## Definition

Euler angles describe any arbitrary rotation using a set of 3 angles, each angle representing a single axis rotation that is applied in an intrinsic sequence. Let \(\mathbf{\Phi}\) be a set of Euler Angles consisting of the angles \(\{\phi, \theta, \psi \}\). The rotation matrix is calculated with:

\[ \mathbf{R}_{ijk}\left(\mathbf{\Phi}\right) = \mathbf{R}_i\left(\phi\right) \mathbf{R}_j\left(\theta\right) \mathbf{R}_k\left(\psi\right) \]

where \( i,j,k = \{x,y,z \} \). There are 12 valid rotation sequences, that can be broken up into * Proper Euler Angles* which have a rotation axis repeated in the sequence and

*which have 3 distinct rotation axes.*

**Tait-Bryan Angles**The Euler sequence * XYZ* is commonly called ‘Cardan Angles’ or ‘Nautical Angles’. It is also the sequence that is typically used when people refer to ‘Euler Angles’. The angles in this sequence \((\phi,\theta,\psi)\) are commonly called Roll, Pitch and Yaw. This is the sequence we will use for the rest of this page.

## Angle Conventions and Ranges

Angles are defined according to the right hand rule. Positive values are clockwise when viewed along the axis direction.

The range for \(\phi\) and \(\psi\) are defined by *modulo* \(2\pi\) so that:
\[ -\pi \le \{ \phi,\psi\} \le \pi \;\;\;\text{or}\;\;\; 0 \le \{ \phi,\psi\} \le 2\pi \]

The range for \(\theta\) is defined between the ranges: \[ -\frac{\pi}{2} \le \theta \le \frac{\pi}{2} \;\;\;\text{or}\;\;\; 0 \le \theta \le \pi \]

## Euler Angle and DCM Conversions

The following are the conversion functions between Euler Angles and Direct Cosine Matrix attitude representations. Let \(c_{(\cdot)}\) and \(s_{(\cdot)}\) be the \(\sin(\cdot)\) and \(\cos(\cdot)\) operations respectively.

\[ \mathbf{R}_{xyz} = \mathbf{R}_x\left(\phi\right) \mathbf{R}_y\left(\theta\right) \mathbf{R}_z\left(\psi\right) \]

\[ \mathbf{R}_{xyz} = \left[ \begin{array}{ccc} c_\theta c_\psi & c_\theta s_\psi & -s_\theta \\ s_\phi s_\theta c_\psi - c_\phi s_\psi & s_\phi s_\theta s_\psi + c_\phi c_\psi & c_\theta s_\phi \\ c_\phi s_\theta c_\psi + s_\phi s_\psi & c_\phi s_\theta s_\psi - s_\phi c_\psi & c_\theta c_\phi \end{array} \right] \]

\[ \mathbf{\Phi}_{xyz} = \left[ \begin{array}{c} \text{arctan2}(r_{23},r_{33}) \\ -\arcsin(r_{13})\\ \text{arctan2}(r_{12},r_{11})\end{array} \right] \]