Quaternions

Inscription on the Broom Bridge in Dublin Ireland

Quaternions can be hard to get your head around, but you'll see they're worth it when you see how much simpler they are to work with than rotation matrices! To get your energy up for this next exercise, check out this awesome music video about William Rowan Hamilton, the inventor of the quaternion.

Quaternion Exercise

In this exercise, you'll write functions to convert from Euler angles to quaternions and back again. The math to do this is given below:

Quaternion to Euler

\begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix} = \begin{bmatrix} \tan^{-1} \frac{2(ab+cd)}{1 - 2(b^2+c^2)} \\ \sin^{-1} 2(ac-db)\\ \tan^{-1} \frac{2(ad+bc)}{1 - 2(c^2+d^2)} \end{bmatrix}

Euler to Quaternion

{\begin{aligned} \begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix} &={\begin{bmatrix}\cos(\psi /2)\\0\\0\\\sin(\psi /2)\end{bmatrix}}{\begin{bmatrix}\cos(\theta /2)\\0\\\sin(\theta /2)\\0\end{bmatrix}}{\begin{bmatrix}\cos(\phi /2)\\\sin(\phi /2)\\0\\0\end{bmatrix}}\\&={\begin{bmatrix}\cos(\phi /2)\cos(\theta /2)\cos(\psi /2)+\sin(\phi /2)\sin(\theta /2)\sin(\psi /2)\\\sin(\phi /2)\cos(\theta /2)\cos(\psi /2)-\cos(\phi /2)\sin(\theta /2)\sin(\psi /2)\\\cos(\phi /2)\sin(\theta /2)\cos(\psi /2)+\sin(\phi /2)\cos(\theta /2)\sin(\psi /2)\\\cos(\phi /2)\cos(\theta /2)\sin(\psi /2)-\sin(\phi /2)\sin(\theta /2)\cos(\psi /2)\end{bmatrix}}\\\end{aligned}}

For additional tips on how you might implement this for your quadcopter in simulation (or in the real world) check out how it's implemented in the Udacidrone API you're using for the projects in this program!

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