Euler Angles

Definition

Figiure 1 shows a 321 rotation of reference frame \(F_\beta\) to object frame \(F_\alpha\) with roll \(\phi_{\beta \alpha}\), pitch \(\theta_{\beta \alpha}\), and yaw \(\psi_{\beta \alpha}\) Euler angles.

Figure 1 321 rotation from left to right (Groves, p34)

The properties table drops the subscripts for simplicity, e.g., \(\phi_{\beta \alpha} \triangleq \phi\).

Properties	321	123
Definition	1. Rotation through the yaw angle \(\psi\) about the common \(z\) axis of the \(F_\beta\) frame and the first intermediate frame 2. Rotation through the pitch angle \(\theta\) about the common \(y\) axis of the first and second intermediate frame 3. Rotation through the roll angle \(\phi\) about the common \(x\) axis of the second frame and the \(F_\alpha\) frame
Type	Intrinsic	Intrinsic
Euler Rotation Vector	Rotation from \(F_\beta\) to \(F_\alpha\): \(\begin{align*}\boldsymbol{\Psi}_{\beta \alpha} = \left[ \begin{array}{c} \phi \\ \theta \\ \psi \\ \end{array} \right] \end{align*}\)
Rotation Composition	\(\mathbf{R}^{\alpha}_{\beta} = \mathbf{R}^{\alpha}_{\theta} \mathbf{R}^{\theta}_{\psi} \mathbf{R}^{\psi}_{\beta} = \mathbf{R}_1(\phi) \mathbf{R}_2(\theta) \mathbf{R}_3(\psi)\)	\(\mathbf{R}^{\alpha}_\beta = \mathbf{R}_3(\psi) \mathbf{R}_2(\theta) \mathbf{R}_1 (\phi)\)
Euler to Rotation Matrix	\(\mathbf{R}^{\beta}_{\alpha} = \left[ \begin{array}{ccc} c\theta c\psi & \left( \begin{array}{c} -c\phi s\psi + \\ s\phi s\theta c\psi \\ \end{array} \right) & \left( \begin{array}{c} s\phi s\psi + \\ c\phi s\theta c\psi \\ \end{array} \right) \\ c\theta s\psi & \left(\begin{array}{c} c\phi c\psi + \\ s\phi s\theta s\psi \\ \end{array} \right) & \left( \begin{array}{c} -s\phi c\psi + \\ c\phi s\theta s\psi \\ \end{array} \right) \\ -s\theta & s\phi c\theta & c\phi c\theta \\ \end{array} \right]\)	\(\mathbf{R}^\beta_\alpha = \left[ \begin{array}{ccc} c\psi c\theta & -s\psi c\theta & s\theta \\ \left(\begin{array}{c} s\psi c\phi + \\ c\psi s\theta s\phi \end{array} \right) & \left(\begin{array}{c} c\psi c\phi - \\ s\psi s\theta s\phi \end{array} \right) & -c\theta s\phi \\ \left(\begin{array}{c} s\psi s\phi - \\ c\psi s\theta c\phi \end{array} \right) & \left(\begin{array}{c} c\psi s\phi + \\ s\psi s\theta c\phi \end{array} \right) & c\theta c\phi \end{array} \right]\)
Rotation Matrix to Euler	\(\begin{align*} \phi &= \text{arctan}_2 \left( \mathbf{R}^{\beta}_{\alpha \ 3,2}, \mathbf{R}^{\beta}_{\alpha \ 3,3} \right) \\ \theta &= -\text{arcsin}\left( \mathbf{R}^{\beta}_{\alpha \ 3,1}\right) \\ \psi &= \text{arctan}_2 \left( \mathbf{R}^{\beta}_{\alpha \ 2,1}, \mathbf{R}^{\beta}_{\alpha \ 1,1} \right) \end{align*}\)
Infinitesimal Rotations	\(\begin{align*} \mathbf{R^{\beta}_{\alpha}} &= \left[ \begin{array}{ccc} 1 & -\psi & \theta \\ \psi & 1 & -\phi \\ -\theta & \phi & 1 \end{array} \right] = \mathbf{I}_{3} + \left[\boldsymbol{\Psi}_{\beta \alpha} \right]_{\times} \\ \mathbf{R}^{\alpha}_{\beta} &= \mathbf{I}_3 - \left[ \boldsymbol{\Psi}_{\beta \alpha} \right]_\times \end{align*}\)

Euler Rates

From Poisson's equation:

\[ \begin{alignat}{2} \boldsymbol{\Omega}^\alpha_{\alpha \beta} &= \dot{\mathbf{R}}^\alpha_\beta \mathbf{R}^\beta_\alpha, \quad \boldsymbol{\Omega}^\alpha_{\beta \alpha} &&= -\dot{\mathbf{R}}^\alpha_\beta \mathbf{R}^\beta_\alpha \\ \boldsymbol{\Omega}^\beta_{\alpha \beta} &= \mathbf{R}^\beta_\alpha \dot{\mathbf{R}}^\alpha_\beta, \quad \boldsymbol{\Omega}^\beta_{\beta \alpha} &&= - \mathbf{R}^\beta_\alpha \dot{\mathbf{R}}^\alpha_\beta. \end{alignat} \]

Then we have:

\[ \begin{align} \left[ \boldsymbol{\omega}^\alpha_{\beta \alpha} \right]_\times &= -\dot{\mathbf{R}}^\alpha_\beta \left( \mathbf{R}^\alpha_\beta \right)^{-1} \\ &= -\dot{\mathbf{R}}^\alpha_\beta \left( \mathbf{R}^\alpha_\beta \right)^{T}. \label{1} \end{align} \]

For each rotation about a principle axis, we have:

\[ \begin{align} -\dot{\mathbf{R}}_3 \mathbf{R}^T_3 &= \left[\mathbf{I}_3 \right]_\times \dot{\psi}_{\beta \alpha}, \\ -\dot{\mathbf{R}}_2 \mathbf{R}^T_2 &= \left[\mathbf{I}_2 \right]_\times \dot{\theta}_{\beta \alpha}, \\ -\dot{\mathbf{R}}_1 \mathbf{R}^T_1 &= \left[\mathbf{I}_1 \right]_\times \dot{\phi}_{\beta \alpha}, \end{align} \]

where \(\mathbf{I}_i\) os column \(i\) of \(\mathbf{I}_{3 \times 3}\).

Proof

The principle rotation matrix about the third axis is given as:

\[ \mathbf{R}_3 = \left[ \begin{array}{ccc} \cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1. \end{array} \right] \]

The time derivative of \(\mathbf{R}_3\) is:

\[ \dot{\mathbf{R}}_3 = \left[ \begin{array}{ccc} -\dot{\psi} \sin \psi & \dot{\psi} \cos \psi & 0 \\ -\dot{\psi} \cos \psi & - \dot{\psi} \sin \psi & 0 \\ 0 & 0 & 0 \end{array} \right]. \]

Then:

\[ -\dot{\mathbf{R}}_3 \mathbf{R}^T_3 = \left[ \mathbf{I}_3 \right]_\times \dot{\psi}. \]

Consider the 3-2-1 Euler angle sequence for frames, \(F_\alpha\) and \(F_\beta\), and its associated rotation matrix:

\[ \mathbf{R}^\alpha_\beta = \mathbf{R}^\alpha_\theta \mathbf{R}^\theta_\psi \mathbf{R}^\psi_\beta = \mathbf{R}_1 \mathbf{R}_2 \mathbf{R}_3. \]

Eq (\(\ref{1}\)) can be written as:

\[ \begin{align} \left[ \boldsymbol{\omega}^\alpha_{\beta \alpha} \right]_\times &= -\overbrace{\mathbf{R}^\alpha_\theta \mathbf{R}^\theta_\psi \mathbf{R}^\psi_\beta}^\cdot \left( \mathbf{R}^\alpha_\theta \mathbf{R}^\theta_\psi \mathbf{R}^\psi_\beta \right)^T \\ &= -\left[\dot{\mathbf{R}}^\alpha_\theta \mathbf{R}^\theta_\psi \mathbf{R}^\psi_\beta + \mathbf{R}^\alpha_\theta \dot{\mathbf{R}}^\theta_\psi \mathbf{R}^\psi_\beta + \mathbf{R}^\alpha_\theta \mathbf{R}^\theta_\psi \dot{\mathbf{R}}^\psi_\beta \right] \mathbf{R}^\beta_\psi \mathbf{R}^\psi_\theta \mathbf{R}^\theta_\alpha \\ &= -\mathbf{R}^\alpha_\theta \mathbf{R}^\theta_\psi \dot{\mathbf{R}}^\psi_\beta \mathbf{R}^\beta_\psi \mathbf{R}^\psi_\theta \mathbf{R}^\theta_\alpha - \mathbf{R}^\alpha_\theta \dot{\mathbf{R}}^\theta_\psi \mathbf{R}^\psi_\theta \mathbf{R}^\theta_\alpha - \dot{\mathbf{R}}^\alpha_\theta \mathbf{R}^\theta_\alpha \\ &= -\mathbf{R}_1 \mathbf{R}_2 \dot{\mathbf{R}}_3 \mathbf{R}^T_3 \mathbf{R}^T_2 \mathbf{R}^T_1 - \mathbf{R}_1 \dot{\mathbf{R}}_2 \mathbf{R}^T_2 \mathbf{R}^T_1 - \dot{\mathbf{R}}_1 \mathbf{R}^T_1 \\ &= \mathbf{R}_1 \mathbf{R}_2 \left[\mathbf{I}_3 \right]_\times \dot{\psi}_{\beta \alpha} \mathbf{R}^T_2 \mathbf{R}^T_1 + \mathbf{R}_1 \left[\mathbf{I}_2 \right]_\times \dot{\theta}_{\beta \alpha} \mathbf{R}^T_1 + \left[\mathbf{I}_1 \right]_\times \dot{\phi}_{\beta \alpha}. \label{2} \end{align} \]

Given:

\[ \left[ \mathbf{R} \mathbf{r} \right]_\times = \mathbf{R} \left[ \mathbf{r} \right]_\times \mathbf{R}^T, \]

for any vector \(\mathbf{r} \in \mathbb{R}^3\), eq (\(\ref{2}\)) becomes:

\[ \begin{align} \left[ \boldsymbol{\omega}^\alpha_{\beta \alpha} \right]_\times &= \left[ \mathbf{R}_1 \mathbf{R}_2 \mathbf{I}_3 \dot{\psi}_{\beta \alpha} \right]_\times + \left[ \mathbf{R}_1 \mathbf{I}_2 \dot{\theta}_{\beta \alpha} \right]_\times + \left[ \mathbf{I}_1 \dot{\phi}_{\beta \alpha} \right]_\times, \end{align} \]

which can be simplified to:

\[ \begin{align} \boldsymbol{\omega}^\alpha_{\beta \alpha} &= \underbrace{\left[ \begin{array}{ccc} \mathbf{I}_1 & \mathbf{R}_1 \mathbf{I}_2 & \mathbf{R}_1 \mathbf{R}_2 \mathbf{I}_3 \end{array} \right]}_{\mathbf{S}(\phi_{\beta \alpha}, \theta_{\beta \alpha})} \left[ \begin{array}{c} \dot{\phi}_{\beta \alpha} \\ \dot{\theta}_{\beta \alpha} \\ \dot{\psi}_{\beta \alpha} \end{array} \right] \\ &= \mathbf{S}(\phi_{\beta \alpha}, \theta_{\beta \alpha}) \dot{\boldsymbol{\Psi}}_{\beta \alpha}, \end{align} \]

which gives the angular velocity in terms of Euler rates. The matrix \(\mathbf{S}\) is:

\[ \mathbf{S}(\phi_{\beta \alpha}, \theta_{\beta \alpha}) = \left[ \begin{array}{ccc} 1 & 0 & -\sin \theta_{\beta \alpha} \\ 0 & \cos \phi_{\beta \alpha} & \sin \phi_{\beta \alpha} \cos \theta_{\beta \alpha} \\ 0 & -\sin \phi_{\beta \alpha} & \cos \phi_{\beta \alpha} \cos \theta_{\beta \alpha} \end{array} \right]. \]

The inverse relationship is:

\[ \begin{align} \left[ \begin{array}{c} \dot{\phi}_{\beta \alpha} \\ \dot{\theta}_{\beta \alpha} \\ \dot{\psi}_{\beta \alpha} \end{array} \right] &= \left[ \begin{array}{ccc} 1 & \sin \phi_{\beta \alpha} \tan \theta_{\beta \alpha} & \cos \phi_{\beta \alpha} \tan \theta_{\beta \alpha} \\ 0 & \cos \phi_{\beta \alpha} & -\sin \phi_{\beta \alpha} \\ 0 & \sin \phi_{\beta \alpha} / \cos \theta_{\beta \alpha} & \cos \phi_{\beta \alpha} / \cos \theta_{\beta \alpha} \end{array} \right] \boldsymbol{\omega}^\alpha_{\beta \alpha}. \end{align} \]

Note that \(\mathbf{S}^{-1}\) does not exist at \(\theta_{\beta \alpha} = \pi / 2\), which is the singularity associasted with Euler series.

Proofs

Proof

From Figure 1, the first rotation will be through the yaw angle \(\psi_{\beta \alpha}\) about the common \(z\) axis of the \(\beta\) frame to yield the first intermediate axes \((x^\psi, y^\psi, z^\psi)\):

\[ \begin{align} & x^{\psi} = x^{\beta} c(\psi_{\beta \alpha}) + y^{\beta} s(\psi_{\beta \alpha}) \\ & y^{\psi} = -x^{\beta} s(\psi_{\beta \alpha}) + y^{\beta} c(\psi_{\beta \alpha}) \\ & z^{\psi} = z^{\beta}, \end{align} \]

which yields to a rotation matrix:

\[ \mathbf{R}^{\psi}_{\beta} = \left[ \begin{array}{ccc} c(\psi_{\beta \alpha}) & s(\psi_{\beta \alpha}) & 0 \\ -s(\psi_{\beta \alpha}) & c(\psi_{\beta \alpha}) & 0 \\ 0 & 0 & 1 \end{array} \right]. \]

The second rotation will be through the pitch angle \(\theta_{\beta \alpha}\) about the common \(y\) axis of the first and second intermediate frames:

\[ \begin{align} & x^{\theta} = x^{\psi} c(\theta_{\beta \alpha}) - z^{\psi} s(\theta_{\beta \alpha}) \\ & y^{\theta} = y^{\psi} \\ & z^{\theta} = x^{\psi} s(\theta_{\beta \alpha}) + z^{\psi} c(\theta_{\beta \alpha}), \end{align} \]

which yields to a rotation matrix:

\[ \mathbf{R}^{\theta}_{\psi} = \left[ \begin{array}{ccc} c(\theta_{\beta \alpha}) & 0 & -s(\theta_{\beta \alpha}) \\ 0 & 1 & 0 \\ s(\theta_{\beta \alpha}) & 0 & c(\theta_{\beta \alpha}) \end{array} \right]. \]

The third and the last rotation will be through the roll angle \(\phi_{\beta \alpha}\) about the common \(x\) axis of the second frame and the \(\alpha\) frame:

\[ \begin{align} & x^{\alpha} = x^{\theta} \\ & y^{\alpha} = y^{\theta} c(\phi_{\beta \alpha}) + z^{\theta} s(\phi_{\beta \alpha}) \\ & z^{\alpha} = -y^{\theta} s(\phi_{\beta \alpha}) + z^{\theta} c(\phi_{\beta \alpha}), \end{align} \]

which yields to a rotation matrix:

\[ \mathbf{R}^{\alpha}_{\theta} = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c(\phi_{\beta \alpha}) & s(\phi_{\beta \alpha}) \\ 0 & -s(\phi_{\beta \alpha}) & c(\phi_{\beta \alpha}) \end{array} \right]. \]