QuaternionsAndDirectionCosineMatrices 2026-08-24 00:20:09 2026-08-24 00:20:09 Topic quaternion direction cosine matrix DCM rotation matrix active rotation passive rotation frame transformation attitude matrix % 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 Direction Cosine Matrices} Unit quaternions and Direction Cosine matrices are two standard ways to encode three-dimensional orientation. The quaternion form is compact and avoids the singularities of Euler angles; the matrix form acts directly on vectors, exposes the geometry of the rotated basis, and interfaces naturally with rigid-body kinematics and linear algebra software. This entry derives the matrix associated with a unit quaternion, explains the relation between active and passive usage, and shows how to move between the two representations. \section{Convention declaration} This article uses the PhysicsLibrary house convention: \begin{enumerate} \item right-handed orthonormal frames; \item Hamilton multiplication, so $\mathbf i\mathbf j=\mathbf k$; \item scalar-first display notation, \begin{equation} q=q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k \end{equation} with vector part \begin{equation} \mathbf q = \begin{bmatrix} q_x\\ q_y\\ q_z \end{bmatrix}; \end{equation} \item active vector rotation by a unit quaternion, \begin{equation} \boxed{\mathbf v' = q\mathbf v q^*.} \end{equation} \end{enumerate} The associated active rotation matrix is denoted by $R(q)$, so that \begin{equation} \boxed{\mathbf v' = R(q)\mathbf v.} \end{equation} The same quaternion written in scalar-last storage order $[q_x,q_y,q_z,q_w]^T$ represents the same physical rotation. \section{From the quaternion sandwich to a matrix} Write the unit quaternion as \begin{equation} q=q_w+\mathbf q, \end{equation} where $q_w$ is the scalar part and $\mathbf q$ is the pure-vector part. From Q07, the quaternion vector-rotation formula is \begin{equation} \mathbf v'= (q_w^2-\mathbf q\cdot\mathbf q)\mathbf v +2\mathbf q(\mathbf q\cdot\mathbf v) +2q_w(\mathbf q\times\mathbf v). \end{equation} This is already linear in $\mathbf v$, so it must have the form $\mathbf v' = R(q)\mathbf v$. The three terms may be written as matrix actions: \begin{equation} (q_w^2-\mathbf q\cdot\mathbf q)\mathbf v = (q_w^2-\mathbf q^T\mathbf q)I_3\,\mathbf v, \end{equation} \begin{equation} 2\mathbf q(\mathbf q\cdot\mathbf v) =2\mathbf q\mathbf q^T\mathbf v, \end{equation} and \begin{equation} 2q_w(\mathbf q\times\mathbf v)=2q_w[\mathbf q]_\times\mathbf v, \end{equation} where \begin{equation} [\mathbf q]_\times= \begin{bmatrix} 0 & -q_z & q_y\\ q_z & 0 & -q_x\\ -q_y & q_x & 0 \end{bmatrix} \end{equation} is the skew-symmetric matrix for the cross product. Therefore, \begin{equation} \boxed{ R(q)= (q_w^2-\mathbf q^T\mathbf q)I_3 +2\mathbf q\mathbf q^T +2q_w[\mathbf q]_\times. } \end{equation} This compact matrix formula is often the cleanest way to derive properties. \section{Component form} Expanding equation (10) gives the familiar scalar-first component formula \begin{equation} \boxed{ R(q)= \begin{bmatrix} 1-2(q_y^2+q_z^2) & 2(q_xq_y-q_wq_z) & 2(q_xq_z+q_wq_y)\\ 2(q_xq_y+q_wq_z) & 1-2(q_x^2+q_z^2) & 2(q_yq_z-q_wq_x)\\ 2(q_xq_z-q_wq_y) & 2(q_yq_z+q_wq_x) & 1-2(q_x^2+q_y^2) \end{bmatrix}. } \end{equation} If $q$ is unit, then $R(q)$ is orthogonal with determinant $+1$, so it is a proper rotation matrix. \section{What the columns mean} The columns of $R(q)$ are the images of the basis vectors under the active rotation. If $\mathbf e_1$, $\mathbf e_2$, and $\mathbf e_3$ are the standard basis vectors, then \begin{equation} R(q)= \begin{bmatrix} R(q)\mathbf e_1 & R(q)\mathbf e_2 & R(q)\mathbf e_3 \end{bmatrix}. \end{equation} Figure~\ref{fig:q09-dcm} shows this geometry for a representative quaternion. \begin{figure}[h] \centering \includegraphics[width=0.87\textwidth]{Q09_quaternion_dcm_geometry.png} \caption{A unit quaternion and its direction cosine matrix. The original basis vectors are the $x$, $y$, and $z$ axes. The columns of $R(q)$ are the rotated basis vectors $R(q)\mathbf e_1$, $R(q)\mathbf e_2$, and $R(q)\mathbf e_3$. The same matrix is produced by $q$ and $-q$.} \label{fig:q09-dcm} \end{figure} \section{Example: quarter-turn about the $z$ axis} Let \begin{equation} q=\cos\frac{\pi}{4}+\mathbf k\sin\frac{\pi}{4} =\frac{1+\mathbf k}{\sqrt2}. \end{equation} Then $q_w=q_z=1/\sqrt2$ and $q_x=q_y=0$, so equation (11) becomes \begin{equation} R(q)= \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}. \end{equation} This is exactly the matrix for a $+90^\circ$ active rotation about the $z$ axis: \begin{equation} R(q) \begin{bmatrix}1\\0\\0\end{bmatrix} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \qquad R(q) \begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}-1\\0\\0\end{bmatrix}. \end{equation} \section{Orthogonality, inverse, and conjugation} For a unit quaternion, \begin{equation} \|q\|=1, \qquad q^{-1}=q^*. \end{equation} The inverse active rotation is therefore \begin{equation} \mathbf v = q^*\mathbf v' q. \end{equation} At the matrix level this means \begin{equation} \boxed{R(q^*)=R(q)^{-1}=R(q)^T.} \end{equation} So quaternion conjugation corresponds to matrix transposition for unit quaternions. \section{The sign ambiguity: $q$ and $-q$} Because equation (10) contains either quadratic terms in the quaternion components or the product $q_w\mathbf q$, replacing $q$ by $-q$ leaves the matrix unchanged. Therefore, \begin{equation} \boxed{R(-q)=R(q).} \end{equation} This is the matrix version of the familiar fact that $q$ and $-q$ represent the same physical orientation. \section{Passive coordinate transformations and DCM notation} In mechanics and aerospace work, the phrase \emph{direction cosine matrix} often refers to a passive coordinate transformation matrix. If ${}^{A}\mathbf v$ and ${}^{B}\mathbf v$ denote the coordinates of the same geometric vector in frames $A$ and $B$, then the passive mapping is written \begin{equation} {}^{A}\mathbf v = C^A_B\,{}^{B}\mathbf v. \end{equation} With the PhysicsLibrary frame quaternion ${}^{A}q_B$, the associated matrix is \begin{equation} \boxed{C^A_B = R({}^{A}q_B).} \end{equation} The frame-chain rule is therefore \begin{equation} \boxed{C^N_B=C^N_A C^A_B,} \end{equation} which matches the quaternion chain ${}^{N}q_B={}^{N}q_A{}^{A}q_B$. \section{Recovering a quaternion from a rotation matrix} Suppose $R=[r_{ij}]$ is a proper rotation matrix. If the trace is not too close to $-1$, a convenient formula is \begin{equation} \boxed{ q_w = \frac{1}{2}\sqrt{1+r_{11}+r_{22}+r_{33}}. } \end{equation} Once $q_w$ is known, the vector components follow from \begin{equation} q_x = \frac{r_{32}-r_{23}}{4q_w}, \qquad q_y = \frac{r_{13}-r_{31}}{4q_w}, \qquad q_z = \frac{r_{21}-r_{12}}{4q_w}. \end{equation} These formulas are simple but become numerically fragile when $q_w$ is small, for example near a $180^\circ$ rotation. In numerical work one then switches to branch formulas based on whichever of $q_w^2$, $q_x^2$, $q_y^2$, or $q_z^2$ is largest. \section{A useful special case} For a unit quaternion written in axis-angle form, \begin{equation} q=\cos\frac{\theta}{2}+\widehat{\mathbf u}\sin\frac{\theta}{2}, \end{equation} substituting into equation (10) yields the matrix Rodrigues formula \begin{equation} \boxed{R(q)=I_3\cos\theta +(1-\cos\theta)\widehat{\mathbf u}\widehat{\mathbf u}^T + [\widehat{\mathbf u}]_\times\sin\theta.} \end{equation} Thus the quaternion-to-matrix map reproduces the standard axis-angle rotation matrix automatically. \section{Common pitfalls} The most common mistakes are: \begin{enumerate} \item using the wrong component order when copying a formula from a source that assumes scalar-last storage; \item mixing active and passive interpretations without taking the transpose or inverse; \item assuming that the rows of the active matrix are the rotated basis vectors rather than the columns; \item forgetting that $q$ and $-q$ give the same matrix; \item recovering a quaternion from a matrix using the trace formula near $180^\circ$ without using a numerically safer branch. \end{enumerate} \section{What comes next} The next entry connects quaternions to Euler angles and shows how the familiar 3-2-1 yaw-pitch-roll formulas fit into the quaternion framework. \section{Sources and historical notes} The matrix formula follows directly from the quaternion rotation law and the scalar-vector product rules. The active/passive distinction and frame-chain interpretation are standard in multibody dynamics, spacecraft attitude work, and inertial navigation. The compact form in equation (10) is especially useful for proofs, while the expanded component form in equation (11) is common in implementations. \section*{License} This article is an original synthesis prepared for PhysicsLibrary and intended for release under CC BY-SA 4.0.