Solution:

Composite Rotations:

Multiplication of the number of fundamental rotation matrices together and the product which contains a sequence of rotations about the unit vectors, such type of multiple rotations is called composite rotations.

Sometimes, a composition of transformations is equivalent to a single transformation. The following is an example of a translation followed by a reflection.

$ \begin{aligned}\\ &Y P R(\theta)_{\text {fixed }}=R P Y(\theta)_{\text {mobile }}=R_3\left(\theta_3\right) R_2\left(\theta_2\right) R_1\left(\theta_1\right)= \\\\ &{\left[\begin{array}{ccc} C \theta_3 & -S \theta_3 & 0 \\\\ S \theta_3 & C \theta_3 & 0 \\\\ 0 & 0 & 1 \\ \end{array}\right]\left[\begin{array}{ccc} C \theta_2 & 0 & S \theta_2 \\\\ 0 & 1 & 0 \\\\ -S \theta_2 & 0 & C \theta_2 \\ \end{array}\right]\left[\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & C \theta_1 & -S \theta_1 \\\\ 0 & S \theta_1 & C \theta_1 \\ \end{array}\right]}\\ \end{aligned}\\ $

$ \begin{aligned}\\ \operatorname{YPR}(\theta) &=\mathrm{R}_3\left(\theta_3\right) \mathrm{R}_2\left(\theta_2\right) \mathrm{R}_1\left(\theta_1\right) \\\\ &=\left[\begin{array}{ccc} \mathrm{C}_3 & -\mathrm{S}_3 & 0 \\\\ \mathrm{~S}_3 & \mathrm{C}_3 & 0 \\\\ 0 & 0 & 1 \\ \end{array}\right]\left[\begin{array}{ccc} \mathrm{C}_2 & 0 & \mathrm{~S}_2 \\\\ 0 & 1 & 0 \\\\ -\mathrm{S}_2 & 0 & \mathrm{C}_2 \\ \end{array}\right]\left[\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & \mathrm{C}_1 & -\mathrm{S}_1 \\\\ 0 & \mathrm{~S}_1 & \mathrm{C}_1 \\ \end{array}\right] \\\\ &=\left[\begin{array}{ccc} \mathrm{C}_3 & -\mathrm{S}_3 & 0 \\\\ \mathrm{~S}_3 & \mathrm{C}_3 & 0 \\\\ 0 & 0 & 1 \\ \end{array}\right]\left[\begin{array}{ccc} \mathrm{C}_2 & \mathrm{~S}_1 \mathrm{~S}_2 & \mathrm{C}_1 \mathrm{~S}_2 \\\\ 0 & \mathrm{C}_1 & -\mathrm{S}_1 \\\\ -\mathrm{S}_2 & \mathrm{~S}_1 \mathrm{C}_2 & \mathrm{C}_1 \mathrm{C}_2\\ \end{array}\right] \\\\ &=\left[\begin{array}{ccc} \mathrm{C}_2 \mathrm{C}_3 & \mathrm{~S}_1 \mathrm{~S}_2 \mathrm{C}_3-\mathrm{C}_1 \mathrm{~S}_3 & \mathrm{C}_1 \mathrm{~S}_2 \mathrm{C}_3+\mathrm{S}_1 \mathrm{~S}_3 \\\\ \mathrm{C}_2 \mathrm{~S}_3 & \mathrm{~S}_1 \mathrm{~S}_2 \mathrm{~S}_3+\mathrm{C}_1 \mathrm{C}_3 & \mathrm{C}_1 \mathrm{~S}_2 \mathrm{~S}_3-\mathrm{S}_1 \mathrm{C}_3 \\\\ -\mathrm{S}_2 & \mathrm{~S}_1 \mathrm{C}_2 & \mathrm{C}_1 \mathrm{C}_2 \\ \end{array}\right] \\ \end{aligned}\\ $