9.1.4.1. Exercise: Frame#

Fig. 9.6 shows inclined local coordinate systems for both the upper arm and forearm. Knowing that the position of the elbow in global coordinates is \((0.34, 0.371, 0)\), and that the forearm is inclined by \(50^\circ\) compared to the global reference frame, construct this 4x4 matrix: \(^\text{global} _\text{forearm} T\).

Fig. 9.6 Local coordinates for both the upper arm and the forearm.#

\[\begin{split} ~^\text{global}_\text{forearm}T = \begin{bmatrix} \cos(50) & -\sin(50) & 0 & 0.34 \\ \sin(50) & \cos(50) & 0 & 0.371 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\= \begin{bmatrix} 0.643 & -0.766 & 0 & 0.34 \\ 0.766 & 0.643 & 0 & 0.371 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{split}\]