6x6 z-u_rotation
Linear parametrization of the (6x6) rotation matrix around the z-axis.
For the 6 d.o.f. body motion (3 translations and 3 rotations), this block implements the LFT parametrization of the $(6 \times 6)$ 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 $\mathcal{R}_2$.
$$ \mathbf{rot} = \left[\begin{array}{cccccc} \cos(\theta) & -\sin(\theta) & 0 & 0 & 0 & 0\\ \sin(\theta) & \cos(\theta) & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \cos(\theta) & -\sin(\theta) & 0 \\ 0 & 0 & 0 & \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] $$
The parametrization is in tangent of the quarter-angle: $\sigma_4=\tan(\theta/4)$ to derive LFT when $\theta$ is a varying or an uncertain parameter (see: 3x3z-u_rotation)
See also: 3x3z-u_rotation, 3x3z-nl_rotation, , 6x6z-nl_rotation
Contents
Description
This block is the concatenation of two blocks 3x3z-u_rotation.
Ports
Inputs: a (6x1) vector $\mathbf{x}$ (3 in translation, 3 in rotation) in the frame $\mathcal{R}_{2}$ (resp. $\mathcal{R}_{1}$)
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
The only parameter is the angle $\theta$ (unit: $rad$). It can also be an uncertain parameter declared with "ureal" with mode "Range".
Example: >> ureal('theta',0,'Range',[-pi pi])
Remarks: when "ureal" us used, "ulinearize" rather than "linmod" is recommanded. Then the uncertain model is parameterized according to the real parameter labelled "tan_theta_div4".
A checkbox is proposed to implement the transposed: $\mathbf{rot}^T$.
Simulink diagram under the mask