Mechanical Periodic Error Correction

A Simple Orthogonal Set-screw Tuning Method to Improve Periodic Error

Michael Blaber

Author biography:

Michael Blaber is an Associate Professor of Biomedical Sciences at Florida State University where he does research in protein engineering. He never owned a telescope until the age of 42, and over the span of a few short years built a roll-off roof observatory in his backyard in Tallahassee. His wife is very understanding.

Abstract

The following article describes a cheap and simple method of minimizing a substantial portion of the periodic error in a typical worm drive. The method makes use of orthogonally positioned set-screws on both ends of the worm. Appropriate pair-wise adjustment of these set-screws permits the addition and subtraction of sine and cosine functions, of variable magnitude, to the worm period. These functions can be applied in combination, like a simple set of Fourier terms, to progressively eliminate a majority of the periodic error.

Introduction

Perhaps my back yard is similar to that of other amateur astronomers. There is an area, overgrown now, where I placed three 12" round aggregate pavers, spaced to accommodate my old 6" reflector's tripod. Next to this is a 4ft² concrete slab that I poured, to permit me to easily reposition my old 6" reflector's tripod (since I erred in positioning the aggregate pavers to permit polar alignment of the telescope). Next to the slab is an 8 x 10ft roll-off roof observatory (which elicited only a single formal complaint to the local homeowner's association) with a half-ton concrete base into which is permanently mounted an 8" diameter steel pier to support my present telescope (a 12" Schmidt Cassegrain). These stages of construction reflect my growing interest in astrophotography (although my wife was concerned that they might also reflect stages of progressive cognitive impairment).

After several years of working with my mass-market 12" Cassegrain I began to appreciate both the strengths and weaknesses of the system. On the one hand, the capability that the telescope was astounding: automated star alignment, periodic error correction, GoTo feature, computer interface, etc., combined with excellent optics, and affordable price. On the other hand, I came to the conclusion that the limitation of the system for astrophotography, was not the optics but the mount; the key limitation being the accuracy of the right ascension drive. In short, errors in the right ascension drive were causing slightly oval star shapes (the long axis being east/west). Despite disassembling the drive base, installing new bearings, cleaning the gears, etc., no amount of simple tinkering could substantially improve matters.

I came to the conclusion that mass market telescope manufacturers probably achieve a desired level of tracking performance by a carefully managed interplay between the mechanical and electronic subsystems. If the system were mechanically precise, there would be no need for electronic measures to correct for tracking error; however, such mechanical precision must be extremely expensive to achieve. Conversely, inexpensive electronic circuitry can only do so much in attempting to compensate for mechanical error. So, there must be an economic balance: reasonable mechanical precision, and reasonable electronic correction, for overall reasonable cost. Although the images I was recording amazed me, the slight oval shape to the stars became more and more irritating; and I realized that I could do no more with the mount that I had. In short, it was time to invest in a more mechanically precise mount.

I had always wondered why some mounts cost more than the optical assembly; now I could appreciate the value of mechanical precision. Unfortunately, purchasing a truly high-end mount was simply not an option. My wife was blissfully unaware of how much I spent on my telescope, and there was no way I was going to tempt fate twice by siphoning our savings for an expensive mount. Furthermore, it is difficult to explain to others why it is essential that images of stars be round instead of slightly oval. I began searching the web for suppliers of accurate (yet reasonably priced) gear and worm sets with which to build a more accurate mount. There was not a lot to choose from. To make matters worse, there were conflicting degrees of apparent user satisfaction with certain suppliers. I settled on a 2" shaft diameter pillow block assembly with a 9" right ascension gear and 1" worm (6" gear with 1/2" worm for the declination drive) from a supplier that quoted a high-degree of accuracy and listed a large number of prior customers (presumably satisfied).

Assembling the mount and drive has been a time-consuming and arduous task that so far has consumed the better part of a year (and is still a work in progress). Among the various issues faced during this past year: 1) a need to find a manufacturer of custom 2" bore gears for 6mm belts to drive shaft encoders; 2) the supplied 1rpm DC motors were wholly inadequate (even after purchasing a regulated power supply and building a sensitive pulse width modulator) - I opted for an installation of Mel Bartel's servo motor system; and 3) design and machining of a spring-tensioned mounting block for the right ascension worm (the solid mount design provided was not precise enough). Finally, when it looked like I could see the light at the end of the tunnel, it turned out to be a lightening strike that fried my computer's serial port, shaft encoder and the servo controller (ah, summer in Florida...). Mel is one heck of a guy and replaced the servo controller; the other parts were, unfortunately, out of warranty.

At long last, the time had arrived to check out the periodic error of my new mount; although it was still summer in Florida, there was the occasional evening when the skies were clear from around 10 p.m. to 4 a.m. I marked a reference position on the worm gear, and after a rough polar alignment began collecting data on the periodic error. I was so excited, anticipating the tracking precision I had long dreamed of, that I my lips were tingling (in retrospect, had applied so much DEET to ward off mosquitoes that I was probably just experiencing early symptoms of nerve poisoning). I plotted the data and could not believe what I was seeing: the periodic error was terrible. A trace of the error was a giant tsunami of a wave, at least 50 arc seconds peak to peak. It was worse than my old mount. I had apparently spent almost a year of my life, and money that took a lot of effort to keep secret, to achieve greater tracking error. I was even more depressed than when the stucco kept falling off the side of the observatory when I was building it.

Several anguished days went by, after which I was finally able to think a little more clearly. I concluded that there must be a good reason why the error was so great (other than I had been savagely ripped-off by the gear manufacturer). The periodic error looked very much like a cosine wave with the peak, curiously, near the exact start of the worm period. After examining the worm gear, and the reference mark I had made, I noted that the mark was, coincidentally, adjacent to one of the two set-screws that secured the worm to its shaft. Was it possible I had applied too much torque while tightening the worm set-screws? Considering that the entire mount assembly (including wedge and OTA) weighs around 400 pounds, some bolts require heavy duty torque when tightening. But the worm set-screws were fairly small and delicate, and perhaps I had been too aggressive when tightening them; I decided to loosen them (what the heck). They did not seem particularly tight, and I loosened them to what seemed to be a bare minimum of torque to retain the worm on its shaft. After another endless wait for clear skies, I plotted the periodic error. A dramatic improvement! The "wave" had subsided noticeably to a height of about 30 arc seconds peak to peak. Things were moving in the right direction, but I could do no more since the set-screws were about as loose as I dared make them. What I needed was a set-screw on the opposite side of the worm that I could tighten.

The Hypothesis

I thought about the problem for several more days. It was possible that the worm shaft was out of round, and that the set-screw issue was merely a way to compensate for this eccentricity. Alternatively, the worm axis might not be exactly parallel to the worm shaft, and the set-screw adjustment provided a means to make fine adjustments to align the axis (the two set-screws on each end of the worm were 180⁰ opposed to each other). Thus, the hypothesis I was contemplating (because of the observed effect of the set-screw adjustments upon the periodic error) was that the worm set-screws not only affix the worm to its shaft, but also determine its alignment with the shaft axis, and misalignment contributes significantly to periodic error. If this hypothesis was correct, then the course of action was obvious: I would need to tap threads for additional set-screws in the worm to permit additional adjustments. In particular, I would need a series of four orthogonal set-screws at each end of the worm. The worm shaft would also need four orthogonal flats filed or machined at each end to accommodate the set-screws. Another agonizing moment of truth: do I drill, tap and grind two of the most holy parts of the mount (using my really cheap drill press), or do I content myself with a periodic error that is going to irritate me in perpetuity?

Figs. 1 and 2 illustrate the ultimate course of action that I took. The worm gear was marked to indicate 0, 90, 180 and 270⁰ reference points (Fig. 1) coincident with the four orthogonal set-screws at each end of the worm (Fig. 2). The eight individual set-screws are identified by their corresponding angle referenced on the worm gear, and which end of the worm they are located (my worm is oriented east/west on the mount and the set-screws are identified as being either "east" or "west").

Figure 1.

The right ascension worm shaft and drive gear. Orthogonal flats have been machined into the shaft for the four set-screws on both ends of the worm. The alignment of the set-screws and their corresponding position in the worm period are marked on the worm shaft drive gear. In this particular mount the end of the shaft nearest the drive gear is "East".

Figure 2.

The worm with four orthogonal set-screws tapped into the hubs at both ends of the worm (and with identical alignment).

The test

To begin the great experiment, each set-screw was tightened to a bare minimum torque to affix the worm to its shaft, and then a healthy torque (I don't own a torque wrench) was applied to the 0⁰W and 180⁰E set-screws. The periodic error (a whopping 50 arc seconds peak to peak) comprised a cosine wave (i.e. a trough at 180⁰ and a peak at 0⁰ (Fig. 3)). I then loosened the set-screws again and tightened the orthogonal 90⁰W and 270⁰E set-screws. The periodic error (a similarly huge 40 arc seconds peak to peak) now comprised a sine wave (i.e. with a trough at 270⁰ and peak at 90⁰ (Fig. 3). My excitement at these results was difficult to contain; the worm set-screws could be used to adjust the worm alignment on the worm shaft in a controlled manner. I had at my disposal the ability to add or subtract sine and cosine functions, in any combination and magnitude, to minimize the periodic error (Fig. 4).

Figure 3.

The right ascension tracking error with the worm initially mounted using only the set-screws at the 0⁰W and 180⁰E positions (magenta trace), or at the 90⁰W and 270⁰E positions (orange).

Figure 4.

A listing of the various orthogonal set-screw adjustments of the worm and the corresponding effect upon the worm tracking error.

Starting over, and with an initial periodic error approximately described by a cosine function (Fig. 5), I applied a negative cosine function by gently tightening the 0⁰E and 180⁰W set-screws. After repeating this (since the initial adjustment was insufficient), the residual periodic error was now approximately described by a sine wave (Fig. 5), and so I applied a negative sine function by tightening the 90⁰E and 270⁰W set-screws. At this stage the periodic error started to resemble a slight negative cosine function and, therefore, a very slight cosine function was applied. At this point, the required adjustments were mere nudges on the set-screws, and it was easy to over-shoot the adjustment. The final periodic error has been reduced to approximately 6.5 arc seconds peak to peak, with a standard deviation of 2.7 arc seconds (Fig. 5); this is substantially better than my previous mount *after* electronic periodic error correction. It may be possible to improve upon the current periodic error by the application of a slight sine wave, but the run-to-run standard deviation is approximately 2.5 arc seconds, so any further improvement will be marginal (but must be attempted nonetheless!).

Figure 5.

Progress of the tracking error as corrective sine and cosine functions are applied using the different set-screws in the worm: 1. Initial periodic error (magenta); 2. After application of an initial (-) cosine function (red); 3. After further application of a (-) cosine function (green); 4. After application of a subsequent (-) sine function (blue); 5. After addition of a corrective (+) cosine function (black).

In short, the right ascension gear and worm assembly do indeed have the accuracy that was stated by the manufacturer (sorry about all the rotten things I was thinking...); however substantial effort was required in achieving it. For systems with accessible, shaft-mounted worms, the orthogonal set-screw tuning method described in this report can substantially reduce periodic error with little to no financial cost.