Periodic Error: Definition, Detection and Correction

Some telescope mounts, including the iOptron iEQ30 Pro, use a worm drive to transfer power from a motor to two powered shafts, allowing us to aim at any point in the sky and track the stars as they wheel overhead from east to west. One problem with the worm drive is that the worm (a helical gear, shaped like a screw or bolt) is always slightly imperfect. In particular, it sometimes turns the the worm gear a little too fast, and sometimes a little too slow. In fact, it does both in a repeating fashion, with the cycle repeating once with every full turn off the worm. This error blurs the images of stars as their light moves across the sensor of the cameras attached to a telescope. In the case of PAN006, the error is a few pixels of movement away from the average position over about 4 and a half minutes, first to one side and then the other.

Astrophotographers often use what’s called an autoguider or guide camera to measure this error and to send commands to the mount to correct for the error. The autoguider needs to be attached to a telescope in place of an eyepiece, so in addition to the cost of the small camera, you also need a small telescope and brackets with which to attach it to your primary telescope (> $300 for a basic model). Autoguiders also help with minor errors in the alignment of the mount or telescope, providing corrections for not only the primary axis (Right Ascension, or RA) of an equatorial mount, but also the Declination (DEC) axis.

As one of the goals of Project PANOPTES is to enable as many people as possible to build telescopes, in large part by keeping the cost down, the design doesn’t include an autoguider. Wilfred has attempted to address the problem of periodic error via the PANOPTES software (POCS), measuring the error after each image is recorded, then making small pointing adjustments before capturing another image. However, the image exposure times are typically 2 minutes long and it can take another minute to do the analysis and adjust the mount, so the total time between exposures is a large fraction of the worm period. That means that we can’t do the adjustments several times a minute, as would be required to eliminate most of the error.

So, what to do?

It happens that I’ve been thinking about a related problem: the lack of absolute encoders in the iEQ30. More expensive mounts include these to report the angle of each axis, which can be used to improve pointing of the telescope (i.e. slewing to a new target). If the RA encoder is very high precision, it can even be used to measure the periodic error in that axis; note that neither encoder can’t help with detecting tracking errors due to misalignment of the mount because they’re only measuring the position of the axis, rather than drift in position of the stars in the recorded images.

An absolute encoder would be useful for making sure that the mount is in the right place when POCS plans to start moving the mount after sunset. For example, let’s say that someone (me) had been working on the scope during the day and moved released the clutches in order to get better access to the camera, and then put the scope back in roughly the same position. This will throw off the iOptron’s ability to point at targets in the sky because it depends on the user to set the zero position: cameras/telescope pointing at the celestial pole with the RA axis as close to vertical as possible. The mount keeps track of the time of day and the last position to which it had commanded the mount to move, from those it can translate the angle of the RA axis relative to the mount to the current right ascension (i.e. where the scope is aimed)… all assuming that the mount doesn’t move without the software tracking it.

DIY Absolute Encoder?

Many commercial encoders are based on circular Gray Codes, a disk with patterned rings of light and dark stripes, and a sensor for each ring indicating whether the stripe under it is dark or light. It occurred to me that we could do something similar with a cheap USB camera. In particular, we could attach a camera to the base of the mount, aimed at a disc (annulus) attached to the moving portion of the RA axis. I bought a USB microscope for about $20 from Amazon, and took some pictures of a millimeter ruler. With some effort I was able to focus in on a single millimeter (from a few millimeters above the ruler):


Given that:

  • The image is 640x480 pixels

  • The radius of such a disc would be at least 78mm (maybe larger to allow for brackets for holding the camera)

  • The image field of view is around 1.1mm across, or about 1.7 microns per pixel

All together, this implies that each pixel would span about 4.5 arcseconds, which is definitely smaller than our image scale of 10+ arcseconds per pixel. So, IF every few seconds we could take a clear picture of the disk and analyze it to determine the angle moved and adjust the speed of the mount accordingly, it would allow us to compensate for periodic error.

The reason for the bold IF is that the depth of field of the microscope I tried out is absolutely tiny. Any slight shift of the target closer or farther renders the image very blurry. It would be exceedingly difficult to mount the disc perfectly perpendicular to the RA axis, so the images wouldn’t stay in focus most of the time. So I’m going to be on the lookout for a closeup camera that has a long working range, at a reasonable price. After all, if the price is above $200 I might as well buy a commercial autoguider

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Interesting write up. Thanks James.

Not to be a party pooper but we would need to consider water proofing this if it were on the outside of the mount which I think is what you are suggesting?

Yup, I’m keeping it in mind as I design the waterproofing for PAN006. I now have a good basic design (after a number of dead ends), but lots of work needs to be done to turn it into a set of 3D parts that can be printed, some glued together, and finally assembled in the field. For example, I need to add ribs to bigger parts to keep them from flexing and screw holes and holes for threaded inserts for fastening parts. I’m making my job extra hard by assuming that I’ll need to put the weatherproofing on and then take it of again quite a few times, so I can’t just glue it all together. After all, I’m sure I’ll forget something like hooking up the DEC cable when putting it together.

Great idea. I’m very optimistic it will work well.

I don’t think we even need an encoder tape. The image shows all sorts of fine structure (texture on the metal surface) that is very unlikely to repeat at multiple positions. The software could store a long image showing the scene across 360 deg, and match each image along this sequence to find where we are. Cross-correlation is the algorithm we can use.

The precision should be much better than one pixel when correlating a large image with many features. I wouldn’t be surprised if we get to 1/100th of a pixel, which would be fantastic (sub-arcsecond).

I believe our accuracy will ultimately be driven by mechanical issues: (1) flexures in mount and between mount and camera, (2) rotation axis centration (if there is a bit of play, the center of rotation for the disk we image can move a bit), (3) thermal expansion effects.
The mechanical issues will mostly affect long-term drift, and most of them are unlikely to play out over ~10mn, which is where the mount periodic error shows up.
Conclusion: I think this will work very well to fix periodic errors. We may not be able to rely on this for keeping stars on the same pixels over long time period (hr), but we have the images to do that. A hybrid scheme combining this trick with astrometry-from-images should work very well.

… testing required to confirm all this.
I can help with software/algorithms.

An autoguider would cost about 350 dollar, but perhaps we can do with a cheaper mount if we use the autoguider ? The iOptron mounts are known to have a well-controlled periodic error ( CEM-25P guarantees less than 10 arcsec), but they have their premium price as well. As I understand from deepsky imaging amateurs, an accuracy of 3 arcsec over long exposures is realistic with an autoguider. As 3 arcsec is also the variance to be expected by seeing, this would be perfect for the Panoptes project.