Euler angle From one position to another.

Question

I am learning about Euler angles and I found that if i want to make some given orientation of the object with fixed reference frame then use the successive rotation on the object $$R_1(\phi)R_2(\theta)R_3(\psi)$$ My question is if I make rotation from $(0,0,0)$ to $(\phi_1,\theta_1 ,\psi_1)$ then use $R_1(\phi_1)R_2(\theta_1)R_3(\psi_1)$ and if another rotation from $(0,0,0)$ to $(\phi_2,\theta_2 ,\psi_2)$ then use $R_1(\phi_2)R_2(\theta_2)R_3(\psi_2)$ Now what will be the rotation if i make rotation from $(\phi_1,\theta_1 ,\psi_1)$ to $(\phi_2,\theta_2 ,\psi_2)$

Answer 1

Denote first rotation as $^0R_A$ and the second one as $^0R_B$.
Now you have ${^0}R_A{{^A}R_B}={^0}R_B$ what is the notation for operation rotating from $0$ to $A$ and from $A$ to $B$ getting this way rotation from $0$ to $B$.

From this you can calculate rotation from $A$ to $B$ as ${{^A}R_B}=({^0}R_A^{-1})({^0}R_B)$.

In your case $({^0}R_A)^{-1}=(R_1(\phi_1)R_2(\theta_1)R_3(\psi_1))^{-1} = R_3(-\psi_1)R_2(-\theta_1)R_1(-\phi_1)$ and ${^0}R_B=R_1(\phi_2)R_2(\theta_2)R_3(\psi_2)$.