Mem. S.A.It. Suppl. Vol. 9, 475c© SAIt 2006

Memorie della

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Integrated modeling in an active optics system

P. Schipani and F. Perrotta

Istituto Nazionale di Astrofisica – Osservatorio Astronomico di Capodimonte, SalitaMoiariello 16, I-80131 Napoli, Italy e-mail: schipani@na.astro.it

Abstract. In all the modern telescopes (medium, large, extremely large) the active opticsis one of the most delicate systems. No assessed simulation tool exists, one of the reasonis that the approach can be different for segmented or monolithic, medium or large mirrors,and so each telescope design staff usually develops its own simulation system. In this paperan integrated modeling approach is proposed, combining finite element analysis, dynamicalsystems simulation methods and optical performance analysis.

Key words. Integrated modeling - Active Optics - Finite Element Analysis - Simulation -Software

1. Introduction

The integrated modeling is emerging as afundamental strategy for the telescopes de-sign (e.g. Angeli et al. (2004), Roberts et al.(2003)). In the framework of the VST (VLTSurvey Telescope) project several simulationshave been carried out to analyze the active op-tics system behavior. Some of the main taskswere the calculation of the deformation im-posed to the primary mirror applying a set ofcalibrated correction forces, the simulation ofthe dynamic response of the mirror (applyingas input to the axial actuators a set of forcesor a wind disturbance) the prediction of theeffects of the gravity at different altitude an-gles. Many software tools were needed: a FEAtool (like Ansys or Nastran), a computationaland simulation environment (e.g. Matlab +Simulink), a ray-tracing software (like Zemaxor Code V). In the end the work has naturallyconverged towards an integrated modeling en-vironment in which the computational envi-

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ronment is the core block. The idea is to useFEA to produce a set of differential equationsrepresenting a nodal system. Then the compu-tational and simulation environment convertsthem to a modal representation and then to a re-duced order space-state model, easy to be ana-lyzed with the traditional control system theoryand tools. Finally the ray-tracing software cangive information on the impact of the deforma-tions of the primary mirror on the performanceof the whole telescope optical system.

2. Primary mirror axial control

The mirror shape is controlled by means of aset of axial actuators. The actuators must bothsupport the weight of the mirror and compen-sate for undesired aberrations of the opticalsurface of the mirror. In order to simulate thedynamic behavior of the mirror it is necessary amathematic model. The starting point is a FEAof the mirror, which produces the input data forall the possible representations (Fig.1).

476 Schipani: Integrated modeling in an active optics system

Fig. 1. 3D FEA mirror model and actuator loca-tions

3. Mirror models

The mechanical structure of a telescope mir-ror can be analyzed through three different de-scriptions representing increasing levels of ab-straction.

3.1. Nodal representation

The nodal representation can be considered asthe physical system, although it already hassome level of approximation and abstractionthrough the choice of nodes. The associatedcoordinate system is defined through the dis-placements and velocities of physical nodes.The nodal model is a set of second order dif-ferential equations characterized by the mass(M), damping (D), and stiffness (K) matrices,the initial and boundary conditions for nodaldisplacements (q) and velocities, and the sen-sor outputs (y). The mass and stiffness ma-trices come from FEA analysis. Damping isnot present in the finite element model butcan be introduced modally to the state-spacemodel. B0, C0q and C0v are the input, output

displacement, and output velocity matrices, re-spectively. The length of q is Nd, the numberof degrees of freedom (DoF) of the system.

M••q +D

•q +Kq = B0u

y = C0qq + C0v•q

3.2. Modal representation

Unlike the nodal one, the modal coordinatesystem is defined through the displacementsand velocities of structural modes. Thesemodes can still be visualized as special shapesof the structure, but they are related rather tothe whole system than to individual parts. Theadvantage of the modal representation is that itdecouples the system differential equations. Soafter the derivation of the nodal model throughFEA analysis, the next step is to convert it ina modal one, decoupling the equations of themodel. This methodology is summarized in thefollowing steps (Hatch (2001)):

- solve the undamped eigenvalue problemwhich identifies the resonant frequencies andmode shapes (eigenvalues and eigenvectors);

- use the eigenvectors to uncouple or diag-onalize the original set of coupled equations,leading to N uncoupled single degree of free-dom problems instead of a single set of N cou-pled equations;

- calculate the contribution of each modeto the overall response; this allow to reducethe size of the problem by eliminating modesthat give little contribution to the system (e.g.high frequency modes have generally a littleeffect at lower frequencies and could be re-moved from the model, accepting some ap-proximation).

The coordinate transformation that decou-ples the second order differential equations ofthe nodal model is:

q = Φqm

where Φ is the eigenvector matrix of the sys-tem containing the eigenvectors as columns.Also the modal representation is a set of sec-ond order differential equations.

Schipani: Integrated modeling in an active optics system 477

Fig. 2. Elastic mode: Symmetry 2, Order 1, dis-placements normal to mirror plane

3.3. Space-state representation

The state-space representation is a first orderset of differential equations obtained describ-ing the system by defining a state vector con-taining not just the displacements but also thevelocities. The state-space model is commonlyused in control engineering and expresses thebasic control characteristics, the inputs, theoutputs, and the dynamics of the system in astandard way. After FEA has produced a nodalrepresentation, and after the conversion to amodal model, the final step is to go to a space-state representation. A space-state representa-tion of a LTI (Linear Time-Invariant) system isa first-order matrix differential equation withconstant coefficients:•x = Ax + Buy = Cx

where u is the input and y is the output vec-tor of the system, and the state of the systemis characterized by the state variable x. A isthe system dynamics matrix. To obtain a statespace representation from the modal represen-

tation, a state vector including displacementsand their speeds must be introduced:

x =

[x1x2

]=

[qm•

qm

]

4. Optical interfaces

Many optical routines have been writtenin Matlab to study the mirror aberrations.Assessed ray-tracing tools exist but they intrin-sically do not deal with the mechanical prop-erties of the mirrors but just with the opti-cal properties. So it is possible to describe theaberrations of an optical system using the stan-dard (or fringe) Zernike polynomials, but notto fit the aberrations to a set of natural vi-bration modes like the one in Fig.2 (Noethe(1991)). Since in the VST case the elasticmodes are used to fit the wavefront and correctthe mirror shape, an interface Matlab-Zemaxhas been written to generate mirror shapesthrough grid sags, in order to analyze the ef-fect on the overall optical system of the defor-mations of the primary mirror at different alti-tude angles obtained by FEA in terms of elas-tic modes. Custom software has been neededto add missing capabilities to conventional ray-tracing, and to implement a bridge betweenFEA and ray-tracing.

References

Angeli, G. et al. 2004, Proceedings of SPIE,5178, 49

Hatch, M. 2001, Vibration simulation usingMatlab and Ansys, Chapman and Hall/CRC

Noethe, L. 1991, Journal Of Modern Optics,38, 1043

Roberts, S. et al. 2003, Proceedings of SPIE,5382, 346