6x6 z-nl_rotation
Non-linear 6x6 rotation matrix around the z-axis.
For the 6 d.o.f. body motion (3 translations and 3 rotations), this block implements the non linear (6x6) rotation matrix -or its transposed- for a given angle $\theta$ around the $3^{rd}$ axis to transform the frame $\mathcal{R}_1$ into the frame $\mathcal{R}_2$.
$$ \mathbf{rot} = \left[\begin{array}{cccccc} \cos(\theta) & -\sin(\theta) & 0 & 0 & 0 & 0 \\ \sin(\theta) & \cos(\theta) & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \cos(\theta) & -\sin(\theta) & 0 \\ 0 & 0 & 0 & \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] $$
It is the non linear version of the linear block named 6x6z-u_rotation.
See also: 3x3z-u_rotation, 6x6z-u_rotation, 3x3z-nl_rotation
Contents
Description
This block is the concatenation of two blocks 3x3 z-nl_rotation.
Ports
Inputs:
- a (6x1) vector $\mathbf{x}$ (3 in translation, 3 in rotation) in the frame $\mathcal{R}_{2}$ (resp. $\mathcal{R}_{1}$)
- the angle $\theta$ (unit: $rd$).
Output: the expression of the (6x1) vector $\mathbf{x}$ 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
A checkbox to implement the transposed: $\mathbf{rot}^T$.
Simulink diagram under the mask
with: $\mathrm{signe}=1$ for direct $\mathbf{rot}$, $\mathrm{signe}=-1$ for $\mathbf{rot}^T$.