I was very frustrated about how difficult it is to get small displacements, under a micrometer range. I guess what I was really frustated was about how expensive it is to get them. So I tried do come up with a cheap, on-the-budget way of doing that.

This was how the gravitic motor was born. The name is very misleading, sorry.

Imagine that you get some PVC pipes, a few PVC connections, and you get everything cut and ready make a tall H. That is two vertical pillars with a thicker tube, and in the middle a horizontal thin pipe crossing it. Use those PVC tubes that are threaded rather than soldable ones. Two caps seal the bottom of the H legs.

Now you prepare the floater. It is a cilinder of some material, with a diameter to fit tighly inside the thinner tube. This is so tight that if you put water in one side it does not leak to the other side, water-tight. This material should be very light, low density.

In the horizontal thin tube, you cut a hole on the lateral surface of the tube, in the middle of it. This is a thin cut, with the lengh as you would like your displacement range to occur. You insert the float inside this tube, and when it is centered in the hole you've made, you put a screw on the middle of the float. Now the screw can go back and forth, as the float moves inside the tube. This is where you get your displacement.

Assemble the H, with the float inside. Put oil lubricant on the float, and move it around a couple of times to get the float all greased up. It should move back and forth with the least friction possible.

Now fill the pillars with water, evenly - don't fill all the way up to the top of the pillars, however.  

Get the finest sand you can possibly get. Like the ones they use for hourglasses. 

Say you drop a grain of sand on the left pillar. The water collumn will raise, and the whole thing will even itself after displacing the float a little to the right. In fact, it will only displace if it beats the fricction, so this machine will probably have a smallest displacement of a bit of sand. The less friction, the better.

So whenever you want movement to the left, you drop sand on the right; movement to right, drop it in the left. For larger displacements, drop more sand. It accumulates at the bottom of the machine. The oil should not only lubricates but prevent water from flowing directly past the float.

And that is it, that is the whole thing. 

Note: the motor has a flaw of characteristic, depends on how you look at it. When the weight of what it pushes becomes too heavy, the system's friction increases and it loses the ability to move in small steps. Therefore, it is only good for light loads. 

Now that you've got the basic idea, some improvements:

• Make a system to drop sand electronically.

• Replace sand with particles of a standard diameter. 

• Make the pillars thick on the bottom, below the crossing pipe, but thin above it. There is a fine balance in autonomy in the gravitic motor.

• Add a second horizontal pipe below the float one, and attach a faucet to it. This is to conveniently remove the water for when you need to get the sand out of the machine, once it is full.

• Use mercury instead of water shrink the machine's size [this is not to be made at your home, and I'm not very sure about it].

Well, it is a crude, rudimentar motor. If you put four motors facing each other, and table on the center, you have a gantry-system like setup, with four degrees of motion.

This is the gravitic motor.