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The Transformation Matrix

Basics of rotation

Let us consider a simple 2D case. In 2D, anti-clockwise rotation by angle $\theta$ can be performed on a vector $P$ in frame $A$ represented as $^{A}P$ to convert to frame $B$ represented as $^{B}P$ using rotation matrix $R^{A}_{B}$.

\[^A P = {R_{B}^{A}} \:{^BP}\\ \text{where, } \:R_{B}^{A}=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\\sin \theta & \cos \theta & 0 \\0 & 0 & 1\end{array}\right]\]


  1. Introduction to Robotics: Mechanics and Control by John J Craig