# Bias instability in Gyroscopes : AVER / ADEV

I am attempting to implement allan variance to determine the amount of bias that is shown in my IMU's gyroscope ( BMI160 ). I am doing this with the following formula, that was captured in the following tutorials: Tutorial 1 and Tutorial 2

Formula for AVAR:

$$\sigma_y^2(\tau) = \frac{1}{2(N-1)} \sum_{i=0}^{N-1}(y(\tau)_{i+1} - y(\tau)_i)^2$$

From my reading, I am implementing the calculation using the following statements:

$$\tau$$ - Bin of time ( 5 seconds of data, 10 seconds, 15 sec, etc ) So if my $$\tau$$ is $$5$$ sec, then I'd multiply $$5$$ by my sample rate to get the number of sample points that I can stuff into a bin. ( $$5*40=200$$, given that my sample rate is $$40$$ )

$$N$$ - Total number of bins that I've split my data into. ( Number of sample points per bin divided by total number of data points, without the remainder )

$$y(\tau)$$ - The average ( mean ) of the data in some bin of time

I then plot the ADEV ( Sqrt of AVAR ), over $$\tau$$, and see the following result:

What I see in that plot is that different axis have different bias instability levels. I did not collect hours of data to let the graph start rising again, like a true allan variance plot would. But for the purpose of this question, let's assume that the graph starts rising for each axis after the point i've selected on the graph. ( in the image )

Questions:

1. When a manufacturer releases a datasheet, and they say that the gyro bias instability is "$$0.55^{\circ} / hr$$ ", are the taking the average of all three axis to come up with one value? Looking at my plot, that would be rather poor, as two of the axis are fairly low as compared to the third.

2. Does it make sense to say that you'd want to see a low bias instability, and for it to be identified through a very small $$\tau$$? i.e Manufacturers claim $$0.55^{\circ} / hr$$ but rarely in a datasheet do i see the result of $$\tau$$ being shown of where that value was found. A sensor with $$0.55^{\circ} / hr$$ at a $$\tau$$ of $$5$$ should have significantly less bias than the one with the $$\tau$$ of $$25$$. No?

Yes you would want to see a low bias instability, and I would think that the same number at a lower $$\tau$$ would actually be worst unless you were only concerned with doing a measurement over that short duration in time (implying the turn-up occurs sooner so that unit would then be drifting sooner and over the longer duration would be a lot worst). You ultimately want the lowest value at the further averaging time indicating the best long term stability, but depending on the process and use, there may be reasons to be concerned with shorter term stability as well such as if you were making difference reading measurements at 1 second intervals (then the 1 sec ADEV would be of interest).