These texts are written with the single aim to clarify the working principles of telescope mounts that are driven with the characteristics of a universal joint with the goal to track celestial bodies, stars for example. We are not discussing disturbing factors that influence accuracy of tracking like atmospheric refraction. Those factors do not distinguish between telescope mounts, they are all the same.

# Axes, angles and angular velocities

The following picture shows you all axes and the moving parts that are involved to retrieve the equatorial rotation. Similar as with a universal joint there are only two axes that make complete rotations, the azimuthal axis and the polar axis. Note that there is no physical polar axis. The rotation of the telescope around the polar axis is the combined result of meticulously controlled motion of the azimuthal -, horseshoe - and fork axes.

First let's name the main parts in this concept.

*azimuthal base*(stator, red).*azimuthal ring*(full rotor, blue)- cradle (green)
- OTA (gold)

The azimuthal base supports the *azimuthal ring*. The azimuthal ring is capable to make full revolutions when the telescope is pointing to a circumpolar object. In that case, it can revolve endlessly as long as there are no limiting parts like cables attached to the mount. The *azimuthal axis* runs vertically through the centre of the azimuthal ring.

The cradle is the part that provides support for the two altitude axes. One axis makes a rectangular intersection of the planes of both horseshoes, so let us call this axis the *horseshoe axis*. The other altitude axis makes a rectangular intersection of the sides of the cradle that connect the horseshoes. This axis acts like a fork for the OTA between the sides of the cradle, so let's call it the *fork axis*.

It is obvious that it is important that these three axis must be mutually somehow rectangular. At least they must have mutual angles of view of 90°, but for our use it is not required to cross each other in a point. It is not hard to create designs in which the azimuthal axis, the fork axis and the horseshoe axis all cross each other rectangular, maybe even in a single point. That is only required in case this mount was equipped with the physical parts of a universal joint. It still is good practice to have axes cross in a point, but it can be beneficial too in order to retrieve an intrinsically balanced design to have to fork axis above the horseshoe axis.

The horseshoe and fork axes are measured relative to the cradle. The horseshoe and fork axes do not rotate, they are tilted only by no more than + or - (90° minus latitude) degrees. The larger arrows on the horseshoe and fork axes mark the first followed directions for a mount on the northern hemisphere that starts at an hour angle of 0°.

What about the *declination axis*? It is a nice property of the declination axis that it must be parallel to the fork axis. In the drafts that I presented here I chose to combine the declination axis with the fork axis to one single axis. By doing so we can avoid using a fourth motor that would be required to drive a separate declination axis. One can think of occasions were it is beneficial to have the functions of the combined fork axis split in a fork axis and a separate declination axis to enable good balance.

# Azimuthal rotation

The next figure shows you the relative azimuthal angle velocity as function of the hour angle for a mount at a latitude of 52°. That this graph is not a sine becomes clear in the third picture. Al these graphs are the result of rotation figures of ellipses. The shape of such an ellipse is easy to imagine by thinking of a cylinder that is filled with a liquid. Hold the axis of the cylinder parallel to your latitude. Now the surface of the liquid forms an ellipse with the shape that belongs to your latitude, with the major axis where the planes of the meridian and the surface level cross each other.

## Comparison of azimuthal angular velocities at different latitudes

The latitude is of great importance when it comes to determine the angles and angular velocities of the azimuthal plane. The lower your latitude, the larger the differences in momentary angular velocities that are required to maintain a uniform rotation of the polar axis. For observatories very close to the equator it even can become impossible to track the sky because of the limited speed of motors. This can be solved by changing the role of two axes: we can drive the horseshoe axis according to the azimuthal graph and the azimuthal plane with the graph that belonged to the horseshoe axis. The only disadvantage of this solution is that meridian flips will be needed.

For astrophotography it is also important to have the azimuthal plane of the mount well levelled horizontally. Speed errors and pointing errors in hour angle will occur when the azimuthal plane isn't level while the mount is driven based on the right latitude setting. There are simple ways to determine these kind of errors, just by taking pictures on certain hour angles. It is not needed to do physical correction of the mount, correction can be done by adjusting parameters in the software.

# Angles and relative angular velocities of the altitude axes at the cradle

At the cradle we find the horseshoe axis and the fork axis as the two altitude axes. The cradle has the same function as the anchor between the shafts of a universal joint. It enables pivoting between the azimuthal and polar axes.

## The horseshoe axis

The next graph shows the way that the cradle needs to be tilted while tracking celestial objects.

## The fork axis

The fork axis is 90° shifted in phase relative to the horseshoe axis, the amplitude is the same.

### Fork and declination axes combined: summation

As explained previously, the fork axis and the declination axis can coincide precisely. In that occasion one motor can do movements for both axes, so we save a motor and we can avoid extra instability that could raise from a separate declination axis. Wherever their positions, functionally they need to be distinguished from each other.

Because of the simplicity we 'll combine the declination and fork axis as one single axis. The next two graphs show two examples of summation for both axis. The first shows a circumpolar example, the second shows the graph for an object with a lower declination. At the northern side the object is below the horizon.