6x6 v-u_rotation

Linear parametrization of the (6x6) rotation matrix around any 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 a given axis $\mathbf{v}$.

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)

It is a generalization of the non linear block "6x6z-u_rotation".

See also: 6x6v-nl_rotation, 3x3v-u_rotation,3x3z-u_rotation

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:

$$ \mathbf{rot} = \left[\begin{array}{cc}\mathbf{DCM} & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & \mathbf{DCM}\end{array}\right] $$

Thus, this block is the concatenation of two blocks 3x3v-u_rotation.

Ports

Input: a (3x1) vector $\mathbf{x}$ in the frame $\mathcal{R}_{2}$ (resp. $\mathcal{R}_{1}$).

Output: the expression of the (3x1) 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 two parameters are

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