Conversion between quaternions and Euler angles

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of "magic squares." For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".

A unit quaternion can be described as:
 * $$\mathbf{q} = \begin{bmatrix} q_0 & q_1 & q_2 & q_3 \end{bmatrix}^T$$
 * $$|\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1$$
 * $$\mathbf{q}_0 = \cos(\alpha/2)$$
 * $$\mathbf{q}_1 = \sin(\alpha/2)\cos(\beta_x)$$
 * $$\mathbf{q}_2 = \sin(\alpha/2)\cos(\beta_y)$$
 * $$\mathbf{q}_3 = \sin(\alpha/2)\cos(\beta_z)$$

where $$\alpha$$ is a simple rotation angle and $$\beta_x$$, $$\beta_y$$, $$\beta_z, $$ are the "direction cosines" locating the axis of rotation (Euler's Theorem). Similarly for Euler angles, we use (in terms of flight dynamics): where the X-axis points forward, Y-axis to the right and Z-axis downward and in the example to follow the rotation occurs in the order yaw, pitch, roll (about body-fixed axes).
 * Roll - $$\phi$$: rotation about the X-axis
 * Pitch - $$\theta$$: rotation about the Y-axis
 * Yaw - $$\psi$$: rotation about the Z-axis

Rotation matrices
The orthogonal matrix corresponding to a rotation by the unit quaternion q is given by
 * $$\begin{bmatrix}

1- 2(q_2^2 + q_3^2) & 2(q_1 q_2 - q_0 q_3) &  2(q_0 q_2 + q_1 q_3) \\ 2(q_1 q_2 + q_0 q_3) & 1 - 2(q_1^2 + q_3^2) &  2(q_2 q_3 - q_0 q_1) \\ 2(q_1 q_3 - q_0 q_2) & 2( q_0 q_1 + q_2 q_3) & 1 - 2(q_1^2 + q_2^2) \end{bmatrix}$$

The orthogonal matrix corresponding to a rotation with Euler angles $$\phi, \theta, \psi$$, is given by
 * $$\begin{bmatrix}

\cos\theta \cos\psi & -\cos\phi \sin\psi + \sin\phi \sin\theta \cos\psi &  \sin\phi \sin\psi + \cos\phi \sin\theta \cos\psi \\ \cos\theta \sin\psi & \cos\phi \cos\psi + \sin\phi \sin\theta \sin\psi & -\sin\phi \cos\psi + \cos\phi \sin\theta \sin\psi \\ -\sin\theta            &  \sin\phi \cos\theta                                          &   \cos\phi \cos\theta \\ \end{bmatrix}$$

Conversion
By comparing the terms in the two matrices, we get:


 * $$ \mathbf{q} = \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}$$

For Euler angles we get:


 * $$\begin{bmatrix}

\phi \\ \theta \\ \psi \end{bmatrix} = \begin{bmatrix} \mbox{arctan} \frac {2(q_0 q_1 + q_2 q_3)} {1 - 2(q_1^2 + q_2^2)} \\ \mbox{arcsin} (2(q_0 q_2 - q_3 q_1)) \\ \mbox{arctan} \frac {2(q_0 q_3 + q_1 q_2)} {1 - 2(q_2^2 + q_3^2)} \end{bmatrix} $$

Singularities
One must be aware of singularities in the Euler angle parametrization when the pitch approaches $$\pm 90^o$$ (north/south pole). These cases must be handled specially.