quaternion series overview and article guide
QuaternionSeriesOverviewAndArticleGuide
2026-08-24 23:53:02
2026-08-28 16:03:51
Topic
Filled out manual link control for all the Article sequences in the table so easy to use article links. Also updated related articles the the sequence ones
quaternion
rotation
attitude
orientation
direction cosine matrix
Euler angles
angular velocity
IMU
attitude estimation
inertial navigation
learning path
article guide
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\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{Article sequence}
\begin{center}
\begin{tabular}{|c|p{0.74\textwidth}|}
\hline
ID & Article\\
\hline
Q00 & \PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}{NotationQuaternionsForPhysicsAndEngineeringOrientation}\\
\hline
Q01 & \PMlinkname{Quaternion Definition and Basic Algebra}{QuaternionDefinitionAndBasicAlgebra}\\
\hline
Q01 & \PMlinkname{Quaternion Definition and Basic Algebra: Examples, Exercises, and Solution}{ExampleOfQuaternionDefinitionAndBasicAlgebra}\\
\hline
Q02 & Quaternion Product in Scalar Vector Form\\
\hline
Q03 & Quaternion Conjugate\\
\hline
Q04 & Quaternion Norm\\
\hline
Q05 & Quaternion Inverse\\
\hline
Q06 & Axis Angle Representation and Unit Quaternion\\
\hline
Q06E & Axis Angle Representation and Unit Quaternion: Examples, Exercises, and Solutions\\
\hline
Q07 & Rotating Vectors with Quaternions\\
\hline
Q07E & Rotating Vectors with Quaternions: Examples, Exercises, and Solutions\\
\hline
Q08 & Composition of Rotations and Quaternion Order\\
\hline
Q08E & Composition of Rotations and Quaternion Order: Examples, Exercises, and Solutions\\
\hline
Q09 & Quaternions and Direction Cosine Matrices\\
\hline
Q09E & Quaternions and Direction Cosine Matrices: Examples, Exercises, and Solutions\\
\hline
Q10 & Quaternions and Euler Angles\\
\hline
Q10E & Quaternions and Euler Angles: Examples, Exercises, and Solutions\\
\hline
Q11 & Quaternion Kinematics and Angular Velocity\\
\hline
Q11E & Quaternion Kinematics and Angular Velocity: Examples, Exercises, and Solutions\\
\hline
Q12 & Relative Attitude and Error Quaternions\\
\hline
Q12E & Relative Attitude and Error Quaternions: Examples, Exercises, and Solutions\\
\hline
Q13 & Numerical Quaternion Propagation and IMU Attitude State Integration\\
\hline
Q13E & Numerical Quaternion Propagation and IMU Attitude State Integration: Examples, Exercises, and Solutions\\
\hline
Q14 & Quaternion Exercises for Physics and Engineering\\
\hline
\end{tabular}
\end{center}
\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{Passive convention and older literature}
The current PhysicsLibrary quaternion series uses the passive frame convention
declared above throughout the convention sensitive engineering articles.
Many books, papers, software libraries, and older PhysicsLibrary entries use
the inverse active viewpoint. Under Hamilton multiplication, the active rotor
for the same positive geometric rotation is the conjugate of the
PhysicsLibrary passive frame quaternion:
\begin{equation}
q_{\mathrm{active}}
=
({}^{B}q_A)^*.
\end{equation}
Consequently, formulas taken from an active source may differ by a quaternion
conjugate, DCM transpose, vector sign, or multiplication side even when the
underlying physical orientation is the same.
Hamilton's algebra itself does not change. The articles on definition,
product, conjugate, norm, and inverse are therefore algebraically stable across
the convention change. Convention sensitive articles explicitly use the
passive frame map for axis angle orientation, vector coordinate
transformation, composition, quaternion DCM conversion, quaternion Euler
conversion, kinematics, relative attitude, error quaternions, and IMU
propagation.
A useful diagnostic is the positive $90^\circ$ frame rotation about
$+\widehat{\mathbf z}$:
\begin{equation}
{}^{B}q_A
=
\frac{1-\mathbf k}{\sqrt2},
\qquad
{}^{B}C_A
=
\begin{bmatrix}
0&1&0\\
-1&0&0\\
0&0&1
\end{bmatrix}.
\end{equation}
A fixed physical vector with $A$ coordinates
$[1,0,0]^T$ therefore has $B$ coordinates $[0,-1,0]^T$.
\section{How the articles fit together}
The material naturally falls into three stages.
\subsection*{1. Conventions and algebra}
Begin with
\emph{Quaternions for Physics and Engineering: Orientation, Notation, and
Conventions}, then read
\emph{Quaternion Definition and Basic Algebra},
\emph{Quaternion Product in Scalar Vector Form},
\emph{Quaternion Conjugate},
\emph{Quaternion Norm}, and
\emph{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
\emph{Axis Angle Representation and Unit Quaternion},
\emph{Rotating Vectors with Quaternions},
\emph{Composition of Rotations and Quaternion Order},
\emph{Quaternions and Direction Cosine Matrices}, and
\emph{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 rotating vectors article 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
\begin{equation}
{}^{B}C_A=C({}^{B}q_A),
\qquad
C(q^*)=C(q)^T,
\qquad
C(-q)=C(q).
\end{equation}
For intrinsic $3$-$2$-$1$ yaw, pitch, roll, with $\psi$ yaw, $\theta$ pitch,
and $\phi$ roll,
\begin{equation}
{}^{B}q_A
=
q_1^P(\phi)
q_2^P(\theta)
q_3^P(\psi),
\end{equation}
\begin{equation}
{}^{B}C_A
=
C_1(\phi)
C_2(\theta)
C_3(\psi).
\end{equation}
Quaternion and DCM sequence products therefore have the same written factor
order.
\subsection*{3. Kinematics, estimation, and IMU propagation}
Finish the main engineering path with
\emph{Quaternion Kinematics and Angular Velocity},
\emph{Relative Attitude and Error Quaternions}, and
\emph{Numerical Quaternion Propagation and IMU Attitude State Integration}.
Let
\[
q={}^Bq_I
\]
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, then
\begin{equation}
\dot q
=
-\frac12
\omega^Bq.
\end{equation}
If the same physical angular velocity is resolved in the inertial frame,
\begin{equation}
\dot q
=
-\frac12
q\omega^I.
\end{equation}
For relative attitude, let $q={}^Bq_I$ be the actual attitude and
$q_d={}^Dq_I$ the desired attitude. The left multiplicative error is the
direct actual to desired passive frame map:
\begin{equation}
\delta q_L
=
q_dq^*
=
{}^Dq_B,
\qquad
q_d=\delta q_Lq.
\end{equation}
A right multiplicative error may instead be defined by
\begin{equation}
\delta q_R=q^*q_d,
\qquad
q_d=q\delta q_R,
\qquad
\delta q_L=q\delta q_Rq^*.
\end{equation}
For a small positive passive frame error,
\begin{equation}
\delta q
\approx
\begin{bmatrix}
1\\
-\frac12\delta\boldsymbol\theta
\end{bmatrix},
\qquad
\delta\boldsymbol\theta
\approx
-2\delta\mathbf q_v.
\end{equation}
For sampled body rate propagation, first correct the measured rate or delta
angle for estimated gyroscope bias. The passive body increment
\[
\delta q_{B,k}
=
{}^{B_{k+1}}q_{B_k}
\]
left multiplies the current inertial to body state:
\begin{equation}
q_{k+1}
=
\delta q_{B,k}q_k,
\qquad
\delta q_{B,k}
=
\exp
\left(
-\frac12
\Delta\boldsymbol\theta_k^B
\right).
\end{equation}
For
$\Delta\theta=\lVert\Delta\boldsymbol\theta^B\rVert\neq0$,
\begin{equation}
\delta\mathbf q_B
=
\begin{bmatrix}
\cos(\Delta\theta/2)\\
-\dfrac{\sin(\Delta\theta/2)}{\Delta\theta}
\Delta\boldsymbol\theta^B
\end{bmatrix}.
\end{equation}
The numerical propagation article develops bias correction, exact exponential
increments, numerical integration, normalization, sign continuity, coning,
time centered attitude use, and the relationship between attitude propagation
and body to navigation vector transformation.
\section{Self study companions}
Beginning with Q06, each convention sensitive engineering article has a
separate companion in which all exercises are stated first and complete worked
solutions follow afterward.
The companion sequence is:
\begin{itemize}
\item \emph{Axis Angle Representation and Unit Quaternion:
Examples, Exercises, and Solutions};
\item \emph{Rotating Vectors with Quaternions:
Examples, Exercises, and Solutions};
\item \emph{Composition of Rotations and Quaternion Order:
Examples, Exercises, and Solutions};
\item \emph{Quaternions and Direction Cosine Matrices:
Examples, Exercises, and Solutions};
\item \emph{Quaternions and Euler Angles:
Examples, Exercises, and Solutions};
\item \emph{Quaternion Kinematics and Angular Velocity:
Examples, Exercises, and Solutions};
\item \emph{Relative Attitude and Error Quaternions:
Examples, Exercises, and Solutions};
\item \emph{Numerical Quaternion Propagation and IMU Attitude State
Integration: Examples, Exercises, and Solutions}.
\end{itemize}
Each companion uses the same passive frame convention as its parent article.
A worked example is therefore also a convention test: its signs,
multiplication side, DCM transpose, and frame labels must agree with the parent
derivation.
\section{Article sequence}
\begin{center}
\begin{tabular}{|c|p{0.74\textwidth}|}
\hline
ID & Article\\
\hline
Q00 & \PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}{NotationQuaternionsForPhysicsAndEngineeringOrientation}\\
\hline
Q01 & Quaternion Definition and Basic Algebra\\
\hline
Q02 & Quaternion Product in Scalar Vector Form\\
\hline
Q03 & Quaternion Conjugate\\
\hline
Q04 & Quaternion Norm\\
\hline
Q05 & Quaternion Inverse\\
\hline
Q06 & Axis Angle Representation and Unit Quaternion\\
\hline
Q06E & Axis Angle Representation and Unit Quaternion: Examples, Exercises, and Solutions\\
\hline
Q07 & Rotating Vectors with Quaternions\\
\hline
Q07E & Rotating Vectors with Quaternions: Examples, Exercises, and Solutions\\
\hline
Q08 & Composition of Rotations and Quaternion Order\\
\hline
Q08E & Composition of Rotations and Quaternion Order: Examples, Exercises, and Solutions\\
\hline
Q09 & Quaternions and Direction Cosine Matrices\\
\hline
Q09E & Quaternions and Direction Cosine Matrices: Examples, Exercises, and Solutions\\
\hline
Q10 & Quaternions and Euler Angles\\
\hline
Q10E & Quaternions and Euler Angles: Examples, Exercises, and Solutions\\
\hline
Q11 & Quaternion Kinematics and Angular Velocity\\
\hline
Q11E & Quaternion Kinematics and Angular Velocity: Examples, Exercises, and Solutions\\
\hline
Q12 & Relative Attitude and Error Quaternions\\
\hline
Q12E & Relative Attitude and Error Quaternions: Examples, Exercises, and Solutions\\
\hline
Q13 & Numerical Quaternion Propagation and IMU Attitude State Integration\\
\hline
Q13E & Numerical Quaternion Propagation and IMU Attitude State Integration: Examples, Exercises, and Solutions\\
\hline
Q14 & Quaternion Exercises for Physics and Engineering\\
\hline
\end{tabular}
\end{center}
\section{Recommended reading paths}
\subsection*{First engineering introduction}
A compact first route is:
\begin{enumerate}
\item Quaternions for Physics and Engineering: Orientation, Notation, and
Conventions;
\item Quaternion Definition and Basic Algebra;
\item Quaternion Conjugate, Quaternion Norm, and Quaternion Inverse;
\item Axis Angle Representation and Unit Quaternion;
\item Rotating Vectors 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 Representation and Unit Quaternion,
Rotating Vectors with Quaternions,
Quaternions and Direction Cosine Matrices,
Quaternion Kinematics and Angular Velocity,
Relative Attitude and 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 gyroscope measurements and IMU
delta angles.
\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
=-\tfrac12q\omega^I
\quad\hbox{for $q={}^Bq_I$},\\
\delta q_L&=q_dq^*,\qquad
\delta q_R=q^*q_d,\\
\delta q&\approx[1,-\tfrac12\delta\boldsymbol\theta^T]^T,\\
\delta q_{B,k}&=\exp(-\tfrac12\Delta\boldsymbol\theta_k^B),\\
q_{k+1}&=\delta q_{B,k}q_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{Series wide verification cases}
Convention sensitive formulas should be checked against a common battery of
simple cases:
\begin{enumerate}
\item identity orientation;
\item positive $90^\circ$ frame rotations about $+x$, $+y$, and $+z$;
\item inverse transformation by quaternion conjugation and DCM transpose;
\item $q$ versus $-q$;
\item two noncommuting successive rotations;
\item quaternion to DCM to quaternion round trip;
\item intrinsic $3$-$2$-$1$ quaternion and DCM agreement;
\item zero angular rate and constant single axis angular rate;
\item left versus right multiplicative attitude error;
\item the passive small error sign
$\delta\boldsymbol\theta\approx-2\delta\mathbf q_v$;
\item body resolved IMU increments left multiplying the inertial to body
attitude state.
\end{enumerate}
These checks are intentionally repetitive across the series. A convention
error that survives symbolic manipulation is often exposed immediately by one
of the $90^\circ$ coordinate tests.
\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 four 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}.
\end{equation}
\begin{equation}
{}^{C}q_A
=
{}^{C}q_B\,{}^{B}q_A,
\qquad
q_{k+1}
=
\delta q_{B,k}q_k.
\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.