3x3 z-nl_rotation

Non-linear 3x3 Direction Cosine Matrix associated to a rotation around the z-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 the $3^{rd}$ axis to transform the frame $\mathcal{R}_1$ into the frame $\mathcal{R}_2$.

$$ \mathbf{DCM} = \mathbf{P}_{2/1} = \left[\begin{array}{ccc}\cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{array}\right] $$

It is the non linear version of the linear block called: 3x3z-u_rotation.

See also: 3x3z-u_rotation, 6x6z-u_rotation, 6x6z-nl_rotation

Contents

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

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 $\mathbf{DCM}^T$.