**This is not about GPS ( Global Positioning System), the mobile phone-sized device that uses satellites to tell us exactly where we are by providing information like coordinates and altitude.**

It is about the GTC's precision encoders, devices that will measure movement and translate it into electronic signals that will determine the position of the telescope or any of its components. This is about a very precise instrument (as its name implies) which is accurate to within nanometers.

**What is a precision encoder and why do you need one?**

Imagine that you are in a circular, well-lit room. The walls are decorated with a thin band of gold-coloured metal tape. As you approach it, you realise that the tape is engraved with numbers and fine lines. The lines are 40 microns apart (a speck of dust is larger than a micron). Imagine that, as you walk around this circular room looking at the tape and counting the lines...you are doing the same thing as the reader heads of a high precision angle encoder.

Fortunately we don't have to worry about carrying out such complicated tasks ourselves. We now have the technology to design and build instruments that can tell us, to within nanometres, the position of objects that we want to control. Broadly then, an encoder is a device that translates movement into one or more electronic signals which are used to determine position.

The subsystems of the Gran Telescopio CANARIAS (GTC), like its mirrors, dome, instruments and the telescope itself, will use several different types of encoder. They will be used to control numerous different parts that will all be moving at the same time. For now though, we are going to focus on the encoders that will be used for the telescope structure. These will be angle encoders and they will be mounted at the azimuth and elevation axes and the Acquisition and Guide system.

The encoders, which are "reticulated optical encoders", have already been built by the German Hendenhein company. "Optical" refers to the photoelectric system they use to read the tape, and "reticulated" to the pattern of fine lines engraved on it.

**THE TAPES**

High precision encoders are made up of tapes and tape head readers. Fine lines known as incremental, absolute or reference marks are engraved on the tapes. Incremental marks are equidistant, there are 1000 of them every 4cm and they are just 40 microns apart. Just beneath the band of incremental marks are the absolute marks: these are small groups of reference marks that occur every 5º. On average, one absolute mark occurs every 500 incremental marks.

The incremental marks tell us how much we have moved and the absolutes where we are now. To understand what each of them is used for we can go back to the circular room, where we can turn through 0º to 360º. Imagine that we start moving at a point x which is to the right of the room. If our starting position is 70º and we move 5º, the incremental encoder will tell us that our current position is 5º: it will only give us the angle we have moved through since we started to move. However, as we started moving at 70º we need to know that our position is now 70º+5º=75º. How do we do this?

This is the job of the absolute marks: these will tell us that we were at 70º to begin with. When we then move 5º, the absolute encoder will register our position as 75º. They can do this because they are all slightly different and are distributed in such a way that, with a small initial movement, they can tell us exactly where we are on the circle.

**READER HEADS**

The reader heads have the job of providing information about speed and position.

Each of the reader heads has an LED (*Light Emitting Diode*) which lights up the part of the tape on which the marks are engraved. The light from the LED shines on the marks and is reflected to photocells which convert it into varying electrical signals. In this case the signals will vary sinusoidally (so called because the shape of the waves is consistent with the mathematical function SENO: see images). These signals are sent to electronic cards which interpolate them - that is, each cycle of the sinusoid is subdivided into up to 4096 points to give greater resolution. The speed at which we are moving is measured using the frequency of these sinusoids.

The reader heads at the azimuth axis need to be very finely adjusted so that they neither come into contact with, nor are too far away from, the tape. They need to be at a constant distance of approximately 2mm from it. The reader heads are located in the mobile section, the rotating platform that supports the telescope structure. The tape, which is some 50 metres long, is in the non-mobile section.

**RESOLUTION AND PRECISION**

In our society the concept of error is seen as negative, but in reality errors are what drive the progress of science. Error leads to correction - you could say that it inspires the search for perfection. From the design and construction phases of the encoders right through to their installation and calibration, every tiny error will be measured so that it can be controlled and so that we will always know what their margin of error is.

What resolution can be obtained using the high precision encoders? Resolution equates to the minimum increase in magnitude that the measuring system is capable of resolving, which in our system is determined by the separation between the incremental marks. In theory, this is approximately 0.2 mili-arcsecs (see the mathematical exercise at the end of this article). The resolution needed at the azimuth axis for the movement of the telescope will be higher than 0.8 mili-arcsecs, a requirement that our encoder will easily meet.

The system can only be as accurate as the measurements taken by the measuring system. In other words, resolution is all well and good, but if our master is inaccurate then we will be too. That is why we will have to calibrate it continuously during the assembly process. One way to improve accuracy is increasing the number of reader heads. At the azimuth axis, a total of 8 positioning heads will be sited at different points. Taking the average of the values provided by each of these 8 heads will give a very precise position reading.

As you can see, with all this resolution, precision and exactness we will be as sure as we can scientifically be of where we are at all times.

**Resolution: a mathematical exercise**

An arcsec is an arcsecond, one sixtieth of an arc minute, which is in turn one sixtieth of a degree. One arcsecond is as big as a 1mm sized flea seen from 206 metres away.

The resolution needed at the azimuth axis for the movement of the telescope will be higher than 0.8 mili-arcsec.. This is how we can be sure that it will be achieved:

The system at the GTC's azimuth axis, which will be 16 metres in diameter, will have a tape **50m** long containing **1.288.000 engraved** marks. If there are **360º x 3.600** in a circle, this will give **1.296.000** arcsecs per mark.

The resolution of the tape is therefore **1.296.000** arcsecs divided by the number of marks, **1.288.000**, which gives **1,00621** arcsecs. Taking into account that the values read by the readers are interpolated by **4.096** we obtain the following theoretical resolution:

**0.00024566 arcsecs -> aprox 0.2 mili-arcsecs.**

Natalia R. Zelman