1 The AM5 mount uses step motors and strain wave gears to drive the R.A. and DEC axes. The strain wave gearing has no backlash, while we use a belt to connect the step motor and the strain wave gear, which may cause some backlash if it isn't fastened appropriately. The backlash may be slightly different due to the tightness of the workers' assembly, which is usually within the range of tens of arc seconds.

2 Meanwhile, according to the analysis of the DEC error principle and the experiences of other users, such backlash won't make the guiding statistics any worse. This is because the DEC axis doesn't need any movement if the polar alignment is very accurate; If the polar alignment is not correct, it will only cause a one-way correction of the DEC axis in the guiding procedure, in which case the backlash will not affect the guiding performance.

3 Even though, if you insist to reduce this backlash, you can follow the guiding manual (we can send you one), open the mount, manually adjust the position of the step motor to fasten the belt. Please be careful not to make the belt too tight. This may affect the regular operation and the lifetime of the belt. If you don't have the necessary gears, the dealer can do it as well.

4 Non-orthogonal calibrations usually occur when using worm gear mounts with unbalanced RA/DEC axis(or both). The AM5 mount is a strain wave gearing mount, which provides pretty much torque and is free of the balance problem. Therefore, the calibrations should be orthogonal. The calibration can sometimes be non-orthogonal with a <10-degree bias which may be caused by poor polar alignment or pointing too close to the polar region.
This calibration still won't make the guiding performance any worse.

If you want to improve the guiding performance, you can try to make a better polar alignment and to redo the calibrations with stars close to the celestial equator.

ASIMount@ZWO If the polar alignment is not correct, it will only cause a one-way correction of the DEC axis in the guiding procedure

The above statement is not true.

The declination drift can change sign sometime during the evening. See this following diagram.

Looking at the coordinate system in the polar projection below, let the sky coordinates (Right Ascension and sky declination) be the blue lines, and let the mount coordinates (Hour Angle and mount declination) be the red lines. I have drawn them so that the altitude polar alignment error has the same error magnitude as the azimuth polar alignment error -- so you can see a symmetry around the 45º angle.

Lets say we start tracking a star at 30º declination on the East (left most side of the diagram). As time passes, the star will move towards the bottom part of the diagram (i.e., RA and HA both rotated).

Notice that when we start with the star on the east (at the left), the mount's declination is closer to the true pole than the sky's declination (red circle is closer to the pole than blue circle)

However, 3 hours later (45º in hour angle), the red circle (mount declination) crosses over the blue circle (true declination), and after that time, the declination has drifted to a different sign (for another 12 hours). The red circle is now farther away from the pole than the blue circle.

So what will happen is that the declination curve on the guide graph will actually flip over. The sign does not stay the same all of the time, and because of this, you can also not use North-only or South-only declination guiding all of the time.

Near that switch over point, the centroid estimation error will be large enough that the estimated declination error will swing in both directions (and thus make backlash important around that moment).

The OP's mount has 33 arc-minutes worth of backlash. I would not be surpised if he sees occasional declination guiding spikes that are around 25 arc-seconds (won't be 33" since backlash is usually gradual and not sudden).

FWIW, here are the curves (I was drawing them to study somethig else [simple geometric explaination for field rotation], so there are some extraneous lines and arrows) for altitude-only polar alignment error and azimuth-only polar aligment error. Notice that the red circles still cross the blue circles, regardless of whether the polar alignment error is in altitude, in azimuth, or both (above case).

Chen

KC_Astro_Mutt The Max DEC/RA duration is simply the maximum pulse that will be sent for correction, and has nothing to do with exposure settings. What is the purpose of this advice?

Absolutely bang on!

The Max pulse durations are to keep PHD2 from wrongly issuing pulses to compensate for large errors that are not caused by the mount itself. For example, if you make a mistake in centroid estimation, a wind gust has come through, an owl or airplane flew past your guide star, etc., you want to ignore those large momentary corrections that has nothing to do with the mount. Or, at least to limit the correction by using a max pulse duration hard limit.

Here is the nitty gritty details on how to estimate the max pulses that you really need, and why the ASIAIR defaults for example, are nonsensically large.

Read on only if you are interested in the technical aspects (I know a lot of astronomy hobbyists are nerds, and have day jobs that are technical, and can follow the math :-).

If not, scroll towards the end to find out how much Max guide pulse durations you really need :-).

First, lets assume that the mount is perfectly smooth, i.e., its periodic error is a sine wave. A single non-zero term in a Fourier Series for those who are mathematically inclined.

If the curve has no high harmonics, it will look like a sine wave with one fundamental frequency. The period of that sine wave is 430.82 seconds for your particular mount. (Equal to one sideral day divided by 200, in case you are interested.)

The slope (first derivative) of that sine wave is basically (A.2.pi/430.82).cos( 2.pi.t/430.82 ) where t is time in seconds, and A is the amplitude of the sine wave (in this case, A is equal to half of the peak-to-peak amplitude to the periodic error).

The worst-case slope are at t = multiple of pi, e.g., at 0. I.e., worst case slope magnitude is A.2.pi/430.82 (in units of "A" and "t"). (The slope can be both positive and negative.)

This above is how much error a smooth mount itself can make. No more, no less.

Now, to put some real world numbers, assume that A is about 25 arc-seconds (i.e., p-p periodic error is 50 arc-seconds), the worse case of the slope from the above equation is about 0.35 arc-seconds per second.

OK, lets say you are guiding at 0.5x sidereal rate. I.e., the mount will move 7.5 arc-second per second that the guide pulse is on. Does something hit you like a lightning bolt already?

What this says is that for a smooth mount, you only need to apply a correction pulse that is 0.35/7.5 seconds long! That is less than 50 milliseconds.

Now, in real life, a mount will have higher harmonics. Especially strain wave geared mounts (and some strain wave gears have smaller high harmonics than others).

Strain wave gears have been around for a while now (if I recall correctly, the original Mars rover used them to directly drive its wheels). In fact here is a paper about it from a NASA conference in 1991 (not the date):

https://articles.adsabs.harvard.edu//full/1991NASCP3113..237N/0000241.000.html

Go to page 241 of the paper under the section named "GENERAL CHARACTERISTICS." Look at Figure 5 in that section (page 242). Does it remind you of something you have seen?

Figure 7 shows the typical spectrum (i.e., harmonic contents) of a strain wave gear, in this case a real gear from Harmonic Drive LLC (the same place that manufactures the strain wave gears that RainbowRobotics use for the RST-150h, RST-135 and RST-300 mounts. I have no idea what stainwave gear Pegasus uses; it could be Harmonic Drives since they calin European manufacturer, and Harmonic Drive also has a plant in Germany.

Harmonic Drive LLC owns the patent for strain wave gears. The patent has since expired and you now see other starin wave gear manufacturers. But they cannot legally call them Harmonic Dive, since the trademark has not expired. Anyone using the name Harmonic Drive, but using strain wave gears from other companies can run into legal trouble.

OK, back to the NASA paper... the fundamental in Figure 7 is at 2 (not 1) since as the paper says, the errors in a starin wave gear occurs at factors of 2 of the rotation!

So, the 430.82 second period I mentioned earlier corresponds to "2" in the abscissa of Figure 7.

In that particlur sample thatthey measure, the fundamental has an amplitude of about 0.0064 and the second harmonic (at "4" in the abscissa) has amplitude of .0025, the third harmonic's amplitude is at 0.0007, etc.

I.e., in percentage terms relative to the fundamental, there is 39% second harmonic distortion (for folks who are more familiar with audio equipment) and 11% third harmonic distortion, etc.

OK, why are harmonics important? Go back to our first derivative again. Notice that by the Chain Rule in calculus, an N-th harmonic term in the argument of the function will pop out ouside the cosine as a linear factor. So, the third harmonic, will for example, cause the first derivatie to be 3 time larger that the fundamental, but its amplitude A will be smaller due to the low third harmonic distortion.

If you apply factor of 3 to the 11% third harmonic distortion, you get 0.33. This is the additional error for the first derivative on top of the fundamental.

But if you look at the numbers, all of the harmonics, won't really be much larger than the error that is caused by the fundamental buy itself (remember that 0.35 arc-second per second number earlier above?).

So, at least for Harmonic Drives, the higher harmonics will only perhaps double the error from the fundamental. If your mount does not use gears from Harmonic Drives (manufaturing plants in USA and Germany), the harmonics could be different. In that case you can use a ruler and pencil to try to eyeball you periodic error curves abx xompare the worst case slope with the smope when the cureve is smoothest. If that factor is more than two, then you will need to correct at a higher rate than the Harmonic Drive in my RST-135.

In any case, I don't think the non-Harmonic Drive strain wave gears have harmonic content that is more than twice that of the real Harmonic Drives from US and Germany.

Anyhow, based on my RST-135, I should not need more than about 0.7 arc-seconds per second of correction. With a 0.5x sidereal guide rate, this means that my max guide pulses need not be more than 100 milliseconds.

I have set my Max RA and Max Declination pulses to 150 milliseconds in ASIAIR (a far cry from the default) and get very good results.

As mention a little earlier, I suspect that the ZWO strain wave gear has more harmonic content -- at least from observing the curves they have published in their ad copies. But, even if you assume that the strain wave gears that ZWO uses has twice the harmonic content as the one in my RST-135, that 150 to 200 milliseconds should still be ample to handle all the errors from the mount.

I really have no idea where the 1000 ms and 2000 ms numbers that ZWO is quoting comes from. They make no sense at all to me. Guiding pulses are there to compensate for the mount errors, and the mount errors should not get anywhere close to 7.5 arc-second per second to need a 1000 ms pulse!

In the current state (if you are using ASIAIR), the dither recovery may take a longer time with the smaller Max values, because ZWO made the mistake of using the Max pulse duration to also control dither recovery too. Confating two very different functions

Chen

For anyone who bothers to read this far... I have earlier discovered that the ideal guiding for a strain wave gear is to issue many small pulses with high rates. Currently, PHD2 can only issue one pulse per camera exposure. That is why we need to keep the exposures short (I use 0.5 second guide exposures) to get high correction rates (2 FPS).

I have started to model a guide system that issues 4 or 8 guide pulses (spead out evenly) per exposure. My simulations has shown so far that that I can comfortably guide a strain wave geared mount with even a 2 second exposure time (the math tells me it can be even longer).

More interestingly even, those familiar will calculus will take the "limit" (i.e., use more and more, but smaller amplitude [e.g., 0.1x sidereal rate] pulses, but repeated more often) until you converge to the limit where the pulses completely overlap oe another to become a continuous pulse that is as long as the exposure duration, but with very small amplitude (like 0.01x sidereal rate). It turns out that my RST-135 guide rate can be controlled in 1/100 of a sidereal rate, so when time comes when I test the "many pulses per exposure" algorithm out (in INDIGO), I will probably also try this "guide by rate" paradigm, instead of "guide by duration" that is done today.

ASIMount@ZWO the user makes the guide move every 2.8-3.1seconds, does it mean almost 0.98-1.09 arc-sec(on one Axis) on average?

The rate is about .35 arc-seconds per second (of time). So, if you do't issue fast enough correction pulses, you will need a longer gulde. However, if you wait 3 seconds before issuing another guide pulse, that means that mount has moved by itself for 1 arc-second before any corrective measure is taken.

Accumulating 3 seconds worth of error on a strain wave gear mount is way too large for good guiding! At that drift rate, and correcting only every 3 seconds will cause that axis' RMS error to be a whopping 0.7 arc seconds! I don't think a user would tolerate that mucg guiding error.

This is why it is imperative for strain wave geared mount to use very rapid correction rates. As I mentioned at the end of my previous post, it is not the camera exposure rate (FPS) that is important, it is the correction rate.

Exposures (i.e., sampling the centroid) can actually be done at a rate even slower than 0.5 FPS (2 second exposures). But the correction rates need to be fast.

It is just that with today's PHD2 (and ASIAIR) guiding, we can only issue one guide pulse per exposure. And because of that we have no choice today but make very short exposures (I need 0.5 second exposures with an RST-135 to guide it at 0.4 arc secnd total RMS).

For example, you can take a 2 second exposure, compute the centroid positionand issue 4 correction pulses that are separated by 0.5 seconds between that exposure and the next exposure. If you also estimate the first derivative of the centroid changes, you can actually issue predictive pulse widths, and get the error to drop below 0.1 arc second between exposures in my simulations.

Ask Andy Meng or Walf Wu to send me an email (they have my email address) if you are interested, and I can send an Xcode project that I have started to work on to simulate such behavior.

Since I never have access to ASIAIR source code, I will be testing this in INDIGO with my RST-135, and perhaps a Nyx-101 if I pick one up.

Chen

ASIMount@ZWO

Here are some figure that I had recently sent to Rumen and Peter (the two principals of Indigo) showing the problem and solution.

The first image shows the typical situation today (one correction pulse per exposure):

As the slope of the PE gets large, the pulses also had to be much longer, and at the same time the actual RA tracking error has a larger error. The main imaging camera will see a star move in RA according to the "actual RA track." Notice that the RA track zig-zag. It "zigs" during the correction pulse, and "zags" because of the mount's PE.

The only way today to improve the RA tracking error in ASIAIR is to shorten the exposure time.

Now, imagine being able to distribute that one pulse per exposure to two pulses per exposure:

Notice that the actual RA tracking error is now smaller (by a factor of about two). The pulses are shorter, but there are more of them. The total area under the curve of the pulses is actually the same as the one pulse per exposure case.

Increasing to 4 pulses per exposure etc will keep reducing the peak RA track error. And 8 pulses per exposure time, even better, etc.

Until in the mathematical limit, we use different guide rates for each exposure duration), instead of using a constant guide rate (0.5x sideral) and different durations:

The RA tracking error is now maximally flattened. The only way to get flatter is to again shorten the exposure duration :-). This ultimate solution ("guide by rate") may not be possible with all mounts (because some of them only have a limited number of guide rates), but in the RST-135, I can control the guide rate in units of 1/100 of the sidereal rate, from 0.01x to 0.99x sidereal rate. But most mounts should be able to handle the "N-pulses per exposure" model. Even with 2 pulses per exposure, we already impoved today's RA tracking error by a factor of two.

Chen

Kevin_A My mount PE error is 16.8/6.0 on RA but it seems to have a secondary harmonic in the graph too.

Many harmonics, not just second. A second harmonic would have looked like a fundamental sine wave where each half cycle is displaced up and down also in a sinusoidal motion.

It is too bad that ZWO does not provide numerical data for the PE curve, just some plots, otherwise we could take a Fourier transform and actually look at the harmonic content.

Notice in the second figure ("partial zoom") that you posted, the cycles don't even repeat. So there are even sub-harmonics present. The peaks of that last cycle is distinctly different from the previous two, and those two don't look perfectly alike either. This is characteristic of strain wave gears, and is caused mostly by the flexible (yep, flexible :-) spline gear component of the strain wave gear. The better machined the gears are, the more repeatable the cycles are. But there is always a slop.

Take a look at the GIF image in this wiki https://en.wikipedia.org/wiki/Strain_wave_gearing . The flexible spline gear is the red gear in the image.

This, by the way, is the reason why you cannot create a fixed periodic error correction (PEC) for strain wave mounts. People are clamoring for PEC, without understanding that it simply is not mathematically possible ("unmöglich", or as the Japanese engineers put it... "verrrrry difficult") to use a single cycle to correct. (You can use some predictive methods, but then, you might as well just guide a strain wave mount the way they need to be guided -- with rapid guide updates).

Since there is no numerical data, we will just bypass the harmonic analysis and just wing it and estimate the worst case slopes directly from the figures as best we can.

The scale (in degrees of the shaft) of the abscissa is not very useful for us, but we know that each cycle of the PE correcponds ro one sidereal day/200 (in the case of your mount; since they had tried to mimic the RainbowAstro RST-150h and RST-135). That is about 480.82 seconds, which I have drawn in red.

Now we try to eyeball some part of the curve that has the largest slope (just don't tell my engineering profs that I am winging it like this, or they would retroactively flunk me)...

The dashed green line is probably a good enough pick.

I am doing this using an application on macOS called EazyDraw, and it is nice enough to show the actual coordinates:

So we know the slope of that line is dy/dx = (2.99/0.59) = 5.07 in the EazyDraw units. To estimate real units, I took the distance between -10 arc seconds and +5 arc-seconds in the ordinate of the graph (dashed red line), and that came to 2.43 in the EazyDraw units. So we know that the vertical scale is 15 arc seconds per 2.43 EazyDraw units, or 6.17 arc-seconds/EazyDraw units.

Similarly, the distance between the 430.82-second lines is 2.18 EazyDraw units, so we know the scale of the abscissa is (430.82/2.18 ) = 198 seconds per EazyDraw units.

We had come up with a slope of 5.07 EazyDraw units, so applying these scales, we get voila, a worst case slope of ( 5.07*6.17/198 ) = 0.16 arc-seconds per second.

A perfect sine wave with 15 arc-second peak-to-peak amplitude would have given you a worse case slope of around 0.105 arc-seconds/second. So, indeed your mount has about what is called 60% total harmonic distortion (THD) in that waveform (the audio amplifier in your TV probaby has a THD of less than 0.1% :-). If this were audible, it would sound rather dissonant -- in fact, you can probably tell when you slew the mount that the sound is not like what you hear from a tuning fork [which is quite sinusoidal] :-)

If you want the peak error (from one axis) to accumulate less than 0.1 arc-seconds before applying a correction pulse, you will need to apply correction puses no longer than 1 per second. And, for PHD2 guiding, that means a guide exposure time no more than 1 second.

You have one of the better ZWO mounts. If you look at their documentation, you will find mounts that have much higher peak-to-peak error than the 15 arc-seconds in your mount. Here is a sampling from their documentation:

https://astronomy-imaging-camera.com/wp-content/uploads/all-kinds-of-periodic-errors.jpg

The one on the top right has a peak-to-peak PE that is almost three times your mount's p-p PE. For people who have those mounts, they would need more correction pulses per second than you do.

However, the devil is in the details. ZWO did not show the per-cycle curve, so we don't know the actual harmonic content.

They do have a picture that showed this:

Notice that the worst case slope looks worse than yours too. And if you stare closely at the scales, it looks like the p-p error is perhaps around 30 arc seconds (much worse than yours, unless the "partial zoom" plot for your mount came from a favorable part of the 360º travel). So, compared to your mount, that particular mount would be harder to guide and would need guide corrections in the 2 per second (0.5 second exposures) region to get anywhere close to 0.5 arc-second total RMS.

With care, and studying and understanding your mount, you can auto guide these strain wave mounts. You just can't use some lazy "what is your settings because I want to copy it" method, because the characteristics are so different from mount to mount that comes off the same assembly line.

So, please folks, don't use the numbers that are derived above if you are not @Kevin_A.

I can get better than 0.5 arcsecond total RMS error with my RST-135, and at times, when the gears are at the smooth part of the curve, I would get 0.25 arcsecond total RMS type guiding. In my case, I use 0.5 second guide exposures, and set my max RA and max declination durations in the 130 ms to 180 ms region (forget the 1000ms to 2000ms nonsense, folks), and setting aggressiveness down in the 35% to 45% region. But not even all RST-135 are alike. I have two RST-135, and they are not alike.

Chen

Kevin_A Once again thank you Chen, your time, knowledge and effort is greatly appreciated!

No sweat, Kevin.

The major take-away should be that the amplitude of the Periodic Error curve for a strain wave geared mount is only important for visual observers, who want to keep a object in the view of their eyepiece.

The perodic errors for these gears are too large for long exposure astrophotography where you must auto-guide anyway.

When auto-guiding, it is the amplitude of the first derivative ("slope") of the periodic error curve that determines how easy the mount is to guide. Not the amplitude of the curve itself -- the amplitude of the curve itself is simply a scale factor of the first derivative of the curve.

The harmonics are more insidious.

A second harmonic that is 50% will contribute as much to the slope as the amplitude of the curve itself. A third harmonic that is just 33% will also contribute as much to the derivative, 25% of 4th harmonic, 20% of the fifth harmonic, etc.

I.e., if you start with a clean perioidic error curve with amplitude A (i.e., peak-to-peak error of 2A), and you add a 50% second harmonic term, it will be as bad as another mount that has no second harmonic, but has a periodic error amplitude that is twice as large to start with.

And the terms accumulate. So, a mount with 33% third harmonic, plus 20% 5th harmonic, will have a worst case first derivative that is 3 times that of a mount that has a clean periodic error curve, when both start with the same fundamental frequency amplitudes.

The more irregular the curve looks to the eye, the scarier it is to guide.

Mounts that have high harmonic distortion will often guide really well, and then suddenly (when the shaft reaches the place with large PE slope) go bonkers for a short while, because the correction pulse rate is not keeping up with the slope.

One day, we will be able to develop better software so that these high first derivative mounts can be tamed (for example, with one of the two schemes that I described earlier). Until then, unless you write your own code, be careful what you buy in the strain wave mount market. I am myself currently on hold waiting for actual numbers from the Pegasus Nyx -- I am looking for a mount with a little more payload capability than my RST-135; the RST-300 is overkill and will kill my back :-).

Chen