We can assume the error will have a random (whether it's actually truly random or merely pseudo-random doesn't matter here, just assume it's indistinguishable from truly random for this discussion) and a non-random component.
The random component I assume to be gaussian (thermal noise, for example) and therefore symmetrical around the real value.
It's obvious we can remove this type of noise through averaging (of which the core operation is integration).
The non-random component I assume to be a skew that can be calibrated out.
With these two assumptions in mind you can see that yes, it's indeed a random walk, but a very well behaved one.
No, you can't remove the random walk error by integrating. The point is that after integrating, what you're left with the random walk error. To make this concrete, if you buy a commercial-grade gyroscope for $10, it will have a random walk error of several º/√h. So after summing the errors for an hour, you're left with several degrees of random error, which is bad. If you spend $100,000 on a navigation-grade gyroscope, you'll get a random walk error < 0.002º/√h, which is much better.
As far as calibrating out the skew, of course you can do that to some extent, but it's not a magic bullet. The Minuteman periodically measures skew and even applies equations for the change in skew with acceleration. The problem is that skew is not constant; it changes with time, changes with temperature, changes with position, and changes randomly, so you can't just calibrate it out. That's one reason why missiles use strategic-grade IMUs for a million dollars rather than a commercial-grade gyro for $10: you're getting drift of .0001º/hour instead of .1º/second.
You are correct, I forgot to separate between long-term and short-term random effects.
Short-term random effects (as in, the part of the gyro's random walk error significantly higher in frequency than the inverse of the integration period) will get cancelled out by integration, assuming they're Gaussian.
Long-term random effects (mainly from time and temperature like you mentioned) will instead tend accumulate with integration aka worsening with time.
P.S. great fan of your many ventures into retro tech, keep them coming!
that was not what you forgot and your summary is still wrong. kens is correct. i suggest programming some simple simulations using a random number generator to get a better feel for the space
try it: take gaussian white noise with zero mean and integrate it twice. You'll see the signal does not stay close to zero, in fact it will drift arbitrarily far away from it over time (it's only necessary to integrate once for this to be true, but doing it twice as an IMU needs to will make it more obvious).
You are correct, I initially did not explicitly separate the noise according to its frequency (my mistake).
Integration only helps with high frequency error and can actually worsen low frequency error, more details in my second reply to kens.
One can't cancel out random errors by integrating. You should take kragen's suggestion and write a quick simulation. To make this concrete, flip a coin 10 times. Take a step to the left for heads and a step to the right for tails. Most of the time you won't end up where you started, i.e. you have residual error.
> One can't cancel out random errors by integrating.
An ideal integrator has a response of 1/s. That's just a 1st order low-pass filter with the pole at 0. Therefore, it will filter out high frequency noise.
> Take a step to the left for heads and a step to the right for tails. Most of the time you won't end up where you started, i.e. you have residual error.
I wrote a quick simulation based on your suggestion [1].
Started by generating 1e6 random points and then applied a high-pass filter.
Calculated the cumulative sum on both the original and the filtered version.
TL;DR: filtered version has small and very fast variations but doesn't feature the much larger amplitude swings seen in the original.
Integration indeed does not help for those large slow swings (I'd call it drift in case of a gyroscope), but that's what I was trying to get at when I started to distinguish between short and long term random effects.
What I was trying to get across originally is that "all the little errors" (which I read to mean tiny fast variations, forgetting that drift is a much bigger issue in gyroscopes) which OP mentioned get filtered/canceled out.
I totally missed to explain that this will vary with frequency, which was my bad.
yes! but also keep in mind that 1/s is never 0 for any finite s, so even at high frequencies the error resulting from random noise is never zero, it's just strongly attenuated
> if you buy a commercial-grade gyroscope for [us]$10, it will have a random walk error of several º/√h. So after summing the errors for an hour, you're left with several degrees of random error, which is bad. If you spend [us]$100,000 on a navigation-grade gyroscope, you'll get a random walk error < 0.002º/√h, which is much better.
if the slope was anything else, the unit of °/√h wouldn't make sense; it would have to be °/h or °/∛h or something. similarly for noise figures given in nanovolts/√Hz
The random component I assume to be gaussian (thermal noise, for example) and therefore symmetrical around the real value. It's obvious we can remove this type of noise through averaging (of which the core operation is integration).
The non-random component I assume to be a skew that can be calibrated out.
With these two assumptions in mind you can see that yes, it's indeed a random walk, but a very well behaved one.