Transformation Operation

• Transform a point in frame A to world frame with the transformation $T_A^W$

$$P^W=T_A^W\ast P^A$$

• Transform a point in world frame to frame B with the transformation $T_B^W$

$$\begin{array}{rl}{P}^B& =T_W^B\ast P^W\\ & =(T_B^W{)}^{-1}\ast P^A\\ & =(R_B^W{)}^T(P^W-t^B)\end{array}$$

• To transform a point by the composition of the transform from A→W and then W→B:

$$\begin{array}{rl}{P}^B& =T_W^B\ast T_A^W\ast P^A\\ & =(T_B^W{)}^{-1}\ast T_A^W\ast P^A\\ & =(R_B^W{)}^T{R}_A^W{p}^A+(R_B^W{)}^T(t_A^W-t_B^W)\end{array}$$

Proof step by step §

https://motion.cs.illinois.edu/RoboticSystems/CoordinateTransformations.html

Remember this Lesson!!!! §