Yaw,Pitch and Roll Multiplication

Question

If I have $(\alpha_1,\beta_1,\gamma_1)$ and $(\alpha_2,\beta_2,\gamma_2)$ as two set of rotation angles in radians.Where $\alpha$ is Yaw of $\alpha$ about $z$ axis, $\beta$ is pitch about $y$ axis and $\gamma$ is roll about $x$ axis. My question is when I multiply the two rotation sets what would be the result? Would it be simple addition of angles if I extract the angle information from result i-e $(\alpha_1+\alpha_2,\beta_1+\beta_2,\gamma_1+\gamma_2)$?

Answer 1

No, this is not true, since it would require that the rotation matrices would be commutative. Matrices in general, including rotation matrices, do not commute.

However, if the second set of angles is very small, say smaller than some $\varepsilon<1°$, then the error committed by this summation is smaller than $ε^2$, which might be negligible.

Answer 2

The short answer, unfortunately, is "no". You can confirm this by selecting distinct non-zero values for all the angles in $(\alpha_1, \beta_1, \gamma_1)$ and $(\alpha_2, \beta_2, \gamma_2)$ and computing the rotation matrices $M_1$ and $M_2$ of these two rotations. The rotation matrix of the first rotation followed by the second is $M_2 M_1$, which you will find is not generally the same as the rotation matrix for the rotation by angles $(\alpha_1+\alpha_2, \beta_1+\beta_2, \gamma_1+\gamma_2)$. In fact, in most cases it matters in which order you perform the rotations, that is, $M_1 M_2$ is not generally equal to $M_2 M_1$, which should tell you right away that simply adding the angles is not correct (because if it were, the rotation sequence wouldn't matter).

Or you can try this example: pitch upward by $\pi/2$ ($90$ degrees), roll by $\pi/4$ ($45$ degrees), then pitch downward by $\pi/2$ ($90$ degrees). The result is a yaw of $\pi/4$ ($45$ degrees) with no pitch or roll. That is, the pitch of your final result is the sum of the pitches of your two rotations, all right, but the roll is not the sum of the two rolls and the yaw is not the sum of the two yaws. And this is only a very simple example.