ExampleOfQuaternionsAndEulerAngles 2026-08-24 02:05:28 2026-08-24 02:05:28 Example quaternions and Euler angles converted to the passive rotation convention quaternion Euler angles yaw pitch roll 321 sequence gimbal lock Tait-Bryan angles exercises worked solutions % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics). % If you use \includegraphics{figure} with no file extension, PhysicsLibrary % will prefer figure.png before falling back to figure.eps. \usepackage{graphicx} \DeclareGraphicsExtensions{.png,.jpg,.jpeg,.pdf,.eps} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \section*{Quaternions and Euler Angles: Examples, Exercises, and Solutions} This companion article belongs to the PhysicsLibrary entry \emph{Quaternions and Euler Angles}. All exercises are stated first. Complete solutions appear only after the exercise section so that the article is self-study friendly. \section{Convention declaration} We use right-handed frames, Hamilton multiplication, scalar-first display notation, and the active 3-2-1 yaw-pitch-roll convention \begin{equation} R = R_z(\psi)R_y(\theta)R_x(\phi), \qquad q = q_z(\psi)q_y(\theta)q_x(\phi). \end{equation} \section{Exercises} \subsection*{Exercise 1} Show that a pure yaw of angle $\psi$ is represented by \begin{equation} q = \cos\frac{\psi}{2} + \mathbf k\sin\frac{\psi}{2}. \end{equation} \subsection*{Exercise 2} Show that a pure pitch of angle $\theta$ is represented by \begin{equation} q = \cos\frac{\theta}{2} + \mathbf j\sin\frac{\theta}{2}. \end{equation} \subsection*{Exercise 3} Show that a pure roll of angle $\phi$ is represented by \begin{equation} q = \cos\frac{\phi}{2} + \mathbf i\sin\frac{\phi}{2}. \end{equation} \subsection*{Exercise 4} Multiply $q = q_z(\psi)q_y(\theta)q_x(\phi)$ and derive the 3-2-1 quaternion components. \subsection*{Exercise 5} For \begin{equation} \phi = 0, \qquad \theta = 0, \qquad \psi = \frac{\pi}{2}, \end{equation} compute the corresponding quaternion. \subsection*{Exercise 6} For \begin{equation} \phi = \frac{\pi}{2}, \qquad \theta = 0, \qquad \psi = 0, \end{equation} compute the corresponding quaternion. \subsection*{Exercise 7} For \begin{equation} \phi = 0, \qquad \theta = \frac{\pi}{2}, \qquad \psi = 0, \end{equation} compute the corresponding quaternion. \subsection*{Exercise 8} Write the 3-2-1 rotation matrix $R = R_z(\psi)R_y(\theta)R_x(\phi)$ explicitly. \subsection*{Exercise 9} Show that the quaternion from Exercise 5 produces the same matrix as the 3-2-1 matrix with $\phi=0$, $\theta=0$, and $\psi=\pi/2$. \subsection*{Exercise 10} Given a unit quaternion \begin{equation} q = \frac{\sqrt3}{2} + \frac{1}{2}\mathbf k, \end{equation} recover the 3-2-1 yaw, pitch, and roll angles. \subsection*{Exercise 11} Given a unit quaternion \begin{equation} q = \frac{\sqrt2}{2} + \frac{\sqrt2}{2}\mathbf i, \end{equation} recover the 3-2-1 yaw, pitch, and roll angles. \subsection*{Exercise 12} Explain why the 3-2-1 Euler angle description becomes singular at $\theta=\pm\pi/2$. \subsection*{Exercise 13} A student uses a scalar-last quaternion array $[q_x,q_y,q_z,q_w]^T$ directly in a scalar-first Euler angle formula. Explain the error. \subsection*{Exercise 14} Why can two different triples $(\phi,\theta,\psi)$ describe the same physical orientation? \subsection*{Exercise 15} What is the practical advantage of propagating attitude with quaternions while still displaying yaw, pitch, and roll to a user? \section{Solutions} \subsection*{Solution 1} A yaw is a rotation about the $z$ axis, so its axis quaternion is \begin{equation} q = \cos\frac{\psi}{2} + \mathbf k\sin\frac{\psi}{2}. \end{equation} \subsection*{Solution 2} A pitch is a rotation about the $y$ axis, so \begin{equation} q = \cos\frac{\theta}{2} + \mathbf j\sin\frac{\theta}{2}. \end{equation} \subsection*{Solution 3} A roll is a rotation about the $x$ axis, so \begin{equation} q = \cos\frac{\phi}{2} + \mathbf i\sin\frac{\phi}{2}. \end{equation} \subsection*{Solution 4} Introduce the half-angle abbreviations $c_\phi$, $s_\phi$, $c_\theta$, $s_\theta$, $c_\psi$, and $s_\psi$. Multiplying $q_z(\psi)q_y(\theta)q_x(\phi)$ gives \begin{equation} q_w = c_\phi c_\theta c_\psi + s_\phi s_\theta s_\psi, \end{equation} \begin{equation} q_x = s_\phi c_\theta c_\psi - c_\phi s_\theta s_\psi, \end{equation} \begin{equation} q_y = c_\phi s_\theta c_\psi + s_\phi c_\theta s_\psi, \end{equation} \begin{equation} q_z = c_\phi c_\theta s_\psi - s_\phi s_\theta c_\psi. \end{equation} \subsection*{Solution 5} Substituting $\phi=0$, $\theta=0$, and $\psi=\pi/2$ gives \begin{equation} q = \cos\frac{\pi}{4} + \mathbf k\sin\frac{\pi}{4} = \frac{1+\mathbf k}{\sqrt2}. \end{equation} \subsection*{Solution 6} Substituting $\phi=\pi/2$, $\theta=0$, and $\psi=0$ gives \begin{equation} q = \cos\frac{\pi}{4} + \mathbf i\sin\frac{\pi}{4} = \frac{1+\mathbf i}{\sqrt2}. \end{equation} \subsection*{Solution 7} Substituting $\phi=0$, $\theta=\pi/2$, and $\psi=0$ gives \begin{equation} q = \cos\frac{\pi}{4} + \mathbf j\sin\frac{\pi}{4} = \frac{1+\mathbf j}{\sqrt2}. \end{equation} \subsection*{Solution 8} The explicit matrix is \begin{equation} R = \begin{bmatrix} c_\psi c_\theta & c_\psi s_\theta s_\phi - s_\psi c_\phi & c_\psi s_\theta c_\phi + s_\psi s_\phi\\ s_\psi c_\theta & s_\psi s_\theta s_\phi + c_\psi c_\phi & s_\psi s_\theta c_\phi - c_\psi s_\phi\\ -s_\theta & c_\theta s_\phi & c_\theta c_\phi \end{bmatrix}. \end{equation} \subsection*{Solution 9} The quaternion of Exercise 5 is $q=(1+\mathbf k)/\sqrt2$. By Q09 its matrix is \begin{equation} R(q)= \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}. \end{equation} Setting $\phi=0$, $\theta=0$, and $\psi=\pi/2$ in the 3-2-1 matrix gives the same result. \subsection*{Solution 10} Comparing with $q = \cos(\psi/2) + \mathbf k\sin(\psi/2)$ shows \begin{equation} \psi = \frac{\pi}{3}, \qquad \theta = 0, \qquad \phi = 0. \end{equation} So the orientation is a pure yaw of $60^\circ$. \subsection*{Solution 11} Comparing with $q = \cos(\phi/2) + \mathbf i\sin(\phi/2)$ shows \begin{equation} \phi = \frac{\pi}{2}, \qquad \theta = 0, \qquad \psi = 0. \end{equation} So the orientation is a pure roll of $90^\circ$. \subsection*{Solution 12} At $\theta=\pm\pi/2$, the yaw and roll axes align. The mapping from orientation to the triple $(\phi,\theta,\psi)$ therefore loses rank and one combination of yaw and roll becomes indistinguishable. This is the Euler angle singularity called gimbal lock. \subsection*{Solution 13} The formulas assume that the first component is the scalar part $q_w$. If the array order is actually $[q_x,q_y,q_z,q_w]^T$, then the symbols are being matched to the wrong numerical entries. The components must be remapped before using the scalar-first formula. \subsection*{Solution 14} Euler angles are not a unique global parameterization. Because rotations are periodic and because of the singular structure of the coordinate chart, different triples can represent the same orientation. For example, adding $2\pi$ to one angle may leave the physical orientation unchanged. \subsection*{Solution 15} Quaternions are numerically robust for propagation, interpolation, and control, and they avoid gimbal-lock singularities. Euler angles, however, are intuitive for humans. A common strategy is therefore to compute internally with quaternions and display yaw, pitch, and roll externally. \section*{License} This article is an original synthesis prepared for PhysicsLibrary and intended for release under CC BY-SA 4.0.