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50\journal{Astroparticle Physics}
55% Graphics at 1000dpi
57\title{The drive system of the\\Major Atmospheric Gamma-ray Imaging Cherenkov Telescope}
65\address[tb]{Universit\"{a}t W\"{u}rzburg, Am Hubland, 97074 W\"{u}rzburg, Germany}
66\address[rw]{Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, 80805 M\"{u}nchen, Germany}
67\cortext[cor1]{Corresponding author:}
69%\input library.def
72\newcommand{\mygtrsim} {{\apprge}}
76The MAGIC telescope is an imaging atmospheric Cherenkov telescope,
77designed to observe very high energy gamma-rays while achieving a low
78energy threshold. One of the key science goals is fast follow-up of the
79enigmatic and short lived gamma-ray bursts. The drive system for the
80telescope has to meet two basic demands: (1)~During normal
81observations, the 72-ton telescope has to be positioned accurately, and
82has to track a given sky position with high precision at a typical
83rotational speed in the order of one revolution per day. (2)~For
84successfully observing GRB prompt emission and afterglows, it has to be
85powerful enough to position to an arbitrary point on the sky within a
86few ten seconds and commence normal tracking immediately thereafter. To
87meet these requirements, the implementation and realization of the
88drive system relies strongly on standard industry components to ensure
89robustness and reliability. In this paper, we describe the mechanical
90setup, the drive control and the calibration of the pointing, as well
91as present measurements of the accuracy of the system. We show that the
92drive system is mechanically able to operate the motors with an
93accuracy even better than the feedback values from the axes. In the
94context of future projects, envisaging telescope arrays comprising
95about 100 individual instruments, the robustness and scalability of the
96concept is emphasized.
100MAGIC\sep drive system\sep IACT\sep scalability\sep calibration\sep fast positioning
109The MAGIC telescope on the Canary Island of La~Palma, located 2200\,m
110above sea level at 28\textdegree{}45$^\prime$\,N and 17\textdegree{}54$^\prime$\,W, is
111an imaging atmospheric Cherenkov telescope designed to achieve a low
112energy threshold, fast positioning, and high tracking
113accuracy~\cite{Lorenz:2004, Cortina:2005}. The MAGIC design, and the
114currently ongoing construction of a second telescope
115(MAGIC\,II;~\cite{Goebel:2007}), pave the way for ground-based
116detection of gamma-ray sources at cosmological distances down to less
117than 25\,GeV~\cite{Sci}. After the discovery of the distant blazars 1ES\,1218+304
118at a redshift of $z$\,=\,0.182~\citep{2006ApJ...642L.119A} and
1191ES\,1011+496 at $z$\,=\,0.212~\citep{2007ApJ...667L..21A}, the most
120recent breakthrough has been the discovery of the first quasar at very
121high energies, the flat-spectrum radio source 3C\,279 at a redshift of
122$z$\,=\,0.536~\cite{2008Sci...320.1752M}. These observational results
123were somewhat surprising, since the extragalactic background radiation
124in the mid-infrared to near-infrared wavelength range was believed to
125be strong enough to inhibit propagation of gamma-rays across
126cosmological distances~\citep{2001MNRAS.320..504S, 2007arXiv0707.2915K, Hauser:2001}.
127%However, it could
128%be shown that the results of deep galaxy surveys with the Hubble and
129%Spitzer Space telescopes are consistent with these findings, if the
130%spurious feature at one micron is attributed to a foreground effect
131%resulting from an inaccurate subtraction of zodiacal
132%light~\citep{Hauser:2001, 2007arXiv0707.2915K, Kneiske:2004}.
134apparent low level of pair attenuation of gamma-rays greatly improves
135the prospects of searching for very high energy gamma-rays from
136gamma-ray bursts (GRBs), cf.~\citep{Kneiske:2004}. Their remarkable similarities with blazar
137flares, albeit at much shorter timescales, presumably arise from the
138scaling behavior of relativistic jets, the common physical cause of
139these phenomena. Since most GRBs reside at large redshifts, their
140detection at very high energies relies on the low level of
141absorption~\citep{1996ApJ...467..532M}. Moreover, the cosmological
142absorption decreases with photon energy, favoring MAGIC to discover
143GRBs due to its low energy threshold.
145Due to the short life times of GRBs and the limited field of view of
146imaging atmospheric Cherenkov telescopes, the drive system of the MAGIC
147telescope has to meet two basic demands: during normal observations,
148the 72-ton telescope has to be positioned accurately, and has to
149track a given sky position, i.e., counteract the apparent rotation of
150the celestial sphere, with high precision at a typical rotational speed
151in the order of one revolution per day. For catching the GRB prompt
152emission and afterglows, it has to be powerful enough to position the
153telescope to an arbitrary point on the sky within a very short time
155and commence normal tracking immediately
156thereafter. To keep the system simple, i.e., robust, both requirements
157should be achieved without an indexing gear. The telescope's total
158weight of 72~tons is comparatively low, reflecting the
159use of low-weight materials whenever possible.
160For example, the mount consists of a space frame of carbon-fiber
161reinforced plastic tubes, and the mirrors are made of polished
164In this paper, we describe the basic properties of the MAGIC drive
165system. In section~\ref{sec2}, the hardware components and mechanical
166setup of the drive system are outlined. The control loops and
167performance goals are described in section~\ref{sec3}, while the
168implementation of the positioning and tracking algorithms and the
169calibration of the drive system are explained in section~\ref{sec4}.
170The system can be scaled to meet the demands of other telescope designs
171as shown in section~\ref{sec5}. Finally, in section~\ref{outlook} and
172section~\ref{conclusions} we draw conclusions from our experience of
173operating the MAGIC telescope with this drive system for four years.
175\section{General design considerations}\label{design}
177The drive system of the MAGIC telescope is quite similar to that of
178large, alt-azimuth-mounted optical telescopes. Nevertheless there are
179quite a few aspects that influenced the design of the MAGIC drive
180system in comparison to optical telescopes and small-diameter Imaging
181Atmospheric Cherenkov telescopes (IACT).
183Although IACTs have optical components, the tracking and stability
184requirements for IACTs are much less demanding than for optical
185telescopes. Like optical telescopes, IACTs track celestial
186objects, but observe quite different phenomena: Optical telescopes
187observe visible light, which originates at infinity and is parallel.
188Consequently, the best-possible optical resolution is required and in
189turn, equal tracking precision due to comparably long integration
190times, i.e., seconds to hours. In contrast, IACTs record the Cherenkov
191light produced by an electromagnetic air-shower in the atmosphere,
192induced by a primary gamma-ray, i.e., from a close by
193(5\,km\,-\,20\,km) and extended event with a diffuse transverse
194extension and a typical extension of a few hundred meters. Due to the
195stochastic nature of the shower development, the detected light will
196have an inherent limitation in explanatory power, improving normally
197with the energy, i.e., shower-particle multiplicity. As
198the Cherenkov light is emitted under a small angle off the particle
199tracks, these photons do not even point directly to the source like in
200optical astronomy. Nevertheless, the shower points towards the
201direction of the incoming gamma-ray and thus towards its source on the
202sky. For this reason its origin can be reconstructed analyzing its
203image. Modern IACTs achieve an energy-dependent pointing resolution for
204individual showers of 6$^\prime$\,-\,0.6$^\prime$. These are the
205predictions from Monte Carlo simulations assuming, amongst other
206things, ideal tracking. This sets the limits achievable in practical
207cases. Consequently, the required tracking precision must be at least
208of the same order or even better. Although the short integration times,
209on the order of a few nanoseconds, would allow for an offline
210correction, this should be avoided since it may give rise to an
211additional systematic error.
213%MAGIC, as other large IACTs, has no protective dome. It is constantly
214%exposed to daily changing weather conditions and intense sunlight, and
215%therefore suffers much more material aging than optical telescopes. A
216%much simpler mechanical mount had to be used, resulting in a design of
217%considerably less stiffness, long-term irreversible deformations, and
218%small unpredictable deformations due to varying wind pressure. The
219%tracking system does not need to be more precise than the mechanical
220%structure and, consequently, can be much simpler and hence cheaper as
221%compared to that of large optical telescopes.
223To meet one of the main physics goals, the observation of prompt and
224afterglow emission of GRBs, positioning of the telescope to their
225assumed sky position is required in a time as short as possible.
226Alerts, provided by satellites, arrive at the MAGIC site typically
227within 10\,s after the outburst~\citep{2007ApJ...667..358A}.
228Since the life times of GRBs show a bimodal distribution~\cite{Paciesas:1999}
229with a peak between 10\,s and 100\,s. To achieve
230a positioning time to any position on the sky within a reasonable
231time inside this window, i.e. less than a minute,
232a very light-weight but sturdy telescope and a fast-acting and
233powerful drive system is required.
236 \includegraphics*[width=0.91\textwidth,angle=0,clip]{figure1.eps}
237\caption{The top picture shows the MAGIC\,I telescope with the major
238components of the drive system. The elevation drive unit, from its back
239side, is enlarged in the bottom left picture. Visible is the actuator
240of the safety holding brake and its corresponding brake disk mounted on
241the motor-driven axis. The motor is attached on the opposite side. The
242picture on the bottom right shows one of the azimuth bogeys with its
243two railway rails. The motor is housed in the grey box on the
244yellow drive unit. It drives the tooth double-toothed wheel gearing
245into the chain through a gear and a clutch.}
246% \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure1a.eps}
247% \hfill
248% \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure1b.eps}
249%\caption{{\em Left}: One of the bogeys with its two railway rails. The
250%motor is encapsulated in the grey box on the yellow drive unit. It drives
251%%the tooth double-toothed wheel gearing into the chain through a gear
252%and a clutch. {\em Right}: The drive unit driving the elevation axis
253%from the back side. Visible is the actuator of the safety holding brake
254%and its corresponding brake disk mounted on the motor-driven axis. The
255%motor is attached on the opposite side. }
260\section{Mechanical setup and hardware components}\label{sec2}
262The implementation of the drive system relies strongly on standard
263industry components to ensure robustness, reliability and proper
264technical support. Its major drive components, described hereafter, are
265shown on the pictures in fig.~\ref{figure2}.
267The azimuth drive ring of 20\,m diameter is made from a normal railway
268rail, which was delivered in pre-bent sections and welded on site. Its
269head is only about 74\,mm broad and has a bent profile. The fixing onto
270the concrete foundation uses standard rail-fixing elements, and allows
271for movements caused by temperature changes. The maximum allowable
272deviation from the horizontal plane as well as deviation from flatness
273is $\pm 2$\,mm, and from the ideal circle it is $\Delta$r\,=\,8\,mm.
274The rail support was leveled with a theodolite every 60\,cm
275with an overall tolerance of $\pm 1.5$\,mm every 60\,cm. In between the
276deviation is negligible. Each of the six bogeys holds two standard
277crane wheels of 60\,cm diameter with a rather broad wheel tread of
278110\,mm. This allows for deviations in the 11.5\,m-distance to the
279central axis due to extreme temperature changes, which can even be
280asymmetric in case of different exposure to sunlight on either side.
281For the central bearing of the azimuth axis, a high-quality ball
282bearing was installed fairly assuming that the axis is vertically
283stable. For the elevation axis, due to lower weight, a less expensive
284sliding bearing with a teflon layer was used. These sliding bearings
285have a slightly spherical surface to allow for small misalignments
286during installation and some bending of the elevation axis stubs under
289The drive mechanism is based on duplex roller chains and sprocket
290wheels in a rack-and-pinion mounting. The chains have a breaking
291strength of 19~tons and a chain-link spacing of 2.5\,cm. The initial
292play between the chain links and the sprocket-wheel teeth is about
2933\,mm\,-\,5\,mm, according to the data sheet, corresponding to much
294less than an arcsecond on the telescope axes. The azimuth drive chain
295is fixed on a dedicated ring on the concrete foundation, but has quite
296some radial distance variation of up to 5\,mm. The elevation drive
297chain is mounted on a slightly oval ring below the mirror dish, because
298the ring forms an integral part of the camera support mast structure.
300% Bosch Rexroth AG, 970816 Lohr am Main, Germany
303% Wittenstein alpha GmbH, Walter-Wittenstein-Stra"se 1, D-97999 Igersheim
306% updated
307Commercial synchronous motors (type designation Bosch
308Rexroth\footnote{\url{}\\Bosch Rexroth AG,
30997816 Lohr am Main, Germany} MHD\,112C-058) are used together with
310low-play planetary gears (type designation
311alpha\footnote{\url{}\\Wittenstein alpha
312GmbH, 97999 Igersheim, Germany} GTS\,210-M02-020\,B09, ratio 20) linked
313to the sprocket wheels. These motors intrinsically allow for a
314positional accuracy better than one arcsecond of the motor axis. Having
315a nominal power of 11\,kW, they can be overpowered by up to a factor
316five for a few seconds. It should be mentioned that due to the
317installation height of more than 2200\,m a.s.l., due to lower air
318pressure and consequently less efficient cooling, the nominal values
319given must be reduced by about 20\%. Deceleration is done operating the
320motors as generator which is as powerful as acceleration. The motors
321contain 70\,Nm holding brakes which are not meant to be used as driving
322brake. The azimuth motors are mounted on small lever arms. In order to
323follow the small irregularities of the azimuthal drive chain, the units
324are forced to follow the drive chain, horizontally and  vertically, by
325guide rolls. The elevation-drive motor is mounted on a nearly 1\,m long
326lever arm to be able to compensate the oval shape of the chain and the
327fact that the center of the circle defined by the drive chain is
328shifted 356\,mm away from the axis towards the camera. The elevation
329drive is also equipped with an additional brake, operated only as
330holding brake, for safety reasons in case of extremely strong wind
331pressure. No further brake are installed on the telescope.
333The design of the drive system control, c.f.~\citet{Bretz:2003drive},
334is based on digitally controlled industrial drive units, one for each
335motor. The two motors driving the azimuth axis are coupled to have a
336more homogeneous load transmission from the motors to the structure
337compared to a single (more powerful) motor. The modular design allows
338to increase the number of coupled devices dynamically if necessary,
341At the latitude of La Palma, the azimuth track of stars can exceed
342180\textdegree{} in one night. To allow for continuous observation of a
343given source at night without reaching one of the end positions in
344azimuth. the allowed range for movements in azimuth spans from
345$\varphi$\,=\,-90\textdegree{} to $\varphi$\,=\,+318\textdegree, where
346$\varphi$\,=\,0\textdegree{} corresponds to geographical North, and
347$\varphi$\,=\,90\textdegree{} to geographical East. To keep slewing
348distances as short as possible (particularly in case of GRB alerts),
349the range for elevational movements spans from
350$\theta$\,=\,+100\textdegree{} to $\theta$\,=\,-70\textdegree{} where
351the change of sign implies a movement {\em across the zenith}. This
352so-called {\it reverse mode} is currently not in use, as it might result
353in hysteresis effects of the active mirror control system, still under
354investigation, due to shifting of weight at zenith. The accessible
355range in both directions and on both axes is limited by software to the
356mechanically accessible range. For additional safety, hardware end
357switches are installed directly connected to the drive controller
358units, initiating a fast, controlled deceleration of the system when
359activated. To achieve an azimuthal movement range exceeding
360360\textdegree{}, one of the two azimuth end-switches needs to be
361deactivated at any time. Therefore, an additional {\em direction
362switch} is located at $\varphi$\,=\,164\textdegree{}, short-circuiting
363the end switch currently out of range.
365\section{Setup of the motion control system}\label{sec3}
367The motion control system similarly uses standard industry components.
368The drive is controlled by the feedback of encoders measuring the
369angular positions of the motors and the telescope axes. The encoders on
370the motor axes provide information to micro controllers dedicated for
371motion control, initiating and monitoring every movement. Professional
372built-in servo loops take over the suppression of oscillations. The
373correct pointing position of the system is ensured by a computer
374program evaluating the feedback from the telescope axes and initiating
375the motion executed by the micro controllers. Additionally, the
376motor-axis encoders are also evaluated to increase accuracy. The
377details of this system, as shown in figure~\ref{figure2}, are discussed
382 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure2a.eps}
383 \hfill
384 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure2b.eps}
385\caption{Schematics of the MAGIC\,I ({\em left}) and MAGIC\,II ({\em
386right}) drive system. The sketches shows the motors, the motor-encoder
387feedback as well as the shaft-encoder feedback, and the motion-control
388units, which are themselves controlled by a superior control, receiving
389commands from the control PC, which closes the position-control loop.
390The system is described in more details in section~\ref{sec3}.}
395\subsection{Position feedback system}
397The angular telescope positions are measured by three shaft-encoders
398(type designation Hengstler\footnote{\url{}\\Hengstler GmbH,
39978554 Aldingen, Germany} AC61/1214EQ.72OLZ). These absolute multi-turn
400encoders have a resolution of 4\,096 (10\,bit) revolutions and 16\,384
401(14\,bit) steps per revolution, corresponding to an intrinsic angular
402resolution of 1.3$^\prime$ per step. One shaft encoder is located on
403the azimuth axis, while two more encoders are fixed on either side of
404the elevation axis, increasing the resolution and allowing for
405measurements of the twisting of the dish (fig.~\ref{figure3}). All
406shaft encoders used are watertight (IP\,67) to withstand the extreme
407weather conditions occasionally encountered at the telescope site. The
408motor positions are read out at a frequency of 1\,kHz from 10\,bit
409relative rotary encoders fixed on the motor axes. Due to the gear ratio
410of more than one thousand between motor and load, the 14\,bit
411resolution of the shaft encoder system on the axes can be interpolated
412further using the position readout of the motors. For communication
413with the axis encoders, a CANbus interface with the CANopen protocol is
414in use (operated at 125\,kbps). The motor encoders are directly
415connected by an analog interface.
417\subsection{Motor control}
419The three servo motors are connected to individual motion controller
420units ({\em DKC}, type designation Bosch Rexroth,
421DKC~ECODRIVE\,03.3-200-7-FW), serving as intelligent frequency
422converters. An input value, given either analog or
423digital, is converted to a predefined output, e.g., command position,
424velocity or torque. All command values are processed through a chain of
425built-in controllers, cf. fig.~\ref{figure4}, resulting in a final
426command current applied to the motor. This internal chain of control
427loops, maintaining the movement of the motors at a frequency of
4281\,kHz, fed back by the rotary encoders on the corresponding motor axes.
429Several safety limits ensure damage-free operation of the system even
430under unexpected operation conditions. These safety limits are, e.g.,
431software end switches, torque limits, current limits or
432control-deviation limits.
434To synchronize the two azimuth motors, a master-slave setup is
435used. While the master is addressed by a command velocity, the
436slave is driven by the command torque output of the master. This
437operation mode ensures that both motors can apply their combined force
438to the telescope structure without oscillations. In principle it is
439possible to use a bias torque to eliminate play. This feature
440was not used because the play is negligible anyhow.
444 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure3.eps}
445\caption{The measured difference between the two shaft-encoders fixed
446on either side of the elevation axis versus zenith angle. Negative
447zenith angles mean that the telescope has been flipped over the zenith
448position to the opposite side. The average offset from zero
449corresponds to a the twist of the two shaft encoders with respect to
450each other. The error bars denote the spread of several measurements.
451Under normal conditions the torsion between both ends of
452the axis is less than the shaft-encoder resolution.}
457\subsection{Motion control}
460 \includegraphics*[width=0.78\textwidth,angle=0,clip]{figure4.eps}
461 \caption{The internal flow control between the individual controllers
462inside the drive control unit. Depending on the type of the command
463value, different controllers are active. The control loops are closed by
464the feedback of the rotary encoder on the motor, and a possible
465controller on the load axis, as well as the measurement of the current.}
470% zub machine control AG, Kastaniensteif 7, CH-6047 Kastanienbaum
473The master for each axis is controlled by presetting a rotational
474speed defined by $\pm$10\,V on its analog input. The input voltage is
475produced by a programmable micro controller dedicated to analog motion
476control, produced by Z\&B\footnote{\url{}\\zub machine control AG, 6074
477Kastanienbaum, Switzerland} ({\em MACS}, type
478designation MACS). The feedback is realized through a 500-step
479emulation of the motor's rotary encoders by the DKCs. Elevation and azimuth movement is regulated by individual
480MACSs. The MACS controller itself communicates with the control
481software (see below) through a CANbus connection.
483It turned out that in particular the azimuth motor system seems to be
484limited by the large moment of inertia of the telescope
485($J_{\mathrm{az}}$\,$\approx$\,4400\,tm$^2$, for comparison
486$J_{\mathrm{el}}$\,$\approx$\,850\,tm$^2$; note that the exact numbers depend
487on the current orientation of the telescope). At the same time, the
488requirements on the elevation drive are much less demanding.\\
490\noindent {\em MAGIC\,II}\quad For the drive system
491several improvements have been provided:\\\vspace{-2ex}
493\item 13\,bit absolute shaft-encoders (type designation Heidenhain\footnote{\url{}\\Dr.~Johannes Heidenhain GmbH, 83301 Traunreut, Germany}
494ROQ\,425) are installed, providing an additional sine-shaped
495$\pm$1\,Vss output within each step. This allows for a more accurate
496interpolation and hence a better resolution than a simple 14\,bit
497shaft-encoder. These shaft-encoders are also water tight (IP\,64), and
498they are read out via an EnDat\,2.2 interface.
499\item All encoders are directly connected to the DKCs, providing
500additional feedback from the telescope axes itself. The DKC can control
501the load axis additionally to the motor axis providing a more accurate
502positioning, faster movement by improved oscillation suppression and a
503better motion control of the system.
504\item The analog transmission of the master's command torque to the
505slave is replaced by a direct digital communication (EcoX)
506of the DKCs. This allows for more robust and precise slave control.
507Furthermore the motors could be coupled with relative angular synchronism
508allowing to suppress deformations of the structure by keeping the
509axis connecting both motors stable.
510\item A single professional programmable logic controller (PLC), in German:
511{\em Speicherprogammierbare Steuerung} (SPS, type designation Rexroth
512Bosch, IndraControl SPS L\,20) replaces the two MACSs. Connection between
513the SPS and the DKCs is now realized through a digital Profibus DP
514interface substituting the analog signals.
515\item The connection from the SPS to the control PC is realized via
516Ethernet connection. Since Ethernet is more commonly in use than
517CANbus, soft- and hardware support is much easier.
520\subsection{PC control}
522The drive system is controlled by a standard PC running a Linux
523operating system, a custom-designed software based on
524ROOT~\citep{www:root} and the positional astronomy library
525{\em slalib}~\citep{slalib}.
527Algorithms specialized for the MAGIC tracking system are imported from
528the Modular Analysis and Reconstruction Software package
529(MARS)~\citep{Bretz:2003icrc, Bretz:2005paris, Bretz:2008gamma} also
530used in the data analysis~\citep{Bretz:2005mars, Dorner:2005paris}.
534Whenever the telescope has to be positioned, the relative distance to
535the new position is calculated in telescope coordinates and then
536converted to motor revolutions. Then, the micro controllers are
537instructed to move the motors accordingly. Since the motion is
538controlled by the feedback of the encoders on the motor axes, not on
539the telescope axes, backlash and other non-deterministic irregularities
540cannot easily be taken into account. Thus it may happen that the final
541position is still off by a few shaft-encoder steps, although the motor
542itself has reached its desired position. In this case, the procedure is
543repeated up to ten times. After ten unsuccessful iterations, the system
544would go into error state. In almost all cases the command position is
545reached after at most two or three iterations.
547If a slewing operation is followed by a tracking operation of a
548celestial target position, tracking is started immediately after the
549first movement without further iterations. Possible small deviations,
550normally eliminated by the iteration procedure, are then corrected by
551the tracking algorithm.
554To track a given celestial target position (RA/Dec, J\,2000.0,
555FK\,5~\citep{1988VeARI..32....1F}), astrometric and misalignment
556corrections have to be taken into account. While astrometric
557corrections transform the celestial position into local coordinates as
558seen by an ideal telescope (Alt/Az), misalignment corrections convert
559them further into the coordinate system specific to the real telescope.
560In case of MAGIC, this coordinate system is defined by the position
561feedback system.
563The tracking algorithm controls the telescope by applying a command
564velocity for the revolution of the motors, which is re-calculated every
565second. It is calculated from the current feedback position and the
566command position required to point at the target five seconds ahead in
567time. The timescale of 5\,s is a compromise between optimum tracking
568accuracy and the risk of oscillations in case of a too short timescale.
570As a crosscheck, the ideal velocities for the two telescope axes are
571independently estimated using dedicated astrometric routines of slalib.
572For security reasons, the allowable deviation between the determined
573command velocities and the estimated velocities is limited. If an
574extreme deviation is encountered the command velocity is set to zero,
575i.e., the movement of the axis is stopped.
577\subsection{Fast positioning}
579The observation of GRBs and their afterglows in very-high energy
580gamma-rays is a key science goal for the MAGIC telescope. Given that
581alerts from satellite monitors provide GRB positions a few seconds
582after their outburst via the {\em Gamma-ray Burst Coordination
583Network}~\cite{www:gcn}, typical burst durations of 10\,s to
584100\,s~\cite{Paciesas:1999} demand a fast positioning of the
585telescope. The current best value for the acceleration has been set to
58611.7\,mrad\,s$^{-2}$. It is constrained by the maximum constant force
587which can be applied by the motors. Consequently, the maximum allowed
588velocity can be derived from the distance between the end-switch
589activation and the position at which a possible damage to the telescope
590structure, e.g.\ ruptured cables, would happen. From these constraints,
591the maximum velocity currently in use, 70.4\,mrad\,s$^{-1}$, was
592determined. Note that, as the emergency stopping distance evolves
593quadratically with the travel velocity, a possible increase of the
594maximum velocity would drastically increase the required braking
595distance. As safety procedures require, an emergency stop is completely
596controlled by the DKCs itself with the feedback of the motor encoder,
597ignoring all other control elements.
599Currently, automatic positioning by
600$\Delta\varphi$\,=\,180\textdegree{} in azimuth to the target position
601is achieved within 45\,s. The positioning time in elevation is not
602critical in the sense that the probability to move a longer path in
603elevation than in azimuth is negligible. Allowing the telescope drive
604to make use of the reverse mode, the requirement of reaching any
605position in the sky within 30\,s is well met, as distances in azimuth
606are substantially shortened. The motor specifications allow for a
607velocity more than four times higher. In practice, the maximum possible
608velocity is limited by the acceleration force, at slightly more than
609twice the current value. The actual limiting factor is the braking
610distance that allows a safe deceleration without risking any damage to
611the telescope structure.
613With the upgraded MAGIC\,II drive system, during commissioning in 2008 August, a
614maximum acceleration and deceleration of $a_{az}$\,=\,30\,mrad\,s$^{-2}$
615and $a_{zd}$\,=\,90\,mrad/s$^{-2}$ and a maximum velocity of
616$v_{az}$\,=\,290\,mrad\,s$^{-1}$ and $v_{zd}$\,=\,330\,mrad\,s$^{-1}$
617could be reached. With these values the limits of the motor power are
618exhausted. This allowed a movement of
619$\Delta\varphi$\,=\,180\textdegree/360\textdegree{} in azimuth within 20\,s\,/\,33\,s.
621\subsection{Tracking precision}
623The intrinsic mechanical accuracy of the tracking system is determined
624by comparing the current command position of the axes with the feedback
625values from the corresponding shaft encoders. These feedback values
626represent the actual position of the axes with highest precision
627whenever they change their feedback values. At these instances, the
628control deviation is determined, representing the precision with which
629the telescope axes can be operated. In the case of an ideal mount this
630would define the tracking accuracy of the telescope.
632In figure~\ref{figure5} the control deviation measured for 10.9\,h of
633data taking in the night of 2007 July 22/23 and on the evening of July
63423 is shown, expressed as absolute deviation on the sky taking both
635axes into account. In almost all cases it is well below the resolution
636of the shaft encoders, and in 80\% of the time it does not exceed 1/8
637of this value ($\sim$10$^{\prime\prime}$). This means that the accuracy of the
638motion control, based on the encoder feedback, is much better than
6391$^\prime$ on the sky, which is roughly a fifth of the diameter of a
640pixel in the MAGIC camera (6$^\prime$, c.f.~\cite{Beixeras:2005}).
643 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure5.eps}
644 \caption{Control deviation between the expected, i.e. calculated,
645position, and the feedback position of the shaft encoders in the moment
646at which one change their readout values. For simplicity, the control
647deviation is shown as absolute control deviation projected on the sky.
648The blue lines correspond to fractions of the shaft-encoder resolution.
649The peak at half of the shaft-encoder resolution results from cases
650in which one of the two averaged elevation encoders is off by one step.
656In the case of a real telescope ultimate limits of the tracking
657precision are given by the precision with which the correct command
658value is known. Its calibration is discussed hereafter.
662To calibrate the position command value, astrometric corrections
663(converting the celestial target position into the target position of
664an ideal telescope) and misalignment corrections (converting it further
665into the target position of a real telescope), have to be taken into
668\subsection{Astrometric corrections}
670The astrometric correction for the pointing and tracking algorithms is
671based on a library for calculations usually needed in positional
672astronomy, {\em slalib}~\cite{slalib}. Key features of this library are
673the numerical stability of the algorithms and their well-tested
674implementation. The astrometric corrections in use
675(fig.~\ref{figure6}) -- performed when converting a celestial position
676into the position as seen from Earth's center (apparent position) --
677take into account precession and nutation of the Earth and annual
679\begin{center} % Goldener Schnitt
680 \includegraphics*[width=0.185\textwidth,angle=0,clip]{figure6.eps}
681\caption{The transformation applied to a given set of catalog source
682coordinates to real-telescope coordinates. These corrections include
683all necessary astrometric corrections, as well as the pointing
684correction to transform from an ideal-telescope frame to the frame of a
685real telescope.
686%A detailed description of all corrections and the
687%calibration of the pointing model is given in section~\ref{sec4}.
692aberration, i.e., apparent displacements caused by the finite speed of
693light combined with the motion of the observer around the Sun during
694the year. Next, the apparent position is transformed to the observer's
695position, taking into account atmospheric refraction, the Earth's
696rotation, and diurnal aberration, i.e., the motion of the observer
697around the Earth's rotation axis. Some of these effects are so small
698that they are only relevant for nearby stars and optical astronomy. But
699as optical observations of such stars are used to {\em train} the
700misalignment correction, all these effects are taken into account.
702\subsection{Pointing model}
704Imperfections and deformations of the mechanical construction lead to
705deviations from an ideal telescope, including the non-exact
706alignment of axes, and deformations of the telescope structure.
709In the case of the MAGIC telescopes the optical axis of the mirror is
710defined by an automatic alignment system. This active mirror control
711is programmed not to change the optical axis once defined, but only
712controls the optical point spread function of the mirror, i.e., it does
713not change the center of gravity of the light distribution.
714This procedure is applied whenever the telescope is observing including
715any kind of calibration measurement for the drive system. The precision
716of the axis alignment of the mirrors is better than 0.2$^\prime$ and can
717therefor be neglected.
720%To assure reliable pointing and tracking accuracy, such effects have to
721%be taken into account.
722Consequently, to assure reliable pointing and tracking accuracy,
723mainly the mechanical effects have to be taken into account.
724Therefore the tracking software employs an analytical pointing model
725based on the {\rm TPOINT}\texttrademark{} telescope modeling
726software~\cite{tpoint}, also used for optical telescopes. This model,
727called {\em pointing model}, parameterizes deviations from the ideal
728telescope. Calibrating the pointing model by mispointing measurements
729of bright stars, which allows to determine the necessary corrections,
730is a standard procedure. Once calibrated, the model is applied online.
731Since an analytical model is used, the source of any deviation can be
732identified and traced back to components of the telescope mount.\\
734Corrections are parameterized by alt-azimuthal terms~\cite{tpoint},
735i.e., derived from vector transformations within the proper coordinate
736system. The following possible misalignments are taken into account:\\\vspace{-2ex}
738\item[Zero point corrections ({\em index errors})] Trivial offsets
739between the zero positions of the axes and the zero positions of the
740shaft encoders.
741\item[Azimuth axis misalignment] The misalignment of the azimuth axis
742in north-south and east-west direction, respectively. For MAGIC these
743corrections can be neglected.
744\item[Non-perpendicularity of axes] Deviations from right angles
745between any two axes in the system, namely (1) non-perpendicularity of
746azimuth and elevation axes and (2) non-perpendicularity of elevation
747and pointing axes. In the case of the MAGIC telescope these corrections
748are strongly bound to the mirror alignment defined by the active mirror
750\item[Non-centricity of axes] The once-per-revolution cyclic errors
751produced by de-centered axes. This correction is small, and thus difficult
752to measure, but the most stable correction throughout the years.
755\noindent{\bf Bending of the telescope structure}
757\item A possible constant offset of the mast bending.
758\item A zenith angle dependent correction. It describes the camera mast
759bending, which originates by MAGIC's single thin-mast camera support
760strengthened by steel cables.
761\item Elevation hysteresis: This is an offset correction introduced
762depending on the direction of movement of the elevation axis. It is
763necessary because the sliding bearing, having a stiff connection with
764the encoders, has such a high static friction that in case of reversing
765the direction of the movement, the shaft-encoder will not indicate any
766movement for a small and stable rotation angle, even though the
767telescope is rotating. Since this offset is stable, it can easily
768be corrected after it is fully passed. The passage of the hysteresis
769is currently corrected offline only.
773Since the primary feedback is located on the axis itself, corrections
774for irregularities of the chain mounting or sprocket wheels are
775unnecessary. Another class of deformations of the telescope-support
776frame and the mirrors are non-deterministic and, consequently, pose an
777ultimate limit of the precision of the pointing.
779% Moved to results
783%\subsubsection{Calibration concept}
785To determine the coefficients of a pointing model, calibration
786data is recorded. It consists of mispointing measurements depending
787on altitude and azimuth angle. Bright stars are tracked with the
788telescope at positions uniformly distributed in local coordinates,
789i.e., in altitude and azimuth angle. The real pointing
790position is derived from the position of the reflection of a bright
791star on a screen in front of the MAGIC camera. The center
792of the camera is defined by LEDs mounted on an ideal ($\pm$1\,mm)
793circle around the camera center, cf.~\citet{Riegel:2005icrc}.
795Having enough of these datasets available, correlating ideal and real
796pointing position, a fit of the coefficients of the model can be made,
797minimizing the pointing residual.
799\subsubsection{Hardware and installations}
801A 0.0003\,lux, 1/2\(^{\prime\prime}\) high-sensitivity standard PAL CCD
802camera (type designation \mbox{Watec}~WAT-902\,H) equipped with a zoom lens (type: Computar) is used
803for the mispointing measurements. The camera is read out at a rate of
80425\,frames per second using a standard frame-grabber card in a standard
805PC. The camera has been chosen providing adequate performance and
806easy readout, due to the use of standard components, for a very cheap price
807($<$\,500\,Euro). The tradeoff for the high sensitivity of the camera is its high
808noise level in each single frame recorded. Since there are no rapidly
809moving objects within the field of view, a high picture quality can be
810achieved by averaging typically 125\,frames (corresponding to 5\,s). An
811example is shown in figure~\ref{figure7}. This example also illustrates
812the high sensitivity of the camera, since both pictures of the
813telescope structure have been taken with the residual light of less
814than a half-moon. In the background individual stars can be seen.
815Depending on the installed optics, stars up to 12$^\mathrm{m}$ are
816visible. With our optics and a safe detection threshold the limiting
817magnitude is typically slightly above 9$^\mathrm{m}$ for direct
818measurements and on the order of 5$^\mathrm{m}$\dots4$^\mathrm{m}$ for
819images of stars on the screen.
825%Model&Watec WAT-902H (CCIR)\\\hline\hline
826%Pick-up Element&1/2" CCD image sensor\\
827%&(interline transfer)\\\hline
828%Number of total pixels&795(H)x596(V)\\\hline
829%Minimum Illumination&0.0003\,lx. F/1.4 (AGC Hi)\\
830%&0.002\,lx. F/1.4 (AGC Lo)\\\hline
831%Automatic gain&High: 5\,dB\,-\,50\,dB\\
832%&Low: 5\,dB\,-\,32\,dB\\\hline
833%S/N&46\,dB (AGC off)\\\hline
834%Shutter Speed&On: 1/50\,-\,1/100\,000\,s\\ (electronic iris)&Off: 1/50\,s\\\hline
835%Backlight compensation&On\\\hline
836%Power Supply&DC +10.8\,V\,-\,13.2\,V\\\hline
837%Weight&Approx. 90\,g\\\hline
840%\caption{Technical specifications of the CCD camera used for
841%measuring of the position of the calibration stars on the PM camera lid.}
846 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure7a.eps}
847 \hfill
848 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure7b.eps}
849 \caption{A single frame (left) and an average of 125 frames (right) of
850the same field of view taken with the high sensitivity PAL CCD camera
851used for calibration of the pointing model. The frames were taken
852with half moon. }
859An example of a calibration-star measurement is shown in
860figure~\ref{figure8}. Using the seven LEDs mounted on a circle around
861the camera center, the position of the camera center is determined.
862Only the upper half of the area instrumented is visible, since
863the lower half is covered by the lower lid, holding a special
864reflecting surface in the center of the camera. The LED positions are
865evaluated by a simple cluster-finding algorithm looking at pixels more
866than three standard deviations above the noise level. The LED position
867is defined as the center of gravity of its light distribution, its
868search region by the surrounding black-colored boxes. For
869simplicity the noise level is determined just by calculating the mean
870and the root-mean-square within the individual search regions below a
871fixed threshold dominated by noise.
875 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure8.eps}
876\caption{A measurement of a star for the calibration of the pointing
877model. Visible are the seven LEDs and their determined center of
878gravity, as well as the reconstructed circle on which the LEDs are
879located. The LEDs on the bottom part are hidden by the lower lid,
880holding a screen in front of the camera.
881%For calibration, the center
882%of gravity of the measured star, as visible in the center, is compared
883%to the center of the circle given by the LEDs, coinciding with the
884%center of the PM camera.
885The black regions are the search regions for
886the LEDs and the calibration star. A few dead pixels in the CCD camera
887can also be recognized.}
892Since three points are enough to define a circle, from all
893possible combinations of detected spots, the corresponding circle
894is calculated. In case of misidentified LEDs, which sometimes
895occur due to light reflections from the telescope structure, the
896radius of the circle will deviate from the predefined radius.
897Thus, any such misidentified circles are discarded. The radius
898determination can be improved further by applying small offsets of
899the non-ideal LED positions. The radius distribution is Gaussian
900and its resolution is $\sigma$\,$\apprle$\,1\,mm
901($\mathrm{d}r/r\approx0.3$\textperthousand) on the camera plane
902corresponding to $\sim$1$^{\prime\prime}$.
904The center of the ring is calculated as the average of all circle
905centers after quality cuts. Its resolution is
906$\sim$2$^{\prime\prime}$. In this setup, the large number of LEDs guarantees
907operation even in case one LED could not be detected due to damage or
908scattered light.
910To find the spot of the reflected star itself, the same cluster-finder
911is used to determine its center of gravity. This gives reliable results
912even in case of saturation. Only very bright stars, brighter than
9131.0$^m$, are found to saturate the CCD camera asymmetrically.
915Using the position of the star, with respect to the camera center, the
916pointing position corresponding to the camera center is calculated.
917This position is stored together with the readout from the position
918feedback system. The difference between the telescope pointing
919position and the feedback position is described by the pointing model.
920Investigating the dependence of these differences on zenith and azimuth
921angle, the correction terms of the pointing model can be determined.
922Its coefficients are fit minimizing the absolute residuals on the celestial
927Figure~\ref{figure9} shows the residuals, taken between 2006 October and
9282007 July, before and after application of the fit of the pointing
929model. For convenience, offset corrections are applied to the residuals
930before correction. Thus, the red curve is a measurement of the alignment
931quality of the structure, i.e., the pointing accuracy with offset
932corrections only. By fitting a proper model, the pointing accuracy can
933be improved to a value below the intrinsic resolution of the system,
934i.e., below shaft-encoder resolution. In more than 83\% of all cases the
935tracking accuracy is better than 1.3$^\prime$
936and it hardly ever exceeds 2.5$^\prime$. The few datasets exceeding
9372.5$^\prime$ are very likely due to imperfect measurement of the
938real pointing position of the telescope, i.e., the center of gravity of
939the star light.
941The average absolute correction applied (excluding the index error) is on
942the order of 4$^\prime$. Given the size, weight and structure of
943the telescope this proves a very good alignment and low sagging of the
944structure. The elevation hysteresis, which is intrinsic to the
945structure, the non-perpendicularity and non-centricity of the axes are
946all in the order of 3$^\prime$, while the azimuth axis
947misalignment is $<$\,0.6$^\prime$. These numbers are well in agreement
948with the design tolerances of the telescope.
952 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure9.eps}
953 \caption{Distribution of absolute pointing residual on the sky between
954the measured position of calibration stars and their nominal position
955with only offset correction for both axes (red) and a fitted pointing
956model (blue) applied. Here in total 1162 measurements where used,
957homogeneously distributed over the local sky. After application of the
958pointing model the residuals are well below the shaft-encoder
959resolution, i.e., the knowledge of the mechanical position of the axes.
968The ultimate limit on the achievable pointing precision are effects,
969which are difficult to correlate or measure, and non-deterministic
970deformations of the structure or mirrors. For example, the azimuth
971support consists of a railway rail with some small deformations in
972height due to the load, resulting in a wavy movement difficult to
973parameterize. For the wheels on the six bogeys, simple, not precisely
974machined crane wheels were used, which may amplify horizontal
975deformations. Other deformations are caused by temperature changes and
976wind loads which are difficult to control for telescopes without dome,
977and which cannot be modeled. It should be noted that the azimuth structure
978can change its diameter by up to 3\,cm due to day-night temperature
979differences, indicating that thermal effects have a non-negligible and
980non-deterministic influence.
982Like every two axis mount, also an alt-azimuth mount has a blind spot
983near its upward position resulting from misalignments of the axis which
984are impossible to correct by moving one axis or the other. From the
985size of the applied correction it can be derived that the blind spot
986must be on the order of $\lesssim$\,6$^\prime$ around zenith.
987Although the MAGIC drive system is powerful enough to keep on track
988pointing about 6$^\prime$ away from zenith, for safety reasons, i.e.,
989to avoid fast movment under normal observation conditions, the observation
990limit has been set to $\theta$\,$<$\,30$^\prime$. Such fast movements
991are necessary to change the azimuth position from moving the telescope
992upwards in the East to downwards in the South. In the case of an ideal
993telescope, pointing at zenith, even an infinitely fast movement would
994be required.
998With each measurement of a calibration-star also the present pointing
999uncertainty is recorded. This allows for monitoring of the pointing
1000quality and for offline correction. In figure~\ref{figure10} the
1003 \includegraphics*[width=0.48\textwidth,angle=0,clip]{figure10.eps}
1004\caption{The distribution of mispointing measurements. The
1005measurement is a direct measurement of the pointing accuracy. The plot
1006shows its time-evolution. Details on the bin edges and the available
1007statistics is given in the caption of table~\ref{table2}. Since the
1008distribution is asymmetric, quantiles are shown, from bottom to top, at
10095\%, 13\%, 32\%, 68\%, 87\% and 95\%. The dark grey region belong to
1010the region between quantiles 32\% and 68\%. } 
1014evolution of the measured residuals over the years are shown. The
1015continuous monitoring has been started in March 2005 and is still
1016ongoing. Quantiles are shown since the distribution can be
1017asymmetric depending on how the residuals are distributed on the sky. The
1018points have been grouped, where the grouping reflects data taken under
1019the same conditions (pointing model, mirror alignment, etc.). It should
1020be noted, that the measured residuals depend on zenith and azimuth
1021angle, i.e., the distributions shown are biased due to inhomogeneous
1022distributions on the sky in case of low statistics. Therefore the
1023available statistics is given in table~\ref{table2}.
10282005/03/20&29\\%&  2005/11/24&38\\
10292005/04/29&43\\%&  2006/03/19&502\\
10302005/05/25&30\\%&  2006/10/17&827\\
10312005/06/08&26\\%&  2007/07/31&87\\
10322005/08/15&160\\%& 2008/01/14&542\\
10332005/09/12&22\\\hline%&  2008/06/18&128\\\hline
1046\caption{Available statistics corresponding to the distributions
1047shown in figure~\ref{figure10}. Especially in cases of low statistics
1048the shown distribution can be influenced by inhomogeneous distribution
1049of the measurement on the local sky. The dates given correspond to dates
1050for which a change in the pointing accuracy, as for example a change to
1051the optical axis or the application of a new pointing model, is known.}
1055The mirror focusing can influence the alignment of the optical axis of
1056the telescope, i.e., it can modify the pointing model. Therefore a
1057calibration of the mirror refocusing can worsen the tracking accuracy,
1058later corrected by a new pointing model. Although the automatic mirror
1059control is programmed such that a new calibration should not change the
1060center of gravity of the light distribution, it happened sometimes in
1061the past due to software errors.
1063The determination of the pointing model also relies on a good
1064statistical basis, because the measured residuals are of a similar
1065magnitude as the accuracy of a single calibration-star measurement. The
1066visible improvements and deterioration are mainly a consequence of new
1067mirror focusing and following implementations of new pointing models.
1068The improvement over the past year is explained by the gain in
1071On average the systematic pointing uncertainty was always better than
1072three shaft-encoder steps (corresponding to 4$^\prime$), most of the
1073time better than 2.6$^\prime$ and well below one shaft-encoder step,
1074i.e.\ 1.3$^\prime$, in the past year. Except changes to the pointing
1075model or the optical axis, as indicated by the bin edges, no
1076degradation or change with time of the pointing model or
1077a worsening of the limit given by the telescope mechanics could be found.
1081With the aim to reach lower energy thresholds, the next generation of
1082Cherenkov telescopes will also include larger and heavier ones.
1083Therefore more powerful drive systems will be needed. The scalable
1084drive system of the MAGIC telescope is suited to meet this challenge.
1085With its synchronous motors and their master-slave setup, it can easily
1086be extended to larger telescopes at moderate costs, or even scaled down
1087to smaller ones using less powerful components. Consequently,
1088telescopes in future projects, with presumably different sizes, can be
1089driven by similar components resulting in a major
1090simplification of maintenance. With the current setup, a tracking
1091accuracy at least of the order of the shaft-encoder resolution is
1092guaranteed. Pointing accuracy -- already including all possible
1093pointing corrections -- is dominated by dynamic and unpredictable
1094deformations of the mount, e.g., temperature expansion.
1098Currently, efforts are ongoing to implement the astrometric subroutines
1099as well as the application of the pointing model directly into the
1100Programmable Logic Controller. A first test will be carried out within
1101the DWARF project soon~\cite{DWARF}. The direct advantage is that the
1102need for a control PC is omitted. Additionally, with a more direct
1103communication between the algorithms, calculating the nominal position
1104of the telescope mechanics, and the control loop of the drive
1105controller, a real time, and thus more precise, position control can be
1106achieved. As a consequence, the position controller can directly be
1107addressed, even when tracking, and the outermost position control-loop
1108is closed internally in the drive controller. This will ensure an even
1109more accurate and stable motion. Interferences from external sources,
1110e.g. wind gusts, could be counteracted at the moment of appearance by
1111the control on very short timescales, on the order of milli-seconds. An
1112indirect advantage is that with a proper setup of the control loop
1113parameters, such a control is precise and flexible enough that a
1114cross-communication between the master and the slaves can also be
1115omitted. Since all motors act as their own master, in such a system a
1116broken motor can simply be switched off or mechanically decoupled
1117without influencing the general functionality of the system.
1119An upgrade of the MAGIC\,I drive system according to the improvements
1120applied for MAGIC\,II is ongoing.
1124The scientific requirements demand a powerful, yet accurate drive
1125system for the MAGIC telescope. From its hardware installation and
1126software implementation, the installed drive system exceeds its design
1127specifications as given in section~\ref{design}. At the same time the
1128system performs reliably and stably, showing no deterioration after
1129five years of routine operation. The mechanical precision of the motor
1130movement is almost ten times better than the readout on the telescope
1131axes. The tracking accuracy is dominated by random deformations and
1132hysteresis effects of the mount, but still negligible
1133compared to the measurement of the position of the telescope axes. The
1134system features integrated tools, like an analytical pointing model.
1135Fast positioning for gamma-ray burst followup is achieved on average
1136within less than 45 seconds, or, if movements {\em across the zenith}
1137are allowed, 30 seconds. Thus, the drive system makes
1138MAGIC the best suited telescope for observations of these phenomena at
1139very high energies.
1141For the second phase of the MAGIC project and particularly for the
1142second telescope, the drive system has been further improved.
1143By design, the drive system is easily scalable from its current
1144dimensions to larger and heavier telescope installations as required
1145for future projects. The improved stability is also expected to meet
1146the stability requirements, necessary when operating a larger number of
1150The authors acknowledge the support of the MAGIC collaboration, and
1151thank the IAC for providing excellent working conditions at the
1152Observatorio del Roque de los Muchachos. The MAGIC project is mainly
1153supported by BMBF (Germany), MCI (Spain), INFN (Italy). We thank the
1154construction department of the MPI for Physics, for their help in the
1155design and installation of the drive system as well as Eckart Lorenz,
1156for some important comments concerning this manuscript. R.M.W.\
1157acknowledges financial support by the MPG. His research is also
1158supported by the DFG Cluster of Excellence ``Origin and Structure of
1159the Universe''.
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