Byrdsfan1948
Hi Al,
First, lets look at the problem at hand. There is so much with hobby stuff (especially in the audio world) where there is so much snake oil being sold, that we need to be careful with "the next newest thing" that comes along, accompanied by ads and Facebook and YouTube shills.
I myself had been offered free things like an ASIAIR from ZWO for example, and have always refused them -- declaring taxes for gifts from foreign counties is something I do not want to deal with). The same is true with the amateur radio hobby, where I was offered gifts of RF modems to include support for them with the software I wrote. The gifts are often masked as "test loaners," but the manufacturers never demand them back. So, yes, there are shills all around.
So, back to the problem -- worm gears. From the days of von Fraunhofer (why a German mount is called "German" mount), the reason for need of worm gears is the extreme ratio that is needed from a drive motor to the drive shaft at Sideral rate. A series of spur gears to achive this kind of ratio will have too much mechanical backlash, etc. As long as you can make the diameter of the worm gear large, or make the pitch of the worm very fine, you can get very large reduction ratios.
However, if you look carefully at a worm gear, undess they are many feet in diameter, only one tooth of the gear is actually touching the worm. This means less torque transferred from the drive motor (Fraunhofer used a weight to mechanically drive his polar axis) to the polar axis.
This lack of torque therefore requires the driven axis to be finely balanced. Thus, you have to come up with a counterweight to balance out the telescope's weight. The telescope itself also needs to be adjusted on its dovetail plate to balance in the other axis (to keep the balance when the declination axis is rotated). And on top of that, if the weight of the stuff that is added to the OTA is not on the same plane, there is a "third axis" balance that needs to be tuned out as both RA and declination axes are rotated.
Musser came up with the strain wave gear back in 1950's (see U.S. Patent 2,906,143; issued 1959). Musser also has patents on the recoiless rifle, etc, but actually patented strain wave gearing while we was working at a shoe company (!). Later, a company called Harmonic Drive™ company was created. So, Harmonic Drive™ gears are strain wave gear, but not all strain wave gears are not Harmonic Drives™.
The stain wave gear has the advantage that multiple teeth of the componets of the gearing are actually in touch, and its behavior allows very high torgue. And because of the grometry, they also have high gear ratios (Harmonic Drive™ sells 100:1 units, and even 200:1, if I recall).
So, the main advantage of a strain wave gear is that it has both high torque and high reduction ratio at the same time.
Because of the high torque, these things are widely used in robotic arms. RainbowAstro's parent company is RainbowRobotics, and they are in cahoots with the UNLV Robotics labs (sharing many facilities and staff in Las Vegas, thus also having a presense outside of Korea) -- RainbowRobotics' CEO is a hobby astronomer, and he had spun off RainbowAstro to produce mounts, including traditional German mounts. The charateristics also caused the strain wave gear to be used to drive the wheels of the NASA Mars Rover (I think that was the rover, could have been a different one.)
But there is a rub.
Unlike a worm gear, where the gears can be machined with as much precision as you are willing to throw money on it, the strain wave gear includes a flexible spline. It is harder to make that as precise all through 360º of rotation. I.e., the priodic error curve is not a sinuoid, but has higher harmonics (see later why higher harmocis are a terrible problem in out application). Additionally, because the outer spline is flexible, each rotation can be dirrent from another, depening on how precise (or imprecise) the maching is, and the choise of material used in the flexible spline (this make its harder to do any form of "predictive PEC" since the periods are displaced by one another at some sub-harmonic.
Now to the actual auto-guiding problem...
For amateur mounts do not have high precision encoders, we auto-guide by using stars as refernce points (i.e. the "encoder is in the sky, free for us to use). By measuring the centroid of a star (or the average centroids of multiple stars), you know exactly were the telescope should be pointed at. You then use any detected deviation to reposition ("pulse guide") the mount to correct for the mount's gear errors.
Lets say, you take an exposure of T seconds to measure the centroids of stars. Lets say the gears are deviating within this T seconds so that the star moves by more than S arc seconds. This means that you cannot tell better than S arc seconds where the centroid has moved to. The star is not a point, but a small S arc-second long line. This is why the rate of change of the PE curve is so important, and missed by neophytes in this business.
This is where the first derivative of the periodic error enters the room. If the first derivative in arc-seconds per second tof time (rate of change of movement) multiplied by T seconds is larger than G arc seconds, you don't stand a chance to guide better than G arc seconds.
By using just high school Math and Physics Fourier Series, any periodic curve is made up of a sum of a series of sine waves, at some fundamental period P. The smoothest curve periodic curve (one with smallest bounded S) is a sine wave containing just a fundamental frequency term.
The rate the curve changes is the time derivative of the curve. I.e., if the sine wave is A.sin( t/P ), then the time derivative is (using high school calculus again),
A/P.cos( t/P )
"A" is half the peak-to-peak error of the periodic error curve. Notice that the "P" term has popped out outside the argument of the cosine.
Notice that the shorter the period, the larger the first deivative! (Most high school math teachers don't teach you how useful calculus is in the real world -- my teacher sure did not). The period P for my RainbowAstro mount is 430.82 seconds, i.e., one Sidereal day divided by 200).
Ah, but the story gets worse...
if the curve has higher harmonics (more bumpy than a pure sine wave), that P term becomes 3P for third harmonic, 5P for fifth harmonic, etc. Your exposre time T has to be really short if you have a bumpy periodic error curve.
My RST135 has a small third harmonic (barely noticable bumps) and I have to use 0.5 second for T to get better than 0.5 arc second type tracking. This is why I simply could not guide with ASIAIR until it added multi-star centroid (PHD2 had multi-star in beta long before). Without multi-star, you need long exposures like 2 to 3 seconds to work around atmospheric turbulence. That produced a centroid estimation (i.e., inability to freeze a guide star) that is too large. And before multi-star centroids, the "seeing" error would be too large, so you are between the frying pan and the fire, and the best I can come up with back then was autoguiding to 0.7" to 1" RMS.
A halfway reasonable worm gear drive has a much smaller "A" term. "A" is about 30" with my RST-135, while a traditional mount that is 1/10 the price produces an "A" of 5". But notice that "A" is by itself not the problem. The problem is
B.n/P, where n is the harmonic, B is the "A" term for that harmonic, and P is the period. And that is the outside term of the first derivative of the periodic error curve. The cheaper the gears, the larger the B term for n=3, 5, etc (i.e., very visible bumps in the periodic error curve.
The A term of the periodic error curve is only important for visual astronomy, where you want the star to be moderately centered over the night -- your eyes can compensate for the movement.
By itself, it is not important when you have to autoguide. To autoguide, you need to freeze the centroids,and that is where the A/P term the amplitude term of the first derivative is the important factor. And even more important are the harmonics (bumps in the curve within a period) since the error becomes B.n/P -- that "n" eats you alive.
So, be careful when you interpret specs. Just because the periodic error is small is not good enough for autoguiding. When you shop for an autoguiding mount, you not only want P to be large and A to be small, and you especially want the "n" amplitudes to be tiny (small harmonic distortion). Manufacturers will often advertise "small PE amplitude," where the more important specs is the amplitudeof the first derivative of the curve. As I have always warned... Caveat Emptor (the onus is one the purchaser to understand what he/she is buying).
I assume this should be easy to grok for someone who wears a Beaver ring. Wink, wink, Al.
Chen
