What is the angular velocity in an inertial frame given the angular velocity in a body fixed frame?

Question

At a given time t, the rotation matrix R has the value:

$$R= \begin{pmatrix} 0.675 & −0.1724 &0.7174\\0.2474 & 0.9689& 0 &\\−0.6951& 0.1775&0.6967. \end{pmatrix}$$

The angular velocity in a body fixed frame $\hat ω^b$ at that same time t is:

$$\hat ω^b=\begin{pmatrix}0& −1 &0.9689&\\1 & 0 &−0.2474\\−0.9689&0.2474&0. \end{pmatrix}$$ What is $\hatω^s$, the angular velocity in an inertial frame?

I know that \begin{align*}\omega^s:\dot q &= \dot RR^Tq\\&=R\hat\omega^b\end{align*}

Therfore I did on Matlab:

>> R = [0.675 -0.1724 0.7174; 0.2474 0.9689 0; -0.6951 0.1775 0.6967];
>> wb= [0 -1 0.9689; 1 0 -0.2474; -0.9689 0.2747 0];
>> R*wb

ans =

   -0.8675   -0.4779    0.6967
    0.9689   -0.2474         0
   -0.4975    0.8865   -0.7174

Which is none of the provided answer:

Answer 1

The angular velocity in a body fixed frame $\hat{\omega}^b$ at that same time $t$ is: [insert matrix here]

No, that's not the angular velocity, that's the matrix version of $\omega\times$, and while we're at it the angular velocity of what in the body frame of what? Can you please explain the setup of the problem? How many bodies are there? What is rotating and what isn't? How is the inertial frame defined?

From what you wrote I would just read off the components of $\omega^b$ (a vector, not a matrix) and multiply them be $R$, trying to see if it agrees with any one of those answers, but without any further details it's tough to give you good advice here.

Answer 2

This is down to the relationship between the angular velocity in the fixed body frame ($w^b$) of a rigid body and in the inertial frame ($w^s$) given a rotation matrix.

You have the wrong equations above for $w^s$ and $w^b$ The points don't come into it.

It should be as below: ($R^T$ is the transpose, $\dot R$ differentiated against time ($t$))

$$ωs=R\dot R^T$$ $$wb=R^T \dot R$$

The missing piece needed to calculate $w^s$ is $\dot R$.

Take the $w^b$ equation and pre-multiply each side of the equation by $inv(RT)$

$$inv(R^T)R^T\dot R= inv(R^T)wb$$

$$\dot R=inv(RT)wb$$

Now you know $R$ and $\dot R$ So you can calculate $ws=\dot RR^T$

You can confirm this by seeing that the answer is one of the 4 options above.