I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the three axis.
For this, I find this PDF document. However, there's something I can't definitely understand; page 10, to results are found: the first is $$\phi_{xyz} = atan\left( \frac{a_{y}}{a_{z}} \right) $$$$ \theta_{xyz} = atan\left( \frac{-a_{x}}{\sqrt{a_{y}^{2} + a_{z}^{2}}}\right) $$ While the second is $$\phi_{yxz} = atan\left(\frac{a_y}{\sqrt{a_x^2 + a_z^2}}\right)$$ $$\theta_{yxz} = atan\left(\frac{- a_x}{a_z}\right)$$ Where $\phi_{abc}$ (resp. $\theta_{abc}$) is the roll (resp. Pitch) obtained after rotating the gravity vector around axis a, then b and finally c, $a$ the measured acceleration. What I can't understand is why we find two different equation for same angle. I know, it comes from matrixes multiplication, but that's truly unintuitive. Moreover, if we take $a_x = a_y = a_z = \frac{\sqrt{3}}{3}$, so $\theta = \phi = \frac{\pi}{4}$, the first equations give us $$ \phi = \frac{\pi}{4} $$ $$\theta \approx -0.62$$ While the second equations lead to $$ \phi \approx 0.62 $$ $$ \theta = \frac{-\pi}{4}$$ So, why do we find to different angles and equations for the same angle ?
