To make an automaton like the Jaquet-Droz artist, you need to program a relatively extensive series of paths.
One mechanically very simple way to do this is with a differential spiral cam gantry, like a couple of Chinese windlasses driving a vertical plotter.
The first step is to wind a couple of threads — one for X, one for Y — around eccentric spools mounted rigidly to a common shaft. The rates at which the threads are taken up by the spools will vary depending on the rotational position of the spools, which can be out of phase. If you hook up the threads to a pen in X and Y, as you wind, the pen will trace a wavy line across the paper as the phases vary.
This is kind of lame, though, because not only does the motion repeat on every turn, but also, in a given direction of windings, the pen is limited to a fairly small range of directions: up, left, or somewhere in between. Moving it right or down is impossible.
The solution to the direction limitation is to make the motion differential: instead of directly driving the pen with the thread, run the thread around a pulley on a long loop between a supply spool and a takeup spool mounted rigidly to the same shaft. Each of the spools can be eccentric or oddly shaped. The difference between their radii at the tangent points of the thread at a particular rotational position results in a net lengthening or shortening of the loop, and thus extension or retraction of the pulley. If they have different overall circumferences, the loop will have a net extension or retraction for each revolution.
Now we have four spools mounted on the same axis and two pulleys connected to the pen; each revolution of the spools moves the pen through a fixed pattern, which may not end where it began, and so repeating the rotation will draw the pattern repeatedly at slightly different positions.
Suppose that we don’t want to simply repeat the pattern, though. Then instead of simple drum-shaped spools, we can use spools with helical grooves to entrain the thread; perhaps a 140-μm thread (Dyneema 10-pound fishing line, say) could run in the bottom of a 2mm-deep wide. Now, each rotation of the spool can vary its radius independently of previous and subsequent rotations, as long as you can confidently shape the 2mm-wide groove “above” or “below” the previous and following windings of the groove. You’re still limited in how quickly you can increase or decrease the radius — you need a quarter turn to ramp all the way up from or down to zero — but hopefully the differential principle keeps that from being a huge problem.
Now let’s suppose we have a reasonable-sized RepRap-printed version of such a doohickey. Let’s say, four spools with 2mm grooves, max 100mm radius, and 200mm allocated among the four, 50mm each, printed at 100-μm precision. 50mm of a 2mm-pitch spiral gives you 25 turns, a total of 7.9 meters at the maximal 100mm radius. It probably makes sense to think of the actual motion as being half of that: about 4 meters. I’m not quite sure how quickly we can change speed, but if we figure that we can control more or less each 100μm unit separately, then we have about 40 000 line segments here to draw with, which is plenty for encoding a Jaquet-Droz-Artist-like picture.
Some final details and variations:
It might make sense to “gear up” the output so that the detail is rougher than 100μm, but the total line length is more than the 6 or 8 meters you get with this design. For example, you might use an extra pulley or two so that the pen moves 2mm or 4mm every time the thread difference changes by 1mm.
Rather than a gantry system (like what you see in a construction crane), you could use the standard vertical-plotter configuration, where the two dimensions are the distances from two reference points.
By using three pairs of spools instead of two, you could overconstrain the pen, keeping it under tighter control; and you could selectively relax that constraint for when, for example, you’d like to lift the pen from the paper. Indeed, this gives you a deltabot, with its attendant three degrees of freedom.
You could print each spool as two separate prints, instead of all as one piece, so that you have four times the length. And you could probably space the grooves closer than 2mm, though 1mm might be pushing it.
Instead of using the threads under tension as linear actuators, you could use them to rotate things. I kind of feel like cords under tension eliminate most of the usual problems with linear motion, though, except for lack of precision.
You could use it to position things other than pens. Markus Kayser used a vaguely similar system in his SunCutter to control the MDF-burning effect of a magnifying glass, although he was using timing-belt-linked big acrylic cams pushing on cam followers on a linear slide gantry moving a table around. You could position machine tools, position squirting nozzles of various materials, trace lines in wet sand, trace lines in wet frosting or other plastic material, move an electro-etching or electrodeposition head around to selectively etch or deposit metal, cut foam with hot wires, cut things with lasers or cutting torches or sparks, etc.
You could use thicker thread. 140-micron Dyneema fishing line is capable of supporting loads of a few kg; if you were to go up to 1500-micron Dyneema braid, that would hold loads of up to 400 kg or more, 800 kg in each dimension considering it’s going around the differential pulley twice. This would also provide greater rigidity, but probably much greater than is necessary; if we want 1mm positioning precision over a 1-meter distance, and our pen or whatever weighs 500g and accelerates at 1 gee, we need to have less than 1 millistrain of deformation at 4.9 N. Doubled-up 140-micron Dyneema with a Young’s modulus of 100 GPa would hit a millistrain at about 3 N, so it’s already close. Doubled-up 1500-micron Dyneema doesn’t stretch a whole millistrain until 350 N; at 5 newtons over a meter, it stretches 28 microns.
The Grail, of course, is to make the device programmable at run-time, instead of requiring a new set of spools for every new picture. This can be accomplished by making the spools cylindrical and not eccentric, with a constant radius, but no longer rigidly fixed to the same axis; instead, their axes should be linked with some kind of continuously variable transmission, probably one of the positive-displacement ratchet-controlled types rather than one of the friction-controlled types. The setting of the CVT then determines whether the thread is being extended or retracted.
You could imagine using a lightly angled cone spool instead of a cylinder, then controlling where along the cone the thread winds onto it. This might work, but I fear that the thread might slip too easily along the cone after being wound; maybe a sufficiently bumpy cone surface would prevent that problem, at the cost of less precise positioning.
Eventually, with any of these variations, you run out of thread in one direction and have to change directions. If you’re driving the spools with a powerful motor, this may require slowing down and stopping before you hit the end and break the thread. In fact, this is likely to be a showstopper unless you have several hundred meters of thread.
By threading the thread through various kinds of paths, we can actuate things other than just a pen floating around in space. For example, three threads running through a universal joint are sufficient to pull the joint into any position.