Derivation of yaw, pitch, roll equations for an accelerometer

Question

I'm struggling to find a good resource that explains how roll, pitch, yaw angles are calculated from the X, Y, Z measurements of an accelerometer. I came across this document but the explanation on page 9 is still very sparse. In particular, I'm unable to understand the following 2 lines:

The accelerometer vector lies on the surface of a sphere with radius 1g. It is not therefore possible to solve for three unique values of the roll φ, pitch θ and yaw ψ angles.

Suggestion for a good book or web resource would be much appreciated.

Answer 1

Yaw, pitch and roll are only well-defined for small angles.

For large angles, the become degenerate. For example, a 90 degree pitch up (ending pointing vertically) followed by a right-down 90 roll is the same as right-down 90 degree roll followed by a left 90 yaw.

The physics reason is a bit subtle. It takes three numbers to orient a body in space, it just turns out that body-centered pitch, roll and yaw are mathematically inconvenient for large angles.

Answer 2

Eqn 25 comes from Eqn 24

$$\tan \phi = \frac{\cos \theta \sin \phi}{\cos \theta \cos \phi} = \frac{G_{py}}{G_{pz}}$$

Eqn 26 is similar. But note that

$$G_{py}^2 + G_{pz}^2 = \cos^2 \theta \times (\sin^2 \phi + \cos^2 \phi) = \cos^2 \theta$$

Answer 3

Regarding the problematic sentence you quote: a spherical surface has only two degrees of freedom. For instance, interpret the yaw angle as longitude and pitch as latitude, leaving the roll angle undefined.