Transformation matrix
In this section we decompose the matrix \( {}^{A}_B T\) into its two main components: translation and rotation.
The transformation between two coordinate systems \(A, B\) can be represented as a 4x4 matrix in homogenous coordinates:
\[ \begin{bmatrix} {}^{A} \vec x \\ 1 \end{bmatrix} = {}^{A}_B \mathbf T \ \begin{bmatrix} {}^{B} \vec x \\ 1 \end{bmatrix} \\ \begin{bmatrix} {}^{A} \vec x \\ 1 \end{bmatrix} = \begin{bmatrix} {}^{A}_B R & {}^{A}t_B \\ \mathbf 0 & 1 \end{bmatrix} \begin{bmatrix} {}^{B} \vec x \\ 1 \end{bmatrix} \]
Note that \( \mathbf 0 = [0, 0, 0]\), i.e. a 3x1 row vector.
The transformation matrix \( {}^{A}_B T\) has two core components (Fig. 1):
- a 3x1 displacement vector \( {}^{A}t_B \), which describes the translation of \(B\)'s origin in the \(A\) system. For a sensor mounted 5m in x-axis direction from the point of reference, the displacement vector is \( [5, 0, 0]^T\).
- a 3x3 rotation matrix \({}^{A}_B R\), which describes the rotation of the axes of coordinate system \(B\) in the \(A\). I.e. the columns are formed from the three unit vectors of B's axes in A: \({}^{A}\vec X_B\), \({}^{A}\vec Y_B\), and \({}^{A}\vec Z_B\).1
Figure 1: Coordinate transformation matrix composed from displacement vector \( {}^{A}t_B \) (red) and rotation matrix \({}^{A}_B R\) (blue).
In the next section we will learn how the rotation matrix is composed.