Rotations

Rotating frames.

A vector whose x,y,z components are $x,y,z$ respectively is represented as the column matrix$\begin{bmatrix}x\\y\\z\end{bmatrix}$

Rotation of vectors in 2D:

In two dimensions, the standard rotation matrix has the following form:-

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The above matrix represents the counterclockwise rotation of a vector through angle θ. The vector in this figure is initially aligned with the x-axis

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If a vector $\begin{bmatrix}x\\y\end{bmatrix}$ is rotated counterclockwise by $\theta$, the resultant vector $\begin{bmatrix}x'\\y'\end{bmatrix}$ is given as

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Rotation of vectors in 3D:

The following matrices represent counterclockwise rotations by angle $\theta$ about the x,y and z axes respectively.

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Their usage is similiar as that explained in the previous section.

For the applications of rotations in UAVs, refer to the following link:

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AE321_EqMotion.pdf

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Flight Mechanics