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\journal{Astroparticle Physics}

\begin{document}
\begin{frontmatter}

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\title{The drive system of the\\Major Atmospheric Gamma-ray Imaging Cherenkov Telescope}

\newcommand{\corref}{\thanksref}
\newcommand{\cortext}{\thanks}

%\tnotetext[t1]{This document is collaborative effort}
%\tnotetext[t2]{This document is collaborative effort more}

\author[tb]{T.~Bretz\corref{cor1}}
\address[tb]{Universit\"{a}t W\"{urzburg}, Am Hubland, D-97074 W\"{u}rzburg, Germany}
%\ead{tbretz@astro.uni-wuerzburg.de}
%\fntext[fn1]{My name is Thomas Bretz}

\author[tb]{D.~Dorner},\author[rw]{R.~M.~Wagner}
%\address[dd]{Integral Science Data Center}
%\ead{dorner@astro.uni-wuerzburg.de}


%\ead[rw]{robert.wagner@mpp.mpg.de}
\address[rw]{Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, D-80805 M\"{u}nchen, Germany}

\cortext[cor1]{Corresponding author: tbretz@astro.uni-wuerzburg.de}

%\input library.def

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\begin{abstract}
The MAGIC telescope is an imaging atmospheric Cherenkov telescope,
designed to observe very high energy gamma-rays achieving a low energy
threshold. One of the key science goals is the fast follow-up of the
enigmatic and short lived gamma-ray bursts. The drive system for the
telescope has to meet two basic demands: (1)~During normal
observations, the 64-ton telescope has to be repositioned accurately,
and has to track a given sky position with high precision at a typical
rotational speed in the order of one revolution per day. (2)~For
successfully observing GRB prompt emission and afterglows, it has to be
powerful enough to reposition the telescope to an arbitrary point on
the sky within a few ten seconds and commence normal tracking immediately
thereafter. To meet these requirements, the implementation and
realization of the drive system relies strongly on standard industry
components to ensure robustness and reliability. In this paper, we
describe the mechanical setup, the drive control and the calibration of
the pointing, as well as present measurements of the accuracy of the
system. We show that within the limits of the mount the drive system is
mechanically able to operate with an accuracy even better than the
feedback values from the axes. In the context of future projects,
envisaging telescope arrays comprising about 100 individual
instruments, the robustness and scalability of the concept is
emphasized. 
\end{abstract}

\begin{keyword}
MAGIC\sep drive system\sep IACT\sep scalability\sep calibration\sep fast repositioning
\end{keyword}

\end{frontmatter}

%\maketitle

\section{Introduction}

The MAGIC telescope on the Canary Island of La~Palma, located 2200\,m
above sea level at 28\textdegree{}45'\,N and 17\textdegree{}54'\,W, is an
imaging atmospheric Cherenkov telescope designed to achieve a low
energy threshold, fast repositioning, and high tracking accuracy,
e.g.~\cite{Lorenz:2004, Cortina:2005}. The MAGIC design, and the
currently ongoing construction of a second telescope
(MAGIC\,II;~\cite{Goebel:2007}), pave the way to the ground-based
detection of gamma-ray sources at cosmological distances down to less
than 80\,GeV. After the discovery of the distant blazars 1ES\,1218+304
at a redshift of $z$\,=\,0.182~\citep{2006ApJ...642L.119A} and
1ES\,1011+496 at $z$\,=\,0.212~\citep{2007ApJ...667L..21A}, the most
recent breakthrough has been the discovery of the first quasar at very
high energies, the flat-spectrum radio source 3C\,279 at a redshift of
$z$\,=\,0.536~\cite{2008Sci...320.1752M}. These observational results
were somewhat surprising, since the extragalactic background radiation
in the mid-infrared to near-infrared wavelength range was believed to
be strong enough to inhibit propagation of gamma-rays across
cosmological distances~\citep{2001MNRAS.320..504S}. However, it could
be shown that the results of deep galaxy surveys with the Hubble and
Spitzer Space telescopes are consistent with these findings, if the
spurious feature at one micron is attributed to a foreground effect
resulting from an inaccurate subtraction of zodiacal
light~\citep{Hauser:2001, 2007arXiv0707.2915K, Kneiske:2004}. The low
level of pair attenuation of gamma-rays greatly improves the prospects
of searching for very high energy gamma-rays from gamma-ray bursts
(GRBs). Their remarkable similarities with blazar flares, albeit at
much shorter timescales, presumably arise from the scaling behavior of
relativistic jets, the common physical cause of these phenomena. Since
most GRBs reside at large redshifts, their detection relies on the low
level of absorption at very high energies~\citep{1996ApJ...467..532M}.
Moreover, the cosmological absorption decreases with photon energy,
favoring MAGIC to discover GRBs due to its low energy threshold.

Due to the short life time of GRBs and the limited field of view of
imaging atmospheric Cherenkov telescopes, the drive system of the MAGIC
telescope has to meet two basic demands: during normal observations,
the 64-ton telescope has to be repositioned accurately, and has to
track a given sky position, i.e., counteract the apparent rotation of
the celestial sphere, with high precision at a typical rotational speed
in the order of one revolution per day. For catching the GRB prompt
emission and afterglows, it has to be powerful enough to reposition the
telescope to an arbitrary point on the sky within a very short time
($\apprle$\,60\,s) and commence normal tracking immediately
thereafter. To keep the system simple, i.e., robust, both requirements
should be achieved without an indexing gear. The telescope's total
weight of 64~tons is a comparatively low measure, reflecting the
construction principle of using low-weight materials whenever possible.
For example, the mount consists of a space frame of carbon-fiber
reinforced plastic tubes, and the mirrors are made of polished
aluminum.

In this paper, we describe the basic properties of the MAGIC drive
system. In section~\ref{sec2}, the hardware components and mechanical
setup of the drive system are outlined. The control loops and
performance goals are described in section~\ref{sec3}, while the
implementation of the repositioning and tracking algorithms and the
calibration of the drive system are explained in section~\ref{sec4}.
The system can be scaled to meet the demands of other telescope designs
as shown in section~\ref{sec5}. Finally, in section~\ref{conclusions}
we draw conclusions from our experience of operating the MAGIC
telescope with this drive system for four years.

\section{General design considerations}

The drive system of the MAGIC telescope is quite similar to that of
large, alt-azimuth-mounted optical telescopes. Nevertheless there are
quite a few aspects that influenced the design of the MAGIC drive
system in comparison to optical telescopes and small-diameter Imaging
Atmospheric Cherenkov telescope (IACT).

The tracking and stability requirements for IACTs are much less
demanding than for optical telescopes. Although IACTs, like optical
telescopes, track celestial objects, they observe quite different
phenomena: Optical telescopes observe visible light, which originates
at infinity and is parallel. The best-possible optical resolution is
required and in turn, equal tracking precision due to comparably long
integration times, i.e., seconds to hours. In ground-based astronomy at
very high gamma-ray energies, IACTs record the Cherenkov light produced
by an electromagnetic air-shower in the atmosphere, induced by a
primary gamma-ray, i.e., from a close by (5\,km\,-\,20\,km) and
extended event with a diffuse transverse extension and a typical
extension of a few km. Due to the stochastic nature of the shower
development, the detected light will have an inherent limitation in
explanatory power, improving normally with the energy, i.e.,
shower-particle multiplicity. As, in this case, the Cherenkov light is
emitted with under a small angle off the particle tracks, these photons
do not even point directly to the source like in optical astronomy.
Nevertheless, the shower points towards the direction of the incoming
gamma-ray and thus towards its source on the sky, and for this reason
its origin can be reconstructed analyzing its image. Modern IACTs
achieve an energy-dependent pointing resolution for individual showers
of 0.1\textdegree\,-\,0.01\textdegree. These are the predictions from
Monte Carlo simulations assuming, amongst other things, ideal tracking.
This sets the limits achievable in practical cases. Consequently, the
required tracking precision must be at least of the same order or even
better. Although the short integration times, on the order of a few
nanoseconds, would allow for an offline correction, this should be
avoided since it may give rise to an additional systematic error.

MAGIC, as other large IACTs, has no protective dome. It is constantly
exposed to daily changing weather conditions and intense sunlight, and
therefore suffers much more material aging than optical telescopes. A
much simpler mechanical mount had to be used, resulting in a design of
considerably less stiffness, long-term irreversible deformations, and
small unpredictable deformations due to varying wind pressure. The
tracking system does not need to be more precise than the mechanical
structure and, consequently, can be much simpler and hence cheaper as
compared to that of large optical telescopes.

To meet one of the main physics goals, the observation of prompt and
afterglow emission of GRBs, repositioning of the telescope to their
assumed sky position is required in a time as short as possible.
Alerts, provided by satellites, arrive at the MAGIC site typically
within 10\,s after the outburst~\citep{2007ApJ...667..358A}. To achieve
a positioning time to any position on the sky within less than a minute
requires a very light-weight but sturdy telescope and a fast-acting and
powerful drive system.
\begin{figure*}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure1a.eps}
 \hfill
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure1b.eps}
\caption{{\em Left}: One of the bogeys with its two railway rails. The
motor is encapsulated in the grey box on the yellow drive unit. It drives
the tooth double-toothed wheel gearing into the chain through a gear
and a clutch. {\em Right}: The drive unit driving the elevation axis
from the back side. Visible is the actuator of the safety holding brake
and its corresponding brake disc mounted on the motor-driven axis. The
motor is attached on the opposite side. }
\label{figure1}
\end{center}
\end{figure*}

\section{Mechanical setup and hardware components}\label{sec2}

The implementation of the drive system relies strongly on standard
industry components to ensure robustness, reliability and proper
support. The azimuth drive ring of 21\,m diameter is made from a normal
railway rail, which was delivered in pre-bent sections and welded on
site. The fixing onto the concrete foundation uses standard rail-fixing
elements, and allows for movements caused by temperature changes. The
maximum allowable deviation from the horizontal plane as well as
deviation from flatness is $\pm 2$\,mm, and from the ideal circle it is
1\,cm. Each of the six bogeys holds two standard crane wheels of 60\,cm
diameter with a rather broad wheel tread. This allows for deviations in
the 11.5\,m-distance to the central axis due to extreme temperature
changes, which can even be asymmetric in case of different exposure to
sunlight on either side. The central bearing of the azimuth axis is a
high-quality ball bearing, while for the elevation axis, due to lower
weight, a less expensive sliding bearing with a teflon layer was
installed. These sliding bearings have a slightly spherical surface to
allow for small misalignments during installation and some bending of
the elevation axis stubs under load.

The drive mechanism is based on duplex roller chains and sprocket
wheels in a rack-and-pinion mounting. The chains have a breaking
strength of 17~tons and a chain-link spacing of 2.5\,cm. The initial
play between the chain links and the sprocket-wheel teeth is about
3\,mm\,-\,5\,mm, according to the data sheet, corresponding to much
less than an arcsecond on the telescope axes. The azimuth drive chain
is fixed on a dedicated ring on the concrete foundation, but has quite
some radial distance variation of up to 3\,cm. The elevation drive
chain is mounted on a slightly oval ring below the mirror dish, because
the ring forms an integral part of the camera support mast structure.

Commercial synchronous motors (type designation Bosch Rexroth
MHD\,112C-058) are used together with low-play planetary gears linked
to the sprocket wheels. These motors intrinsically allow for a
positional accuracy better than one arcsecond of the motor axis. Having
a nominal power of 11\,kW, they can be overpowered by up to a factor
five for a few seconds. It should be mentioned that due to the
installation height of more than 2200\,m a.s.l., due to lower air
pressure and consequently less efficient cooling, the nominal values
given must be reduced by about 25\%. The azimuth motors are mounted on
small lever arms. In order to follow the small radial irregularities of
the azimuthal drive chain, the motors are pressed against it by
springs. The elevation-drive motor is mounted on a nearly 1\,m long
lever arm to be able to compensate the oval shape of the chain and the
fact that the center of the circle defined by the drive chain is
shifted 50\,cm away from the axis towards the camera.

The design of the drive system control, c.f.~\citet{Bretz:2003drive},
is based on digitally controlled industrial drive units, one for each
motor. The two motors driving the azimuth axis are coupled to have a
more homogeneous load transmission from the motors to the structure
compared to a single (more powerful) motor. The modular design allows
to increase the number of coupled devices dynamically if necessary,
c.f.~\citet{Bretz:2005drive}.

At the latitude of La Palma, the azimuth track of stars can exceed
180\textdegree{} in one night. To allow for continuous observation of a
given source at night without reaching one of the end positions in
azimuth. the allowed range for movements in azimuth spans from
$\varphi$\,=\,-90\textdegree{} to $\varphi$\,=\,+318\textdegree, where
$\varphi$\,=\,0\textdegree{} corresponds to geographical North, and
$\varphi$\,=\,90\textdegree{} to geographical East. To keep slewing
distances as short as possible (particularly in case of GRB alerts),
the range for elevational movements spans from
$\theta$\,=\,+100\textdegree{} to $\theta$\,=\,-70\textdegree{} where
the change of sign implies a movement {\em across the zenith}. This
so-called {\it reverse mode} is currently not in use, as it might imply
hysteresis effects of the active mirror control system, still under
investigation, due to shifting of weight at zenith. The accessible
range in both directions and on both axes is limited by software to the
mechanically accessible range. For additional safety, hardware end
switches are installed directly connected to the drive controller
units, initiating a fast, controlled deceleration of the system when
activated. To achieve an azimuthal movement range exceeding
360\textdegree{}, one of the two azimuth end-switches needs to be
deactivated at any time. Therefore, an additional {\em direction
switch} is located at $\varphi$\,=\,164\textdegree{}, short-circuiting
the end switch currently out of range.

\section{Setup of the motion control system}\label{sec3}

Like for the mechanical drive system, also for the motion-control system
standard industry components were used. The drive is controlled by the
feedback of encoders measuring the angular positions of the motors and
the telescope axes. The encoders on the motor axes provide
information to microcontrollers dedicated for motion control,
initiating and monitoring every movement. Professional build-in servo
loops take over the suppression of oscillations. The correct pointing
position of the system is ensured by a computer program evaluating the
feedback from the telescope axes and initiating the motion executed by
the micro controllers. Additionally, the motor-axis encoders are also
evaluated to increase accuracy. The details of this system, as shown in
figure~\ref{figure2}, are discussed below.

\begin{figure*}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure2a.eps}
 \hfill
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure2b.eps}
\caption{Schematics of the MAGIC\,I ({\em left}) and MAGIC\,II ({\em
right}) drive system. The sketches shows the motors, the motor-encoder
feedback as well as the shaft-encoder feedback, and the motion-control
units, which are themselves controlled by a superior control, receiving
commands from the control PC, which closes the position-control loop.
The system is described in more details in section~\ref{sec3}.}
\label{figure2}
\end{center}
\end{figure*}

\subsection{Position feedback system}

The angular telescope positions are measured by three shaft-encoders
(Hengstler AC61/1214EQ.72OLZ). These absolute multi-turn encoders have
a resolution of 4\,096 (10\,bit) revolutions and 16\,384 (14\,bit)
steps per revolution, corresponding to an intrinsic angular resolution
of 1.3\,arcmin per step. One shaft encoder is located on the azimuth
axis, while two more encoders are fixed on either side of the elevation
axis, increasing the resolution and allowing for measurements of the
twisting of the dish (fig.~\ref{figure3}). All shaft encoders used are
watertight (IP\,67) to withstand the extreme weather conditions
occasionally encountered at the telescope site. The motor positions are
read out at a frequency of 1\,kHz from 10\,bit relative rotary encoders
fixed on the motor axes. Due to the gear ratio of more than one
thousand between motor and load, the 14\,bit resolution of the shaft
encoder system on the axes can be interpolated further using the
position readout of the motors. For communication with the axis
encoders, a CANbus interface with the CANopen protocol is in use
(operated at 125\,kbps). The motor encoders are directly connected by
an analog interface.

\subsection{Motor control}

The three servo motors are connected to individual motion controller
units ({\em DKC}, type designation Bosch Rexroth,
DKC~ECODRIVE\,03.3-200-7-FW), serving as intelligent frequency
converters and power supplies. An input value, given either analog or
digital, is converted to a predefined output, e.g., command position,
velocity or torque. All command values are processed through a chain of
build-in controllers, cf. fig.~\ref{figure4}, resulting in a final
command current applied to the motor. This internal chain of control
loops, maintaining the movement of the motors, works at a frequency of
1\,kHz fed back by the rotary encoders on the corresponding motor axes.
Several safety limits ensure damage-free operation of the system even
under unexpected operation conditions. These safety limits are, e.g.,
software end switches, torque limits, current limits or
control-deviation limits.

To synchronize the two azimuth motors, a master-slave setup is
used. While the master is addressed by a command velocity, the
slave is driven by the command torque output of the master. This
operation mode ensures that both motors can apply their combined force
to the telescope structure without oscillations.

\begin{figure}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure3.eps}
\caption{The measured difference between the two shaft-encoders fixed
on either side of the elevation axis versus zenith angle. Negative
zenith angles mean that the telescope has been flipped over the zenith
position to the opposite side. The average offset from zero
corresponds to a the twist of the two shaft encoders with respect to
each other. Under normal conditions the torsion between both ends of
the axis is less than the shaft-encoder resolution.}
\label{figure3}
\end{center}
\end{figure}

\subsection{Motion control}
\begin{figure*}[htb]
\begin{center}
 \includegraphics*[width=0.6\textwidth,angle=0,clip]{figure4.eps}
 \caption{The internal flow control between the individual controllers
inside the drive control unit. Depending on the type of the command
value, different controllers are active. The control loops are closed by
the feedback of the rotary encoder on the motor, and a possible
controller on the load axis, as well as the measurement of the current.}
\label{figure4}
\end{center}
\end{figure*}

The master DKC for each axis is controlled by presetting a rotational
speed defined by $\pm$10\,V on its analog input. The input voltage is
produced by a programmable microcontroller dedicated to analog motion
control, produced by Z\&B ({\em MACS}, type designation MACS). The
feedback is realized through a 500-step emulation of the motor's rotary
encoders by the DKCs for each axis seperately. Elevation and azimuth
movement is regulated by individual MACSs. The MACS controller itself
communicates with the control software (see below) through a CANbus
connection.

It turned out that in particular the azimuth motor system seems to be
limited by the large moment of inertia of the telescope
($J_{\mathrm{az}}\approx4400$\,tm$^2$, for comparison
$J_{\mathrm{el}}\approx850$\,tm$^2$; note that the exact numbers depend
on the current orientation of the telescope). At the same time, the
requirements on the elevation drive are much less demanding.\\

\noindent {\em MAGIC\,II}\quad For the drive system currently under commissioning for
MAGIC\,II, several improvements have been provided:\\
\begin{itemize}
\item 13\,bit absolute shaft-encoders (type designation Heidenhain
ROQ\,425) are installed, providing an additional sine-shaped
$\pm$1\,Vss output within each step. This allows for a more accurate
interpolation and hence a better resolution than a simple 14\,bit
shaft-encoder. These shaft-encoders are also water tight (IP\,64), and
they are read out via an EnDat\,2.2 interface.
\item All encoders are directly connected to the DKCs, providing
additional feedback from the telescope axes itself. The DKC can control
the load axis additionally to the motor axis providing a more accurate
positioning, faster movement by improved oscillation suppression and a
better motion control of the system.
\item The analog transmission of the master's command torque to the
slave is substituted by a direct digital bi-directional communication
of the DKCs.
\item A single professional programmable logic controller (PLC), in German:
{\em Speicherprogammierbare Steuerung} (SPS, type designation Rexroth
Bosch, IndraControl SPS L\,20) replaces the two MACSs. Connection between
the SPS and the DKCs is now realized through a digital Profibus DP
interface substituting the analog signals.
\item The connection from the SPS to the control PC is realized via
Ethernet connection. Since Ethernet is more commonly in use than
CANbus, soft- and hardware support is much easier.
\end{itemize}

\subsection{PC control}

The drive system is controlled by a standard PC running a Linux
operating system, a custom-designed software based on
ROOT~\citep{www:root} and the positional astronomy library
{\em slalib}~\citep{slalib}.

Algorithms specialized for the MAGIC tracking system are imported from
the Modular Analysis and Reconstruction Software package
(MARS)~\citep{Bretz:2003icrc, Bretz:2005paris, Bretz:2008gamma} also
used in the data analysis~\citep{Bretz:2005mars, Dorner:2005paris}.

\subsubsection{Positioning}

Whenever the telescope has to be repositioned, the relative distance to
the new position is calculated in telescope coordinates and then
converted to motor revolutions. Then, the micro-controllers are
instructed to move the motors accordingly. Since the motion is
controlled by the feedback of the encoders on the motor axes, not on
the telescope axes, backlash and other non-deterministic irregularities
cannot easily be taken into account. Thus it may happen that the final
position is still off by a few shaft-encoder steps, although the motor
itself has reached its desired position. In this case, the procedure is
repeated up to ten times. After ten unsuccessful iterations, the system
would go into error state. In almost all cases the command position is
reached after at most two or three iterations.

If a slewing operation is followed by a tracking operation of a
celestial target position, tracking is started immediately after the
first movement without further iterations. Possible small deviations,
normally eliminated by the iteration procedure, are then corrected by
the tracking algorithm.

\subsubsection{Tracking}
To track a given celestial target position (RA/Dec, J\,2000.0,
FK\,5~\citep{1988VeARI..32....1F}), astrometric and misalignment
corrections have to be taken into account. While astrometric
corrections transform the celestial position into local coordinates as
seen by an ideal telescope (Alt/Az), misalignment corrections convert
them further into the coordinate system specific to the real telescope.
In case of MAGIC, this coordinate system is defined by the position
feedback system.

The tracking algorithm controls the telescope by applying a command
velocity for the revolution of the motors, which is re-calculated every
second. It is calculated from the current feedback position and the
command position required to point at the target five seconds ahead in
time. The timescale of 5\,s is a compromise between optimum tracking
accuracy and the risk of oscillations in case of a too short timescale.

As a crosscheck, the ideal velocities for the two telescope axes are
independently estimated using dedicated astrometric routines of slalib.
For security reasons, the allowable deviation between the determined
command velocities and the estimated velocities is limited. If an
extreme deviation is encountered the command velocity is set to zero,
i.e., the movement of the axis is stopped.

\subsection{Fast repositioning}

The observation of GRBs and their afterglows in very-high energy
gamma-rays is a key science goal for the MAGIC telescope. Given that
alerts from satellite monitors provide GRB positions a few seconds
after their outburst via the {\em Gamma-ray Burst Coordination
Network}~\cite{www:gcn}, typical burst durations of 10\,s to
100\,s~\cite{Paciesas:1999} demand a fast repositioning of the
telescope. The current best value for the acceleration has been set to
11.7\,mrad\,s$^{-2}$. It is constrained by the maximum constant force
which can be applied by the motors. Consequently, the maximum allowed
velocity can be derived from the distance between the end-switch
activation and the position at which a possible damage to the telescope
structure, e.g.\ ruptured cables, would happen. From these constraints,
the maximum velocity currently in use, 70.4\,mrad\,s$^{-1}$, was
determined. Note that, as the emergency stopping distance evolves
quadratically with the travel velocity, a possible increase of the
maximum velocity would drastically increase the required braking
distance. As safety procedures require, an emergency stop is completely
controlled by the DKCs itself with the feedback of the motor encoder,
ignoring all other control elements.

Currently, automatic repositioning by
$\Delta\varphi$\,=\,180\textdegree{} in azimuth to the target position
is achieved within 45\,s. The repositioning time in elevation is not
critical in the sense that the probability to move a longer path in
elevation than in azimuth is negligible. Allowing the telescope drive
to make use of the reverse mode, the requirement of reaching any
position in the sky within 30\,s is well met, as distances in azimuth
are substantially shortened. The motor specifications allow for a
velocity more than four times higher. In practice, the maximum possible
velocity is limited by the acceleration force, at slightly more than
twice the current value. The actual limiting factor is the braking
distance that allows a safe deceleration without risking any damage to
the telescope structure.

\subsection{Tracking precision}

The intrinsic mechanical accuracy of the tracking system is determined
by comparing the current command position of the axes with the feedback
values from the corresponding shaft encoders. These feedback values
represent the actual position of the axes with highest precision
whenever they change their feedback values. At these instances, the
control deviation is determined, representing the precision with which
the telescope axes can be operated. In the case of an ideal mount this
would define the tracking accuracy of the telescope.

In figure~\ref{figure5} the control deviation measured for 10.9\,h of
data taking in the night of 2007 July 22/23 and on the evening of July
23 is shown, expressed as absolute deviation on the sky taking both
axes into account. In almost all cases it is well below the resolution
of the shaft encoders, and in 80\% of the time it does not exceed 1/8
of this value ($sim$10\,arcsec). This means that the accuracy of the
motion control, based on the encoder feedback, is much better than
1\,arcmin on the sky, which is roughly a fifth of the diameter of a
pixel in the MAGIC photomultiplier (PM) camera (0.1\textdegree{},
c.f.~\cite{Beixeras:2005}).

In the case of a real telescope ultimate limits of the tracking
precision are given by the precision with which the correct command
value is known. Its calibration is discussed in the following.

\begin{figure}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure5.eps}
 \caption{Control deviation between the expected, i.e. calculated,
position, and the feedback position of the shaft encoders in the moment
at which one change their readout values. For simplicity, the control
deviation is shown as absolute control deviation projected on the sky.
The blue lines correspond to fractions of the shaft-encoder resolution.
}
\label{figure5}
\end{center}
\end{figure}

\section{Calibration}\label{sec4}

To calibrate the position, command value-astrometric corrections,
(converting the celestial target position into the target position of
an ideal telescope) and misalignment corrections (converting it further
into the target position of a real telescope), have to be taken into
account.

\subsection{Astrometric corrections}

The astrometric correction for the pointing and tracking algorithms is
based on a library for calculations usually needed in positional
astronomy, {\em slalib}~\cite{slalib}. Key features of this library are
the numerical stability of the algorithms and their well-tested
implementation. The astrometric corrections in use
(fig.~\ref{figure6}) -- performed when converting a celestial position
into the position as seen from Earth's center (apparent position) --
take into account precession and nutation of the Earth and annual
aberration, i.e., apparent displacements caused by the finite speed of
light combined with the motion of the observer around the Sun during
the year. Next, the apparent position is transformed to the observer's
position, taking into account atmospheric refraction, the Earth's
rotation, and diurnal aberration, i.e., the motion of the observer
around the Earth's rotation axis. Some of these effects are so small
that they are only relevant for nearby stars and optical astronomy. But
as optical observations of such stars are used to {\em train} the
misalignment correction, all these effects are taken into account.

\begin{figure}[htb]
\begin{center} % Goldener Schnitt
 \includegraphics*[width=0.185\textwidth,angle=0,clip]{figure6.eps}
\caption{The transformation applied to a given set of catalog source
coordinates to real-telescope coordinates. These corrections include
all necessary astrometric corrections, as well as the pointing
correction to transform from an ideal-telescope frame to the frame of a
real telescope. A detailed description of all corrections and the
calibration of the pointing model is given in section~\ref{sec4}.}
\label{figure6} 
\end{center} 
\end{figure}

\subsection{Pointing model}

Imperfections and deformations of the mechanical construction lead to
deviations from an ideal telescope. These deviations include the non-exact
alignment of axes, and deformations of the telescope structure. To
assure reliable pointing and tracking accuracy, such effects have to be
taken into account. Therefore the tracking software employs an
analytical pointing model based on the {\rm TPOINT}\texttrademark{}
telescope modeling software~\cite{tpoint}, also used for
optical telescopes. This model, called {\em pointing model},
parameterizes deviations from the ideal telescope. Calibrating the
pointing model by mispointing measurements of bright stars, which
allows to determine the necessary corrections, is a standard procedure.
Once calibrated, the model is applied online. Since an analytical model
is used, the source of any deviation can be identified and traced back
to components of the telescope mount.\\

Corrections are parameterized by alt-azimuthal terms~\cite{tpoint},
i.e., derived from vector transformations within the proper coordinate
system. The following possible misalignments are taken into account:\\

\begin{description}
\item[Zero point corrections ({\em index errors})] Trivial offsets
between the zero positions of the axes and the zero positions of the
shaft encoders.
\item[Azimuth axis misalignment] The misalignment of the azimuth axis
in north-south and east-west direction, respectively. For MAGIC these
corrections can be neglected.
\item[Non-perpendicularity of axes] Deviations from right angles
between any two axes in the system, namely (1) non-perpendicularity of
azimuth and elevation axes and (2) non-perpendicularity of elevation
and pointing axes. In the case of the MAGIC telescope these corrections
are strongly bound to the mirror alignment defined by the active mirror
control.
\item[Non-centricity of axes] The once-per-revolution cyclic errors
produced by de-centered axes. This correction is small, and thus difficult
to measure, but the most stable correction throughout the years.
\end{description}

\noindent{\bf Bending of the telescope structure}
\begin{itemize}
\item A possible constant offset of the mast bending.
\item A zenith angle dependent correction. It describes the camera mast
bending, which originates by MAGIC's single thin-mast camera support
strengthened by steel cables.
\item Elevation hysteresis: This is an offset correction introduced
depending on the direction of movement of the elevation axis. It is
necessary because the sliding bearing, having a stiff connection with
the encoders, has such a high static friction that in case of reversing
the direction of the movement, the shaft-encoder will not indicate any
movement, even though the telescope is rotating.
\end{itemize}
\vspace{1em}

Since the primary feedback is located on the axis itself, corrections
for irregularities of the chain mounting or sprocket wheels are
unnecessary. Another class of deformations of the telescope-support
frame and the mirrors are non-deterministic and, consequently, pose an
ultimate limit of the precision of the pointing correction.

As discussed below, the size of the total correction (excluding the
index error) is on the order of 0.1\textdegree{}. Therefore all individual
correction terms are smaller or of similar size.

\subsection{Determination}

%\subsubsection{Calibration concept}

To determine the coefficients of a pointing model, calibration
data is recorded, consisting of mispointing measurements depending
on altitude and azimuth angle. Bright stars are tracked with the
telescope at positions uniformly distributed in local coordinates,
i.e.\, in altitude and azimuth angle. To measure the real pointing
position of the telescope, the reflection of a bright star on a
screen in front of the MAGIC PM camera is determined. The center
of the camera is defined by LEDs mounted on an ideal ($\pm$1\,mm)
circle around the camera center, cf.~\citet{Riegel:2005icrc}.

Having enough of these datasets available, correlating ideal and real
pointing position, a fit of the coefficients of the model can be made,
minimizing the pointing residual.

\subsubsection{Hardware and installations}

A 0.0003\,lux, 1/2\(^{\prime\prime}\) high-sensitivity standard PAL CCD
camera (\mbox{Watec}~WAT-902\,H, technical details given in
table~\ref{table}) equipped with a zoom lens (type: Computar) is used
for the mispointing measurements. The camera is read out at a rate of
25\,frames per second using a framer-grabber card in a standard PC. The
tradeoff for the high sensitivity of the camera is its high noise level
in each single frame recorded. Since there are no rapidly moving
objects within the field of view, a high picture quality can be
achieved by averaging typically 125\,frames (corresponding to 5\,s). An
example is shown in figure~\ref{figure7}. This example also
illustrates the high sensitivity of the camera, since both pictures of
the telescope structure have been taken with the residual light of less
than a half-moon. In the background individual stars can be seen.
Depending on the installed optics, stars up to 12$^\mathrm{m}$ are
visible. With our optics and a safe detection threshold the limiting
magnitude is typically slightly above 9$^\mathrm{m}$ for direct
measurements and on the order of 5$^\mathrm{m}$\dots4$^\mathrm{m}$ for
images of stars on the screen.

\begin{table}[htb]
\begin{center}
\small
\begin{tabular}{|l|l|}\hline
Model&Watec WAT-902H (CCIR)\\\hline\hline
Pick-up Element&1/2" CCD image sensor\\
&(interline transfer)\\\hline
Number of total pixels&795(H)x596(V)\\\hline
Minimum Illumination&0.0003\,lx. F/1.4 (AGC Hi)\\
&0.002\,lx. F/1.4 (AGC Lo)\\\hline
Automatic gain&High: 5\,dB\,-\,50\,dB\\
&Low: 5\,dB\,-\,32\,dB\\\hline
S/N&46\,dB (AGC off)\\\hline
Shutter Speed&On: 1/50\,-\,1/100\,000\,s\\ (electronic iris)&Off: 1/50\,s\\\hline
Backlight compensation&On\\\hline
Power Supply&DC +10.8\,V\,-\,13.2\,V\\\hline
Weight&Approx. 90\,g\\\hline
\end{tabular}
\end{center}
\caption{Technical specifications of the CCD camera used for
measuring of the position of the calibration stars on the PM camera lid.}
\label{table}
\end{table}

\begin{figure*}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure7a.eps}
 \hfill
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure7b.eps}
 \caption{A single frame (left) and an average of 125 frames (right) of
the same field of view taken with the high sensitivity PAL CCD camera
used for calibration of the pointing model. The frames were taken
with half moon. }
\label{figure7}
\end{center}
\end{figure*}

\subsubsection{Algorithms}

An example of a calibration-star measurement is shown in
figure~\ref{figure8}. Using the seven LEDs mounted on a circle around
the camera center, the position of the camera center is determined.
Only the upper half of the area instrumented with PMs is visible, since
the lower half is covered by the lower lid, holding a special
reflecting surface in the center of the camera. The LED positions are
evaluated by a simple cluster-finding algorithm looking at pixels more
than three standard deviations above the noise level. The LED position
is defined as the center of gravity of its light distribution. The
search region is defined by the surrounding black-coloured boxes. For
simplicity the noise level is determined just by calculating the mean
and the root-mean-square within the individual search regions below a
fixed threshold dominated by noise.

\begin{figure}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure8.eps}
\caption{A measurement of a star for the calibration of the pointing
model. Visible are the seven LEDs and their determined center of
gravity, as well as the reconstructed circle on which the LEDs are
located. The LEDs on the bottom part are hidden by the lower lid,
holding a screen in front of the PM camera. For calibration, the center
of gravity of the measured star, as visible in the center, is compared
to the center of the circle given by the LEDs, coinciding with the
center of the PM camera. The black regions are the search regions for
the LEDs and the calibration star. A few dead pixels in the CCD camera
can also be recognized. }
\label{figure8}
\end{center}
\end{figure}

Since three points are enough to define a circle, from all
possible combinations of detected spots, the corresponding circle
is calculated. In case of misidentified LEDs, which sometimes
occur due to light reflections from the telescope structure, the
radius of the circle will deviate from the predefined radius.
Thus, any such misidentified circles are discarded. The radius
determination can be improved further by applying small offsets of
the non-ideal LED positions. The radius distribution is Gaussian
and its resolution is $\sigma\,\apprle$\,1\,mm
($\mathrm{d}r/r\approx0.3$\textperthousand) on the camera plane
corresponding to $\sim$1\,arcsec.

The center of the LED ring is calculated as the average of all circle
centers after quality cuts. The resolution of the center is
$\sim$2\,arcsec. In this setup, the large number of LEDs guarantees
operation even in case one LED is damaged or could not be detected due
to scattered light.

To find the spot of the reflected star itself, the same cluster-finder
is used to determine its center of gravity. This gives reliable results
even in case of saturation. Only very bright stars, brighter than
1.0$^m$, are found to saturate the CCD camera asymmetrically.

Using the position of the star, with respect to the camera center, the
pointing position corresponding to the camera center is calculated.
This position is stored together with the readout from the position
feedback system. The difference between the telescope pointing
position and the feedback position is described by the pointing model.
Investigating the dependence of these differences on zenith and azimuth
angle, the correction terms of the pointing model can be determined.
Its coefficients are fit minimizing the absolute residuals on the celestial
sphere.

\subsection{Results}

Figure~\ref{figure9} shows the residuals, taken between 2006 October and
2007 July, before and after application of the fit of the pointing
model. For convenience, offset corrections are applied to the residuals
before correction. Thus, the red curve is a measurement of the alignment
quality of the structure, i.e., the pointing accuracy with offset
corrections only. By fitting a proper model, the pointing accuracy can
be improved to a value below the intrinsic resolution of the system,
i.e., below shaft-encoder resolution. In more than 83\% of all cases the
tracking accuracy is better than 0.02\textdegree{} ($\sim$1.3\,arcmin)
and it hardly ever exceeds 0.04\textdegree. The few datasets exceeding
0.04\textdegree{} are very likely due to imperfect measurement of the
real pointing position of the telescope, i.e.\, the center of gravity of
the star light.

\begin{figure}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure9.eps}
 \caption{Distribution of absolute pointing residual on the sky between
the measured position of calibration stars and their nominal position
with only offset correction for both axes (red) and a fitted pointing
model (blue) applied. Here in total 1162 measurements where used,
homogeneously distributed over the local sky. After application of the
pointing model the residuals are well below the shaft-encoder
resolution, i.e., the knowledge of the mechanical position of the axes.
\label{figure9}
}

\end{center}
\end{figure}

\subsubsection{Limitations}

The ultimate limit on the achievable tracking precision are effects,
which are difficult to correlate or measure, and non-deterministic
deformations of the structure or mirrors. For example, the azimuth
support consists of a railway rail with some small deformations in
height due to the load, resulting in a wavy movement difficult to
parameterize. For the wheels on the six bogeys, simple, not precisely
machined crane wheels were used, which may amplify horizontal
deformations. Other deformations are caused by temperature changes and
wind loads which are difficult to control for telescopes without dome,
and which cannot be modelled. It should be noted that the azimuth structure
can change its diameter by up to 3\,cm due to day-night temperature
differences, indicating that thermal effects have a non-negligible and
non-deterministic influence.

\subsubsection{Stability}

With each measurement of a calibration-star also the present tracking
uncertainty is recorded. This allows for monitoring of the tracking
quality and for offline correction. In figure~\ref{figure10} the
evolution of the measured residuals over the years are shown. The
continuous monitoring has been started in March 2005 and is still
ongoing. Quantiles are shown since the distribution can be highly
asymmetric. The points have been grouped, where the grouping reflects
data taken under the same conditions (pointing model, mirror alignment,
etc.) It should be noted, that the measured residuals depend on zenith
and azimuth angle, i.e., the distributions shown are biased due to
inhomogeneous distributions on the sky in case of low statistics.
Therefore the available statistics is given in table~\ref{table2}.
%, for convenience, together with the average measured mispointing.

\begin{figure}[htb]
\begin{center}
 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure10.eps}
 \caption{The distribution of mispointing measurements, which is an
exact measure of accuracy of the command values send to the drive
system. The plot shows its time-evolution. Details on the separation
and the available statistics is given in the caption of
table~\ref{table2}. Since the distribution is highly asymmetric,
quantiles are shown, from bottom to top, at 5\%, 13\%, 32\%, 68\%, 87\%
and 95\%. The dark grey region belong to the region between quantiles
32\% and 68\%.
}
\label{figure10}
\end{center}
\end{figure}

The mirror focusing can influence the alignment of the optical axis of
the telescope, i.e., it can modify the pointing model. Therefore a mirror
refocusing can worsen the tracking accuracy, later corrected by a new
pointing model. Also the determination of the pointing model relies on
a good statistical basis, because the measured residuals are of a
similar magnitude as the accuracy of a single calibration-star
measurement. The visible improvements and deterioration are mainly a
consequence of new mirror focusing and following implementations of new
pointing models. The improvement over the past year is explained by the
gain in statistics.

On average the systematic pointing uncertainty was always better than
three shaft-encoder steps (corresponding to 4\,arcmin), most of the
time better than 2.6\,arcmin and well below one shaft-encoder step,
i.e.\ 1.3\,arcmin, in the past year. Except changes to the pointing
model or the optical axis, as indicated by the bin edges, no
degradation or change with time of the pointing model could be found.

\section{Scalability}\label{sec5}

With the aim to reach lower energy thresholds the next generation of
Cherenkov telescopes will become larger and heavier. Therefore more
powerful drive systems will be needed. The scalable drive system of the
MAGIC telescope is suited to meet this challenge. With its synchronous
motors and their master-slave setup it can easily be extended to larger
telescopes at moderate costs or even scaled down using less powerful
components. A motion accuracy at least of the order of the
shaft-encoder resolution is guaranteed. Real tracking accuracy --
already including all possible pointing corrections -- is dominated by
dynamic and unpredictable deformations of the mount, e.g., temperature
expansion.

\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|c|}\hline
Begin&Counts\\\hline\hline
2005/03/20&29\\
2005/04/29&43\\
2005/05/25&30\\
2005/06/08&26\\
2005/08/15&160\\
2005/09/12&22\\
2005/11/24&38\\
2006/03/19&502\\
2006/10/17&827\\
2007/07/31&87\\
2008/01/14&542\\
2008/06/18&128\\\hline
\end{tabular}
\end{center}
\caption{Available statistics corresponding to the distributions
shown in figure~\ref{figure10}. Especially in cases of low statistics
the shown distribution can be influenced by inhomogeneous distribution
of the measurement on the local sky. The dates given correspond to dates
for which a change in the pointing accuracy, as for example a change to
the optical axis or the application of a new pointing model, is known.}
\label{table2}
\end{table}

\section{Outlook}

Currently, efforts are ongoing to implement the astrometric subroutines
as well as the application of the pointing model directly into the
Programmable Logic Controller. A first test will be carried out within
the DWARF project soon~\cite{DWARF}. The direct advantage is that the
need for a control PC is omitted. Additionally, with a more direct
communication between the algorithms, calculating the nominal position
of the telescope mechanics, and the control loop of the drive
controller, a real time, and thus more precise, position control can be
achieved. As a consequence, the position controller can directly be
addressed, even when tracking, and the outermost position control-loop
is closed internally in the drive controller. This will ensure an even
more accurate and stable motion. Interferences from external sources,
e.g. wind gusts, could be counteracted at the moment of appearance by
the control on very short timescales, on the order of milli-seconds. An
indirect advantage is that with a proper setup of the control loop
parameters, such a control is precise and flexible enough that a
cross-communication between the master and the slaves can also be
omitted. Since all motors act as their own master, in such a system a
broken motor can simply be switched off or mechanically decoupled
without influencing the general functionality of the system.

\section{Conclusions}\label{conclusions}

The scientific requirements demand a powerful, yet accurate drive
system for the MAGIC telescope. From its hardware installation and
software implementation, the installed drive system exceeds its design
specifications. At the same time the system performs reliably and
stably, showing no deterioration after five years of routine operation.
The mechanical precision of the motor movement is almost ten times
better than the readout on the telescope axes. The tracking accuracy is
dominated by random deformations and hysteresis effects of the mount,
but still within limits within which the position of the telescope axes
can be measured. The system features integrated tools, like an
analytical pointing model. Fast repositioning for gamma-ray bursts
followup is achieved on average within less than 45 seconds, or, if
movements {\em across the zenith} are allowed, within less than 30
seconds. Thus, the drive system makes MAGIC the best suited telescope
for observations of gamma-ray burst at very high energies.

For the second phase of the MAGIC project and particularly for the
second telescope, the system has been further improved by replacing the
partially analog communication with a completely digital one.
By design, the drive system is easily scalable from its current
dimensions to larger and heavier telescope installations as required
for future projects. The improved stability is also expected to meet
the stability requirements, necessary when operating a larger number of
telecopes.

\section[]{Acknowledgments}
The authors acknowledge the support of the MAGIC collaboration, and
thank the IAC for providing excellent working conditions at the Roque
de los Muchachos Observatory in La Palma. The MAGIC telescope project
is mainly supported by BMBF (Germany), MCI-NN (Spain), INFN and MUR
(Italy). We thank the construction department of the MPI for Physics,
particularly Peter Sawallisch, for their help in the design and
installation of the drive system. We are grateful for Eckart Lorenz's
help with the hardware installations and also for some important
comments concerning this manuscript. R.M.W.\ acknowledges financial
support by the Max Planck Society. His research is also supported in
part by the DFG Cluster of Excellence ``Origin and Structure of the
Universe''.

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\end{document}
