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ARGOS testbed: study of multidisciplinarychallenges of future spaceborne interferometricarrays

Soon-Jo Chung, MEMBER SPIEDavid W. Miller, MEMBER SPIEOlivier L. de WeckMassachusetts Institute of TechnologySpace Systems LaboratoryCambridge, Massachusetts 02139E-mail: [email protected]

Abstract. Future spaceborne interferometric arrays must meet stringentoptical performance and tolerance requirements while exhibiting modu-larity and acceptable manufacture and integration cost levels. The Mas-sachusetts Institute of Technology (MIT) Adaptive ReconnaissanceGolay-3 Optical Satellite (ARGOS) is a wide-angle Fizeau interferometerspacecraft testbed designed to address these research challenges. De-signing a space-based stellar interferometer, which requires tight toler-ances on pointing and alignment for its apertures, presents unique mul-tidisciplinary challenges in the areas of structural dynamics, controls,and multiaperture phasing active optics. In meeting these challenges,emphasis is placed on modularity in spacecraft subsystems and opticsas a means of enabling expandability and upgradeability. A rigoroustheory of beam-combining errors for sparse optical arrays is derived andflown down to the design of various subsystems. A detailed elaborationon the optics system and control system is presented based on the per-formance requirements and beam-combining error tolerances. Thespace environment is simulated by floating ARGOS on a frictionless air-bearing that enables it to track both fast and slow moving targets. © 2004Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/1.1779232]

Subject terms: synthetic apertures; apertures; space optics; imaging systems;optical systems.

Paper 030610 received Dec. 2, 2003; accepted for publication Mar. 5, 2004. Thispaper is a revision of a paper presented at the SPIE conference on AstronomicalTelescopes and Instrumentation, Waikoloa, Hawaii, August 2002. The paperpresented there appears (unrefereed) in SPIE Proceedings Vol. 4849.

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1 Introduction

1.1 Sparse Aperture Interferometric Arrays

The quest for finer angular resolution in astronomy inetably leads to larger aperture since the resolution is proptional to the wavelength over the diameter of a circuaperture. Unfortunately, the primary mirror diameter fspace telescopes is limited by volume and mass constrof current launch vehicles~ca. 4 to 5 m! as well as thescaling laws of manufacturing costs.1 Since the cost ofmonolithic optics increases faster than their diamesquared, and mirrors such as the Hubble Space Telescoare already at the edge of what is financially feasible,forts are ongoing to break this trend by employing brethrough technologies such as deployed segmented mtelescopes and sparse aperture optics using interferomWhereas long baseline stellar Michelson2 interferometersfeed light from independent collectors to a beam combito obtain interference fringes over a period of time~pupilplane beam combination!, Fizeau2 interferometers producea direct image with full instantu-v coverage~image-planebeam combination!. Hence, Fizeau interferometers are suable for optical imaging of extended objects and rapichanging targets. In contrast to the long baselines of Mielson interferometers, Fizeau interferometry systems tto have compact telescope arrays. An optimal imaging c

2156 Opt. Eng. 43(9) 2156–2167 (September 2004) 0091-3286/2004/

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figuration designed for sparse arrays was first proposedGolay.3 Sparse arrays are promising for applications thatnot require extremely high sensitivity~bright sourcepresent! and allow for a rather limited field of view4,5

~FOV!. Diffraction-limited performance has been demostrated for sparse or dilute-aperture telescopes with acphasing control.6 A notable project in the area of phasetelescope arrays is the Multipurpose Multiple TelescoTestbed7 ~MMTT ! built by the Air Force Research Laboratory ~AFRL!. The MMTT consists of four 20-cm-aperturtelescopes phased together with a 15-arcmin FOV. TMMTT employs a complex laser interferometer metroloto sense wavefront error~WFE!. The Multi-Aperture Imag-ing Array8 built by Lockheed Martin demonstrated phadiversity computation techniques for WFE sensing. Tsparse array consists of afocal telescopes arranged inYformation that are combined to a common focus in a Fizeinterferometer configuration. It demonstrated the firstsults of a broadband multiple-telescope imaging arphased over a significant FOV using an extended improjector in the lab. Research into WFE sensing and conhas been extensively conducted for the Next GeneraSpace Telescope~renamed the James Webb SpaTelescope!.9

The Massachusetts Institute of Technology~MIT ! SpaceSystems Laboratory~SSL! successfully completed the Mid

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Chung, Miller, and de Weck: ARGOS testbed . . .

deck Active Control Experiment10 ~MACE!, a SpaceShuttle flight experiment that flew on STS-67 in Mar1995. The primary goal of the MACE project was to invetigate approaches for achieving high-precision pointing avibration control of future spacecraft. The MIT-SSL alsdeveloped The Origins Testbed11 ~OT!, a laboratory testarticle to capture the dynamics and control problems anpated for future space telescopes. Built on the heritagthe MACE and the OT, a ground-based satellite testwith a more sophisticated active optical payload systempresented in this paper. Most of the previous sparaperture systems do not deal with real-world problems sas the vibrational coupling between a spacecraft strucand the wavefront errors propagating through the whsystem. The presented testbed explores such issuesimplemented on a nonrotationally constrained platform.

1.2 Overview of ARGOS

To better understand the technological difficulties involvin designing and building a sparse aperture array, the clenge of building a Golay-3 interferometer system wasdertaken. The MIT Adaptive Reconnaissance Golay-3 Otical Satellite12 ~ARGOS! exploits wide-angle Fizeauinterferometer technology with an emphasis on modulain the optics and spacecraft subsystems. The objectivthe ARGOS project is to demonstrate the feasibility omodular architecture for space-based optical systems.

To demonstrate a complete spacecraft in a 1-g environ-ment, the ARGOS system is mounted on a frictionlessbearing, as shown in Fig. 1, and has the ability to track forbiting satellites like the International Space Station~ISS!as well as slow moving objects such as point stars. Tmodular architecture emphasizes the use of replicated cponents and standard interfaces. The system consisthree identical apertures arranged in a Golay-3 distributThe light from these telescopes is combined in a cenmodule and transmitted to a charge-coupled device~CCD!.

Fig. 1 Golay-3 ARGOS system with the three attitude control sys-tem (ACSs) shown at the bottom.

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Wavefront sensing techniques are explored to mitigatetial misalignment and to feed back real-time aberratiointo the optical control loop. The goal is to obtain an imaof similar quality as the image received from a monolithtelescope using a single comparable aperture. The primfunctional requirements of ARGOS are prescribed in Ta1.

The structure of this paper follows the step-by-stepsign procedures of ARGOS. A theory of interferometrbeam combining errors is developed in the processconnected to the actual design and implementation of AGOS. The paper also details the design of the optics baon the performance requirements and beam-combiningerances. In addition, it includes the control avionics astrategies for a large-angle slewing and active interferomric beam combining.

2 Physics of Sparse-Aperture InterferometricArrays

2.1 Determination of Array Configuration

Traditional image quality criteria such as resolution aencircled energy~EE! are inadequate for many sparsaperture or interferometric array applications.13 When look-ing at extended objects such as the Moon or faint distnebulae, evaluation of an optical system involves more tsimply looking at a point source response@point spreadfunction~PSF!#. The modulation transfer function~MTF! isa better metric to evaluate the contrast~modulation! trans-fer characteristic of an extended object. Figure 2 showscrucial relationship between the two most common imaquality criteria—PSF and MTF. The PSF is the squarmodulus of the Fourier transform of the complex pufunction. The optical transform function~OTF! is a Fouriertransform of the PSF, and the MTF is an absolute va~magnitude! of the OTF. In Fig. 3, the PSF and MTF plotof a Golay-3-like array withD50.21 m~ARGOS subaper-ture diameter! and the array radius,L50.12 m~solid!, 0.19m ~dash!, 0.3 m ~dot! are shown. A perfect monolithic aperture, free of optical aberrations, has a linearly decreasMTF contrast characteristic~see the solid MTF line in Fig.3!. In case of a sparse array, the MTF suffers a contrastin the mid spatial frequency range, as shown in Fig. 3.can observe that the MTF plot withL50.3 m exhibits twozero MTF values rather than one. The first zero denotedFr is the practical spatial cutoff frequency, and defines‘‘practical resolution limit.’’ TheFc is the cutoff frequency,

Table 1 Key functional requirements.

Key Requirements

Angular resolution 0.35 arcsec at operating wavelength

Operating wavelength 400–700 nm (visible)

FOV 3 arcmin33 arcmin

Field of regard (FOR) 120 deg

Signal-to-noise ratio (SNR) 100

ACS pointing accuracy 61 arcmin

Image acquisition time 20 images/h (max)

Autonomous operation time up to 1 continuous h

2157Optical Engineering, Vol. 43 No. 9, September 2004


2158 Optical Engi

Fig. 2 The relationship between the PSF and OTF.

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whose inverse indicates an angular resolution undernormal condition that there is noFr ~another zero point!beforeFc . Thus, the largerFc or Fr is, the better is theangular resolution a sparse array will achieve.

In contrast with monolithic apertures, the method of fwidth at half maximum~FWHM! is not sufficient to deter-mine the angular resolution. Assuming that angular resotion is fully determined by the array size, the PSF canveal the highest achievable angular resolution. Tassumption holds especially for very large baseline Mielson interferometers. Figure 3 shows that the width ofmainlobe of the PSF plot is getting smaller, indicating improving angular resolution as we increase the array sizeL.The Fizeau interferometer, however, requires an instaneous fullu-v coverage, which limits the practical resolution. In addition, as discussed, considerable contrast losthe mid-spatial-frequency range should be avoided. Thegular resolution from the MTF plot of a Golay-3 typsparse imaging array is approximated as12

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As L is increased, the array becomes sparser, which,result, boosts the heights of sidelobes in the PSF. The Mplot atL50.3 m~dotted line in Fig. 3! has two zeros whileothers have only one. In that case, the resolution is liminot by Fc but by Fr , whose inverse is the angular resoltion of a single aperture (1.22l/D). Thus, in that case, thesparse array has no advantage over a single subtelesexcept for increased light-gathering power. The array cfiguration of ARGOS is selected asL50.19185 m forD50.21 m subtelescopes. This results in a better theoreangular resolution of 0.35 arcsec for the 550-nm walength rather than 0.55 arcsec of a single aperture, commising with a reasonable MTF characteristic.

2.2 Beam Combining Errors

There are three major wavefront errors that must be ctrolled at the beam combiner to achieve phased beam cbining. Those errors are optical path difference~OPD!, tip/tilt error, and pupil-mapping error.

Fig. 3 PSF and MTF plots when D50.21 m and L50.12 m (solid), 0.19 m (dash), and 0.3 m (dot).The corresponding array configurations are shown to the right of PSF-MTF plot. The black-gray figureis an MTF plot. Gray circles indicate the practical cutoff frequency.

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2.2.1 OPD (piston) error

We can plot the effects of OPD errors using the interferoetry equation given by Mennesson and Mariotti14

I}UpD@11cos~r !#

l U2UJ1~pD sinr /l!

pD sinr /l U2

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n

exp@ j 2p(Lkr /l)cos(dk2u)#ej fkU2

, ~2!

wherer is the off-axis angular direction,u is the azimuthangle,D is a subaperture diameter, (Lk ,dk) are the polarcoordinates of the array configuration, andfk is the phaseshift.

Numerical simulations using Eq.~2! were performed toinvestigate the effects of OPD errors, as shown in Fig. 4.the piston error increases, two major deviations deveover the envelope of the PSF. First, the main envelshifts in the direction of the piston error. The resultantrection of the envelope shift is the vector sum of pisterror directions weighted by the amount of error. Secothe peak intensity is reduced compared to the nominal Pwithout any piston errors resulting in a reduced Strehl ra~SR!. The size of the mainlobe also expands showingdegraded angular resolution@see Fig. 4~c!#. When the pis-ton error is 0.1l between apertures, the peak intensity98% of the normal intensity. The beam combining pisterror tolerance is chosen to be 0.1l555 nm for most ofinterferometric beam combining applications.

2.2.2 Tip/tilt error

The approach employed here to analyze tilt errors isfurther segment each aperture into smaller elements

Fig. 4 PSF plot of Golay-3 array with (a) 0 OPD, (b) 0.5l OPD, and(c) 1.0l OPD.

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analysis~finite element method!. The PSF is calculated bysumming up the interference from each element. The phdifference at the central point due to tilt errors are addedthe interference term of Eq.~2!:

I}Upd@11cos~r !#

l U2UJ1~pd sinr /l!

pd sinr /l U2

3U(k51

m

exp@ j 2p(Lkr /l)cos(dk2u)#

3exp@ jdk sin(tiltn)2p/l#U2

, ~3!

where D in Eq. ~2! is replaced with the smaller elemendiameterd, dk is the distance from the aperture centerthe k’ th element center, tiltn denotes the tilt in radians fothe n’ th aperture,k ranges from 1 tom, andm equals thenumber of apertures (n) multiplied by the number ofsmaller elements of each aperture. By reading the SRues from the PSF plots under the influence of tilt errors,maximum allowable tilt error at the beam-combining stais determined. This numerical method predicts that thetolerance between each beam entering the beam combshould be less than 20mdeg (0.072 arcsec50.35mrad).

2.2.3 Pupil mapping error

If coherent imaging is to be achieved over any significaFOV, the pupil mapping process must be performed sthat the exit pupil is an exact~scaled! replica of the en-trance pupil.4 This constraint is commonly called th‘‘golden rule’’ of beam combining. From the geometry oFig. 5, Faucherre et al.5 derived the net OPD error due tthe incorrect pupil mapping,

OPDnet5uB sina2b sinbu5UB sina2B

mssin~maa!U, ~4!

wherema is the aperture magnification factor (D/d), ms isthe baseline magnification factor (B/b), a is half FOV ofthe target, andb is magnified field angle through the subtelescope array.

Fig. 5 The golden rule of beam combining, pupil mapping.


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According to the discussion in Sec. 2.2.1, 1/10l phasingis desired to meet the SR and angular resolution requments. Thus, equating Eq.~4! with 1/10l and small angleapproximation sina>a results in

BaU12ma

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10. ~5!

A calculation of shear error tolerance is performed withassumption that there is no magnification error~subtele-scopes are strictly afocal with exactly the same magnifition!. Then, the correct magnification isma5D/d5B/(b1Db). The incorrect mappingms is still B/b. Insertingma

andms into Eq. ~5! results in

auB2bmau5baUBb 2B

b1DbU5aBDb

b1Db5

l

10. ~6!

Solving for Db, gives

Db5l

10maa. ~7!

Equation ~7! means that the allowable shear error isversely proportional to both aperture magnificationma andoff-axis target anglea ~FOV/2!. However, the assumptioof no magnification error used for Eq.~7! is inadequatesince perfectly replicating any subaperture is impossibleterms of manufacturability. There usually exists some pformance window specified by optics manufacturers. Tobjective of this calculation is to find out how precise tmagnification tolerance level must be in order to achiecoherent beam combining. If the magnification tolerancetoo stringent to be manufactured, methods of relaxingtolerance will be explored.

For both incorrect ma(5D/d) and incorrect ms

(5B/b), the acceptable magnification ratios are definedfollows:

macorrect5ma1Dma5

B

b1Db5ms

correct. ~8!

Inserting these results into Eq.~5!, gives

abUBb 2maU5abUBb 2B

b1Db1DmaU

5aU DbB

b1Db1DmabU5 l

10. ~9!

We can removeb by using Eq.~8!:

aUDbmacorrect1DmaS B

macorrect2DbDU5 l

10. ~10!

Setting a5FOV/2, macorrect5m and B5)L and for a

Golay-3 array leads to

2160 Optical Engineering, Vol. 43 No. 9, September 2004

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FOVUmD l 1DmL

m2DmD lU5 l

5). ~11!

From Eq.~11!, we can find the FOV as a function of magnification (m) and pupil mapping tolerances (D l andDm):

FOV5l

5)umD l 1 ~DmL/m! 2DmD l u. ~12!

We can also represent the shear error toleranceD l as afunction of FOV, magnification (m) and magnification tol-erance (Dm) for a positiveDm:

D l 5l

5)FOV~m2Dm!2

DmL

m22mDm. ~13!

2.2.4 Pupil mapping error for ARGOS

These results are graphically plotted in Figs. 6 and 7. Atpreliminary design phase of ARGOS, 12mm was suggestedfor the shear error tolerance from Eq.~7! with an assump-tion of no magnification error (FOV54 arcmin,magnification510). However, 12-mm shear error cannoproduce the FOV requirement of ARGOS~3 arcmin! forany range of magnification error in Fig. 6. We can tightshear error tolerance to meet the FOV requirement orcan relax the FOV requirement by shrinking the regioninterest within the whole FOV. Therefore, the pupil maping process is the primary limiting factor deciding the resonable FOV of a sparse aperture interferometric array. AGOS’s subtelescope collimators are designed withtolerance of60.0095 with a nominal magnification factoof 10. However, the ARGOS subtelescopes have a focuknob that can control the distance between the primary mror and the secondary mirror, thereby controlling the sizethe beam more precisely. It is usually considered tha1/1000 magnification tolerance requirement is too expsive to manufacture. Figure 7 describes how the magnifition of a subaperture affects the shearing tolerance o

Fig. 6 FOV versus pupil mapping tolerances for ARGOS withachievable magnification tolerance.

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sparse aperture imaging system. This plot implies that saperture magnification can be tuned to maximize allowashear errorD l ~lateral pupil mapping error! thereby reduc-ing control complexity. For a magnification of 10, we caincrease the shear tolerance value sufficiently high atexpense of the reduced FOV.

2.3 Optics Design and Implementation

2.3.1 Subaperture

The overall cost of the ARGOS optics system couldsignificantly reduced by selecting one of the highest presion optics commercially off-the-shelf~COTS! telescope.However, it was necessary to customize a collimating lto convert the Dall-Kirkham-type focal telescope to an acal telescope with a magnification ratio of 10. The colliming lens is placed into the baffle of the telescope to mthe system compact. A trade-off analysis was performedseveral different cemented doublets that were optimizedtensively by ZEMAX. One drawback of the cemented doblet is that it has bonded glasses, therefore if therechange of temperature, the doublet may fail. Althoughdoublet with CaF2 performs best in reducing chromatic aerrations, the high coefficients of thermal expansion~CTEs!of CaF2 ~18.3! forced to find another glass combination fefficient achromatic doublet design. Smith15 suggests FK51~as a crown element! with a KzFS or LaK glass~as a flint!.Although the maximum focal shift range can be minimizto 247mm with FK51-KzFS11, it was not the best choicdue to the residual aberrations@root mean square~rms!wavefront errors predicted by ZEMAX#. The final FK51-BaK2 design achieves 271.6mm chromatic focal shiftrange.

2.3.2 Design of OPD and tilt/tip controller

When strictly looking at two designs of optical delay lin~ODL! in Fig. 8, a perpendicular design seems to offermost benefits. For the perpendicular design, a multiaxis

Fig. 7 Magnification versus shear tolerance with magnificationerror50.001.

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steering mirror~FSM! could be used to generate the rquired tip/tilt actuations as well as fine optical path diffeence control. FSMs have very fine resolutions so dependon them for fine OPD control will enable us to have a moreasonable coarse control for the ODL. When the paraODL design is coupled with a FSM~Fig. 8!, the resultingdesign is very simple and more cost-effective than the ppendicular design. This design cannot perform coarse Ocontrol, but this design is more compact resulting in easintegration with the structural design. In addition, thereno need for a translational stage or rooftop mirror, whireduces the cost and control complexity, and there is alsgreater total reflectance since there are fewer mirroredfaces. Fewer mirrored surfaces also lead to fewer structmisalignment errors. In both the perpendicular ODL desand the parallel ODL design, fine OPD control is couplwith shear control so thatDFine OPD5DShear. The effectthat a change in fine OPD would have on shear, is not vsignificant and could be ignored for adjustments,10mm.This is because we have a much tighter tolerance on pierror ~50 nm! than the 12-mm shear error. A fine resolutionmultiaxis FSM, capable of controlling the tilt/tip as well apiston motion, was selected as the ODL-FSM actuatorARGOS. The FSM has to be able to compensate forerrors in its mounting. Therefore a high-precision mouwith a range up to 7 deg was selected for the FSM actua~see Fig. 10 and Table 2 in Sec. 2.3.3!.

2.3.3 Pyramidal mirror and beam combiner

The pyramidal mirror turns all three beams 45 deg intobeam combiner. A custom pyramidal mirror is chosen dto the cost of making one out of regular mirrors. The mareason for the high cost is thin mirrors that cost $10001,and then mount them to an accuracy of60.001 deg(63.6 arcsec). The pyramid cannot be made from the relar thickness mirrors since they constrain the beam diaeter. We customized a pyramidal mirror with a surfacecuracy ofl/10 peak to valley,63 arcsec angle error, and50-mm clear aperture. The substrate material is BK7 witcoating of AlSiO ~aluminum with silicon monoxide!. Thereflectance will be approximately 90% in the visible rangThe two PSFs shown in Fig. 9 demonstrate how the Fcan compensate for the63 arcsec errors in the pyramid. Tcompensate for pyramid errors the FSM has to align itsso that the two reflecting surfaces are parallel. The pyradal mount is composed of two stages. The first stage pvides all of the angular adjustments and the second hanX andY translation in the entrance pupil of the beam cobiner. TheX-Y translation stage is small enough to fit bhind the tip/tilt rotation stage and has the load capacity

Fig. 8 Perpendicular ODL design with FSM (left) and parallel ODLdesign coupled with FSM (right).


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hold both the second stage and the mirror. Any additionaZdirectional error can be offset by the FSM mounts. Tspecifications of the ARGOS optical actuators and mouare tabulated in Table 2 and their pictures are shown in10.

Two options available for the image-plane beam cobiner are either reflecting or refracting optics. The reflectbeam combiner is compact when compared to a refracUnfortunately, the secondary mirror of a Cassegrain tescope would partially block the three incoming beamsany possible configurations to obey the golden rule dcussed in Sec. 2.2.3. A single parabolic mirror was conered, however, there was not enough space betweenpyramidal mirror. Had we used reflecting optics, thewould be nothing available as COTS, therefore it wouhave to be custom manufactured increasing cost sigcantly. A reflector would also complicate the relay optisignificantly since we would not be able to use the pyradal mirror. In contrast, a refracting combining telescopemany advantages. It allows for very simple relay opticsis available as COTS with high-quality optics such as Tkahashi FSQ-106N~see Fig. 10!. This telescope has significantly less chromatic aberration than other COTS tescopes. This is due to its four-element design, two of whare fluorite. It has a diameter of 106 mm, and a 530-mfocal length, resulting in a total system focal length of 53mm.

2.3.4 CCD system design

The optics performance requirements driving the CCcamera selection are SNR and FOV. It was pointed outthe pupil mapping process is the primary limiting factdeciding the reasonable FOV of a sparse aperture interfmetric array~see Sec. 2.2!. This means that the pupil mapping tolerances become expensive as we increase the

Fig. 9 PSF simulation of ARGOS with 3-arcsec tilt error of the py-ramidal mirror (SR50.444, left). The PSF with FSM correction(SR50.960, right).

Table 2 Optical actuators and mounts specifications.

Model Angular RangeAngular

ResolutionLinearRange

LinearResolution

FSM 6600 mrad 60.05 mrad 12 mm 0.2 nm

FSM mount 67 deg 60.0008 deg(614 mrad)

1 cm 1 mm

Pyramid mirrormount

64 deg 62 arcsec(69.6 mrad)

13 mm 3 mm


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In addition, the off-axis imaging aberrations such as disttion and field curvature become dominant as we increthe FOV. Therefore, the CCD system of a sparse aperimaging system should be carefully designed to meetoptics performance requirements while maintaining the rsonable FOV large enough to capture the targets of interThe following relationship is used to find an appropriaCCD pixel size and the number of CCD pixels:

FOV5d

fn52.44

l

DeffQn, ~14!

wheren is the number of pixels along thex axis or theyaxis of a CCD matrix,d is the pixel size,f is the systemeffective focal length, andQ is a quality factor. The Ny-quist sampling criterion tells us that two pixels are genally required to properly record a star’s image (Q50.5).

A sparse aperture can cause a substantial depressiothe MTF of the system, resulting in a low-contrast aunsatisfactory SNR image. This can be compensatedimage reconstruction, such as by using a Wiener-Helstfilter, but at the expense of greater noise sensitivity copared to a monolithic aperture.16 Images acquired by modern CCD cameras may be contaminated by a varietynoise sources. Three primary noise sources are discussthis section: photon noise, read-out noise, dark current.

When the physical signal detected by a CCD is basedlight, the quantum nature of light plays a significant roPhoton noise is derived by calculating the number of ptons received through an imaging system, and is therenot independent of light source. The spectral energy disbution of a blackbody is given by Planck’s law:17

E~l!52phc2

l5

1

exp~ch/kTl!21, ~15!

whereE(l) is the energy per unit wavelength~also calledthe spectral irradiance measured in Wm22 mm21), l is thewavelength, h is Planck’s constant (6.626075310234 W s2), T is the absolute temperature of a targetcis the speed of light, andk is Boltzmann’s constan(1.380658310223 W s/K).

Fig. 10 Pictures of optical components.

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Then the input power at the entrance to a sparse apearray is calculated by

Pin5El1

l2E~l!dlS D

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N, ~16!

where the integration term integratesE(l) from Eq. ~15!over the operating wavelengths,N is the number of subapertures,D is the diameter of a subaperture, andDeff is theeffective diameter of a sparse aperture array.

Using Eq.~16!, the number of available photons candetermined as follows:

Np5Pint0Ti

l

hc, ~17!

whereTi is the integration time,t0 is the optical transmis-sion factor~typically 0.75!, c is the speed of light, andh isthe Planck constant defined in Eq.~15!.

The number of signal electrons is computed by muplying Np with the CCD quantum efficiency:

Nsignal5PinS t0

l

hcQEDTi

5El1

l2E~l!dlS N

D2

Deff2 D S 1.22l

4 D 2S t0

l

hcQEDTi ,

~18!

where QE is the quantum efficiency of a CCD. The teN(D2/Deff

2 ) is commonly called the filling factor of an interferometric array.

The probability distribution for the average numberphotons received by a CCD is known to be Poisson. Tstandard deviation of the Poisson distribution is the squroot of the average value. Accordingly, the total numberphoton noise electrons is

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.

~19!

By cooling the CCD chip it is possible to efficiently reducthe number of thermal electrons that produce thermal noor dark current. As the integration time increases, the nuber of thermal electrons increases:

NDC5RDCTi , ~20!

whereRDC is the dark current rate andTi is the integrationtime.

Readout noise~on-chip electronic noise! originates inthe process of reading the signal from the sensor throthe field effect transistor~FET! of a CCD chip. This noisecan become a significant component in the overall SNRvery low signal levels.

Finally, we can calculate the SNR of a sparse apertarray based on the noise characteristics of a CCD sysas already discussed. In general, photon noise, dark cunoise, and readout noise are considered as dominant ncontributors:

SNR5Nsignal

~NPN2 1NDC1NR

2 !1/2, ~21!

whereNsignal is the number of signal electrons in Eq.~18!,NPN is the number of photon noise electrons in Eq.~19!,NDC is the number dark current noise electrons in Eq.~20!,and NR is the number of readout noise electrons specifiby a manufacturer.

The SNR can be defined again expanding all the eqtions for noise electrons:

SNR5*l1

l2E~l!dl@N ~D2/Deff2 !#~1.22l/4!2@t0 ~l/hc!QE#Ti

$*l1

l2E~l!dl@N~D2/Deff2 !#~1.22l/4!2@t0 ~l/hc!QE#Ti1~RDCTi !1NR

2%1/2. ~22!

weheb-ovehe

dedaof

Using Eq.~22!, the optimal frame rates and sensitivities cbe obtained to achieve the required SNR for a given tar

2.3.5 Structural misalignment tolerancing

Using the mode of nonsequential ray tracing of ZEMAXcomplete ARGOS optics layout is constructed based onoptical specifications of a subaperture, pyramidal mirrand the beam combining telescope, as shown in Fig. 11.intentionally perturb the subtelescope or pyramidal mir

.to determine allowable structural misalignment, andcompensate the tilt error by changing the tilt angle of tfold mirror attached to the FSM. At 0.01-deg tilt of a suaperture, a pure FSM motion cannot restore the SR ab0.8. But the addition of FSM piston motion can restore tSR value to 0.859. We could achieve a SR of 0.859~whichis above diffraction limited! over a 0.01-deg tilt. But due toa magnification factor 10, the FSM compensation exceeits max range (0.6 mrad50.034 deg). Since we mountedFSM onto a precision tip-tip mount, which is capable


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several arcsecond adjustment~see Table 2!, this static errordoes not limit the FSM performance. However, it is musafer to have a FSM within a range of eliminating a posible maximum alignment error, 0.005 deg or 15 arcsecsubtelescope structural misalignment is suggested.

By assuming that all other optical components are pfectly aligned and the FSM can compensate for all thesidual tilt errors, the tilt errors for each surface of the pramidal mirror are calculated. When the tilt error of thpyramidal mirror unit equals the tilt compensation ofFSM, the aberration loss due to the tilt is completely elimnated. Therefore there is no theoretical tilt tolerance forpyramidal mirror as long as it does not exceeds the mmum compensation range~0.01 deg!.

The beam combiner was tilted along thex and y axeswhile leaving other optical components perfectly align~Fig. 12!. This beam combiner misalignment is not correable by optical actuators like FSMs. However, it turns othat we can tolerate up to 0.2 deg for the beam combiwhich is less stringent than other misalignment toleranc

3 Control System

3.1 Attitude Control System Overview

The final ARGOS structure and the major ACS componeare depicted in Fig. 1. The FOV of the main CCD isarcmin and thus to give us 1/2 arcmin of margin on eithside, the ACS subsystem is required to provide a pointaccuracy of61 arcmin. The period of operation of the ARGOS system without human intervention must be 60 mingreater~see Table 1!, meaning that the ACS system wihave to either not saturate its actuators, or have someof desaturating them within this given time span. The s

Fig. 11 ZEMAX 3-D nonsequential ray tracing.

Fig. 12 PSF plots when the beam combiner has tilt errors. From leftto right, X tilt: 0.2, Y tilt: 0.4; X: 0.3 Y: 0.3; X: 0.25 Y: 0.25 (deg).


,.

y

tem must be able to slew at a rate of at least 1.5 deplacing a minimum requirement on the capabilities of treaction wheels to slew the spacecraft. Due to the naturthe air bearing system chosen to simulate the space-boperation of ARGOS, the center of gravity and the centerotation of the body will not necessarily be at the samposition. Three perpendicular precision linear slides ofactive balancing system12 ~ABS! correct for small center ofmass offsets that would hurt the closed-loop performanThe ABS can also be used to remove momentum fromwheels by intentionally causing an offset in the appropridirection ~see Fig. 1!.

The sensor suite is composed of three integral elemeFirst, the TCM-2-50 electronic compass, with a three-amagnetometer, a two-axis tilt sensor, and facilities to pvide temperature information. Inclinometers/electroncompasses measure the relative angles between the incoordinate frame and the body fixed frame. That is, theyused to give relative elevation, roll, and azimuth informtion between these two coordinate systems. Tinclinometer/electronic compass is an essential componof the coarse pointing sensor suite. Second, there is antermediary sensor, which takes the form of a scope. It isintermediate wide FOV CCD that provides sufficient ovelap ~greater than 4! between the TCM~Fig. 13! and themain imager. Third, there is a three-axis rate gyroscopeangular rate measurement.

3.2 Control Hardware and Avionics

The HEPC8 digital signal processing module houses aC6701 digital signal processor~DSP!, two 12-bit eight-channel digital-to-analog~D/A! ~three signals for eachFSM and three RWA! and one 14-bit eight-channel analoto digital ~A/D!. Each reaction wheel speed command frothe DSP is sent through smoothing filters with a cutoff frquency of 280 Hz. The internal analog motor controlemploys a proportional integral derivative~PID! logic us-ing the motor encoder sensor outputs. A set of three pieelectric stacks for a single aperture FSM~a total of ninepiezolinear stacks! is driven by the D/A signals, which arefiltered by a 303-Hz cutoff frequency low-pass filter anthen amplified by the compact voltage amplifier~PIE0669.OE! located in the right side of ARGOS. The ragyroscope measures the rotational rates of ARGOSfeedbacks the voltages to the A/D with a resolution50 mV/~deg/s!. An antialiasing low-pass filter with a 48-Hcutoff frequency was implemented to reduce the higfrequency noise sent to the A/D channels. The electrocompass is directly connected to the PC motherboard u

Fig. 13 Multistaged ACS sensors.


Fig. 14 System identification of ARGOS-R/W 3 to gyros 1, 2, and 3 outputs (frequency in hertz).

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a standard RS232 serial port. The PC-DSP first in first~FIFO! designated for direct communication betweenhost PC memory and the DSP memory, sends the improcessing data12 such as the centroids of the target frothe intermediate CCD, and the compass readings toDSP controller in real time. To meet the requirementsimultaneous automatic control of various subsystems,DSP modules, the ABS controller and the wireless cardthe communication with a remote laptop control groustation, all reside in the peripheral component interconntion ~PCI! slots of the Athlon 1.4-GHz PC system wit1024 MB RAM. The graphic user interface~GUI! PC con-trol application sends the reference slew speed and poin~azimuth and elevation! for a specific target to the DSPwhile performing real-time image processing from both tintermediate ACS CCD and the main imager~spot-basedwavefront sensing12!.

3.3 Pointing and Tracking Control Design

The objective of system identification is to determine tdynamic properties of ARGOS and correctly design anpropriate control system to capture all present issueswould deteriorate the performance. The 333 transfer func-tion matrix between the three D/A signals from the Dand the three gyro outputs to the A/D port for the reactwheel assembly~RWA! control is obtained. In this case, thsystem identification will give us a transfer function withe ACS filter board reference command as the input,smoothing low-pass filter, motor controller, reactiowheels, structure, rate gyros, and antialiasing low-pass fias the plant, and the filtered gyro signal from the ACS filboard as the output.

The transfer function of the plant is then found by dviding the cross spectral density of the input and output

e

g

t

r

the power spectral density of the input. Then, tmeasurement-based system identification algorithm caintegrated frequency domain observability range spacetraction and Least Square parameter estimatalgorithm10,18 ~IFORSELS!, is performed to derive a statespace representation of the system dynamics. IFORSalgorithm integrates the frequency domain observabirange space extraction~FORSE! identification algorithm,the balanced realization~BR! model reduction algorithm,and the logarithmic and additive least squares~LLS! modalparameter estimation algorithms for low-order highly accrate model identification.

The FORSE algorithm is a slight variation of the suspace identification algorithm derived by De Moor et al19

and its objective is to minimize the following cost to obtathe system matrices,A, B, C, D,

J5 (k51

K

iG~vk!2@C~exp~ j vkDt !I 2A!#21B1D)i22, ~23!

whereG(vk) is the frequency response samples from eperiments.The BR model reduction20 algorithm transforms a statspace model to a balanced coordinate and reducesmodel by deleting the states associated with the smabalanced singular values. The LLS estimation algorithimprove the fitting of reduced models to experimental dby updating state space parameters in modal coordinat

From the experimental transfer function measuremea 0.01- to 40-Hz chirp signal generated the sampled tranfunctions in Fig. 14~solid line!. Using the FORSE algo-rithm, an 80th-order state-space model was selected toproximate the experimental data and then BR and LLS


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gorithms were used to get a 60th-order IFORSELS mo~shown by dotted lines in Fig. 14!. The low-frequencypeaks below 0.1 Hz is the pendulum mode of ARGOwhile the first lightly damped structural mode occurs at 89 Hz. Based on the linear model obtained, the pointingtracking controller was successfully designed and impmented performing a large angle slew maneuver.21 Thecontroller calculates the error quaternions22 from the rategyroscope, the electronic compass/inclinometer andviewfinder centroiding CCD mixing the mechanical senswith the optical sensor~the viewfinder CCD in Figs. 1 and13!.

4 Conclusion

The ARGOS is successfully integrated into the full struture ready to operate. The ARGOS testbed is the fisparse-aperture array simulating a space-borne observin a 1-g environment representing real-world problemsuch as the vibrational coupling between a spacecraft stture and the wavefront errors propagating throughwhole system. In this paper, the physical nature of intferometric arrays was studied and the design trades to mthe performance requirements were explored. A preliminassessment showed that the beam combining problem imost challenging aspect of sparse optical arrays. The nfor optical control is paramount due to tight beam combing error tolerances. The attitude controller design in tpaper is focused on the attitude control system incorpoing the coarse optical sensor and general ACS sensor sThe interferometric beam-combining controller utilizinmodel-based control and neural network is under devement as well as software-based wavefront sensing teniques. The wavefront sensing/control requirements appto be a major technology and cost driver. We anticipate tARGOS will provide an experimental platform for the dvelopment of many phasing and vibration control tecniques for future interferometers. Follow-on work will focus on actual versus expected imaging performance frange of targets.

Acknowledgments

This research was supported by the National Reconnsance Office~NRO! Director’s Innovation Initiative~DII !under contracts number NRO-000-01-C-0207. The authgratefully acknowledge the contributions of Dr. Carl Blarock at MideTechnology Corporation and Dr. Alice K. Liuat the NASA Goddard Space Flight Center.

References

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2. P. R. Lawson et al.,Principles of Long Baseline Stellar Interferometry, Course Notes from the 1999 Michelson Summer School, PLawson, Ed., NASA Jet Propulsion Laboratory~2000!.

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7. C. R. De Hainaut et al., ‘‘Wide field performance of a phased artelescope,’’Opt. Eng.34~3!, 876–880~1995!.

8. V. Zarifis et al., ‘‘The multi aperture imaging array,’’ in ASP ConSeries 194,Working on the Fringe: Optical and IR Interferometrfrom Ground and Space, Unwin and Stachnik, Eds., pp. 278~1999!.

9. D. C. Redding et al., ‘‘Wavefront sensing and control for a NeGeneration Space Telescope,’’Proc. SPIE3356, p. 758–772~1998!.

10. D. W. Miller et al., ‘‘The middeck active control experimen~MACE!: summary report,’’ Space Engineering Center, MISERC#7–96~1996!.

11. G. J. W. Mallory, A. Saenz-Otero, and D. W. Miller, ‘‘Origins testbecapturing the dynamics and control of future space-based telescopOpt. Eng.39~6!, 1665–1676~2000!.

12. S.-J. Chung, ‘‘Design, implementation and control of a sparse apeimaging satellite,’’ MS Thesis, Department of Aeronautics and Astnautics, MIT~2002!.

13. J. E. Harvey and R. A. Rockwell, ‘‘Performance characteristicsphased array and thinned aperture optical telescopes,’’Opt. Eng.27~9!, 762–768~1988!.

14. B. Mennesson and J. M. Mariotti, ‘‘Array configurations for a spainfrared nulling interferometer dedicated to the search for earthextrasolar planets,’’Icarus 128, 202–212~1997!.

15. W. J. Smith,Modern Optical Engineering, 3rd ed., McGraw-Hill,New York ~2000!.

16. J. R. Fienup, ‘‘MTF and integration time versus fill factor for sparsaperture imaging systems,’’ inImaging Technology and Telescope,Proc. SPIE4091A-06, 43–47~2000!.

17. J. R. Wertz and W. J. Larson,Space Mission Analysis and Design, 3rded., Microcosm Press, El Segundo, CA~1999!.

18. K. Liu, R. N. Jacques, and D. W. Miller, ‘‘Frequency domain strutural system identification by observability range space extractioASME J. Dyn. Syst. Meas. Control118~2!, 211–220~1996!.

19. B. De Moor et al., ‘‘A geometrical approach for the identificationstate space models with singular value decomposition,’’ inProc.,IEEE ICASSP,vol. 4, pp. 2244–2247, New York, NY~1998!.

20. K. Zhou, J. C. Doyle, and K. Glover,Robust and Optimal Control,Prentice Hall, Upper Saddle River, NJ~1996!.

21. S.-J. Chung, D. Lobosco, D. W. Miller, and C. Blaurock, ‘‘Multidisciplinary control of a sparse interferometric array satellite test-bein Proc. AIAA Guidance, Navigation and Control Conf., Austin, TX~2003!.

22. J. L. Marins, X. Yun, E. R. Bachman, R. B. McGhee, and M. J. Zy‘‘An extended Kalman filter for quaternion-based orientation estimtion using MARG sensors,’’ inProc. 2001 IEEE/RSJ Int. Conf. onIntelligent Robots and Systems, Maui, HI ~2001!.

Soon-Jo Chung received his BS degree inaerospace engineering in 1998 from theKorea Advanced Institute of Science andTechnology (KAIST) and his MS degree inaeronautics and astronautics in 2002 fromthe Massachusetts Institute of Technology(MIT), where he is currently a PhD candi-date. During 2002, he was a research en-gineer for the Giant Segmented MirrorTelescope project in the National OpticalAstronomical Observatories (NOAO), Tuc-

son, Arizona. His research focuses on the control and estimation oflarge complex space telescopes and interferometers.

David W. Miller is an associate professorand the director of the Space SystemsLaboratory in the Department of Aeronau-tics and Astronautics at the MassachusettsInstitute of Technology (MIT). His researchfocus is on dynamics, controls, and sys-tems engineering as applied to distributedsatellite systems and precision optical tele-scopes. He has developed a series ofground testbed and Shuttle flight facilitiesfor the conduct of research into dynamic

modeling and control synthesis for precision optical systems envi-sioned under the National Aeronautics and Space Administration’s(NASA) Origins Program. He is also working with the Air Force onsystem design, aperture synthesis, and formation flying for a space-based synthetic aperture radar concept. He is a member of theNASA Terrestrial Planet Finder (TPF) science working group.


Olivier L. de Weck is the Robert N. NoyceAssistant Professor of Aeronautics Astro-nautics Engineering Systems at the Mas-sachusetts Institute of Technology (MIT).His research interests are multidisciplinarydesign optimization (MDO) and system ar-chitecture. He earned his SM degree in1999 and his PhD degree in 2001 from MITand a diplomingenieur degree in industrialengineering in 1993 from ETH Zurich, Swit-zerland. He was the Engineering Program

Manager for the Swiss F/A-18 Program at McDonnell Douglas, St.Louis, Missouri, from 1993 to 1997.


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