QuaternionSeriesOverviewAndArticleGuide 2026-08-24 23:53:02 2026-08-26 02:20:14 Topic clean up and update to other article updates quaternion rotation attitude orientation direction cosine matrix Euler angles angular velocity IMU attitude estimation inertial navigation learning path article guide % 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*{Quaternion series overview and article guide} The PhysicsLibrary quaternion collection is organized as an engineering and physics learning path rather than as a purely algebraic development. It begins with notation and convention discipline, establishes the quaternion algebra needed for computation, develops three-dimensional orientation and frame to frame coordinate transformations, and then proceeds to angular rate kinematics, relative attitude, estimation errors, and inertial measurement unit propagation. The collection is intended for students and practitioners in mechanics, rigid-body dynamics, spacecraft attitude determination and control, inertial navigation, robotics, simulation, and related fields. The deeper algebraic structure of quaternions remains important, but the first objective is to make quaternion formulas unambiguous and usable in physical applications. The quaternion and Euler angle series share one geometric convention: \emph{passive coordinate transformations} are the primary attitude maps, and named rotation sequences are interpreted \emph{intrinsically}, meaning that successive rotations are about axes of the current, already-rotated frame. \section{PhysicsLibrary quaternion convention} Unless an article explicitly states otherwise, PhysicsLibrary uses Hamilton multiplication, scalar-first mathematical notation, right-handed orthonormal frames, positive frame rotation according to the right-hand screw rule, passive frame to frame coordinate transformations, and intrinsic moving axis interpretation for named Euler sequences. The canonical attitude quaternion is written ${}^{B}q_A$. It maps coordinates from frame $A$ into frame $B$, in the same direction as the passive direction cosine matrix ${}^{B}C_A$: \begin{equation} {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^* = {}^{B}C_A\,{}^{A}v. \end{equation} If frame $B$ is obtained from frame $A$ by a positive right-handed rotation of the axes through angle $\theta$ about unit axis $\widehat{\mathbf u}$, then the passive axis angle quaternion is \begin{equation} {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}. \end{equation} The minus sign is the passive counterpart of the familiar positive angle active rotor. For example, a positive $90^\circ$ frame rotation about $+z$ gives ${}^{B}q_A=\sqrt{2}/2-(\sqrt{2}/2)\mathbf k$, and a fixed physical vector with coordinates $[1,0,0]^T$ in $A$ has coordinates $[0,-1,0]^T$ in $B$. The inverse map is ${}^{A}q_B=({}^{B}q_A)^*$. Hamilton multiplication then chains passive quaternion maps in the same written order as passive DCMs: \begin{equation} {}^{C}q_A = {}^{C}q_B\,{}^{B}q_A, \qquad {}^{C}C_A = {}^{C}C_B\,{}^{B}C_A. \end{equation} This common frame direction is a central design choice of the revised PhysicsLibrary orientation series. The detailed convention discussion belongs in Quaternions for Physics and Engineering: Orientation, Notation, and Conventions. \section{Convention migration status} Older PhysicsLibrary quaternion entries were written around an active vector interpretation. The revised series instead uses the passive frame map declared above. During the cleanup, an older linked article may temporarily retain an explicit active convention. Its local declaration controls until that article is converted. New and revised quaternion entries should use the passive convention of this overview and of the convention reference article. The algebraic articles on quaternion definition, multiplication, conjugation, norm, and inverse require little or no mathematical change because Hamilton's algebra is unchanged. Convention sensitive entries require explicit revision, particularly those dealing with axis angle orientation, vector or coordinate transformation, composition, quaternion DCM conversion, quaternion Euler conversion, kinematics, attitude error, and IMU propagation. \section{How the articles fit together} The material naturally falls into three stages. \subsection*{1. Conventions and algebra} Begin with Quaternions for Physics and Engineering: Orientation, Notation, and Conventions, then read quaternion, quaternion product in scalar vector form, quaternion conjugate, quaternion norm, and quaternion inverse. These entries establish the Hamilton basis rule $\mathbf i\mathbf j=\mathbf k$, scalar/vector decomposition, the Hamilton product, conjugation, norm, and the identity $q^{-1}=q^*/\lVert q\rVert^2$. For a unit quaternion, $q^{-1}=q^*$. Storage order is treated separately from algebra. A program may store $[q_w,q_x,q_y,q_z]$ or $[q_x,q_y,q_z,q_w]$ while still using the same Hamilton multiplication. \subsection*{2. Frame transformations and attitude representations} Continue with axis-angle representation and unit quaternion, coordinate transformations with quaternions, composition of rotations and quaternion order, quaternions and direction cosine matrices, and quaternions and Euler angles. The axis angle article develops the fundamental passive parameterization and its quaternion exponential, \begin{equation} {}^{B}q_A = \exp\left(-\frac{\theta}{2}\widehat{\mathbf u}\right). \end{equation} The coordinate-transformation article then derives the passive quaternion sandwich and its Rodrigues equivalent. The composition article shows why finite transformations do not commute and why explicit frame labels determine factor order more reliably than verbal mnemonics. The DCM article establishes the direct correspondence \begin{equation} {}^{B}C_A=C({}^{B}q_A), \qquad C(q^*)=C(q)^T, \qquad C(-q)=C(q). \end{equation} The Euler angle article connects the quaternion series to the revised PhysicsLibrary passive/intrinsic Euler convention. For intrinsic $3$--$2$--$1$ yaw--pitch--roll, with $\psi$ yaw, $\theta$ pitch, and $\phi$ roll, define passive elementary quaternions $q_1^P(\phi)$, $q_2^P(\theta)$, and $q_3^P(\psi)$. Then \begin{equation} {}^{B}q_A = q_1^P(\phi)\,q_2^P(\theta)\,q_3^P(\psi), \qquad {}^{B}C_A = C_1(\phi)\,C_2(\theta)\,C_3(\psi). \end{equation} Thus quaternion and DCM sequence products have the same factor order. \subsection*{3. Dynamics, estimation, and IMU propagation} Finish the main engineering path with quaternion kinematics and angular velocity, relative attitude and error quaternions, and numerical quaternion propagation and IMU attitude state integration. Let $q={}^{B}q_I$ map inertial coordinates into body coordinates. If $\omega^B$ is the angular velocity of $B$ relative to $I$, resolved in the body frame and embedded as a pure quaternion, the passive kinematic equation is \begin{equation} \dot q=-\frac12\omega^B q. \end{equation} If the same physical angular velocity is resolved in the inertial frame, the equivalent form is $\dot q=-\tfrac12 q\omega^I$. For relative attitude, if $q={}^{B}q_I$ is the actual attitude and $q_d={}^{D}q_I$ is the desired attitude, then the passive coordinate map from the actual body frame $B$ into the desired frame $D$ is \begin{equation} {}^{D}q_B=q_dq^*, \qquad {}^{B}q_D=qq_d^*. \end{equation} For a small positive passive frame rotation error, $\delta q\approx[1,-\tfrac12\delta\boldsymbol\theta^T]^T$ in scalar-first notation. Estimation articles must always identify which relative frame map is being represented before assigning a sign to the local error vector. For sampled body rate propagation, the incremental passive quaternion $\delta q_k={}^{B_{k+1}}q_{B_k}$ left multiplies the current inertial to body state: \begin{equation} q_{k+1} = \delta q_k q_k, \qquad \delta q_k = \begin{bmatrix} \cos(\Delta\theta/2)\\ -\dfrac{\sin(\Delta\theta/2)}{\Delta\theta}\, \Delta\boldsymbol\theta \end{bmatrix}. \end{equation} The numerical propagation article develops bias correction, exact exponential increments, numerical integration, normalization, sign continuity, coning, and the relationship between attitude propagation and body/navigation vector transformation. \section{Self-study companions} Beginning with the axis angle article, each main engineering entry has a separate self study companion. The exercises are stated first and complete worked solutions follow afterward. The companion articles are: \begin{itemize} \item axis-angle representation and unit quaternion: examples, exercises, and solutions; \item coordinate transformations with quaternions: examples, exercises, and solutions; \item composition of rotations and quaternion order: examples, exercises, and solutions; \item quaternions and direction cosine matrices: examples, exercises, and solutions; \item quaternions and Euler angles: examples, exercises, and solutions; \item quaternion kinematics and angular velocity: examples, exercises, and solutions; \item relative attitude and error quaternions: examples, exercises, and solutions; \item numerical quaternion propagation and IMU attitude state integration: examples, exercises, and solutions. \end{itemize} During the active to passive convention migration, a companion should be revised together with, or immediately after, its parent article so that its examples do not silently use an obsolete sign or map direction. \section{Recommended reading paths} \subsection*{First engineering introduction} A compact first route is: \begin{enumerate} \item the convention reference article; \item quaternion; \item conjugate, norm, and inverse; \item axis-angle representation and unit quaternion; \item coordinate transformations with quaternions; \item quaternions and direction cosine matrices. \end{enumerate} This path establishes enough algebra and geometry to construct and apply passive attitude quaternions without immediately requiring rigid-body kinematics. \subsection*{Spacecraft attitude, robotics, and rigid-body simulation} After the transformation articles, continue with quaternions and Euler angles, quaternion kinematics and angular velocity, and relative attitude and error quaternions. \subsection*{Inertial navigation and IMU propagation} For strapdown applications, emphasize axis angle, coordinate transformations, quaternion DCM conversion, quaternion kinematics, relative attitude/error quaternions, and numerical quaternion propagation and IMU attitude state integration. The kinematics article supplies the continuous passive rate equation, the relative attitude article supplies the local multiplicative error concept, and the numerical article applies both to sampled gyro measurements. \section{Core formula map} The main formulas of the engineering sequence can be summarized compactly as \begin{align*} q&=q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k,\\ q^{-1}&=q^*/\lVert q\rVert^2,\\ {}^{B}q_A&=\cos(\theta/2)-\widehat{\mathbf u}\sin(\theta/2),\\ {}^{B}v&={}^Bq_A\,{}^Av\,({}^{B}q_A)^*,\\ {}^{C}q_A&={}^Cq_B\,{}^Bq_A,\\ {}^{B}C_A&=C({}^{B}q_A),\\ {}^{B}q_A&=q_1^P(\phi)q_2^P(\theta)q_3^P(\psi) \quad\hbox{for intrinsic $3$--$2$--$1$},\\ \dot q&=-\tfrac12\omega^Bq \quad\hbox{for $q={}^Bq_I$},\\ {}^{D}q_B&=q_dq^*,\\ \delta q&\approx[1,-\tfrac12\delta\boldsymbol\theta^T]^T,\\ q_{k+1}&=\delta q_kq_k. \end{align*} This map is a guide, not a substitute for the individual derivations. \section{Conceptual relationships} Several relationships recur throughout the series. \begin{enumerate} \item \textbf{Quaternion algebra and quaternion attitude convention are different layers.} Hamilton multiplication remains unchanged when the application convention is changed from active vector rotation to passive coordinate transformation. \item \textbf{Storage order is not multiplication convention.} Scalar-first and scalar-last arrays can represent the same Hamilton quaternion. \item \textbf{The PhysicsLibrary attitude quaternion is a passive coordinate map.} The physical vector is held fixed while its components are transformed between reference frames. \item \textbf{Quaternion and DCM frame directions are deliberately aligned.} Both ${}^{B}q_A$ and ${}^{B}C_A$ map $A$ coordinates into $B$ coordinates. \item \textbf{Intrinsic describes a sequence construction.} For a named Euler sequence, successive elementary rotations are about axes of the moving frame. \item \textbf{Positive passive and active axis angle quaternions are conjugates.} For the same positive geometric angle, the passive frame quaternion has the opposite vector part sign from the positive active rotor. \item \textbf{Euler angles are local coordinates.} They remain valuable for interpretation but have coordinate singularities; quaternions and DCMs provide globally nonsingular orientation propagation. \item \textbf{A global quaternion and a local three component error are complementary.} The nominal state may remain a unit quaternion while an estimator linearizes a small relative frame transformation. \item \textbf{IMU propagation is repeated frame composition.} Each gyro sample produces a small passive body frame increment whose sign, multiplication order, and bias correction must agree with the declared attitude state. \end{enumerate} \section{Where deeper mathematics fits} The engineering sequence intentionally reaches applications before pursuing the deeper algebra. A later mathematical branch can develop division algebras, the geometry of the unit three sphere, $SU(2)$, the double covering of $SO(3)$, Lie groups and Lie algebras, and exponential and logarithmic maps. Those subjects explain why the engineering formulas have the structure they do, but they are not prerequisites for beginning quaternion attitude work. \section{Summary} The PhysicsLibrary quaternion series proceeds from convention discipline and Hamilton algebra to passive frame transformations, direction cosine matrices, intrinsic Euler angles, angular rate kinematics, attitude errors, and sampled IMU propagation. Its convention can be remembered from three statements: \begin{equation} {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*, \qquad {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}, \qquad {}^{C}q_A = {}^{C}q_B\,{}^{B}q_A. \end{equation} A reader who follows the collection should be able not only to quote quaternion formulas, but also to determine what a formula means, identify its convention, derive multiplication order from frame labels, test a transformation with a simple coordinate example, and use the result consistently in a physical or numerical application. \section*{License} This article is an original synthesis prepared for PhysicsLibrary and intended for release under CC BY-SA 4.0.