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 ?
Best Answer
The solutions these equations give for the angles are different because they are for different rotation sequences. The first set of equations has the subscript xyz and the second has the subscript yxz, so these angles depend on the rotation sequence.