# Angles of rotation

I am haunted by a problem of angles of rotation. Here's my nightmare. At the origin of an inertial frame $R_0(XYZ)$, there is a rigid ball. Another reference frame $R(xyz)$ is fixed to the center of this rigid ball. At the beginning, the origin of $R_0$ coincides the origin of $R$, and $x\parallel X$, $y\parallel Y$, $z\parallel Z$ and in the same orientation. Then the ball begins to roll. Firstly, it rotates by an angle of $\psi$ around the axe $Z$ in $R_0$; secondly, it rotates by an angle of $\theta$ around the 'new' axe $x$ in $R$; finally, it rotates by an angle of $\phi$ around the 'final' axe $y$ in $R$. My question is could we express the angles of rotation ($\theta_X$,$\theta_Y$,$\theta_Z$)of the ball with respect to the 3 axes $X$,$Y$ and $Z$ of inertial frame $R_0$? I tried different ways, but it seems to be too complexe to express them without using integration. Is there any existing mathematical formulation to deal this problem? Thank you for taking a look!

• You're describing Euler angles. Feb 18 '16 at 12:57
• Exactly, this combination of rotation is one of Euler angles. But, I could not find any formula about the angles of rotation respecting to the inertial frame. Feb 18 '16 at 13:00
• I want to understand better your question, and after that maybe add an answer. Do you want to express the rotation as the result of three different rotations along the axes? If this is so, remember that the result will depend on the order of the rotations. If you want, you can express the final rotation as a single rotation along a particular axis. Is this an acceptable answer for you?
– GCLL
Feb 18 '16 at 16:20
• Thanks a lot. My questions is could we describe this combination of rotation from the view of the $R_0$, imagine that there are 3 sensors of angle of rotation fixed on the axes of $R_0$ to detect the rolling of the ball. What will these outputs look like? Could we establish the relations with the Euler angles? Feb 18 '16 at 16:39

When you rotate around a given axis, which can be identified by a versor $\hat{n}$, a sensor which measures the total rotation angle around $i=X,Y,Z$ gives $$\Theta_i = \hat{e}_i\cdot\hat{n}\, \alpha$$ where $\alpha$ is the total rotation angle. So the final result will be $$\Theta_i = \hat{e}_i\cdot\hat{Z} \psi + \hat{e}_i\cdot \left(R_Z(\psi)\hat{X}\right) \theta + \hat{e}_i\cdot \left(R_X(\theta) R_Z(\psi)\hat{Y}\right)\phi$$ where $R_i(\alpha)$ is a rotation matrix around the axis $i=X,Y,Z$. By inserting the expressions for the matrix the explicit result can be obtained.