I believe when implemented correctly either of these two methods can practically eliminate backlash and be DIY'd for less than $50 per axis.

I like the idea of direct drive most, shown here as truly direct however step-down reduction could be used to increase resolution or power if needed. It's more about the idea of motion transferred by rolling. The 'infinite gear'. Obviously the load is limited by the friction between the surfaces. Im thinking of a knurled roller on hard flat rubber (cheap and easily replaceable too). Will require some experimenting. The spring drawn represents some preload with the 'idle' wheel below. I believe with the right combination of materials and preload, this could route wood no problem.

The other method is Cable. Im thinking large cable here, about half inch wire rope as the loads we generate would not cause that cable to distort. I prefer the idea of the fixed cable. I didnt show a tensioner on the fixed but it should be used as having the cable correctly tensioned is a must to minimize backlash. Both show a shaft connecting the other side of the axis which should negate the need twin drives.

Nivea, I'll throw in another. (not my origin, you just brought it to mind) Similar to your second graphic. Cable attached firmly one end (A), loop around drum(pulley), cable tensioned to end opposite of (A). Tension enough to prevent slippage.

Drum diameter calculation includes pitch line of cable for dimension/movement.

Drum can be powered by small gearhead servo/stepper.

This assumes axis being positioned is not "loose", that is secure from motion other than intended.

Richard, your method is better yet as it maximises contact surface between the drive and the cable. I have seen it used often to drive the larger automated reflecting telescopes.
What's also nice about using a cable system is it's relatively insensitive to dust, chips, sparks and will never wear!

Nivea, I suppose that drum thing will work on most any low force application. Two applications I'm familiar with are moving the welding gun along the "Z" axis of a welding lathe used to build up the diameter of cylinders and moving a laser on a similar type of machine.

Another one is your Number 2 graphic again but using a metal tape instead of a cable. The tape had perferations similar to that on the edge of 35 mm photagraphic film and the drum had tapered "cogs" to engage the perferations. Again, light load, low force.

DZASTR's method is the same one used on many manual surface grinders. It's very smooth and would work well with light loads and low accelerations.

Do note though that unlike R&P and ballscrews, where backlash is relatively fixed (predictably the same amount regardless of screw length and load), a cable drive's backlash/stretch is directly proportional to the length of the cable. An 8 foot long cable will stretch twice as much as a 4 foot long cable given the same diameter cable and the same applied force.

Some simple Hooke's law calculations will let you predict how much the cable will stretch and whether the accuracy will suit your needs.

I am really interested to see where you go with this. I am studying components for a belt drive build similar to your picture #2 - a trolley I think we call it.

One thing I have learned while studying this arrangement is that many of the stepper / servo motors are not designed for significant side load (radial) - often about 0.5 Kg or so, which is why they are often set up with the mechanical transfer connections. This is an important design consideration as version #2 will put 2 x the radial load on the motor as your stretched cable is tightened.

After some searching, and very useful guidance from CNCZONE people, I found that pacific scientific and anaheim automation both produce special versions of their stepper motors designed for side loads up to 30 + Kg.

Interestingly, the approach of wrapping the cable around the drum produces much lower radial loads than version #2 - almost zero under zero motion if my vector mechanics thinking hat is still working, but still equal to your driving force under load.

Your non - linearity will be directly affected by cable slip and non- concentric rotation of the drum / support bearings, so I am not sure if this is a $ 50 solution or not, but it is interesting. Please keep us posted.

Your machine #1 is similar to the paper drive in modern large format plotters. I think this was first introduced by Hewlett Packard in a small desktop plotter. They use a drive roller that's metal impregnated with grit, opposing a resilient idler. The grit makes tiny indentations in the paper, and these help to prevent slip as the paper makes multiple passes through the drive.

If the drive roller is directly attached to the motor shaft, your step size will be proportional to both the step angle and the diameter of the drive roller. For a 200 step motor and a 1" dia roller, your steps would be about 0.016 inches. Perhaps a servo with a fine encoder would be better.

Your machine #2 unnecessarily adds the mass of the motor to the moving mass. Also, how will you get the motor to grip the cable? Use a synchromesh cable? But then you might as well use a timing belt. I don't think you will be able to bend 1/2" dia wire rope enough for this application. Even 1/16" wire rope is limited to about 1/2" minimum radius.

The drum idea solves the grip problem. But it introduces another problem.. that the wire walks across the drum as it rotates, so the smaller the dia of the drum, the longer it has to be. Do you have any idea how to make the drum, and couple it to the motor?

Your machine #3 makes the most sense to me, but it might as well use timing belts instead of cable. I assume you intend here a motor with a shaft at each end, and shaft couplings.

Using cable for the other axis could make sense. It requires 8 pulleys and a drum, but gives the advantage of very low moving mass.

As stated above, these are used for low force and typically low accel rates. Yes the cable can stretch. If you require more accuracy you can use a seperate measuring device, readily available and relatively inexpensive.

I doubt a $50/axis machine will be expected to deliver extreme accuracy.

The radial (side) force can be canceled with an outboard radial bearing.

These cable drives aren't intended for high precision, heavy machines or fast machines unless they are very light.

Besides, this isn't new, there are many applications out there already, some many years old. Some even older than me and I'm older than Adobe and he's older than dirt. lol