Question

Exercise (5) Points (25) The figure below shows a frame (B) that is rotated relevant to a frame (A) about z by 30°, where z is pointing out of the page Dr. Mykoniatis, ENGT/IME 410 Robotics Technology, Spring 2018 A) Calculate the Rotation matrix of B with respect to A. 0.0 3 B) Given that the vector 2.0calculate the vector AP, B)

Answer 1

a) The angle of rotation is 30 degrees. Hence the rotation matrix is

$R=\begin{pmatrix} \cos 30^o & -\sin 30^o & 0\\ \sin 30^o & \cos 30^o & 0\\ 0&0 &1 \end{pmatrix}= \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ \frac{1}{2} & \frac{\sqrt{3}}{2} & 0\\ 0&0 &1 \end{pmatrix}$

b) The vector is frame A is

${}^Ap=R\,{}^Bp= \begin{pmatrix} \frac{\sqrt{3}}{2} &-\frac{1}{2} & 0\\ \frac{1}{2} & \frac{\sqrt{3}}{2} & 0\\ 0&0 &1 \end{pmatrix}\begin{pmatrix} 0\\ 2\\ 0 \end{pmatrix}= \begin{pmatrix} -1\\ \sqrt{3} \\ 0 \end{pmatrix}= \begin{pmatrix} -1\\ 1.732 \\ 0 \end{pmatrix}$