December 23, 2012
December 23, 2012
When observe celestial objects with astronomical telescope or camera, a moving mount mechanism is required to follow the rotation of the sky. The viewing angle of the astronomical telescope is pretty narrow; therefore the object goes out of the viewing area quickly if the telescope is fixed on the static mount. The photographs of a star also requires to follow the rotation of the sky for a time because a long exposure time is needed to take the dark sky.
The moving mount mechanism to follow the the rotation of the sky is called Equatorial Mount. It is driven by hand wheel or motor. Of course a moter driven equatorial mount is required for astrophotography to exactly follow the the rotation of the sky for a long time. Recently, small and inexpensive products are appearing in the market. But the mechanism is not that complicated and many articles on building the equatorial mount are found on the web, so that I tried to built it for simple astrophotography.
Of cource the rotation of the sky is an effect by the Earth's rotation. It can be easily canceled by rotating the equipment in the axis parallel to the Earth's rotation axis but opposit direction (Figure 1). The angular velocity of the Sun, Moon and a fixed star in the sky a bit differs. The angular velocity of the fixed star is 2.92E-6[rad/s].
The angle of Earth's axis of rotation with the ground varies depends on the latitude of the site, so that the axis of equatorial mount must be set to the proper angle and orientation. To set the axis of equatorial mount, most equatorial mount has a small telescope called polar finder. The polar finder is mounted on the equatorial mount in parallel to the axis of equatorial mount, where the axis of equatorial mount can be set parallel to the Earth's axis of rotation by catching the polar star in the polar finder.
To follow the rotation of the sky by equatorial mount, the polar axis must be driven very slowly and accurate in velocity by a synchronous motor and a reduction drive with high gear ratio. Figure 2a shows the typical driving mechanism of the equatorial mount as manufactured products. It has a large worm wheel for the final gear. But the worm wheel is not a standardized component. This makes obtaining the component difficult. To avoid this problem, the lever driven method shown in Figure 2b is often used for home built equatorial mount. I built a portable equatorial mount in this method and named it Star Tracker.
Photo 2 shows the close up views of the built portable equatorial mount. The dimensions of the body is 120mm x 100mm x 35mm and 450g in depth x width x height and weight. The effective length of the lever is 85mm and is driven by a silder with a screw in pitch of 0.5mm, where it is equivalent to a worm wheel with 1068 teeth. The silder screw is driven by a geared stepper motor in 960 steps per a rotation. The overall stepping resolution is 1025416 steps per a rotation but the lever rotates only a limited angle.
There are two considerations on design of the mechanism. The one is to be accurate the length of the lever, between center of polar axis and point of effort by slider. The other is to keep rigidly of the each mechanical components against load weight. Especially the polar axis should be held by two bearings and add some reinforcement to the holding parts.
The controller board has a motor drive output and a limit switch to detect home position of the lever. The OLED display on the controller board indicates operation time from origin. The acceleration sensor senses the tilt angle of the polar axis but it could not achieve a sufficient accuracy to set the polar alignment. The polar alignment is very impotant to follow the celestial objects with accuracy, so that a consideration to set the polar axis is left.
The function of the controller is driving the stepper moter and notihing else, however lever driven equatorial mount needs to return the lever to home position. The motor drive timing is generated in DDS algorithm to set it to an arbitrary rate depends on the mechanical design. The step rate should be slightly changed to compensate an error due to accuracy of the mechanism.
There are some errors that should be considered at lever driven mechanism. The slider moves linearly and lever moves circularly, so that the point of effort on the lever must be movable freely by a folk and a pin like shown in Figure 2b. However it causes a linearity error on the relationship between movement of slider and lever. Figure 3 illustrates the errors on the lever driven mechanism. When the slider moves from B to C at constant velocity, the lever should rotate Θ0 if it retains a constant angle velocity at B. However the lever actually rotates to Θ1 if the folk is attached on the lever and Θ2 if the folk is on the slider. Note that the unit of Θ in this figure is radian.
This Excel sheet is to estimate the overall guide error appearing on the sensor in unit of pixel that caused by some factors. It calculates on the assumption of asin that the folk is on the slider and pin is on the lever. First of all, enter the specifications, focal length and pixel pitch, of camera to be used in the cell. For example, 25mm in focal length of the lens and 4.3μ in pixel pitch on the sensor. This pixel pitch corresponds to 12M pixels on M4/3 sensor or 18M pixels on APS-C sensor. And the graph will express blur vs elapsed time. Of course this unlinearity due to asin/atan can be easily compensated by software. But it is less than 0.2 pixel over 20 minutes of exposure time, so that asin/atan error is negligible at this viewing angle and exposure time.
Accuracy of lever length also affects guide error. When the lever length A-B is off by 0.05mm from 85mm in designed length, a significant error is resulted and the slope of the graph will be filpped. Because 0.05mm in accuracy is very tight for hand work and it will cause more error than this. It should be removed off by software error compensation.
Accuracy of system clock is a factor of guide error. Typical ceramic resonator has ±5000ppm of error in frequency. It can be compensated by software but temperature coefficient is also large, so that a crystal resonator is recommended for the system clock oscillator.
There is a factor of guide error other than angle velocity of polar axis that described above. An alignment error of polar axis causes a guide error. The graph at lower part of the sheet shows the worst tracking error due to alignment error of polar axis in angle. The amount of guide error varies by direction of alignment error under error in angle is fixed. You will find that the polar alingment is very important.
