How precisely can you measure position? How precise can a servo be? What if it’s bodged together from mass-market junk?
It’s practical to illuminate a fused-quartz optical scale with 365-nm-wavelength LEDs. A modern CMOS imaging sensor can fairly easily reach the Abbe limit; if we take our numerical aperture as 1.4, the Abbe limit is λ/2.8; at 365 nm that’s 130 nm. We can reasonably expect to estimate the centers of blobs in the image to within about a quarter of the resolvable feature size, so we should be able to estimate our absolute position along the scale to within about 35 nm by this method.
This is only about an order of magnitude worse than the recently best available optical position sensors, such as Renishaw’s 5-nm-resolution “RELM” scales that limited the precision of Awtar and Parmar’s XY HiPER NAP from 2008, and it should be achievable with cheap off-the-shelf parts. The EPSRC Center for Innovative Manufacturing in Ultra Precision at Cranfield University routinely achieves better precision than this, 10 nm in many cases and down to 1 nm in some cases, but I don’t know how.
Fused quartz’s linear thermal coefficient of expansion is about 0.5 ppm/K, so a meter-long fused-quartz scale will lengthen or shorten by 35 nm every time the temperature changes by about 70 millikelvins. I don’t think infrared thermometers can measure temperature with this precision, so compensating for temperature variations would probably require the use of some kind of contact thermometer; I’ve argued in my note on Thermistors that this level of temperature measurement precision is fairly easily achievable without super-precision circuitry. Note that precise temperature calibration is not necessary, but the absence of drift is.
But contact thermometers are slow, at least the macroscopic ones I was considering in that note, so it seems likely that a more viable approach is to prevent temperature variations of more than 70 mK or so. To do this, you would use PID thermostatic control of a stream of coolant, such as water, air, or hydrogen, continuously pumped past the fused-quartz scale to prevent thermal variation; and you can calibrate the impulse response of the scale’s thermal expansion to the error signal from the PID control system, so you can to some extent compensate for smaller variations.
500-millikelvin-precision temperature control has been a sine qua non for precision manufacturing since Michelson’s first grating engines.
A different and possibly more difficult problem is that, if you’re measuring motion along such a scale, you may actually be interested in the relative positions of things attached to the scale and the sensor. And those things will almost certainly have a much larger TCE than fused quartz, as well as being subject to some side loading that causes them to deform elastically. It may be feasible to use the above-described techniques to take measurements from different points of them to quantify these deformations.
Consider the Samsung Galaxy Note 20 rear camera, which is 108 megapixels (presumably 9000×12000) with, if I understand correctly, 0.8 μm pixel pitch; it is capable of delivering 720p video at 960 frames per second. (0.8 μm would give us a sensor physical size of 7.2 mm × 9.6 mm, which is quite reasonable.) For the pixels of such a sensor to correspond to 130-nm regions of the surface being focused on, the optics need to magnify it by only about 6.1×, which is to say that the focal-plane sensor would need to be 6.1× further from the lens plane than the surface is.
You might think that this technique would require a specially printed pattern on the glass scale, and indeed that is the standard approach, but that is not necessary; you can simply store high-resolution photographs of the whole scale and match the image against them. Relatively efficient algorithms for this kind of thing are used for precise orientation by star tracking in spacecraft, but that really isn’t important; a particle filter would probably be perfectly adequate, since most of the time your new position and velocity can be nearly extrapolated from the old ones. A meter is only about 6 million pixels long at 130-nm resolution, so probably a few tens of megabytes would suffice for a long enough map.
As I wrote in Dercuano in a note on sparkle servos, resolution can be improved with moiré effects. In this case, though, the “moiré grating”, which obscures part of the field of view except for some subpixel-sized holes, needs to be in the optical near field of the glass scale to work, say positioned within 200 nm or so of it. But if another transparent object is scraping against the fused-quartz scale, not only can it deform it elastically, but also it will either scratch the glass, be scratched itself, or both. So I think you need either a lubricating film, an air bearing with high-quality filtered gas, or some other kind of arrangement to hold the two within a few dozen enanometers without making contact.
Hard disks routinely servo back to the same 80-nm-wide track reliably within a few milliseconds; the heads float on an air bearing tens of nanometers away from the constantly-moving disk surface. I’m not sure there’s any practical way to reuse this amazing feat of engineering for nanopositioning.
To achieve higher positioning resolution than an optical system can provide, a scanning tunneling microscope (“STM”) may be an option. These can provide subatomic precision (on the order of 0.01 nm) but are not normally used as positioning servos. Also, they are sensitive to shock and vibration. However, they can be homebrewed with a few hundred hours of work and can operate in air.
Capacitive sensors routinely achieve nanometer precision, but only over submillimeter distances. I’m not sure if there’s a way to improve this.
Maybe a holographic approach with light has some hope of working?
Metamaterials similar to gridiron pendulums could potentially take the place of fused quartz and Zerodur for applications like this.
You could imagine, for example, three concentric metal tubes, the inner and outer tube of steel and the intermediate tube of zinc, with the inner tube being interrupted at 0 cm, 2 cm, 4 cm, and every other even centimeter; the outer tube being interrupted at 1 cm, 3 cm, 5 cm, and every other odd centimeter; and the zinc tube being interrupted at every centimeter, and soldered to the interrupted steel tube at that point. In this way thermal expansion of the zinc shortens the tube, while expansion of the steel lengthens it, but by a slightly smaller factor (about 12 ppm/K against about 30–35 ppm/K for zinc.) By adjusting the kerf width at the interruptions, the expansion and contraction can be perfectly balanced.
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Consider the section from 0 cm to 2 cm with an effective kerf width k of 1 mm. We have a total of 38 mm of steel pipe in the “kinematic chain”: 9.5 mm on the outside, 19 mm on the inside, and another 9.5 mm on the outside, 40 mm - 2k. And we have 18 mm of zinc pipe, in two 9-mm lengths, 20 mm - 2k. Suppose its TCE is 32 ppm/K. If the temperature increases by 100° the zinc pipe will lengthen by 0.32%, to 18.0576 mm, thus shortening this section by 57.6 μm, and the steel pipe will lengthen by 0.12%, to 18.0456 mm, thus lengthening it by 45.6 μm. This already gives us a much reduced thermal expansion coefficient, but by widening the “kerfs” to a bit more than 3 mm to diminish the zinc with respect to the steel, we can reduce it to zero.
However, that zero depends on the precision of our effective kerf width (which extends roughly to the middle of the solder joint), on the precision of our estimation or measurement of the thermal coefficients involved (which vary with temperature), and on the precision of equality of temperature of the different components. Still, it should be straightforward to excel invar’s 1.5 ppm/K performance.
The resulting object, if properly soldered, has strength similar to a single layer of the zinc tubing, if it were a continuous pipe, and a lower stiffness — the compliance is roughly the sum of the compliances of the three pipes.
There are a number of common materials with smaller TCE than steel, but nearly all of them are brittle: soda-lime glass (9 ppm/K), limestone (8), granite (7.9–8.4), sapphire (5.3 or 8.1, depending on which part of that table you believe), tungsten carbide (4.9), brick masonry (4.7–9), graphite (4–8), industrial porcelain (4), borosilicate glass (4), wood parallel to the grain (3–5), silicon (3–5), mica (3, presumably parallel to the grain), carborundum (2.77), and diamond (1.1–1.3). Common materials in the table with substantially larger TCEs include basically all organic chemicals (with some polymers up into the hundreds), plaster (of Paris?) (17 ppm/K), 304 stainless (17.4), aluminum (21–24), fluorite (19.5), magnesium and its alloys (25–27), lead (29), wood across the grain (30), and rock salt (40.4). Kapton is notable in this table among organic polymers for having a TCE of only 20, though a different source gives 55 for “polyimide”. Unfilled epoxies are claimed to be in the 45–65 range.
You can do this kind of trick with materials whose TCE is arbitrarily close together, but you need more layers of pipe; that’s why traditional gridiron pendulums had seven vertical bars rather than five, because in the Victorian age they just had brass and steel, no aluminum or magnesium, and even zinc was comparatively exotic. To get by with just three layers you need materials whose TCE differs by more than a factor of 2.
By using materials further apart in TCE, such as brick masonry and magnesium alloys or rock salt, you may be able to get by with much smaller “compensatory pieces”. Brick masonry has the additional advantage that it is cheap enough and hard enough that you can achieve very high stiffness with it at a reasonable cost.
A different metamaterial approach would be to use leverage to amplify small differences in thermal expansion into larger compensating contractions.