Active Optics for Large Segmented Mirrors: Scale Effects

Transcription

1 Active Optics for Large Segmented Mirrors: Scale Effects A. Preumont U.L.B., Active Structures Lab., Brussels, Belgium R. Bastaits U.L.B., Active Structures Lab., Brussels, Belgium G. Rodrigues U.L.B., Active Structures Lab., Brussels, Belgium ABSTRACT: This paper is concerned with the extrapolation of the active optics of current 10-meter class telescopes (Keck, VLT) to the next generation of 30 m to 40 m Extremely Large Telescopes (ELT). Using the scaling laws for the structural response and the control requirements, the paper shows that the current baseline design of ELT is likely to bring a strong control-structure interaction, which could deteriorate significantly the image quality. Two options are discussed to alleviate this situation: (1) building the support structure in a material with high specific modulus, like for example carbon reinforced composites, and (2) enhancing the structural damping, possibly by active means. This discussion is intended to be generic rather than targeted to a specific telescope. 1 INTRODUCTION Figure 1 shows the primary mirrors (M1) of the largest optical telescopes [1], the existing ones: Hubble Space Telescope (HST), ESO's VLT at Paranal and Keck in Hawaï, and the future ones under design: The James Webb Space Telescope (JWST) and the two Extremely Large Telescopes (ELT), the American TMT [2] and the European E-ELT [3], due to be built within the next decade. All future large telescopes will be segmented. Note that the size of the earthbased telescopes is one order of magnitude larger than the space telescopes. Note also that there seems to be a huge gap between the largest existing segmented telescope (Keck) and the future ones. The gap is so big that one can reasonably wonder if the past experience with Keck is sufficient to warrant a sound design and optimum operation of the future ELTs. Table 1 gives more data on Keck and the future E-ELT. Figure 2 shows a view of the primary mirror of E- ELT [4]; it consists of 984 aspherical segments, each of them equipped with 6 edge sensors and 3 two-stage position actuators. As the size of the telescopes increases, they become increasingly sensitive to external disturbances such as thermal gradients, gravity and wind, and also to internal disturbances from support equipments such as pumps, cryocoolers, fans, etc... As a result, their shape stability relies more and more on active control means: the control system involves larger loop gains, and therefore a significantly larger bandwidth. At the same time, the natural frequency of future ELTs is expected to be substantially lower than operating telescopes (Table 1). So far, the classical method for minimizing control-structure interaction relies on having a wide separation between the lowest frequency resonance and the control bandwidth [5,6]. However, the joint effect of increasing the control bandwidth and reducing the natural frequency of the structure, and the very low inherent damping of welded steel structures, poses unprecedented challenges

2 Figure 1: Primary mirrors (M1) of current and future optical and infrared telescopes. Table 1: Keck vs. E-ELT Keck E-ELT φ M1 11 m 42 m φ segments 1.8 m 1.4 m Collecting Area 76m m 2 # Segments (N) # Actuators # Edge Sensors f segment (+ Whiffle Tree) 30 Hz ~ 60 Hz f 1 (M 1 ) ~ 10 Hz ~ 2.5 Hz f 2 (M 2 ) ~ 5 Hz ~ 1-2 Hz Adaptive Optics #d.o.f. (S 0.5) ~ 1000 ~ to the design of new giant telescopes and calls for innovative ways to alleviate control-structure interaction and avoid control instability. This paper is organized as follows: section 2 briefly describes the control architecture of large telescopes and large segmented primary mirrors; section 3 illustrates the influence of the structural vibration on the optical performances of the telescope; section 4 discusses some scaling laws for truss supported segmented mirrors; section 5 compares the gravity compensation systems of VLT and E-ELT in relation with the control-structure interaction; two avenues for improvement are suggested: (1) building the structure in a material with high specific modulus and (2) increasing the structural damping. Section 6 summarizes the results and gives some conclusions.

3 Figure 2: Primary mirrors (M1) of the future E-ELT telescope; it consists of 984 segments, each of them equipped with 6 edge sensors and 3 two-stage position actuators 2 CONTROL ARCHITECTURE Figure 3 describes the temporal and spatial frequency distribution of the various layers of the control system involved in the wavefront correction of a large telescope [8]; the spatial frequency is expressed in terms of Zernike modes. The amplitudes involved in the adaptive optics is generally small, typically a few microns. Our discussion is focused on M1; the amplitudes to be corrected by the active optics are typically several hundred microns [7]. The co-phasing strategy of segmented mirrors is illustrated in Fig.4; m is the mass of the segment, k and c refer to the stiffness and damping of the whiffle tree, m a, k a, c a and F a describe the position actuators (free actuator displacement: a=f a /k a ). The resonance frequencies of the supporting truss are f i. For E-ELT, the local modes of the segments are one order of magnitude larger than the first mode of the supporting structure (Table 1). Every segment is provided with three position actuators and six edge sensors measuring the displacement of the segment with respect to its six neighbors. The quasi-static relationship between the actuator displacements a and the edge sensor output y is y = J a (1) where J is the Jacobian of the segmented mirror. The pseudo-inverse of the Jacobian J + is best obtained by singular value decomposition (SVD): J = U Σ V T (2) where the column of U are the orthonormalized sensor modes, the column of V are the orthonormalized actuator modes, and Σ contains the singular values on its diagonal. The control system works according to Fig.5, where a set of filters H(s) provide (hopefully) adequate disturbance rejection and stability margins. Note that piston, tilt and defocus are unobservable from the edge sensors and must be taken care of by other sensors.

4 Figure 3: Temporal and spatial frequency distribution of the various control layers of a large telescope (adapted from [8]). Figure 4: Co-phasing strategy of segmented mirrors. Every segment is equipped with 3 position actuators. Figure 5: Block diagram of the co-phasing control system.

5 3 DEFLECTION OF THE SUPPORTING TRUSS The global deformations of the supporting truss, either static due to gravity, or the lowest vibration modes, introduce optical aberrations. The details of these aberrations depend on the exact boundary conditions of the supporting truss, but they tend to be dominantly in the lowest optical modes such as defocus and astigmatism; precisely those which are not, or only weakly observable from the edge sensors [they correspond to the lowest singular values in the decomposition (2)]. This is illustrated in Fig.6 on an hypothetical free flying truss structure supporting segmented mirrors (space telescope). For this free-free boundary conditions, the first vibration mode is mostly astigmatism. The figure also shows the Point Spread Function (PSF) corresponding to the largest circular aperture inside the mirror, resulting from a vibration amplitude of λ/2 peak to valley (250 nm in this case). In order to be able to compensate for the deflection of the supporting truss, the set of edge sensors must be supplemented by a Shack-Hartman sensor (or another) measuring the normal to the segment (one normal by segment). In fact, the Shack-Hartman array does not measure the normal to the segments, but rather the normal to the wavefront, in which the atmospheric turbulence appears as noise, and must be filtered out as well as possible. This leads to the control architecture of Fig.7. Figure 6: First vibration mode of a free flying truss supporting a flat segmented mirror. The diameter of the segments is d = 2 m. The first mode is essentially astigmatism. The figure also shows the PSF corresponding to the largest circular aperture inside the mirror, when it is flat and when the vibration amplitude is λ/2 (peak to valley).

6 Figure 7: Active optics control flow for large segmented mirrors. 4 SCALING LAWS This section discusses some scaling laws for truss supported segmented mirrors. 4.1 Static deflection under gravity A spring mass system subjected to gravity (Fig.8) undergoes a deflection = Mg/K = g/ω 1 2. More generally, for given boundary conditions, the gravity-induced deflection of a truss structure scales according to f 1-2 (3) where f 1 is the lowest natural frequency of the structure. Referring to Table 1, this means that the primary mirror of E-ELT will undergo gravity disturbances 16 times larger than Keck; the control gains will have to be increased accordingly. Figure 8: Static deflection under gravity.

7 4.2 Wind response of M1 Figure 9: (a) Turbulent wind spectrum. (b) Structural response to turbulent wind. Depending on the wind flow distribution and the mirror orientation, large segmented mirrors may act as lifting surfaces (i.e. producing lift and drag forces proportional to the square of the mean wind), or respond like a bluff body (i.e. with drag forces proportional to the turbulent velocity), e.g. see [9]. The reality is a mix of the two. Here, we examine the scaling law for the turbulent response of a bluff body, with the classical assumption that the turbulent wind velocity is small compared to the mean wind, and is distributed according to Davenport's spectrum [10], Fig.9(a). In all cases, the first mode of the supporting truss is located in the tail of the wind spectrum, where the power spectral density Φ(f) behaves according to Φ(f) ~ f -5/3 (4) Because the turbulent wind is small compared to the mean wind, the turbulent forces are essentially proportional to the turbulent wind velocity and the spectral density of the structural response has a shape similar to that of Fig.9(b): at low frequencies, the quasi-static response has the same shape as the turbulent wind and there is a resonant response in the vicinity of the natural frequencies of the structure. Due to the decaying shape of (4), the structural response is often dominated by the first mode. The mean square (MS) value of the resonant response to the wind force applied to one segment may be estimated assuming that the segment is subjected to a white noise of intensity Φ(f 1 ) equal to the value of the spectral density of the excitation evaluated at the natural frequency f 1 of the structure. According to random vibration theory (e.g. see [11]), the MS response is σ 2 ~ Φ(f 1 ) / ξ f 1 3 (5) where ξ is the damping ratio of the structural mode. Combining with (4), one finds σ 2 ~ 1 / ξ f 1 14/3 (6) Finally, since the correlation length of the wind turbulence is of the same order of magnitude as the segment size, the forces acting on the various segments can be considered as statistically independent, leading to a global response σ 2 ~ N / ξ f 1 14/3 (7) where N is the number of segments. Using the data of Table 1, for identical wind conditions and assuming the same damping ratio, the RMS resonant response of E-ELT may be expected to be magnified by (984/36) 1/2. (2.5/10) -7/3 ~ 130. Equ.(7) also shows the role of damping in the attenuation of the resonant response.

8 4.3 First natural frequency Figure 10: Geometry of the truss supported reflector. The foregoing sections have shown the central role played by the first natural frequency in the static deflection as well as the wind response of the segmented primary mirror. It is interesting to investigate the scaling law for the first natural frequency of a truss supported segmented reflector. According to [12,13], the first natural frequency of a free flying truss follows η D D h. E (8) f 1 ~ ( ) ρ where η is the structural mass fraction Truss Mass η = (9) Truss Mass + Reflector Mass (using lighter reflectors increases η). In Equ.(8), h and D are respectively the thickness and the diameter of the supporting truss (Fig.10); E is the Young modulus and ρ the material density of the truss. The coefficient refers to the free-free boundary conditions, but this result also applies to other boundary conditions with another coefficient. This formula shows clearly the advantage of building the supporting truss with a material of high specific modulus E/ ρ. Table 2 compares the mechanical properties of traditional structural materials for telescope structures (steel and aluminum) with carbon fiber reinforced composites (CFRP) [14]. Observe that the latter have a specific modulus 4 times larger than either steel or aluminum, which doubles the natural frequency f 1 if everything else is equal. As a side effect, the outstanding thermal stability of CFRP is worth noting; the thermal expansion coefficient α of CFRP given in Table 2 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 optics control system. E (Gpa) ρ (g/cm³) E / ρ α (10-6 C -1 ) Steel Aluminium CFRP 180 / / / / 0.1 Table 2: Mechanical and thermal expansion properties of Steel, Aluminium and carbon fiber reinforced plastics (CFRP).

9 5 CONTROL-STRUCTURE INTERACTION At present, the active optics controllers do not include a dynamic model of the support structure; the control system assumes the supporting truss rigid. In practice, however, the structural vibrations and the control system interact, and a strong interaction may lead to control instability. The potential for control-structure interaction is measured by the frequency gap between the controller bandwidth and the first flexible mode of the structure; for lightly damped structures, a frequency gap larger than one decade is often necessary to avoid control-structure interaction. Taking the example of gravity compensation, Figure 11 compares the active optics of VLT (which works very well) 1 with that of the future E-ELT. The bandwidth of the control system of VLT is f c = 0.03 Hz [15] and the first flexible mode of the structure is f 1 = 10 Hz; thus, the frequency gap is f 1 / f c 300 (which, of course, eliminates any possibility of control-structure interaction). Figure 11: Comparison of the gravity compensation of VLT and ELT. According to control theory, the disturbance rejection is governed by y d = GH (10) where GH is the open-loop transfer function of the control system (e.g.[17]). Assuming an average decay rate of -20 db/decade in the vicinity of crossover 2, a bandwidth of f c =0.03 Hz leads to an open-loop amplitude of 68 db at the earth rotation frequency ( Hz). According to the foregoing formula, this reduces a static deformation of 110 µm to 40 nm. If one assumes that the static deflections of E-ELT are 16 times larger (as we have seen above in the comparison with Keck which is about the same size as VLT), achieving the same accuracy 1 The control approach for the active optics of VLT is based on [16]. 2 This gives a phase margin of 90 ; note, however, that the following discussion is to a larger extent independent of the assumed decay rate.

10 on the controlled shape will require that the disturbance rejection, and therefore GH be 16 times larger (+24 db); with the same decay rate, this increases the bandwidth from f c =0.03 Hz to f c =0.46 Hz. At the same time, the first natural frequency of the support structure has been reduced from f 1 = 10 Hz for VLT to f Hz for E-ELT, reducing the frequency ratio to f 1 / f c =5, well under the limit where control-structure interaction is expected. Figure 12: Position control of a two-mass system. k w refers to the whiffle tree and k s to the support structure. Model used to study the control-structure interaction by reducing k s. The danger of ignoring the control-structure interaction in the controller design can be illustrated with the example of Fig.12, which is a simplified model of Fig.7. m is the mass of a segment and the subscripts w and s refer to the whiffle tree and the support structure, respectively; the damping ratio is fixed to ξ = 0.01 for all modes. The control objective is to keep the segment position x fixed in spite of the low frequency disturbance d applied to it. We begin with a fairly stiff structure (large k s ) and the following controller (integral plus low-pass filter): g H( s) = s (11) τ s In the initial design, the controller parameters g and τ are adjusted to obtained the open-loop frequency response function GH represented in full line in Fig.13, corresponding closely to Fig.11; the peak near 100 Hz corresponds to the resonance of the whiffle tree [(k w / m) 1/2 ]. The stability margins are comfortable: 33 db of gain margin (GM) and 90 of phase margin (PM). Next, the stiffness k s of the support structure is gradually reduced (the damping constant is adjusted simultaneously to keep the modal damping equal to ξ = 0.01) and the corresponding frequency response functions are also represented (in dotted lines) in Fig.13 for various values of the natural frequency of the support structure, respectively f 1 =32 Hz and 2.5 Hz. One sees that the stability margin decreases when f 1 decreases, and when f 1 approaches the bandwidth f C, the control system becomes unstable.

11 Figure 13: Position control of a two-mass system. Open loop transfer function for various frequencies of the supporting structure, respectively f 1 = 422 Hz (rigid), 32 Hz and 2.5 Hz. Figure 14 shows the evolution of the gain margin as a function of the frequency ratio f 1 / f c, for various values of the damping; one sees that, for ξ = 0.01, the stability limit f 1 / f c is significantly larger than 10. This graph shows clearly the benefit of bringing additional damping to the vibration of the supporting structure. However, passive devices like Dynamic Vibration Absorbers (DVA) [18] may be difficult to tune, because of the change of modal properties with the elevation angle of the telescope. Figure 14: Position control of a two-mass system. Evolution of the gain margin with the frequency ratio f 1 / f c for various values of the structural damping ξ.

12 6 SUMMARY AND CONCLUSION This study has been devoted to the control-structure interaction in the active optics of giant segmented mirrors. After a review of the intended control strategy and a discussion of the influence of the deflection of the supporting truss on the optical performance, the paper discusses the scaling laws for the static deflection and the turbulent response to the wind. It has been shown that the magnitude of the structural response is controlled by the first natural frequency of the structure, f 1 which should be kept as large as possible. The paper also analyzes the dependence of f 1 with respect to the geometry and the material properties of the supporting truss. It is shown that using CFRP instead of steel or aluminum would double the natural frequency of the structure. A comparison of the gravity compensation control system of VLT and E-ELT has been conducted and, assuming identical disturbance rejection performance, it has been shown that the current baseline design of E-ELT is likely to be prone to control-structure interaction. The analysis suggests that this can be improved by using a composite structure which would increase the frequency ratio f 1 / f C between the structural frequency and the control bandwidth. Additionally, the thermal stability of composites could be exploited. Finally, the analysis shows that increasing the structural damping is beneficial, by reducing the dynamic response to the wind and also by reducing the control-structure interaction (increase of the gain margin). It follows that various options for damping enhancement should be considered seriously, including active damping of the supporting truss with active struts or a cable network [19]. 7 ACKNOWLEDGEMENTS The authors wish to thank FCT (Portugal) and FRIA (Belgium) for supporting the two junior authors. 8 REFERENCES [1] Wilson R. N., Reflecting Telescope Optics I and II, Springer-Verlag, [2] Nelson, J., Sanders G., The status of the Thirty Meter Telescope project, In Ground-based and Airborne Telescopes II - SPIE 7012 (2008), Larry M. Stepp, Ed. [3] Gilmozzi, R., Spyromilio, J., The 42m European ELT: status, In Ground-based and Airborne Telescopes II SPIE 7012 (2008), Larry M. Stepp, Ed. [4] Dimmler, M. E-ELT Programme: M1 Control Strategies, E-TRE-ESO , Issue 1, April [5] Aubrun, J.N., Lorell, K.R., Mast, T.S. & Nelson, J.E., Dynamic Analysis of the Actively Controlled Segmented Mirror of the W.M. Keck Ten-Meter Telescope, IEEE Control Systems Magazine, 3-10, December [6] Aubrun, J.N., Lorell, K.R., Havas & T.W., Henninger, W.C., Performance Analysis of the Segment Alignment Control System for the Ten-Meter Telescope, Automatica, Vol.24, No 4, , [7] European Southern Observatory, The VLT White Book, ESO, [8] Angeli, G.Z., Cho, M.K., Whorton, M.S., Active optics and architecture for a giant segmented mirror telescope, in Future Giant Telescopes (Angel, and Gilmozzi, eds.), Proc. SPIE 4840, Paper No , pages [9] Scanlan, R.H. & Simiu, E., Wind Effects on Structures, Wiley, [10] Davenport, A.G., The application of statistical concepts to the wind loading of structures, Proc. Inst. Civ. Eng., Vol.19, , Aug [11] Crandall, S.H. & Mark, W.D., Random Vibration in Mechanical Systems, Academic Press, [12] Lake, M.S., Peterson, L.D. & Levine, M.B., Rationale for Defining Structural Requirements for Large Space Telescopes, AIAA Journal of Spacecraft and Rockets, Vol.39, No 5, September-October, 2002.

13 [13] Lake, M.S., Peterson, L.D. & Mikulas, M.M., Space Structures on the Back of an envelope: John Hedgepeth's Design Approach, AIAA Journal of Spacecraft and Rockets, Vol.43, No 6, November- December, [14] Agarwal, B.D. & Broutman, L.J., Analysis and Performance of Fiber Composites, Wiley, 2nd Ed., [15] Bely, P.Y. (Editor), The Design and Construction of Large Optical Telescopes, Springer, 2003, p [16] Wilson, R., N., Franza, F., and Noethe, L., Active Optics I: A system for optimizing the optical quality and reducing the costs of large telescopes, Journal of Modern Optics, 34, 4 (1987), [17] Franklin, G.F., Powell, J.D., Emani-Naemi, A., Feedback Control of Dynamic Systems, Addison- Wesley, [18] Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill, [19] Preumont, A., Vibration Control of Active Structures, An Introduction, 2nd Edition, Kluwer, 2002.