.70.01 45 31.5 26.2 21.80.02 51 27.3 20.9 15.1

34.3 3.1 0.099 0.005 20 77.4 67.8 64.50.01 26 58.6 47.2 42.20.02 32 49.8 36.0 29.3

42.9 2.5 0.155 0.005 7 128 110 1060.01 13 95.3 76.1 70.80.02 19 79.9 56.6 49.3

Table 6.3: Synthesis of the results obtained with the set 3 (support #3), for threevalues of the damping ratio, ξi = 0.005, 0.01 and 0.02. σ is the RMS responsewithout control, σ∗ is the RMS response with active control and σ∗[1,∞] is theresonant response, for f > 1 Hz.

1

10

100

8 10 20 30 40

D [m]

50

s* [µm]

x=0.005

x=0.01

x=0.02

Figure 6.12: RMS wind response as a function of the diameter, for various valuesof the damping. The solid lines correspond to the full frequency range, σ∗. Thedashed lines correspond to the resonant response, σ∗[1,∞]. The resonant response

scales according to ξ−1/2 and D7/3.

6.6 Effect of mean wind velocity 127

6.6 Effect of mean wind velocity

According to our wind model, the turbulent force acting on the structure scalesas u

7/310 , the corner frequency (at which the turbulent PSD starts to drop) scales

as u10 and the spatial correlation length scales as u−110 , where u10 is the refer-

ence velocity 10m above ground of Davenport’s spectrum. All the results pre-sented above correspond to a reference velocity u10 = 10m/s representative for aproject without enclosure such as OWL. For telescopes with enclosures, the meanwind velocity is reduced by one order of magnitude and a reference velocity ofu10 = 2 m/s may be regarded as more representative. The models of sets 1 to 3have been reanalyzed for a reference velocity of 2 m/s. The results are synthe-sized in Tables 6.4 and 6.5.

8 10 20 30 40

D [m]0

1

2

#1#2#3

Support

Set

1 2 3

sR

s

50

Figure 6.13: Open-loop reduced RMS response σ to the wind excitation, normal-ized to the reference structure, as a function of the diameter D. The referencevelocity is u10 = 2 m/s.

Fig.6.13 shows the reduced RMS response σ to the wind excitation in open-loop,normalized to the reference structure, as a function of the diameter D (the defi-nition of σ follows that of section 6.3). The differences with respect to the resultsobtained for u10 = 10 m/s (Fig.6.7) are very small.


u10 = 10 m/s u10 = 2 m/sM N D Support f1 σx σx

[m] [Hz] [µm] [µm]3 36 11 #1 10 7.34 0.26

#2 10 7.29 0.21Ref.#3 10 8.44 0.25

Set 1

2 18 7.8 #1 13.2 6.28 0.22#2 13.7 5.91 0.25#3 12.9 7.04 0.29

4 60 14 #1 6.7 9.76 0.34#2 7.6 9.42 0.32#3 7.8 10.51 0.36

5 90 17.1 #1 5.4 12.46 0.43#2 5.9 12.28 0.41#3 6.2 13.27 0.44

6 126 20.3 #1 4.2 16.21 0.56#2 4.7 15.88 0.52#3 5 16.64 0.54

7 168 23.4 #1 3.5 20.25 0.69#2 3.8 20.28 0.66#3 4.2 20.62 0.66

8 216 26.5 #1 2.9 25.7 0.87#2 3.2 25.54 0.82#3 3.5 25.44 0.8

Set 2

5 90 17.1 #2 5.5 14.17 0.476 126 20.3 #2 4.6 16.1 0.537 168 23.4 #2 4 18.5 0.608 216 26.5 #2 3.4 21.26 0.69

Table 6.4: Synthesis of the numerical results obtained in open-loop with themodels belonging to set 1 and set 2, for two values of the wind reference velocity,u10 = 10 m/s and 2 m/s; σx is the RMS wind response of the mirror (temporaland spatial average).


u10 = 10 m/s u10 = 2 m/sD Support f1 fc σx σ∗x σx σ∗x[m] [Hz] [Hz] [µm] [µm] [µm] [µm]8.6 #1 10.9 0.009 3.21 3.16 0.12 0.09

#2 11.8 0.01 3.09 3.01 0.11 0.08#3 12.4 0.008 3.27 3.28 0.11 0.09

17.1 #1 5.4 0.032 12.46 10.37 0.43 0.25#2 5.9 0.037 12.28 10.32 0.41 0.25#3 6.2 0.026 13.26 11.75 0.44 0.28

25.7 #1 3.6 0.071 28.89 21.78 0.98 0.51#2 3.9 0.082 28.9 22.75 0.94 0.53#3 4.1 0.056 31.52 26.17 1.01 0.62

34.3 #1 2.7 0.124 53.01 38.3 1.78 0.90#2 2.9 0.145 53.65 41.2 1.71 0.96#3 3.1 0.099 58.63 47.16 1.84 1.10

42.9 #1 2.2 0.194 85.37 61.21 2.82 1.43#2 2.4 0.225 87.19 68.79 2.74 1.61#3 2.5 0.155 95.33 76.06 2.95 1.78

Table 6.5: Synthesis of the results obtained with the models belonging to set 3,for two values of the wind reference velocity, u10 = 10 m/s and 2 m/s. fc is thebandwidth of the control system, σ∗x is the RMS (spatial and temporal average)wind response with control.


10-4

10-3

10-2

10-1

100

101

sRMS

[µm]

f [Hz]

0

0

20

40

60

80

10-20

10-15

10-10

0.5

1

1.5

2

2.5

3

open-loop 10=10m/s

closed-loop

u-

10=10m/su-

open-loop

closed-loop

10= 2m/su-

10= 2m/su-

sRMS

[µm]

F (f) [m2/Hz]X

Figure 6.14: Wind response with and without control of a mirror of 43m, for twovalues of the mean wind reference velocity, u10 = 10 m/s and u10 = 2 m/s. Theupper figure shows the PSD of the spatial average of the mirror displacement, withand without control. The two lower figures show the cumulative RMS response,σRMS(f) = [

∫∞f Φ(ν)dν]1/2.

The upper part of Fig.6.14 shows the PSD of the spatial average of the mirrordisplacement for the two reference velocities, with and without control (the samecontroller is used). The cumulative RMS response diagrams show that the controlcompensates for the quasi-static part of the wind response, but does not reducethe resonant contribution.


10-8

10-7

u107/3

*

s

D7/3

Set 3, closed- loop

2m/s 10m/s

#1#2#3

Support

8 10 20 30 40 50

D [m]

Figure 6.15: Closed-loop RMS response of the models of set 3, normalized withrespect to the mean wind reference velocity, σ∗/u

7/310 , as a function of the diameter

D.

Fig.6.15 shows the evolution of the closed-loop RMS displacement of M1 (spatialand temporal average), normalized with respect to the mean wind reference veloc-ity, σ∗/u

7/310 , as a function of the diameter of the primary mirror, for u10 = 10 m/s

and u10 = 2 m/s; the results for the two velocities are almost superimposed. Thiscurve supports reasonably the scaling of the closed-loop RMS response accordingto σ∗ ∝ /u

7/310 D7/3, but recall that the exponent in D results from the particular

assumption of the model of set 3 (full correlation on segments of increasing size,as explained in appendix F) which was done for computational reasons and is notfully representative of actual conditions. As far as the mean wind is concerned,Eq.(5.19) can be modified into

σ ∝ u7/310 D5/6ξ−1/2. (6.7)

This result is consistent with (MacMynowski et al., 2006); it clearly indicates theimportance of reducing the mean wind speed by a careful design of the enclosure.According to this equation, going from E-ELT to OWL (approximately doublingthe diameter and removing the enclosure, which increases the wind speed by oneorder of magnitude) would magnify the RMS response by a factor between 50and 200.

132 References

6.7 Conclusions

Based on an extensive numerical analysis of truss supported segmented mirrorsof various sizes and with various boundary conditions, one can say that the an-alytical scaling rules suggested in chapter 5 for (1) the static deflection, (2) thenatural frequency, (3) the control bandwidth of the gravity compensation con-trol and (4) the RMS response to wind are supported by the numerical analysis.These rules can therefore be used with confidence in preliminary design and para-metric studies.

Concerning the effect of damping and the mean wind velocity, this study confirmsthat the wind response of extremely large mirrors is dominated by the resonantpart, and that the RMS displacement (spatial and time average) scales accordingto u

7/310 ξ

−1/2i . The gain margin with respect to spillover scales according to ξ−1.

6.8 References

Bastaits, R. and Preumont, A. On the structural response of Extremely LargeTelescopes. AIAA Journal of Guidance, Control and Dynamics (accepted forpublication), 2010.




MacMynowski, D. G., Blaurock, C., Angeli, G. Z., and Vogiatzis, K. Modelingwind-buffeting of the Thirty Meter Telescope. In Cullum, M. J. and Angeli,G. Z., editors, Modeling, Systems Engineering, and Project Management forAstronomy II - SPIE 6271, July 2006.



Chapter 7

Conclusions

7.1 Original aspects of the work

This thesis is articulated around three innovative axes:

(1) The development of a numerical model of a large segmented mirror and itssupporting truss; the complexity of this model is kept minimum to allow extensiveparametric studies while preserving the quasi-static behavior of the segmentedmirror and the low-frequency resonances.

(2) The application of the general theory of multivariable feedback systems tothe control of large segmented mirrors and to evaluate the robustness marginswith respect to control-structure interaction.

(3) The development of scaling laws establishing trends and orders of magnitudesin the evolution of the structural response and of the control-structure interactionas a function of the diameter and other key parameters.

133

134 7 Conclusions

7.2 Scaling laws

A set of scaling rules have been developed and confirmed by an extensive numer-ical study. They allow to extrapolate the data from one telescope to a larger onebelonging to the same family. The scaling rules are summarized in Table 7.1.Once the diameter D and the aspect ratio h/D are fixed, there are only 3 freeparameters: The specific modulus of the structural material, E/%, the structuraldamping ξ and the average decay rate a near the earth rotation frequency, whichdepends on the control system design.

(1) Static deflection under gravity : ∆ ∝ f−21

(2) First resonance frequency : f1 ∝ D−1√

E/%

(3) Control bandwidth of the gravity compensation system: fc ∝ f−2/a1

(4) RMS response to wind gust : σ ∝ u7/310 D−3/2 f

−7/31 ξ−1/2

or, for given truss geometry and material properties: σ ∝ u7/310 D5/6 ξ−1/2

(5) Control-structure interaction is governed by the damping ξ and thefrequency ratio: f1/fc ∝ f

1+2/a1

Table 7.1: Scaling laws: D is the diameter of the primary mirror, E the Youngmodulus and % the material density of the truss, a the average decay rate of thecontroller (number of poles) near the earth rotation frequency (0 < a < 2; a = 1for integral control), u10 is the mean wind reference velocity.

7.3 Future perspectives

The natural continuation of the work presented in this thesis would be to broadenthe scope of the scaling laws and of the numerical model by including the sec-ondary mirror and its supporting structure.

In this study, optical models have been developed in ZEMAX while structuralmodels are first built in SAMCEF and then imported in MATLAB, in which thecontrol models are implemented. Combining optics, control and structure in asingle optomechanical module can be envisaged.

The numerical model is currently limited by the way MATLAB uses the memoryof the computer. Several solutions could be investigated to push these limitationsfurther and allow the study of mirrors made up of 1000 segments and beyond.

7.3 Future perspectives 135

Publications in the fields of active and adaptive optics

Journal papers

• Rodrigues G., Bastaits R., Roose S., Stockman Y., Gebhardt S., Schoenecker A.,Villon P., Preumont A., Modular bimorph mirrors for adaptive optics, OpticalEngineering, Vol. 48, No 3, March, 2009.

• Bastaits R., Rodrigues G., Mokrani B., Preumont A., Active Optics of LargeSegmented Mirrors: Dynamics and Control, AIAA Journal of Guidance, Dynamicsand Control, Vol. 32, No 6, 1795-1803, Nov.-Dec., 2009.

• Preumont A., Bastaits R., Rodrigues G., Scale Effects in Active Optics of LargeSegmented Mirrors, Mechatronics, Vol. 19, No 8, 1286-1293, December, 2009.

• Bastaits R., Preumont A., On the Structural Response of Extremely Large Tele-scopes, AIAA Journal of Guidance, Dynamics and Control, in press.

Conference proceedings

• Rodrigues G., Bastaits R., Stockman Y., Gebhardt S., Preumont A., AdaptiveMirror, International Symposium on Piezocomposite Applications, Dresden, Ger-many, September 27, 2007.

• Gebhardt S., Schonecker A., Bruchmann C., Beckert E., Rodrigues G., Bastaits R.,Preumont A., Active Optical Structures by Use of PZT Thick Films, InternationalConference and Exhibition on Ceramic Interconnect and Ceramic MicrosystemsTechnologies, Munich, Germany, April 21-24, 2008.

• Rodrigues G., Bastaits R., Roose S., Stockman Y., Gebhardt S., Schoenecker A.,Villon P., Preumont A., Large lightweight segmented mirrors for adaptive optics,SPIE Astronomical Telescopes and Instrumentation, Marseille, France, June 23-28,2008.

• Bastaits R., Rodrigues G., Mokrani B., Preumont A., Future Giant SegmentedMirrors: Scale Effects in Active Optics and the Use of Composites in ELTs Struc-tures, 8th National Congress on Theoretical and Applied Mechanics, Bruxelles,Belgique, May 28, 2009.

• Rodrigues G., Bastaits R., Preumont A., Adaptive Optics for ELTs with Low-Costand Lightweight Segmented Deformable Mirrors, Adaptive Optics for ExtremelyLarge Telescopes, Paris, France, June 22, 2009.

• Preumont A., Bastaits R., Mokrani B., Rodrigues G., Active Optics For LargeSegmented Mirrors, invited lecture, Smart Structures And Materials (SMART09)IV ECCOMAS Thematic Conference, Porto, Portugal, July 13, 2009.

• Rodrigues G., Bastaits R., Preumont A., Low-cost And Light-weight DeformableMirrors For High Order Adaptive Optics, Smart Structures And Materials (SMART09)IV ECCOMAS Thematic Conference, Porto, Portugal, July 13, 2009.

136 7 Conclusions

• Bastaits R., Rodrigues G., Mokrani B., Preumont A., Control-structure interactionin active optics of large segmented mirrors, 7th EUROMECH Solid MechanicsConference, Lisbon, Portugal, September 7-11, 2009.

• Bastaits R., Rodrigues G., Mokrani B., Preumont A., Large Segmented Mirrors :Scale Effects, Dynamics and Control, 1st ESA Multibody Dynamics Workshop onMultibody Dynamics for Space Applications, Noordwijk, Netherlands, February2, 2010.

• Preumont A., Bastaits R., Active Optics Of Extremely Large Telescopes: A Chal-lenge in Precision Mechatronics, keynote lecture, The 12th Mechatronics Forum,Biennial International Conference, Zurich, Suisse, June 28, 2010.

• Preumont A., Bastaits R., Rodrigues G., Extremely Large Telescopes: Dynamics,Active Optics, Scale Effects. MoSS2010: Mechanics of Slender Structures, SanSebastian, Espagne, July 7, 2010.

• Bastaits R., Mokrani B., Preumont A., Control-Structure Interaction in Activeoptics of Future Large Segmented Mirrors, International Conference of Noise andVibration Engineering (ISMA 2010), Leuven, Belgium, September 20, 2010.

• Rodrigues G., Bastaits R., Preumont A., Intelligent Segmented Deformable Mirrorsfor Adaptive Optics, International Conference of Noise and Vibration Engineering(ISMA 2010), Leuven, Belgium, September 20, 2010.

Oral presentations

• Rodrigues G., Bastaits R., Stockman Y., Gebhardt S., Preumont A., AdaptiveMirror, International Symposium on Piezocomposite Applications, Dresden, Ger-many, September 27, 2007.

• Preumont A., Bastaits R., Deraemaeker A., De Marneffe B., El Ouni M., RodriguesG., Structural Intelligence: Self Monitoring, Actuation and Adaption, invited lec-ture, 3rd Conf. on Smart Materials and Structures Systems, Acireale, Italy, June8-13, 2008.

• Rodrigues G., Bastaits R., Uhlig S., Schonecker A., Preumont A., Modular Bi-morph Mirrors for High-Order Adaptive Optics, International Symposium on Piezo-composite Applications, Dresden, Germany, Sept. 25, 2009.

• Hutsebaut X., Joannes L., Rodrigues G., Bastaits R., Preumont A., Deflectometriede Schlieren a decalage de phase en reflexion : mesure de miroirs deformables poly-morphes, Dixieme colloque international francophone sur les Methodes et Tech-niques Optiques pour l’Industrie, Reims, France, November 18, 2009.

• Bastaits R., Rodrigues G., Preumont A., Challenges in Modelling and Control ofExtremely Large telescopes, European Conference on Computational Mechanics,Paris, France, May 16, 2010.

Appendix A

Definitions of optical designparameters

M2

F

a)

V1 1 VV2

1

F2

M1M1

M2

VV2

1

F3

M1

M3V3

b)

c)

Figure A.1: Examples of the definition of the (paraxial) effective focal length(EFL): (a) EFL = V1F1 -(b) EFL = V1V2+V2F2 -(c) EFL = V1V2+V2V3+V3F3

The effective focal length (EFL) of an optical system can be defined as the dis-tance over which collimated rays are brought to a focus [(Geary, 2002), p.38]. Inthe case of a reflecting telescope made up of several mirrors, it can be calculated,to a first approximation, as the sum of the distances on the optical axis betweenthe vertices of the successive mirrors and of the focal surface (see Fig.A.1). Animportant design parameter is derived from the definition of the EFL, namely the

137

138 A Definitions of optical design parameters

f-number (also referred to as the focal ratio, f-ratio, or relative aperture, denotedby f/#):

f/# = EFL/EPD, (A.1)

where EPD stands for effective pupil diameter, namely the diameter of the ele-ment that effectively limits the extent of the incoming light flux [see e.g. (Geary,2002), p.43], for more details)1. The f/# can be defined for a whole telescope,as well as only for a single mirror; in this case, it can be written

f/# = F/D, (A.2)

where D is the diameter of the mirror and the F is its focal length2. Mirrors withdifferent f/# are compared in Fig.A.2. In the case of a single mirror, the smallerthe f/#, the more curved the mirror and the closer the focus to its vertex.

f/#

fast slow1 10 100

f/100f/10f/2.5f/1

Figure A.2: Comparison of mirrors with different f/# and a constant diameter.

1Other definitions exist for f/#, but the small differences in their results are not relevant forthe remainder of this text that focuses on trends and orders of magnitudes, see [(Geary, 2002),p.19] for a deeper discussion.

2Mirrors different from paraboloids do not bring collimated rays to a single focus; for thosecases, [(Schroeder, 2000), p.49], establishes the formula relating the paraxial focal length to theradius of curvature and conic constant.

A.1 References 139

The field of view (FOV), or more simply field, denoted by θ, is an angular mea-surement of the portion of the skies that can be imaged when the telescope ispointing in a fixed direction. A bundle of parallel incoming rays exhibiting anangle larger than the FOV will be partially or totally blocked before attainingthe focal surface (Fig.A.3). The field can be physically limited by various ele-ments: Mirrors, baffles, diaphragms, instrument aperture (most general case), . . .

M1

M2

qa)

M1

M2

qb)

M1

M2

qc)

CCD plate

Figure A.3: Different examples of elements limiting the field of view: (a) M2, (b)central hole in M1, (c) aperture of the scientific camera.

Moreover, the observation of off-axis objects (i.e. at a field angle different from0) causes aberrations in the reflected wavefront. Consequently, design specifica-tions often define one or several FOV’s (encompassed by the physical limitationsset by the telescope). Their specifications are based on the amount of off-axisaberrations that are tolerated over them.

A.1 References



140 References

Appendix B

Primary aberrations

This discussion assumes parallel rays coming from a source at an infinite distancefrom the mirror.

a b c d

a b c d

Figure B.1: Left, ray trace of a mirror subjected to positive spherical aberration.Right, corresponding spot diagrams at various locations: (a) corresponds to theparaxial focus (the focus of all rays if no aberration was present) and (c) cor-responds to the location of the image plane where the blur would be minimum[adapted from (Geary, 2002), p.69 and (Walker, 1998), p.131].

When a mirror is subjected to spherical aberration, depicted in Fig.B.1, it nolonger has a single focus. Instead, one can divide the mirror surface in infinitesi-mal annular zones that each reflect the rays hitting them at a different focus; thelarger the ring, the shorter the focus with respect to the vertex. Consequently,the distribution of light energy associated with the ray density changes accordingto where the image plane is located1. As its name suggests, spherical aberrationis generated when using a spherical mirror to focus a collimated wavefront (in-stead of a paraboloid with the same focal length). More generally, is results from

1The spherical aberration is positive when the rays are brought to a focus closer to the mirror,and negative when the rays focus further away from it.

141

142 B Primary aberrations

a mismatch in the conic constants of the mirrors of a telescope.

The analytic expression of Λ, the difference between the surface of a paraboloidand that of a sphere with equal radii of curvature Rc in function of the aper-ture radius r is given in Eq.(B.1) [derived from (Nelson and Sheinis, 2006)] anddepicted in Fig.B.2.

0 20 40 60 80

0.2

0.4

0.6

0.8

r [mm]

Ll [

@ 6

32nm

]

f/6.66

f/8.33

f/10

f/12.25

f/13.33

f/16.66

f/33.33

20 400

0.2

0.4

0.6

0.8

1

f/#[/]

S[/]

Figure B.2: Difference between the surfaces of spheres and paraboloids for var-ious focal lengths, projected along an aperture radius of 75mm. The dashedcurves correspond to diffraction-limited mirrors, the red curve corresponds to thediffraction limit (for S = 0.8, λ = 632nm), and the continuous curves correspondto aberration-limited mirrors. The vignette shows the corresponding evolution ofS with the f/#.

Λ ' r4

8.Rc3 +

r6

16.Rc5 +

5.r8

128.Rc7 + . . . . (B.1)

Using r = D/2, Rc/2 as the focal length of the mirror2 and the definition off/# [Eq.(A.2)], one obtains Eq.(B.2) showing that the spherical aberration isroughly proportional to f/#−3.

Λ ∝ (f/#)−3 +12(f/#)−5 +

516

(f/#)−7 + . . . (B.2)

2This approximation is exact for paraboloids and induces small errors for spheres [(Schroeder,2000), p.49]; however, they are negligible in Eq.(B.2).

143

1

2

3

4

60°

1

2

3

4

Figure B.3: Left, ray trace of a wavefront subjected to coma; each number definesan annular zone in the mirror. Center, characteristic spot diagram of a wavefrontsubjected to coma imaged at the paraxial focus. Right, schematic representationof the same spot diagram; the numbers associated with each circle match thoseof the annular zones on the left [adapted from (Geary, 2002), pp.72-73].

When a mirror is subjected to coma, one can again divide the mirror surfacein infinitesimal annular zones generating different foci. However, unlike sphericalaberration, rays from a common ring do not all come to a focus at the samepoint, only ray pairs from opposite sides of the vertex do. Consequently, eachring is imaged as another ring: The larger the object ring, the larger the imagering and the further it is located laterally from the nominal focus (Fig.B.3). Mosttypically, coma can be produced by deformations of the reflecting surface but alsoby the observation of off-axis objects with somewhat perfect optical surfaces asis the case depicted in Fig.B.3, or it can be due to the design, manufacturing andmisalignments of the optical elements. Such off-axis coma is amplified in mirrorswith low f/#, as the regions far from the center of the mirror bend more thelight rays.

A mirror subjected to astigmatism sees its radius of curvature varying from aminimum along one of its diameters, to a maximum along the diameter perpen-dicular to it. Accordingly, rays contained in a common plane with the chief ray3

focus at the same point, but the axial position of that focus changes accordingto the radius of curvature intercepted by that plane on the mirror. Therefore,the image spot has an elliptic shape, the large axis of which rotates accordingto the orientation of the plane. That ellipse degenerates to a straight line whenthe image surface is located at the focus corresponding either to the minimum ormaximum radius of curvature, or to a circle if it is located halfway between them.

3The ray that passes through the center of the mirror in this case; for a more general defini-tion, see [(Geary, 2002) p.46] e.g.

144 References

Again, astigmatism can either arise from a deformation of the optical surface, orfrom the observation of off-axis objects, that causes the effective reflective surfaceas seen by the wavefront to be distorted (however, off-axis aberrations are dom-inated by coma, unless the telescope is aplanatic). Again, off-axis astigmatismincreases when the f/# of the mirror is smaller.

a

b

c

d

e

a b c d e

mirror

Figure B.4: Left, rays trace of a wavefront subjected to astigmatism after re-flection on a mirror. Right, spot diagrams corresponding to particular locationsalong the axis of propagation. Location (c) corresponds to the minimum blur;when the image plane il located elsewhere, the spot has a characteristic ellipticshape that degenerates in straight lines at the focuses corresponding to the mini-mum and maximum radii of curvature of the mirror [adapted from (Geary, 2002),p.74 and (Walker, 1998), p.136].

B.1 References





Appendix C

Shack-Hartmann sensors

Shack-Hartmann sensors measure the gradient of a wavefront over an array ofdiscrete points. It consists in an array of microlenses and in a Charge CoupledDevice (CCD) detector located in their common focal plane. Each microlensconstitutes a subaperture over which the wavefront is sampled (Fig.C.1):

• A collimated wavefront, propagating parallel to the optical axis of the mi-crolens, is focused on its focal point.

• The focus of a collimated wavefront that is slightly tilted with respect tothe optical axis of the microlens is shifted laterally on the focal plane. Thisshift is linearly proportional to the tilt angle.

• Local curvature of the wavefront results in a longitudinal shift of the focuswith respect to the focal plane (upward or downward, depending on thesign of the curvature). It results in a spreading of the image at the level ofthe CCD detector. The centroid of that spot is used to measure the localslope similarly as above.

145

146 C Shack-Hartmann sensors

incomingwavefront

microlensesarray

CCDcamera

F

plane(unperturbed)

puretilt

purecurvature

Tilt andcurvature

bijDy ij

Figure C.1: Convergence of beams within the subapertures of the Shack-Hartmann sensor for different shapes of the wavefront.

147

The CCD camera consists in an array of semiconducting elements that producean electrical charge proportional to the intensity of the incident light. A centroidalgorithm analyzes the image frames captured by the CCD to determine thelateral deviations of each spot with respect to the nominal foci of an unperturbedwavefront. This allows the determination of the local slopes of the wavefront overeach microlens, projected along two directions (Fig.C.2):

αij =∆xij

F(C.1)

βij =∆yij

F(C.2)

where F is the focal length of the microlenses and ∆xij and ∆yij are the lateral de-viations of the centroid with respect to the nominal foci. Various post-processingalgorithms then allow the reconstruction of the shape of the wavefront from thearray of local slopes.

microlensesarray

CCDcamera

Incomingwavefront

CCDpixel

Dy ij

Dx ij

Figure C.2: Due to the local tilts and curvatures of the aberrated wavefrontover each microlens, the image of the Shack-Hartmann appears as an array ofspots of various sizes shifted laterally from the nominal foci of the microlenses.A centroid algorithm determines those lateral deviations from the signals of theCCD camera.

148 C Shack-Hartmann sensors

Appendix D

Small-gain theorem

D.1 General formulation

e

+

+u1

G ( )s1

1

G ( )s2

+

+

u2

e2

Figure D.1: Small gain theorem.

The small gain theorem plays an important role in the analysis of robust stabilityof MIMO systems. Referring to Fig.D.1, if G1(s) and G2(s) are stable systems,a sufficient condition for closed-loop stability is the small gain condition:

σ[G1(jω)G2(jω)] < 1 (D.1)

where σ stands for the maximum singular value. An alternative, more conserva-tive condition is

σ[G1(jω)]σ[G2(jω)] < 1 (D.2)

D.2 Stability robustness tests

From the small gain theorem, it is possible to derive sufficient conditions forstability which have useful applications:

149

150 D Small-gain theorem

+

-

(a)DG

z

+

+

G

v

z v

DG

- ( + )I G-1

(b)

Figure D.2: (a) Additive uncertainty. (b) Seen from the break points z and v.

D.2.1 Additive uncertainty

Consider the perturbed unit feedback system with additive uncertainty of Fig.D.2.a,where G is the nominal plant and ∆G is the uncertainty. If one breaks the loop atz and v, one finds that z = (I + G)−1v and the block diagram can be recast intothat of Fig.D.2.b. According to the small gain theorem, a sufficient condition forstability is that

σ(∆G). σ[(I + G)−1] < 1 ω > 0 (D.3)

or

σ(∆G).1

σ(I + G)< 1

which is equivalent to

σ(∆G) < σ(I + G) ω > 0 (D.4)

D.2.2 Multiplicative uncertainty

z v

L

-( + )I G G-1

(c)

(b)

+

-

Lz

+

+

v

G

(a)

+

-

GI L+

Figure D.3: (a) and (b) Multiplicative uncertainty. (c) Seen from the break pointsz and v.

D.3 Residual dynamics 151

Consider the perturbed unit feedback system with multiplicative uncertainty ofFig.D.3.a., or equivalently Fig.D.3.b, where G is the nominal plant and L is theuncertainty. Upon breaking the loop at z and v, one finds z = −(I + G)−1Gvand the block diagram can be redrawn as in Fig.D.3.c. According to the smallgain theorem, a sufficient condition for stability is

σ(L). σ[(I + G)−1G] < 1 ω > 0 (D.5)

or

σ(L).1

σ[G−1(I + G)]< 1

which is equivalent to

σ(L) < σ(I + G−1) ω > 0 (D.6)

D.3 Residual dynamics

Primary

H

+ +

- +

Residual

Control

OG

RG

(a)

G G H=O

DG = G HR

+ +

- +

(b)

G

DG

G HG=O

L G G=O R

-1

+

-

(c)

GI L+

Figure D.4: (a) System with residual dynamics. (b) Additive uncertainty. (c)Multiplicative uncertainty.

A frequent form of uncertainty encountered in structural control is the residualdynamics: The control model includes only the primary modes, G0(s) and theresidual (high frequency) modes are considered as uncertainty, GR(s) (Fig.D.4).Because many controlled structures are lightly damped, the residual modes maybe destabilized by the controller H(s); this phenomenon is known as spillover ; itwill be studied in detail in the next chapter. In this section, we use the foregoinginequalities to establish a lower bound to the stability margin.

The system of Fig.D.4 may be recast in the standard form of additive uncertaintyby taking G = G0H and ∆G = GRH. It follows that the stability robustnessbecomes:

152 D Small-gain theorem

σ(GRH) < σ(I + G0H) ω > 0 (D.7)

Alternatively, the system may also be recast in the standard form for multiplica-tive uncertainty by taking G = HG0 and L = G−1

0 GR. Accordingly, the stabilityrobustness test (D.6) becomes

σ(G−10 GR) < σ[I + (HG0)−1] ω > 0 (D.8)

Note that these tests being sufficient conditions, they are both conservative, withdifferent degrees of conservatism. Typical representations of these tests are shownin Fig.D.5. In the inequality (D.8), the left hand side does not involve the con-troller at all; it measures the relative size of the contribution of the residualdynamics in the global response as a function of ω. The peaks in the uncertaintycurves depend on the damping of the residual modes. The minimum distancebetween the two curves may be regarded as a lower bound to the gain margin.

10

1

0.1

0.01

GM

s [ + ( ) ]I HG0-1

A

f1

structuraluncertainty

s [ ]G G0 R

-1

10

1

1

0.1

0.01 100.1

[Hz]0.01

s [ + ( ) ]I G H0

f1

GM

A

s [ ]G HRuncertainty

(a)

(b)

Figure D.5: System with residual dynamics. (a) Robustness tests based on mul-tiplicative uncertainty. (b) Robustness test based on additive uncertainty. Theminimum distance between the two curves may be regarded as a lower bound tothe gain margin.

Appendix E

Mode shapes of segmentedmirrors with supports 1-3

Figures E.1 to E.3 show the first four eigen modes of mirrors (models of set 1)with a growing number of rings (M = 2-8) for each of the three support conditionsconsidered in the numerical study of the scale effects.

153

154 E Mode shapes of segmented mirrors with supports 1-3

13.2 Hz 14.4 Hz 18.8 Hz 18.9 Hz

10 Hz 10.1 Hz 12.3 Hz 14.3 Hz

6.7 Hz 8.4 Hz 8.9 Hz 10.8 Hz

5.4 Hz 6.2 Hz 6.6 Hz 8.7 Hz

4.2 Hz 5.3 Hz 5.3 Hz 7.1 Hz

3.5 Hz 4.1 Hz 4.2 Hz 6 Hz

2.9 Hz 3.5 Hz 3.7 Hz 5.2 Hz

M=8

M=7

M=6

M=5

M=4

M=3

M=2

Figure E.1: Modes 1 to 4 of a primary mirror with 2 to 8 rings for support #1.

155

13.7 Hz 19.8 Hz 19.9 Hz 35.6 Hz

10 Hz 15.6 Hz 15.6 Hz 23.9 Hz

7.6 Hz 12.2 Hz 12.2 Hz 17.3 Hz

5.9 Hz 9.7 Hz 9.7 Hz 13.2 Hz

4.7 Hz 7.8 Hz 7.8 Hz 10.5 Hz

3.8 Hz 6.4 Hz 6.4 Hz 8.6 Hz

3.2 Hz 5.3 Hz 5.3 Hz 7.2 Hz

M=8

M=7

M=6

M=5

M=4

M=3

M=2


156 E Mode shapes of segmented mirrors with supports 1-3

12.9 Hz 14.4 Hz 34.5 Hz 34.9 Hz

10 Hz 10.5 Hz 24 Hz 26.4 Hz

7.8 Hz 8 Hz 18 Hz 20.5 Hz

6.2 Hz 6.4 Hz 14.2 Hz 16.4 Hz

5 Hz 5.2 Hz 11.6 Hz 13.5 Hz

4.2 Hz 4.4 Hz 9.8 Hz 11.3 Hz

3.5 Hz 3.7 Hz 8.4 Hz 9.6 Hz

M=8

M=7

M=6

M=5

M=4

M=3

M=2


Appendix F

Wind response of Set 3

0

10

20

30

40

50

60

= 9mD

= 17mD

= 25mD

= 34mD

= 43mD

Figure F.1: Geometry of the mirrors of set 3: The geometry is scaled up by aconstant factor 0.5 < γ < 2.5.

In chapter 6, the models of set 3 have been constructed to analyze extremely largetelescopes with a limited numerical effort, by limiting the number of segmentsto N = 90 (M = 5 rings). The geometry is scaled up by a constant factor0.5 < γ < 2.5 [in contrast with the assumption of segments of equal size in

157

158 F Wind response of Set 3

developing Eq.(5.18)]. In these models, the segment size is proportional to γ.Because our wind model assumes a full correlation of the wind pressure over onesegment, the PSD of the force acting on one segment scales as γ4. On the otherhand, the models have been constructed assuming a constant areal density andthe same structural mass fraction η; this means that the mass of the primarymirror scales as m ∝ γ2. Following the same reasoning as in chapter 5, Eq.(5.10),the MS random response to the excitation over one segment is

σ2 ∝ γ4Φ0(f1)m2 ξ f3

1

∝ Φ0(f1)ξ f3

1

∝ 1

ξ f14/31

(F.1)

where the shape of the Davenport’s spectrum tail has been used, Φ0(f1) ∝ f−5/31 .

Finally, the number of segments being the same in all cases, one finds that theglobal RMS response scales according to

σ ∝ f−7/31 ξ−1/2 (F.2)

This formula must be used instead of Eqs.(5.13/5.18) when dealing with modelsbelonging to set 3.

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Documents

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