ExampleOfEulerAnglesTaitBryanAngles 2026-08-29 15:16:16 2026-08-29 15:16:16 Example Euler angles: Tait Bryan angles Euler angles Tait Bryan angles Cardan angles yaw pitch roll passive transformation intrinsic rotation direction cosine matrix gimbal lock exercises % 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*{Euler Angles: Tait Bryan Angles Examples, Exercises, and Solutions} This entry is the self study companion to \emph{Euler angles: Tait Bryan angles}. The exercises emphasize the six distinct-axis sequences, their passive intrinsic matrix products, the universal Tait Bryan singularity, aerospace $3$-$2$-$1$ yaw pitch roll, and consistency with passive quaternion representations. All exercises are stated first. Complete worked solutions follow afterward. \section{Convention summary} PhysicsLibrary uses passive coordinate maps: \begin{equation} {}^B\mathbf v = {}^BC_A\,{}^A\mathbf v. \end{equation} For intrinsic $i$-$j$-$k$, \begin{equation} {}^BC_A = C_k(\gamma)C_j(\beta)C_i(\alpha). \end{equation} For Tait Bryan angles, \begin{equation} i\neq j, \qquad j\neq k, \qquad i\neq k. \end{equation} The six intrinsic Tait Bryan sequences are \begin{equation} 123,\quad132,\quad213,\quad231,\quad312,\quad321. \end{equation} Their generic middle-angle singularity is \begin{equation} \cos\beta=0. \end{equation} For the aerospace intrinsic $3$-$2$-$1$ sequence, \begin{equation} \alpha=\psi, \qquad \beta=\theta, \qquad \gamma=\phi, \end{equation} and \begin{equation} {}^BC_A = C_1(\phi)C_2(\theta)C_3(\psi). \end{equation} \section{Visual reference} \begin{center} \includegraphics[width=0.88\textwidth]{EA05_six_tait_bryan_sequences.png} \vspace{0.35em} \textbf{Figure.} The six intrinsic Tait Bryan sequences. \end{center} \begin{center} \includegraphics[width=0.86\textwidth]{EA05_tait_bryan_middle_angle_singularity.png} \vspace{0.35em} \textbf{Figure.} The generic Tait Bryan singularity illustrated with intrinsic $3$-$2$-$1$. \end{center} \section{Exercises} \begin{enumerate} \item \textbf{Recognize a Tait Bryan sequence.} Which of the following are Tait Bryan sequences? \[ 123,\quad 121,\quad 312,\quad 323,\quad 231,\quad 313. \] \item \textbf{Generate the six sequences.} Starting from the statement that each coordinate axis is used exactly once, derive the six possible Tait Bryan axis orders. \item \textbf{Write all six passive products.} Write the passive intrinsic DCM product for each of the six Tait Bryan sequences using generic angles $(\alpha,\beta,\gamma)$. \item \textbf{Sequence from a matrix product.} A passive intrinsic DCM is written \[ {}^BC_A = C_2(\gamma)C_1(\beta)C_3(\alpha). \] Identify the chronological Tait Bryan sequence. \item \textbf{Another sequence from a matrix product.} A passive intrinsic DCM is written \[ {}^BC_A = C_3(\gamma)C_2(\beta)C_1(\alpha). \] Identify the sequence. \item \textbf{Single-angle checks for 2-3-1.} For intrinsic $2$-$3$-$1$, \[ {}^BC_A = C_1(\gamma)C_3(\beta)C_2(\alpha). \] Find the remaining matrix when: \begin{enumerate} \item $\beta=\gamma=0$; \item $\alpha=\gamma=0$; \item $\alpha=\beta=0$. \end{enumerate} \item \textbf{Universal singularity condition.} Show that every Tait Bryan sequence has the same generic singularity condition \[ \cos\beta=0. \] What are the singular values on the usual principal branch? \item \textbf{Interpret the singularity geometrically.} For intrinsic $3$-$2$-$1$, explain what happens to the first and third physical rotation axes at \[ \theta=+\frac{\pi}{2} \] and at \[ \theta=-\frac{\pi}{2}. \] \item \textbf{Why the physical orientation remains valid.} Explain why gimbal lock does not mean that the rigid body orientation itself becomes undefined. \item \textbf{Aerospace notation.} For intrinsic $3$-$2$-$1$, identify which generic angle is yaw, pitch, and roll. Write the passive DCM product. \item \textbf{Expand the first row of the 3-2-1 DCM.} Starting from \[ {}^BC_A = C_1(\phi)C_2(\theta)C_3(\psi), \] derive the first row of the matrix. \item \textbf{Extract yaw pitch roll.} Suppose a nonsingular passive $3$-$2$-$1$ DCM is \[ C= \begin{bmatrix} C_{11}&C_{12}&C_{13}\\ C_{21}&C_{22}&C_{23}\\ C_{31}&C_{32}&C_{33} \end{bmatrix}. \] Write the principal extraction formulas for \[ \psi,\qquad\theta,\qquad\phi. \] \item \textbf{Numerical yaw pitch roll matrix.} Compute the passive $3$-$2$-$1$ DCM for \[ \psi=40^\circ,\qquad \theta=-15^\circ,\qquad \phi=25^\circ. \] \item \textbf{Numerical round trip.} Using the matrix from Exercise 13, apply the extraction equations and recover the original principal yaw pitch roll angles. \item \textbf{Equivalent extrinsic description.} Find the extrinsic sequence and chronological angle order equivalent to \[ \text{intrinsic }3\text{-}2\text{-}1 (\psi,\theta,\phi). \] \item \textbf{Another intrinsic/extrinsic conversion.} Find the extrinsic sequence equivalent to \[ \text{intrinsic }1\text{-}3\text{-}2 (\alpha,\beta,\gamma). \] \item \textbf{Tait Bryan triples are not vectors.} Explain why \[ (\psi_2-\psi_1,\theta_2-\theta_1,\phi_2-\phi_1) \] is not generally the exact relative attitude between two finite $3$-$2$-$1$ orientations. \item \textbf{Quaternion counterpart.} Write the passive quaternion product for intrinsic $3$-$2$-$1$ yaw pitch roll. \item \textbf{Small pitch case.} For intrinsic $3$-$2$-$1$, suppose \[ \phi=0, \qquad \psi=0, \qquad |\theta|\ll1. \] Write the first-order DCM approximation. \item \textbf{Convention audit.} A textbook says only: \begin{quote} Yaw pitch roll is represented by $R_z(\psi)R_y(\theta)R_x(\phi)$. \end{quote} List at least five convention questions that must be answered before comparing that formula with PhysicsLibrary. \end{enumerate} \section{Solutions} \subsection*{Solution 1: recognize a Tait Bryan sequence} A Tait Bryan sequence uses all three axes exactly once. Therefore \[ 123,\qquad312,\qquad231 \] are Tait Bryan. The sequences \[ 121,\qquad323,\qquad313 \] are proper Euler because the first and third axis labels match. \subsection*{Solution 2: generate the six sequences} Choose a first axis in three ways. Then arrange the remaining two axes in two possible orders. Therefore \[ 3\times2=6. \] The six sequences are \begin{equation} 123,\quad132,\quad213,\quad231,\quad312,\quad321. \end{equation} \subsection*{Solution 3: write all six passive products} Using \[ {}^BC_A=C_k(\gamma)C_j(\beta)C_i(\alpha), \] the six products are \begin{equation} C_{123} = C_3(\gamma)C_2(\beta)C_1(\alpha), \end{equation} \begin{equation} C_{132} = C_2(\gamma)C_3(\beta)C_1(\alpha), \end{equation} \begin{equation} C_{213} = C_3(\gamma)C_1(\beta)C_2(\alpha), \end{equation} \begin{equation} C_{231} = C_1(\gamma)C_3(\beta)C_2(\alpha), \end{equation} \begin{equation} C_{312} = C_2(\gamma)C_1(\beta)C_3(\alpha), \end{equation} and \begin{equation} C_{321} = C_1(\gamma)C_2(\beta)C_3(\alpha). \end{equation} \subsection*{Solution 4: sequence from a matrix product} Compare \[ C_2(\gamma)C_1(\beta)C_3(\alpha) \] with \[ C_k(\gamma)C_j(\beta)C_i(\alpha). \] Thus \[ i=3,\qquad j=1,\qquad k=2. \] The chronological intrinsic sequence is \begin{equation} 3\text{-}1\text{-}2. \end{equation} \subsection*{Solution 5: another sequence from a matrix product} For \[ C_3(\gamma)C_2(\beta)C_1(\alpha), \] we identify \[ i=1,\qquad j=2,\qquad k=3. \] Therefore the sequence is \begin{equation} 1\text{-}2\text{-}3. \end{equation} \subsection*{Solution 6: single-angle checks for 2-3-1} From \[ {}^BC_A = C_1(\gamma)C_3(\beta)C_2(\alpha), \] we obtain \begin{equation} \beta=\gamma=0 \quad\Longrightarrow\quad {}^BC_A=C_2(\alpha), \end{equation} \begin{equation} \alpha=\gamma=0 \quad\Longrightarrow\quad {}^BC_A=C_3(\beta), \end{equation} and \begin{equation} \alpha=\beta=0 \quad\Longrightarrow\quad {}^BC_A=C_1(\gamma). \end{equation} \subsection*{Solution 7: universal singularity condition} For every Tait Bryan sequence, the Jacobian from Euler coordinate rates to angular velocity loses rank when the first and third physical rotation axes become collinear. That occurs when the middle rotation turns through a right angle: \begin{equation} \cos\beta=0. \end{equation} On the usual principal branch, \begin{equation} -\frac{\pi}{2} \leq\beta\leq \frac{\pi}{2}, \end{equation} so the singular values are \begin{equation} \beta=\pm\frac{\pi}{2}. \end{equation} \subsection*{Solution 8: interpret the singularity geometrically} For intrinsic $3$-$2$-$1$, the first rotation axis is $z_A$. The third rotation is about $x_2$. At \[ \theta=+\frac{\pi}{2}, \] the axis $x_2$ becomes anti-aligned with $z_A$. At \[ \theta=-\frac{\pi}{2}, \] the axis $x_2$ becomes aligned with $z_A$. Thus the first and third rotations are about the same physical line, so yaw and roll can no longer be determined independently. \subsection*{Solution 9: why the physical orientation remains valid} The rigid body still has a perfectly valid orientation in three-dimensional space. Only the chosen three-coordinate parameterization loses local uniqueness. A quaternion or DCM remains nonsingular at the same physical attitude. Therefore gimbal lock is a coordinate singularity, not a disappearance of physical orientation. \subsection*{Solution 10: aerospace notation} PhysicsLibrary uses \[ \alpha=\psi=\text{yaw}, \] \[ \beta=\theta=\text{pitch}, \] and \[ \gamma=\phi=\text{roll}. \] Thus \begin{equation} {}^BC_A = C_1(\phi)C_2(\theta)C_3(\psi). \end{equation} \subsection*{Solution 11: expand the first row of the 3-2-1 DCM} First multiply \[ C_2(\theta)C_3(\psi). \] The first row of that product is \[ \begin{bmatrix} c_\theta c_\psi & c_\theta s_\psi & -s_\theta \end{bmatrix}. \] Premultiplication by $C_1(\phi)$ leaves the first row unchanged because the first row of $C_1(\phi)$ is \[ \begin{bmatrix} 1&0&0 \end{bmatrix}. \] Therefore the first row of the full DCM is \begin{equation} \begin{bmatrix} c_\theta c_\psi & c_\theta s_\psi & -s_\theta \end{bmatrix}. \end{equation} \subsection*{Solution 12: extract yaw pitch roll} On the nonsingular principal branch, \begin{equation} \theta=\arcsin(-C_{13}), \end{equation} \begin{equation} \phi=\operatorname{atan2}(C_{23},C_{33}), \end{equation} and \begin{equation} \psi=\operatorname{atan2}(C_{12},C_{11}). \end{equation} \subsection*{Solution 13: numerical yaw pitch roll matrix} With \[ \psi=40^\circ, \qquad \theta=-15^\circ, \qquad \phi=25^\circ, \] the passive DCM is \begin{equation} {}^BC_A \approx \begin{bmatrix} 0.73994&0.62089&0.25882\\ -0.66635&0.63247&0.40822\\ 0.08906&-0.46246&0.88211 \end{bmatrix}. \end{equation} \subsection*{Solution 14: numerical round trip} From the matrix, \[ C_{13}\approx0.25882. \] Thus \[ \theta = \arcsin(-0.25882) \approx -15^\circ. \] Next, \[ \phi = \operatorname{atan2}(0.40822,0.88211) \approx25^\circ, \] and \[ \psi = \operatorname{atan2}(0.62089,0.73994) \approx40^\circ. \] Therefore the principal round trip recovers \begin{equation} (\psi,\theta,\phi) = (40^\circ,-15^\circ,25^\circ). \end{equation} \subsection*{Solution 15: equivalent extrinsic description} Reverse both axis order and associated angle order: \begin{equation} \text{intrinsic }3\text{-}2\text{-}1 (\psi,\theta,\phi) \equiv \text{extrinsic }1\text{-}2\text{-}3 (\phi,\theta,\psi). \end{equation} \subsection*{Solution 16: another intrinsic/extrinsic conversion} For intrinsic $1$-$3$-$2$, \begin{equation} \text{intrinsic }1\text{-}3\text{-}2 (\alpha,\beta,\gamma) \equiv \text{extrinsic }2\text{-}3\text{-}1 (\gamma,\beta,\alpha). \end{equation} \subsection*{Solution 17: Tait Bryan triples are not vectors} Euler coordinates are nonlinear sequence-dependent coordinates. The second and third rotations occur about intermediate axes whose directions depend on earlier rotations. Therefore subtracting coordinate triples does not reproduce exact finite rotation composition. The exact relative attitude should instead be computed from DCMs or quaternions. \subsection*{Solution 18: quaternion counterpart} For intrinsic $3$-$2$-$1$ yaw pitch roll, \begin{equation} {}^Bq_A = q_1^P(\phi) q_2^P(\theta) q_3^P(\psi). \end{equation} The quaternion and DCM must satisfy \begin{equation} {}^BC_A = C({}^Bq_A). \end{equation} \subsection*{Solution 19: small pitch case} With \[ \phi=0,\qquad\psi=0, \] the DCM reduces to \[ {}^BC_A=C_2(\theta). \] For small $|\theta|$, \[ \cos\theta\approx1, \qquad \sin\theta\approx\theta. \] Thus \begin{equation} {}^BC_A \approx \begin{bmatrix} 1&0&-\theta\\ 0&1&0\\ \theta&0&1 \end{bmatrix}. \end{equation} \subsection*{Solution 20: convention audit} Before comparing the textbook formula with PhysicsLibrary, one should ask: \begin{enumerate} \item Is the rotation active or passive? \item Is the sequence intrinsic or extrinsic? \item What map direction does the matrix represent? \item Does the source use right hand positive angles? \item Does ``yaw pitch roll'' mean chronological $3$-$2$-$1$ intrinsic rotation? \item Are column vectors or row vectors being used? \item Is the written product chronological order or operator composition order? \item Are the frame labels and basis conventions the same? \end{enumerate} Without those declarations, identical-looking symbols can represent different transformations. \section{Compact review} Tait Bryan sequences use the three coordinate axes exactly once: \begin{equation} 123,\quad132,\quad213,\quad231,\quad312,\quad321. \end{equation} For intrinsic $i$-$j$-$k$, \begin{equation} {}^BC_A=C_k(\gamma)C_j(\beta)C_i(\alpha). \end{equation} Their generic singularity is \begin{equation} \cos\beta=0. \end{equation} For aerospace intrinsic $3$-$2$-$1$, \begin{equation} {}^BC_A=C_1(\phi)C_2(\theta)C_3(\psi), \end{equation} with \begin{equation} \theta=\arcsin(-C_{13}), \end{equation} \begin{equation} \phi=\operatorname{atan2}(C_{23},C_{33}), \end{equation} and \begin{equation} \psi=\operatorname{atan2}(C_{12},C_{11}). \end{equation} \section{Sources and exercise provenance} The exercises and solutions in this companion are newly written for PhysicsLibrary to reinforce the Tait Bryan framework developed in \emph{Euler angles: Tait Bryan angles}. \begin{thebibliography}{9} \bibitem{Henderson1977} D. M. Henderson, \emph{Euler Angles, Quaternions, and Transformation Matrices: Working Relationships}, JSC-12960, NASA Johnson Space Center, 1977. \PMlinkexternal{NASA Technical Reports Server} {https://ntrs.nasa.gov/citations/19770024290} \bibitem{Diebel2006} J. Diebel, ``Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,'' Stanford University, 2006. \PMlinkexternal{Online PDF} {https://www.astro.rug.nl/software/kapteyn-beta/_downloads/attitude.pdf} \bibitem{Goldstein2002} H. Goldstein, C. Poole, and J. Safko, \emph{Classical Mechanics}, 3rd ed., Addison Wesley, 2002. \PMlinkexternal{Publisher search} {https://www.pearson.com/en-us/search.html?aq=Classical+Mechanics+Goldstein} \end{thebibliography} \section*{License} Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution ShareAlike 4.0 International license.