Accelerometer and angles calculation

Question

I was reading on https://www.analog.com/en/app-notes/an-1057.html about the accelerometer and how it is used to calculate the pitch, roll, and yaw. However, I have several issues to understand the formulas. For example, in the picture attached I am not understanding how can the angles in b) be different then each other? isn't it a rotation around the y axis? and the same apply for the other pictures? So the first issue with me is understanding why the angles are not equal thus is affecting my understanding on the equations 11 to 13.

Thank you for your help

Image

Answer 1

As the paragraph just preceding your image says,

The reference position is taken as the typical orientation of a device with the $x$- and $y$-axes in the plane of the horizon (0 g field) and the $z$-axis orthogonal to the horizon (1 g field). This is shown in Figure 12 with $\theta$ as the angle between the horizon and the $x$-axis of the accelerometer, $\psi$ as the angle between the horizon and the $y$-axis of the accelerometer, and $\phi$ as the angle between the gravity vector and the $z$-axis. When in the initial position of 0 g on the $x$- and $y$-axes and 1 g on the $z$-axis, all calculated angles would be $0°$.

Instead of just looking at the images and equations, it is useful to read the text of the appnote as well.