Question

I have a tricky one for you guys. See problem #1 in the attached PDF file.

Answer 1

It's a review of rigid body kinematics, but I'm having trouble getting the correct rotation matrix.

Answer 2

Problem #1 is really tricky, do you have the solution?

Answer 3

Yes. I figured it out.

We can track the motion of the box from the origin. That is to say that body i,j, and k are aligned with ground I, J, and K.

First, make a rotation about body k by angle \(\psi\). \[R_z(\psi) = \left[\begin{matrix}\cos(\psi) & -\sin(\psi) &0 \\ \sin(\psi) & \cos(\psi) & 0 \\ 0 & 0 & 1\end{matrix}\right]\]
Then, make a rotation about body j by angle \(\theta\). \[R_y(\theta) = \left[\begin{matrix}\cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta)\end{matrix}\right]\]

We can multiply both rotation matrix together. Let's call the new matrix R\[R = \left[\begin{matrix}a_1 & b_1 & c_1\\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{matrix}\right]\]
The components of the body frame in the ground frame can be found as\[\hat i = a_1 \hat I + a_2 \hat J + a_3 \hat K\]\[\hat j = b_1 \hat I + b_2 \hat J + b_3 \hat K\]\[\hat k = c_1 \hat I + c_2 \hat J + c_3 \hat K\]