The vocabulary of measurement

May 20 , 2011 7:57:21 PM CDT

The goal of fabrication is to make things correctly. The trick comes in defining what 'correctly' means.

It's more involved than you might think. There are five closely-related terms that come into play:

  1. Error
  2. Accuracy
  3. Tolerances
  4. Precision
  5. and Repeatability

Error is most easily defined as "the distance between where you are and where you wanted to be."

Accuracy is the inverse of error. High accuracy means small error and vice versa. The term 'degree of accuracy' more or less means the order of magnitude in the error. An error of .001" is one degree of accuracy better than an error of .01".

The hardest part of understanding accuracy as it applies to fabrication is "knowing when to stop."

Perfect accuracy is impossible.. once you get down below a few angstroms (one ten-millionth of a millimeter), you're trying to trim off half an atom -- a process which is difficult, messy, and loud. To make matters worse, all materials expand and contract as they change temperature, so the practical limit on accuracy depends on the range of operating temperatures you're willing to maintain. For people who work to really high accuracy (a millionth of an inch or so) it isn't a race to build the best cutting tools.. it's a race to build the best thermostat.

Machinist jokes like this make more sense when you understand the relationship between temperature and accuracy.

Tolerances are an explicit statement of how much accuracy you want, usually expressed in terms of acceptable error. For machinists, the usual "don't worry about it" tolerance is around .015", meaning the actual position should be within fifteen thousandths of the 'perfect' spot. At the other end of the scale, NASA's JPL routinely makes microwave blocks the size of a sugar cube, with grooves smaller than a human hair that have to line up within 100 millionths of an inch.

Precision is a bugger because it's two numbers rolled into one. It's "the smallest step you can make with a given tolerance." The trouble is that the step size can be completely different from the tolerance.

For example: I have some cheap stepper motors that make 20 steps per revolution.. ideally, 18 degrees per step. Thing is, the motors are guaranteed to be accurate within half a degree at each step.. no position will be more than half a degree from the nearest multiple of 18 degrees. That makes them low-precision motors (18 degrees per step) with reasonably good accuracy (within half a degree at each step). It does not make them motors with half-degree precision though.

For the sake of contrast, imagine another motor that makes 200 steps per revolution (ideally 1.8 degrees per step), but is only accurate to plus or minus half a step (.9 degrees). That motor would be ten times as precise as the ones I have, but only about half as accurate.

You can see how 'precision' offers plenty of opportunity for use as a weasel word.

Which brings us to the greatest weasel-word of them all: Repeatability.

Okay, I'm being harsh. Repeatability is a valuable concept when used in an appropriate context. It's just used inappropriately 99.95% of the time.

In theory, repeatability is simple: it's the error between two attempts at doing the same thing. I start my robot at position 0, tell it to move to position X, and mark the place where it stops. I move the robot back to position 0, tell it to go to position X again, and mark the stopping point again. The distance between the two marks is my robot's repeatability.

In some ways, that number is more useful than a straight measurement of accuracy.

Let's say I've designed my robot to use one of those 20-step motors I mentioned earlier, and that one revolution of the motor should produce one inch of travel. That means my robot should have a precision of 20 steps per inch, and each step should be accurate to a tolerance of .0014".

Of course, that assumes my drive wheel is exactly the right size, and we already know that perfect accuracy is impossible. Let's say it's off by a few thousandths, and that the actual distance per revolution is 1.01". If I tell my robot to travel 100 inches, its final position will be 101 inches.. an error of a full inch. Thing is, my second attempt should land within .002" of the first. The accuracy is off, but the repeatability is good.

Instead of rebuilding the robot, I can just tell the software that runs the motor to scale every measurement down by about 1%. That will give me a robot with both good repeatability and good accuracy.

That's the theory at least.

In practice, 'repeatability' implies 'repetition'. Taking two different paths to the same point does not count as repetition. If I tell my robot to move from 0 to 50 it will land at one point. If I tell it to move from 100 to 50 it may land at a different point. Even if I get repeatability to within .002" on each operation, it doesn't guarantee that the two points will be within .002" of each other.

Repeatability is great in factories where a machine performs the same operation a few million times. The first part you get out of the machine might be completely inaccurate, it may take fifty tries to get a part that's within tolerances, but that's okay. As long as the machine is truly repeatable, every tweak in the pattern will carry over to all the parts that follow. The fifty pieces of scrap you made while tuning the pattern are negligible compared to the amount of good product you'll get over the run.

Outside that context, repeatability figures are pretty much worthless.

Marketers use repeatability as a sort of hybrid value that implies the best of both precision and accuracy without actually having to deliver either. My hypothetical robot above has a precision of .05", its accuracy is 1% off, and it has a repeatability of .002".. guess which number will go in big type on the front of the brochure, and which ones will be buried in a footnote of an exceedingly dull report somewhere on the company website.

If you're doing short runs in a small shop:

  • you want your parts to be accurate, meaning the errors are less than the specified tolerances.
  • you want your tools and jigs (devices which guide a tool) to be precise.
  • and you want your fixtures (devices that hold a workpiece in place) to be repeatable.