3x3 v-nl_rotation

Non-linear 3x3 Direction Cosine Matrix associated to a rotation around any axis.

This block implements the non linear (3x3) direction cosine matrix (DCM) -or its transposed- associated to the rotation of a given angle $\theta$ around a given axis $\mathbf{v}$ to transform the frame $\mathcal{R}_1$ into the frame $\mathcal{R}_2$.

It is the non linear version of the linear block 3x3v-u_rotation and a generalization of the non linear block 3x3z-nl_rotation.

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

Contents

Description

The block named 3x3z-nl_rotation gives the non linear direction cosine matrix ($\mathbf{DCM}_\mathbf{z}$) associated to the rotation of a given angle $\theta$ around the $3^{rd}$ axis. To exploit this result, an intermediate frame $\mathcal{R}_i=(\mathbf{x}_i,\mathbf{y}_i,\mathbf{z}_i)$ is used such that $\mathbf{z}_i$ is along $\mathbf{v}$. Then: $$\mathbf{DCM}=\mathbf{P}_{2/1}=\mathbf{P}_{i/1}\underbrace{\left[\begin{array}{ccc}\cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{array}\right]}_{\mathbf{DCM}_\mathbf{z}} \mathbf{P}_{i/1}^T\,.$$

with:

Ports

Inputs:

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

$\left[\mathbf{x}\right]_{\mathcal{R}_{1}} = \mathbf{DCM} \, \left[\mathbf{x}\right]_{\mathcal{R}_{2}}$ (resp. $\left[\mathbf{x}\right]_{\mathcal{R}_{2}} = \mathbf{DCM}^T \, \left[\mathbf{x}\right]_{\mathcal{R}_{1}}$)

Parameters

The only parameters are the coordinates of the (3x1) rotation axis vector $\mathbf{v}$ in $\mathcal{R}_{1}$ (or $\mathcal{R}_{2}$) and a checkbox to implement the transposed: $\mathbf{DCM}^T$.

Simulink diagram under the mask

with: $\mathrm{signe}=1$ for direct $\mathbf{DCM}$, $\mathrm{signe}=-1$ for $\mathbf{DCM}^T$ and $\mbox{pass}=\mathbf{P}_{i/1}$.