6x6 DCM
(6x6) rotation matrix for a given (3x3) Direction Cosine Matrix (DCM).
For a 6 d.o.f. body motion (3 translations and 3 rotations), this block implements the $(6 \times 6)$ rotation matrix -or its transpose- for a $(3 \times 3)$ given direction cosine matrix (DCM).
Contents
Description
Considering the 2 frames $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$, the $\mathbf{DCM}=\mathbf{P}_{2/1}$ between $\mathcal{R}_{1}$ and $\mathcal{R}_2$ is defined as the $(3\times 3)$ matrix where each column is the component vector of each $\mathcal{R}_{2}$ orthogonal unit vector expressed in $\mathcal{R}_{1}$.
$$ \left[\mathbf{v}\right]_{\mathcal{R}_{1}} = \mathbf{DCM}\;\left[\mathbf{v}\right]_{\mathcal{R}_{2}} $$
In the 6 d.o.fs case (3 translations and 3 rotations), the rotation matrix is:
$$ \mathbf{rot} = \left[\begin{array}{cc}\mathbf{DCM} & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & \mathbf{DCM}\end{array}\right] $$
Ports
Input: a $(6 \times 1)$ vector $\mathbf{x}$ (3 in translation, 3 in rotation) in the frame $\mathcal{R}_{2}$ (resp. $\mathcal{R}_{1}$)
Output: the expression of $\mathbf{x}$ in the frame $\mathcal{R}_{1}$ (resp. $\mathcal{R}_{2}$)
$\left[\mathbf{x}\right]_{\mathcal{R}_{1}} = \mathbf{rot} \,\, \left[\mathbf{x}\right]_{\mathcal{R}_{2}}$ (resp. $\left[\mathbf{x}\right]_{\mathcal{R}_{2}} = \mathbf{rot}^T \, \, \left[\mathbf{x}\right]_{\mathcal{R}_{1}}$)
Parameters
The only parameters are the $(3 \times 3)$ matrix $\mathbf{DCM}$ and a checkbox to implement the transposed: $\mathbf{rot}^T$.
Simulink diagram under the mask