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    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm \zihao{6}\songti suppl.\hfill R. Nan et al.: Adaptive cable-mesh
    reflector for the FAST  %%（作者：题目名称）
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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf Adaptive Cable-mesh Reflector for the FAST    %%（文题）
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\vs\vs{\normalsize Rendong Nan\up{1},\hs\hs Gexue Ren\up{2},\hs\hs
Wenbai Zhu\up{1}\hs\&\hs Yingjie Lu\up{2}}  %%（本文作者）

\vs
(\footnotesize 1~\nao) \\%%（本文作者单位）
 ({\footnotesize 2~{\it Department of Engineering Mechanics, Tsinghua
 University, Beijing}~ 100084})

\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
The adaptive cable-mesh reflector is an Arecibo-type reflector
with the additional capability to adjust the illuminated area to
activate the antenna surface. The key knowledge in this newly
proposed engineering concept is to use the equilibrium shape of
the cables under gravity, catenary, to form the parabola. The
adaptive control for the cable mesh reflector is to allocate the
nodal points of the mesh to appropriate position on the
illuminated parabola so that the parabola is formed within given
precision. To test the feasibility of this idea, a finite element
software is developed to simulate the concept. The preliminary
results are encouraging.

\vs\vs\no {\bf Key words:}~~Radio telescope, Spherical reflector

\end{quotation}

\para{\large\bf{1}~~ Brief Introduction to FAST}

\vs\vs China has embarked on a project to build the world's
largest radio telescope FAST, Five hundred-meter Aperture
Spherical Telescope, in a karst depression in southwest of the
country (Nan,1996). The FAST is an Arecibo-type telescope with a
number of innovations. The optical geometry and 3-D computer image
of FAST are shown in Figure 1. The main reflector is a spherical
cap with a radius of $\sim $300 m and an opening up to 500 m in
diameter. The effective aperture of $\sim $300 m is illuminated by
the feed moving on the focus surface halfway from reflector to its
center. The telescope is pointed by moving the focus cabin and
simultaneously adjusting the shape of the illuminated area. The
geometrical configuration together with the offset illumination at
the edge of the reflector enables FAST to observe at larger zenith
angle. FAST is expected to continuously cover frequency range
200-2000MHz, with capability up to C-band or even X-band.

\begin{figure}[h]
\center {\includegraphics [bb=14 14 828 411,scale=0.4]{c1.eps}}

   \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,1~~ The full view of the FAST. (a) The optic geometry of the FAST.
The parabola surface deformed is shown by the dashed line. (b)The
3-D computer image of the FAST.}

    \end{figure}

The active main reflector (Qiu 1998) corrects the spherical aberration on
the ground, which breaks the bandwidth and polarization limits without
involving a complex feeding system. A so-called integrated
optical-mechanical-electronic design (Duan 1996) for FAST feed support has
been intensively investigated to replace the heavy platform in air (see 3-D
image in Fig.1b). The feed cabin is supported and driven by sets of cables
and servomechanisms. Feasibility study on these critical technologies for
FAST as well as science case, site surveying, layout of the focus package,
complicated measurement and control has been selected as one of the key
projects and supported by the Chinese Academy of Sciences in 1999.

\para{\large\bf{2}~~ Summary on Scaled Model Experiments for FAST Main Reflector}

\vs\vs To deform the reflector, it is necessary to divide it into
small elements. Each element is a small part of a spherical
surface whose curvature optimized carefully for FAST is R=335 m,
which is slightly different from the one of the neutral curve of
the main reflector in Figure 1. One proposed way of dividing it is
to have $\sim $1800 hexagons as shown in Figure 2. Each element
has three actuators to fix its position and connect it with
adjacent ones, and there would be an average of one actuator per
element. As FAST is tracking and forming a real-time parabola, the
actuators move along the radial direction with a maximum throw of
67 cm.

\begin{figure}[h]
\center {\includegraphics [bb=31 133 397 468,scale=0.6]{c2.eps}}

   %\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,2~~ The FAST main reflector is divided into about 1800 hexagons}

    \end{figure}

Four pieces of elements at moderate altitude of the spherical cap
as marked in Figure 2 were selected for scale modeling. Figure 3
shows the fieldwork of the experiment while the last element was
being mounted. The up-to-date field bus technology, LonWorks, was
employed to serve as the actuator control network. This photo also
illustrates that future FAST main reflector will be a buildup of 4
layers -- surface and its back structure, self--adaptive
connector, actuator, and civil engineering in depression. This
model experiment basically has approved the feasibility of the
engineering concept of the FAST main reflector on the whole.

Many questions, as many as those positive results, were raised from the
experiment. Investigations addressed these questions have made notable
progresses. These include a new scheme of segmentation to largely reduce the
species number of hexagons for mass production, kinematics study on the
adaptive connector to modify its controllability, innovated tense grid
design for the reflector element to lighten it and to reduce bearing force
of the actuators at the tangent direction, test on reliability and
life-time, layout of the civil engineering structure in depression, and etc.
In this paper, a completely different realization for the active main
reflector of FAST is proposed based on our learning and understanding of the
FAST from prophase feasibility study.

\begin{figure}[h]
\center {\includegraphics [bb=14 14 828 518,scale=0.4]{c3.eps}}

   \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,3~~ The scaled model of the FAST active reflector.
      The marked numbers indicate the 4 layers of the main
      reflector construction: 1-surface and its back structure, 2-self-adaptive connector
      , 3-actuator, 4-civil engineering between actuators and depressions
      (here, experiment mount).}

    \end{figure}

\para{\large\bf{3}~~The Engineering Concept of Adaptive Cable-mesh Reflector}

\vs\vs As a piece of massless and ideally stretched rope is slightly
pulled away from the tense position at its middle, the total
length varies almost invisibly. The increment in length of
500-meter-long rope, for an example, is about 4 mm, less than
10$^{ - 5 }$of the total, while the central offset is 1 m as shown
in Figure 4. This hints us that an antenna surface supported by
cable net as Arecibo telescope could be activated in some extend
without extra-servos controlling the lengths of those supporting
ropes.

\begin{figure}[h]
\center {\includegraphics [bb=105 312 373 418,scale=1]{c4.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize

      Fig.\,4~~ The increment of long rope}

    \end{figure}

The central part of a spherical surface is very close to a
paraboloid of revolution when a proper focal length is chosen. The
focal length of the FAST could be accurately determined as
f=0.4665\,R according the formulae given by Li (1959), which
minimizes the peak deviation of the spherical surface from the
paraboloid of revolution across the $\sim $300\,m of illuminated
aperture. The integrated length of the dashed line (Fig.1a)
indicating the deformed parabola is only 0.36\,m shorter than the
spherical curve, about one in a thousand of the total length. This
small difference required by deforming could be easily achieved
within the elasticity of ordinary wire ropes, although the
mechanical analysis is much more intricate. Instead of elastic
deformation, adjusting the meshes outside the illustrated area can
also compensate this small difference in length.

\begin{figure}[h]
\center {\includegraphics [bb=14 14 828 399,scale=0.4]{c5.eps}}

  % \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,5~~ The cable-mesh reflector with control points and
      the control cable for the adaptive reflector.}

    \end{figure}

Ignoring the active control, the newly proposed FAST surface
adopts a similar structure of Arecibo telescope. The cable-mesh
consists of two sets of cables respectively in two orthogonal
directions, as shown in Figure 5. Above the cable network aluminum
plates, expanded and flattened metal mesh, or welded stainless
steel mesh are attached. The crossing-nodal points of the cable
mesh are used as the control points. The neutral equilibrium of
the reflector is taken as the optimized spherical cap. The full
3-D controllability around the equilibrium is assumed for the
control points. In practical implementation, three downward cables
are to be used to realize the control along the radial and tangent
direction, see figure 5b. Motors adjust these cables with feedback
from the ranging system for the control points. The proposed
ranging system will be consisted of 6 rangefinders which have been
equipped on the Green Bank Telescope(100\,m). We plan to have
$\sim $300 targets attached to the control points on the
cable-mesh reflector of FAST, and monitor positions of $\sim $140
targets located within illuminated area in observing time. For
individual target, the position data with an accuracy up to
100\,$\mu$m will be obtained by the ranging system with a flash
period of 15 seconds, during which the maximum movement of each
target is estimated as 3\,mm and the real time position could be
worked out by combining the discrete measurement and interpolation
based on kinematic and dynamic analysis.

\newpage
\vspace*{1mm}
\para{\large\bf{4}~~Simulating}

In order to test the feasibility of the concept, a finite element
software is developed for simulating. Grating is firstly made
across the aperture of 500\,m in diameter, following by vertical
projection on reflector considering that the equilibrium position
of a cable under gravity is in a vertical plane. Grating intervals
are 30\,m in this experiment, which configures two orthogonal sets
of cables and 305 control points on the spherical surface.
Diameter of cables is 3\,cm and typical characteristics of
ordinary wire ropes are supposed.

\begin{figure}[h]
\center {\includegraphics [bb=14 14 809 582,scale=0.4]{c6.eps}}

   \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,6~~ Preliminary simulation results with the conditions: Diameter
      of the cable: 0.03\,m; Horizontal distance between cables: 30\,m;
      Total number of control points: 305. (a)Elevation 0\dg, Azimuth 0\dg,
      actual control points: 137, RMS error: 0.0034. (b)Elevation 20\dg, Azimuth 0\dg,
      actual control points: 132, RMS error: 0.014\,m. (c)Elevation 15\dg, Azimuth 45\dg,
      actual control points: 135, RMS error: 0.008\,m. (d)Elevation 30\dg, Azimuth 45\dg,
      actual control points: 122, RMS error: 0.031\,m.}

    \end{figure}

The deformation is realized by reallocating those control points
within illuminated area and its vicinity from equilibrium
positions on spherical curve onto the parabola. The whole process
starts from the area center and radiates out to the whole area to
be deformed. There are two conditions to be held during the
interactive process --- minimizing the displacement of each nodal
point from its equilibrium position on spherical curve and
equalizing the distances on the two curves between the adjacent
points. The equilibrium shapes of cables under gravity are
calculated and the fit errors are estimated while the reallocation
process converges. Figure 6a through Figure 6d show the simulation
results at different elevation and azimuth angles. The RMS fit
error is 3.4\,mm as FAST points to the zenith (Fig.6a), which
meets the accuracy request at highest frequency for FAST. The
errors increase as the illuminated area goes to high altitudes,
reaching 30\,mm at the edge where control points are sparse and
the mechanical structure of the system becomes more asymmetric,
and the continuation of deformation loses at the up edge due the
lack of control points in the vicinity. These drawbacks could be
easily overcome by further optimization.

\newpage
\vspace*{1mm}
\para{\large\bf{5}~~Remarks}

\vs\vs The preliminary results from the simulating are
encouraging. If the feasibility of this concept, adaptive
cable-mesh reflector for FAST, is finally approved, it will
benefit future construction of the telescope in several aspects:
(1) Simplify the structure of FAST main reflector, reducing the
number of layers from 4 to 2 and the number of control points from
$\sim $2000 to $\sim $400; (2) Largely cut down the number of
moveable parts, e.g. bearings, screws and joints, easing machinery
work and improving reliability; (3) Need not to divide the surface
into solid elements with fixed curvature, relaxing appeals to
fabrication accuracy and leaving space for further telescope
upgrading; (4) Almost avoid the civil engineering between
actuators and ground. These possible advantages mentioned above
may be potentially beneficial to FAST in construction price, time
limit for the project, future operating reliability and telescope
maintenance.

The simulation shows that the adaptive cable-mesh reflector is feasible from
the point of mechanics. From the present understanding, better controls can
also be found to reduce the RMS errors of the illuminated parabola. Further
research on the mechanics and control of the adaptive cable-mesh reflector
is definitely required. Many questions, moreover, should be answered in
details, e.g. material and techniques of the surface, attachment between
surface and adaptive cable-mesh, ranging system of high accuracy and high
sampling rates. A scaled model is necessary for finally confirming the
feasibility of this new realization of the FAST reflector.

\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \nref Nan R D, Nie Y P, Peng B, et al. Site Surveying for the LT in
 Guizhou Province of China, In: Strom R, et al. ed. Proc of the
 LTWG-3 \& W-SRT,Beijing: International Academic Publishers, 1996.
 59-66
 \nref Qiu Y H, MNRAS, 1998, 301: 827-830
 \nref  Duan B.Y, Zhao Y.Z, Wang J.L, et al. Study of the Feed System
for a Large Radio Telescope from the Viewpoint of Mechanical and
Structural Engineering, In: Strom R, et al. ed. Proc of the LTWG-3
\& W-SRT. Beijing: International Academic Publishers, 1996 85-102
 \nref Li T Y, IEEE Trans. 1959, AP-7 No.3, 223-226


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