high precision stepper gearing

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i have been building a small specialzed cnc machine, that is up till now i have used floppy steppers for program testing.
so before i go out and spend a thousand bucks on equipment i thought i would check my math and conclsion here on WP

i have been looking for some affordable gearboxes for some 0.9 degree steppers for a rotational stage and i am limited to 1/8th microstepping,

now as far as i can calculate the correct gearing to aproximate precise 360 degree motion on the sub milimeter level is through either a 15 or 30 to one gearing

360 / ((400 * 8 ) * 30) = 0,00375

1 / 0,00375 = 266,666667
266,666667 * 360 = 96 000,0001

now in the above i have divided the full 360 degrees of rotation with the (step count * microstepping) * gearing.
then i find the steps pr single degree of motion and finally i find the step count for a full rotation.
am i correct in assuming that bar any backlash and general theory vs. practicality i should end up with a theoretical accuracy of around +/- one full step, ie one 244th of a degree.
the reported backlash for my stepper of choice including gearing is 0.05-0.08 degrees.

is there anything i have missed?
is my math wrong?
will this even allow for sub milimeter precvison in practicality?
any help or input is greatly apreciated.

this rotary stepper will be mounted on another rotary stepper that is in turn mounted to a vertical stepper axis using a precision lead ball screw.
the moving gantry itself should be relatively stable with 4 sliders in the corners and almost 100% even wheight distribution, the stepper mounted there however does have to be mounted slightly off center to allow for the lead screw passthrough.
the gantry will be cnc machined with a mounting bracket for the front rotary stage.

Okay, this is my specialist subject

A slightly easier way of working this out is steps/rev * microsteps * gearbox ratio
400 * 8 * 30 = 96000 microsteps per rev. So far we agree.

Ignoring backlash in the gearbox your accuracy should be step angle / ratio
0.9 / 30 = .03 degrees

That sounds about right - at least until the gearbox wears a bit.

It depends on the distance from the center of rotation. Without more information it is difficult to say.

What exactly are you trying to achieve here? Do you have any diagrams or pictures?

i have some ever changing CAD models.

it is a cnc lens and mirror grinder capable of producing lenses in a decent quality(i have some experience working in crystals and stones and i recently saw some decently cheap optical blanks) the z stage is still a bit fuzzy since i am looking at several intergrated and home built lead screw assemblies, i would like to go down to around 0.005mm accuracy, since this is the actual feed axis.
thing is hand grinding lenses, while fun and cheap is a long and hard process and the results take years to master, yet the theory and the optics are fairly simple to calculate so i want a cnc machine that is able to do the work.

all linkages will be close to direct, ie, the lens is mounted "directly"(no gearing, only cnc'ed metal) on the geared stepper i found, with its torgue rating plus the gearbox it should be doable, again i might have to rethink that and use a worm gear with a precision slot for the dop(rod the lens will be mounted on with resin) instead, i am however afraid of the inherent backlash in worm gears when changing direction.
i have also contemplated wire pulleys with pattern wire, they should have very little play and are used for laser positioning.

as said i have access to some cnc equipment to create the pieces, i mainly started the program for fun but now that it runs with floppy steppers i would like to take it all the way, for the research and experience as much as the product.

currently i am trying to tie together the cost of the gearing and stepper motors with that of the mechanical parts, when i know what i can do with the money i have i can start thinking about finalizing the design,

as for gearbox wear, true, but no matter how good a gearbox you get that will be an issue.

0.005mm is pretty tight. I think you will have a job getting that sort of repeatability.
How are lenses ground by hand? Could you find a way of automating that process instead of trying to directly machine the shape?

there is vibration forming(vibrating against an abrasive, requires near human coordination because of the rocking and varied pressure one apllies) but ultimately to reach the finish i am looking for (laser mirror flats, pure quartz prisms and such) you need to employ several diamond pastes and grind them against a copper tin or fine ceramic surface.

in essence it is lapidary but on a very hard and brittle material, in fact the grinder we will use is one used for lapidary.(i have already cut quartz, agate, tourmalins and opals by hand, diamond is possible to grind like this, in fact its how most stones in the world are made for the jewelry trade, bar the mass produced cnc worked stones but since they are sized by computers they usually have a much lower yield than this method would bring if used)
now the general idea is that in essence a lens is nothing but a very precise cabochon cut, the instructions for doing it by hand is already in a CNC compatible format, ie two radial measurements a sequence and the depth can be synthesized by using a drop force keyholed dop, it will rest the lens against the grind stone untill contacting a hall interrupt and ledge preventing further grinding, the interrupt will trigger the next step, now mind you it wont be an ultra fast process, i estimate a coupleof hours of grinding per lens, it would take longer for these sized lenses by hand at least if i do it.
now i have worked in a prototype mould maker when younger and i learned cnc cand and cam there and still have access to the machines for a modest fee,
so producing actual high precision parts is a possibility, i would however like to minimize the amount of parts as metal aint that cheap and the machines could be busy so it might be hard to fit in a large courtesy order despite the payment.

now i actually found a finished lead screw assembly with stepper encoder(allows for zeroing) and driver in one and it had a precision of 0.005mm, it came with anti backlash leaded ball flanges so it should provide next to no backlash.(the precision of course does vary with distance from the bottom but since the cutting action is in a relatively narrow band 20-50mm vertically, it would mean that the only time i suffer from a lack of precision is when i disengange the grinding wheel anyway).
the tower itself would need support and i need a tunable counterwheight to balance the off center stepper (wheight on a slider with a locking nut, or perhaps a linear slide opposite the outhanging arm, use tension isntead of wheight, i have some decent precision slide rail (about 4 meters, only for z axis,)

but yes it is tight, it isnt strictly neccesary but i might as well take as much into account as i can, better do it right with a little need of tuning than stand with a limited machine.
now in essence my biggest trouble doesnt seem to be the z axis since if wheight permits i will buy and use one of the above mentioned lead screw assemblies, they will also customize the other 2 steppers with built on gearing with a 30:1 ratio, however i am worried that i will need encoders as i am unsure of the repeatability of the rotary steppers, one would take roughly equal sequences of each direction adn the other only has a limited area of travel about 90 degrees or a bit less, doesnt matter that much, because of the gearing that will of course mean quite a lot of microsteps but still.

meh wall of text and special interest, thanks for bringing some well thought out criticism to the debate, much apreciated

btw. i lvoe your turners cube in your profile

CNC machines are capable of far exceeding humans for coordination but they aren't very good at making judgements on how much force to apply.
Sounds good. I'm not a great fan of integrated electronics as they are a lot more expensive to repair but if the unit is otherwise ideal, that isn't a big enough reason not to use it.

The weight may be a good thing. It will help preload the screw and slide. As your motor is going to be geared down a lot I don't think you will have any shortage of force to lift the weight.
I have to admit I don't like steppers but in this application I think they will work well and you shouldn't need encoders. As long as you keep the speed down and make sure the steps are smooth you shouldn't have any problems with lost steps. How are you intending to generate the steps?Thanks!

i will be using a uC attiny with an arduino bootloader (also why im limited to 8 times microstepping, even if i use seperate drivers there is a frequency discrepancy)
i havent decided if i will use my current darlingtion array setup or to go for discrete drivers, i like the idea of using i2c with drivers as it allows for a much simpler command structure and higher speeds but no matter what my current components wont hold up to the actual task.


the pulses will be generated by the uC but the command sequencing will be handled over a serial connection with a pc running the actual script that converts the hand instructions to cnc data and sequences them for use, so far it has worked perfectly with the floppy steppers, that is the software runs and the steppers are driven fairly well(there have been lost steps but i can see the full pulse train on my scope, i suspect its the low quality floppy steppers when at too high frequency), there will be a ton of finetuning and maybe rewrites when i change steppers, i also need to test it with the huge amounts of steps required for the new steppers

what type of motor would you use for these kinds of precision tasks if you had no constraints?
is there any specific axis (such as the limited rotation axis) that could be switched to something more ideal at a similar price?

i was guessing at servos and i have worked with my share of rc but i never thought of them as high precision motors, that might only be the quality though and my misperceptions.

Do you mean you are directly driving the steppers with a darlington array? A proper switching stepper driver will run the motors much faster and you just need to output simple step/direction command signals. Something like the Xylotex 3 axis board should do the job nicely.

Yes, floppy steppers are pretty useless at speed. They are generally rated at 12V. You want steppers that are rated at around 3 - 5V. When running steppers with a switching drive you generally use a supply voltage 5 to 10 times the motor's rated voltage to get good high speed response. The drive takes care of regulating the motor's current so you won't overload it.

You could use a servo motor with something like a DeskCNC driver. If you use a 500 line encoder you should be able to easily achieve motor positioning within 0.5 degrees. You would still have to use a reduction gearbox. However for this application I think servos are overkill. Without going for exotic solutions like harmonic drive gearboxes I think a stepper with a preloaded worm and wheel gearbox is probably the most practical solution.CNC type servos are very different to RC servos. As you say, RC servos are pretty inaccurate.

discrete drivers it is,
that said the darlington array was never meant for anything but testing my step sequencing, it wouldnt handle any stepper large enough to do the job, i was contemplating building a driver myself but since the pricing of discrete drivers seem to be within acceptable limits it would save a lot of work and research others have already done.
i might even get their individual power supplies as they are priced at 35 usd,

as for angular precision of the 90 degree axis it should be higher than 0.5 degrees using a 0,9 degree stepper at 30:1 gearing and 8 times microstep.
i wonder what sort of drivetrain the positioning system of comercial 5 axis machines are made, i never quite got to see anything but the spindle disassembled.
i did see some linear servos with a ridicoulous precision as in nanometer scale and if it werent for the price and the weakest link problem i ight have gotten one, 400 dollars(i would need timed laser distance sensing or similar to get a high enough precision zeroing when the high precision stage is connected to less precise stages)

the piezo drive technology truly is amazing but also amazingly expensive, especially considering the thrust and travel distance of the system, they also had rotary stages but their prices were even more prhibitive, sometimes several thousand dollars.

As far as I can find out, most use servos. AC servos are available with very high resolution but also a high price tag. As you say there are some other very interesting technologies but the prices are staggering.