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note: I'm just a day or so into the use of inertial measurements and trying to learn everything at once, this may be a noob question (it's my first here).

I have seen this image of "Figure 8" in several sites geekmomprojects, makerworkshop, husstechlabs, aros.se, but I don't understand it or how the equations are derived.

If I set an accelerometer on a table so that $a_x$ and $a_y$ are nulled and rotate it around the $\hat{z}$ axis, I'm changing the yaw of the accelerometer but of course there is no change in the value of $\theta$ away from zero as given by the last equation.

So how is this a measurement of yaw? Or am I missing something obvious? What do these three expression actually mean? I'm beginning to think that the first two expressions are simply the "tilt angles" and the third is some geometrical angle that is not actually independent of the other two.

enter image description here

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Not sure if I understand your question exactly, but the scenario you point out is the blind spot of imus: they can't tell you the rotation about the gravity vector. It's nothing specific to yaw per se; if you put the imu on edge then you couldn't get rotation about the x or y axis, depending on which was aligned with the gravity vector.

Basically, when the imu is sitting still is tells you which direction gravity points, but this is only 1 axis of a coordinate system so doesn't fully define your orientation. This is why systems will frequently add another measurement such as a magnometer which can generally be used to know which direction is north. With these two directions you can fully define the orientation.

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  • $\begingroup$ Thanks. So the first two expressions give what can be called as the "tilt angles" around the $x$ and $y$ axes, correct? My question is really about the equations in this often used image. What does the third expression represent then? $\endgroup$
    – uhoh
    Commented Sep 26, 2017 at 13:18
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    $\begingroup$ The third equation is also the "tilt" angle about z but when you are near horizontal it does not mean much. If you are on your side then it makes more sense. Depending on your application this might not matter. $\endgroup$
    – ryan0270
    Commented Sep 26, 2017 at 13:23

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