quaternion series overview and article guide
QuaternionSeriesOverviewAndArticleGuide
2026-08-24 23:53:02
2026-08-25 03:55:47
Topic
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, 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.
This revision adopts a common geometric convention with the PhysicsLibrary
Euler-angle series: \emph{passive coordinate transformations} are the primary
attitude maps, and sequences of named rotations are interpreted
\emph{intrinsically}, meaning that each successive rotation is about an axis of
the current, already-rotated frame.
\section{PhysicsLibrary convention used throughout the collection}
Quaternion literature contains several independent convention choices. The
collection therefore establishes a common convention before applying any
formula. Unless an article explicitly states otherwise, PhysicsLibrary uses:
\begin{enumerate}
\item right-handed orthonormal physical frames;
\item positive frame rotation according to the right-hand screw rule;
\item Hamilton multiplication,
\begin{equation}
\mathbf i\mathbf j=\mathbf k,
\qquad
\mathbf j\mathbf k=\mathbf i,
\qquad
\mathbf k\mathbf i=\mathbf j;
\end{equation}
\item scalar-first display notation,
\begin{equation}
q=
\begin{bmatrix}
q_w&q_x&q_y&q_z
\end{bmatrix}^{T};
\end{equation}
\item passive frame-to-frame coordinate transformation by a unit quaternion,
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,
{}^{A}v\,
({}^{B}q_A)^*,
\end{equation}
where ordinary three-dimensional vectors are embedded as pure quaternions;
\item frame quaternion notation in which
\begin{equation}
{}^{B}q_A
\end{equation}
maps coordinates from frame $A$ to frame $B$;
\item intrinsic, moving-axis interpretation for named Euler or axis sequences
unless an article explicitly declares another interpretation.
\end{enumerate}
Thus ${}^{B}q_A$ and ${}^{B}C_A$ have the same coordinate-map direction:
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*
=
{}^{B}C_A\,{}^{A}v.
\end{equation}
This alignment is intentional. It makes frame chains read identically in
quaternion and direction-cosine-matrix notation:
\begin{equation}
{}^{C}q_A
=
{}^{C}q_B\,{}^{B}q_A,
\end{equation}
and
\begin{equation}
{}^{C}C_A
=
{}^{C}C_B\,{}^{B}C_A.
\end{equation}
Scalar-last storage such as $[q_x,q_y,q_z,q_w]^T$ is treated as a common data
layout, not as a different quaternion algebra. Likewise, the informal terms
``right-handed quaternion'' and ``left-handed quaternion'' are not used as the
primary definition of a multiplication convention because those labels are
historically overloaded.
The detailed convention discussion belongs in
\PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}
{NotationQuaternionsForPhysicsAndEngineeringOrientation}.
\subsection*{Positive passive axis-angle quaternion}
Let frame $B$ be obtained from frame $A$ by a positive right-handed rotation of
the coordinate axes through angle $\theta$ about the unit axis
$\widehat{\mathbf u}$. Under the PhysicsLibrary passive convention, the
coordinate transformation from $A$ to $B$ is represented by
\begin{equation}
{}^{B}q_A=\cos\frac{\theta}{2}-\widehat{\mathbf u}\sin\frac{\theta}{2}.
\end{equation}
The minus sign in the vector part is essential. It appears because the
physical vector is held fixed while the basis rotates positively.
For example, if frame $B$ is obtained by a $+90^\circ$ rotation about
$+\widehat{\mathbf z}$, then
\begin{equation}
{}^{B}q_A=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathbf k.
\end{equation}
A fixed physical vector with coordinates
\begin{equation}
{}^{A}\mathbf v
=
\begin{bmatrix}
1\\0\\0
\end{bmatrix}
\end{equation}
therefore has coordinates
\begin{equation}
{}^{B}\mathbf v
=
\begin{bmatrix}
0\\-1\\0
\end{bmatrix}.
\end{equation}
This is the quaternion counterpart of the passive elementary direction-cosine
matrix used in the PhysicsLibrary Euler-angle series.
\section{Convention migration status}
The present article establishes the new collection-level default. During the
cleanup of the existing quaternion collection, some older linked articles may
still contain an explicit active-vector convention such as
\begin{equation}
v'=qvq^*.
\end{equation}
Such an article should be regarded as a legacy convention until it is revised.
Its local convention declaration remains necessary for interpreting that
version, but new and revised PhysicsLibrary quaternion articles should use the
passive frame-map convention of this overview.
The algebraic articles on quaternion definition, multiplication, conjugation,
norm, and inverse require little or no mathematical change because Hamilton's
algebra itself is unchanged. Convention-sensitive articles require explicit
conversion, particularly those on:
\begin{enumerate}
\item axis-angle orientation;
\item vector and coordinate transformation;
\item composition order and frame chains;
\item quaternion--DCM conversion;
\item quaternion--Euler conversion;
\item quaternion kinematics;
\item relative attitude and error quaternions; and
\item numerical IMU propagation.
\end{enumerate}
\section{How the articles fit together}
The material naturally falls into three stages.
\begin{enumerate}
\item \textbf{Conventions and algebra.} Begin with
\PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}
{NotationQuaternionsForPhysicsAndEngineeringOrientation},
then read
\PMlinkname{quaternion}{Quaternion},
\PMlinkname{quaternion product in scalar-vector form}{QuaternionProductInScalarVectorForm},
\PMlinkname{quaternion conjugate}{QuaternionConjugate},
\PMlinkname{quaternion norm}{QuaternionNorm}, and
\PMlinkname{quaternion inverse}{QuaternionInverse}.
\item \textbf{Frame transformations and attitude representations.} Continue
with
\PMlinkname{axis-angle representation and unit quaternion}{AxisAngleRepresentationAndUnitQuaternion},
\PMlinkname{coordinate transformations with quaternions}{RotatingVectorsWithQuaternions},
\PMlinkname{composition of rotations and quaternion order}{CompositionOfRotationsAndQuaternionOrder},
\PMlinkname{quaternions and direction-cosine matrices}{QuaternionsAndDirectionCosineMatrices},
and
\PMlinkname{quaternions and Euler angles}{QuaternionsAndEulerAngles}.
\item \textbf{Dynamics, estimation, and IMU propagation.} Finish the main
engineering path with
\PMlinkname{quaternion kinematics and angular velocity}{QuaternionKinematicsAndAngularVelocity},
\PMlinkname{relative attitude and error quaternions}{RelativeAttitudeAndErrorQuaternions},
and
\PMlinkname{numerical quaternion propagation and IMU attitude-state integration}
{NumericalQuaternionPropagationAndIMUAttitudeStateIntegration}.
\end{enumerate}
Beginning with the axis-angle article, each main article also has a separate
self-study companion. Those companions state the exercises first and place the
complete worked solutions afterward.
\section{Conventions and quaternion algebra}
\subsection*{Quaternion conventions for physics and engineering}
\PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}
{NotationQuaternionsForPhysicsAndEngineeringOrientation}
should normally be read before applying quaternion formulas from an unfamiliar
source. It separates convention choices that are often conflated: Hamilton
versus flipped multiplication, scalar-first versus scalar-last storage,
active versus passive interpretation, frame direction, and handedness
terminology.
The practical diagnostic is equation-level rather than terminology-level. For
example,
\begin{equation}
\mathbf i\mathbf j=+\mathbf k
\end{equation}
identifies Hamilton multiplication in the convention used by this collection,
while the coordinate-map equation
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*
\end{equation}
identifies the passive frame interpretation.
\subsection*{Quaternion}
\PMlinkname{Quaternion}{Quaternion} is the canonical PhysicsLibrary definition
entry for the mathematical object. It defines
\begin{equation}
q=q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k
\end{equation}
and develops the Hamilton basis rules, scalar and vector parts, distributivity,
associativity, and noncommutativity.
The quaternion algebra is independent of whether a unit quaternion is later
used actively or passively. Consequently the switch to a passive attitude
convention does not change Hamilton's multiplication table.
\subsection*{Quaternion product in scalar-vector form}
\PMlinkname{Quaternion product in scalar-vector form}
{QuaternionProductInScalarVectorForm}
connects quaternion algebra directly to familiar vector operations. If
$p=(p_w,\mathbf p)$ and $q=(q_w,\mathbf q)$, then
\begin{equation}
(p_w,\mathbf p)(q_w,\mathbf q)
=
\left(
p_wq_w-\mathbf p\cdot\mathbf q,
p_w\mathbf q+q_w\mathbf p+\mathbf p\times\mathbf q
\right).
\end{equation}
For pure-vector quaternions this reduces to
\begin{equation}
\mathbf a\mathbf b
=
-\mathbf a\cdot\mathbf b+\mathbf a\times\mathbf b.
\end{equation}
This entry is especially useful for engineers because it exposes the dot and
cross products hidden inside the Hamilton product.
\subsection*{Quaternion conjugate, norm, and inverse}
Three separate definition entries build the algebraic tools needed by
orientation formulas:
\begin{itemize}
\item \PMlinkname{quaternion conjugate}{QuaternionConjugate},
\item \PMlinkname{quaternion norm}{QuaternionNorm},
\item \PMlinkname{quaternion inverse}{QuaternionInverse}.
\end{itemize}
Their central relationship is
\begin{equation}
qq^*=q^*q=\|q\|^2,
\end{equation}
which leads to
\begin{equation}
q^{-1}=\frac{q^*}{\|q\|^2}
\end{equation}
for every nonzero quaternion. For a unit quaternion,
\begin{equation}
q^{-1}=q^*.
\end{equation}
For frame transformations, conjugation reverses the map direction:
\begin{equation}
{}^{A}q_B
=
({}^{B}q_A)^*.
\end{equation}
The definitions are kept separate so later articles can link precisely to the
concept they require.
\section{Quaternions as passive three-dimensional frame transformations}
\subsection*{Axis-angle representation and unit quaternion}
\PMlinkname{Axis-angle representation and unit quaternion}
{AxisAngleRepresentationAndUnitQuaternion}
develops the fundamental passive frame parameterization
\begin{equation}
{}^{B}q_A=\cos\frac{\theta}{2}-\widehat{\mathbf u}\sin\frac{\theta}{2},
\end{equation}
where $\theta$ is the positive right-handed rotation of frame $B$ relative to
frame $A$ about $\widehat{\mathbf u}$.
The corresponding active rotor for rotating a physical vector through the same
positive angle is the conjugate quaternion. Keeping these two operations
separate prevents an implicit change of sign when moving between frame
kinematics and vector geometry.
The article also explains the half-angle, connects the formula to the
quaternion exponential, discusses the equivalent pair $q$ and $-q$, and treats
small-angle behavior.
Its self-study companion is
\PMlinkname{axis-angle representation and unit quaternion: examples, exercises, and solutions}
{AxisAngleRepresentationAndUnitQuaternionExamplesExercisesAndSolutions}.
\subsection*{Coordinate transformations with quaternions}
The article currently linked under the canonical object
\PMlinkname{coordinate transformations with quaternions}
{RotatingVectorsWithQuaternions}
develops the sandwich operation in frame notation:
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*.
\end{equation}
For a positive frame rotation, this passive transformation produces the
coordinate form corresponding to a rotation through the negative active angle.
Equivalently, if $q_{\rm act}$ is the active rotor for the same positive
geometric angle, then
\begin{equation}
{}^{B}q_A=q_{\rm act}^*.
\end{equation}
The article should retain Rodrigues' formula as an important comparison, but it
must distinguish clearly between the active physical-vector form and the
passive coordinate-transformation form rather than using the active form as the
PhysicsLibrary default.
Its companion is
\PMlinkname{coordinate transformations with quaternions: examples, exercises, and solutions}
{RotatingVectorsWithQuaternionsExamplesExercisesAndSolutions}.
\subsection*{Composition of frame transformations and quaternion order}
\PMlinkname{Composition of rotations and quaternion order}
{CompositionOfRotationsAndQuaternionOrder}
shows why finite rotations do not commute and why frame labels are the safest
way to determine factor order.
If
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*
\end{equation}
and
\begin{equation}
{}^{C}v
=
{}^{C}q_B\,{}^{B}v\,({}^{C}q_B)^*,
\end{equation}
then substitution gives
\begin{equation}
{}^{C}q_A
=
{}^{C}q_B\,{}^{B}q_A.
\end{equation}
Thus Hamilton quaternion chaining has the same written order as the passive DCM
chain
\begin{equation}
{}^{C}C_A
=
{}^{C}C_B\,{}^{B}C_A.
\end{equation}
For a named intrinsic sequence, each successive stage rotates about an axis of
the current moving frame. The corresponding passive stage quaternion
left-multiplies the accumulated frame map.
Its companion is
\PMlinkname{composition of rotations and quaternion order: examples, exercises, and solutions}
{CompositionOfRotationsAndQuaternionOrderExamplesExercisesAndSolutions}.
\subsection*{Quaternions and direction-cosine matrices}
\PMlinkname{Quaternions and direction-cosine matrices}
{QuaternionsAndDirectionCosineMatrices}
derives the direction-cosine matrix associated with a passive frame quaternion.
For
\begin{equation}
q=(q_w,\mathbf q),
\end{equation}
define
\begin{equation}
C(q)=
(q_w^2-\mathbf q^T\mathbf q)I_3
+2\mathbf q\mathbf q^T
+2q_w[\mathbf q]_{\times}.
\end{equation}
Under the PhysicsLibrary convention,
\begin{equation}
{}^{B}C_A=C({}^{B}q_A),
\end{equation}
so that
\begin{equation}
{}^{B}\mathbf v
=
{}^{B}C_A\,{}^{A}\mathbf v.
\end{equation}
Because the passive axis-angle quaternion carries a negative vector part for a
positive frame rotation, the matrix above has the passive signs required by the
Euler-angle series.
Conjugation and transposition agree:
\begin{equation}
C(q^*)=C(q)^T,
\end{equation}
and the double representation remains
\begin{equation}
C(-q)=C(q).
\end{equation}
Its companion is
\PMlinkname{quaternions and direction-cosine matrices: examples, exercises, and solutions}
{QuaternionsAndDirectionCosineMatricesExamplesExercisesAndSolutions}.
\subsection*{Quaternions and Euler angles}
\PMlinkname{Quaternions and Euler angles}{QuaternionsAndEulerAngles}
uses the intrinsic $3$-$2$-$1$ yaw--pitch--roll sequence as the primary
engineering example.
Let
\begin{equation}
\psi=\text{yaw},
\qquad
\theta=\text{pitch},
\qquad
\phi=\text{roll}.
\end{equation}
Using passive elementary quaternion transformations,
\begin{equation}
q_1^{P}(\phi)=\cos\frac{\phi}{2}-\mathbf i\sin\frac{\phi}{2},
\end{equation}
\begin{equation}
q_2^{P}(\theta)=\cos\frac{\theta}{2}-\mathbf j\sin\frac{\theta}{2},
\end{equation}
and
\begin{equation}
q_3^{P}(\psi)=\cos\frac{\psi}{2}-\mathbf k\sin\frac{\psi}{2}.
\end{equation}
The intrinsic $3$-$2$-$1$ passive frame map is
\begin{equation}
{}^{B}q_A
=
q_1^{P}(\phi)\,
q_2^{P}(\theta)\,
q_3^{P}(\psi).
\end{equation}
Its DCM has the identical factor order:
\begin{equation}
{}^{B}C_A
=
C_1(\phi)\,
C_2(\theta)\,
C_3(\psi).
\end{equation}
This is the central convention alignment between the PhysicsLibrary quaternion
and Euler-angle series. The article gives forward and inverse conversion
formulas and explains the $3$-$2$-$1$ coordinate singularity at
\begin{equation}
\theta=\pm\frac{\pi}{2}.
\end{equation}
Its companion is
\PMlinkname{quaternions and Euler angles: examples, exercises, and solutions}
{QuaternionsAndEulerAnglesExamplesExercisesAndSolutions}.
\section{Attitude dynamics, errors, and inertial sensors}
\subsection*{Quaternion kinematics and angular velocity}
\PMlinkname{Quaternion kinematics and angular velocity}
{QuaternionKinematicsAndAngularVelocity}
connects the passive attitude quaternion to measured or modeled angular rate.
Let
\begin{equation}
q={}^Bq_I
\end{equation}
map inertial-frame coordinates into body-frame coordinates. If
$\omega^B$ is the angular velocity of frame $B$ relative to frame $I$,
resolved in the body frame and embedded as a pure quaternion, then the passive
kinematic equation is
\begin{equation}
\dot q
=
-\frac12\omega^B q.
\end{equation}
If the same angular velocity is resolved in the inertial frame, then
\begin{equation}
\dot q
=
-\frac12q\omega^I.
\end{equation}
Both the minus sign and the multiplication side follow from the declared
passive map. They are not arbitrary implementation choices.
The article derives both forms, explains their relationship to the conjugate
body-to-inertial quaternion, and presents matrix forms useful for software and
estimation.
Its companion is
\PMlinkname{quaternion kinematics and angular velocity: examples, exercises, and solutions}
{QuaternionKinematicsAndAngularVelocityExamplesExercisesAndSolutions}.
\subsection*{Relative attitude and error quaternions}
\PMlinkname{Relative attitude and error quaternions}
{RelativeAttitudeAndErrorQuaternions}
develops transformations between an actual attitude and a desired attitude
using explicit frame labels.
Let
\begin{equation}
q={}^Bq_I
\end{equation}
describe the actual body frame and
\begin{equation}
q_d={}^Dq_I
\end{equation}
describe the desired body frame. The passive coordinate transformation from
the actual frame $B$ to the desired frame $D$ is
\begin{equation}
{}^{D}q_B
=
q_d q^*.
\end{equation}
The inverse relative transformation is
\begin{equation}
{}^{B}q_D
=
q q_d^*.
\end{equation}
For a small positive frame-rotation vector
$\delta\boldsymbol\theta$ under the passive axis-angle convention,
\begin{equation}
\delta q
\approx
\begin{bmatrix}
1\\[1mm]
-\dfrac12\delta\boldsymbol\theta
\end{bmatrix}.
\end{equation}
Later estimation articles must state explicitly which relative frame map is
used for the error state before assigning a sign to
$\delta\boldsymbol\theta$.
Its companion is
\PMlinkname{relative attitude and error quaternions: examples, exercises, and solutions}
{RelativeAttitudeAndErrorQuaternionsExamplesExercisesAndSolutions}.
\subsection*{Numerical quaternion propagation and IMU attitude-state integration}
\PMlinkname{Numerical quaternion propagation and IMU attitude-state integration}
{NumericalQuaternionPropagationAndIMUAttitudeStateIntegration}
turns the continuous passive kinematic equation into an executable sensor-driven
state update.
Let
\begin{equation}
q_k
=
{}^{B_k}q_I.
\end{equation}
A body-gyro propagation chain is
\begin{equation}
\boldsymbol\omega_m^B
\longrightarrow
\boldsymbol\omega_c^B
\longrightarrow
\Delta\boldsymbol\theta
\longrightarrow
\delta q_k
\longrightarrow
q_{k+1}
=
\delta q_k q_k.
\end{equation}
The incremental quaternion is the passive map from the old body frame to the
new body frame:
\begin{equation}
\delta q_k
=
{}^{B_{k+1}}q_{B_k}.
\end{equation}
For one constant body-rate sample,
\begin{equation}
\delta q_k
=
\begin{bmatrix}
\cos(\Delta\theta/2)\\[1mm]
-\dfrac{\sin(\Delta\theta/2)}{\Delta\theta}
\Delta\boldsymbol\theta
\end{bmatrix}.
\end{equation}
For a very small sample increment,
\begin{equation}
\delta q_k
\approx
\begin{bmatrix}
1\\[1mm]
-\dfrac12\Delta\boldsymbol\theta
\end{bmatrix}.
\end{equation}
Frame chaining then gives
\begin{equation}
q_{k+1}
=
\delta q_k q_k.
\end{equation}
The article distinguishes this propagation increment from the local
multiplicative error used in estimation. It also covers bias correction,
Euler and Runge--Kutta propagation, exact exponential updates, coning, sign
continuity, and use of the propagated passive attitude map to transform
accelerometer specific force between body and navigation coordinates.
Its companion is
\PMlinkname{numerical quaternion propagation and IMU attitude-state integration: examples, exercises, and solutions}
{NumericalQuaternionPropagationAndIMUAttitudeStateIntegrationExamplesExercisesAndSolutions}.
\section{Recommended reading paths}
The complete article sequence is appropriate for a reader who wants a
systematic treatment. Shorter paths are useful for specific goals.
\subsection*{First engineering introduction}
A compact route is:
\begin{enumerate}
\item \PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}
{NotationQuaternionsForPhysicsAndEngineeringOrientation};
\item \PMlinkname{quaternion}{Quaternion};
\item \PMlinkname{quaternion conjugate}{QuaternionConjugate},
\PMlinkname{quaternion norm}{QuaternionNorm}, and
\PMlinkname{quaternion inverse}{QuaternionInverse};
\item \PMlinkname{axis-angle representation and unit quaternion}
{AxisAngleRepresentationAndUnitQuaternion};
\item \PMlinkname{coordinate transformations with quaternions}
{RotatingVectorsWithQuaternions};
\item \PMlinkname{quaternions and direction-cosine matrices}
{QuaternionsAndDirectionCosineMatrices}.
\end{enumerate}
This route establishes conventions, unit quaternions, passive coordinate
transformation, and the matrix connection without immediately requiring
dynamics or estimation.
\subsection*{Spacecraft attitude, robotics, or rigid-body simulation}
After the transformation articles, continue with
\PMlinkname{quaternions and Euler angles}{QuaternionsAndEulerAngles},
\PMlinkname{quaternion kinematics and angular velocity}
{QuaternionKinematicsAndAngularVelocity},
and
\PMlinkname{relative attitude and error quaternions}
{RelativeAttitudeAndErrorQuaternions}.
This adds human-readable Euler angles, angular-rate kinematics, and relative
attitude error while retaining the same passive frame-map direction.
\subsection*{Inertial navigation and IMU propagation}
The most direct application path emphasizes
\PMlinkname{axis-angle representation and unit quaternion}
{AxisAngleRepresentationAndUnitQuaternion},
\PMlinkname{coordinate transformations with quaternions}
{RotatingVectorsWithQuaternions},
\PMlinkname{quaternions and direction-cosine matrices}
{QuaternionsAndDirectionCosineMatrices},
\PMlinkname{quaternion kinematics and angular velocity}
{QuaternionKinematicsAndAngularVelocity},
\PMlinkname{relative attitude and error quaternions}
{RelativeAttitudeAndErrorQuaternions},
and
\PMlinkname{numerical quaternion propagation and IMU attitude-state integration}
{NumericalQuaternionPropagationAndIMUAttitudeStateIntegration}.
The kinematics article supplies the continuous passive rate equation, the
relative-attitude article supplies the local multiplicative error
representation, and the numerical propagation article applies both ideas to
sampled gyro measurements and a propagated inertial-to-body quaternion state.
\subsection*{Self-study}
Beginning with
\PMlinkname{axis-angle representation and unit quaternion}
{AxisAngleRepresentationAndUnitQuaternion},
read each main entry and then immediately work its companion before moving on.
The companion articles are:
\begin{itemize}
\item \PMlinkname{axis-angle representation and unit quaternion: examples, exercises, and solutions}
{AxisAngleRepresentationAndUnitQuaternionExamplesExercisesAndSolutions};
\item \PMlinkname{coordinate transformations with quaternions: examples, exercises, and solutions}
{RotatingVectorsWithQuaternionsExamplesExercisesAndSolutions};
\item \PMlinkname{composition of rotations and quaternion order: examples, exercises, and solutions}
{CompositionOfRotationsAndQuaternionOrderExamplesExercisesAndSolutions};
\item \PMlinkname{quaternions and direction-cosine matrices: examples, exercises, and solutions}
{QuaternionsAndDirectionCosineMatricesExamplesExercisesAndSolutions};
\item \PMlinkname{quaternions and Euler angles: examples, exercises, and solutions}
{QuaternionsAndEulerAnglesExamplesExercisesAndSolutions};
\item \PMlinkname{quaternion kinematics and angular velocity: examples, exercises, and solutions}
{QuaternionKinematicsAndAngularVelocityExamplesExercisesAndSolutions};
\item \PMlinkname{relative attitude and error quaternions: examples, exercises, and solutions}
{RelativeAttitudeAndErrorQuaternionsExamplesExercisesAndSolutions};
\item \PMlinkname{numerical quaternion propagation and IMU attitude-state integration: examples, exercises, and solutions}
{NumericalQuaternionPropagationAndIMUAttitudeStateIntegrationExamplesExercisesAndSolutions}.
\end{itemize}
\section{Core formula map}
The following equations form a compact map of the collection:
\begin{align*}
q&=q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k,
&&\text{definition},\\
(p_w,\mathbf p)(q_w,\mathbf q)
&=(p_wq_w-\mathbf p\cdot\mathbf q,
p_w\mathbf q+q_w\mathbf p+\mathbf p\times\mathbf q),
&&\text{Hamilton product},\\
q^{-1}&=q^*/\|q\|^2,
&&\text{inverse},\\
{}^{B}q_A
&=\cos(\theta/2)-\widehat{\mathbf u}\sin(\theta/2),
&&\text{positive passive frame rotation},\\
{}^{B}v
&={}^Bq_A\,{}^Av\,({}^{B}q_A)^*,
&&\text{passive coordinate map},\\
{}^{C}q_A
&={}^Cq_B\,{}^Bq_A,
&&\text{frame-chain composition},\\
{}^{B}C_A
&=C({}^{B}q_A),
&&\text{quaternion--DCM map},\\
{}^{B}q_A
&=q_1^P(\phi)q_2^P(\theta)q_3^P(\psi),
&&\text{intrinsic 3-2-1 yaw--pitch--roll},\\
\dot q
&=-\tfrac12\omega^B q,
\quad q={}^Bq_I,
&&\text{body-rate passive kinematics},\\
{}^{D}q_B
&=q_dq^*,
&&\text{actual-to-desired relative map},\\
\delta q
&\approx[1,-\tfrac12\delta\boldsymbol\theta^T]^T,
&&\text{small passive frame error},\\
q_{k+1}
&=\delta q_k q_k,
&&\text{body-gyro passive IMU propagation}.
\end{align*}
This formula map is not a substitute for the individual derivations. Its
purpose is to show how the definitions and applications are connected under
one frame convention.
\section{Conceptual relationships}
Several relationships recur throughout the collection:
\begin{enumerate}
\item \textbf{A quaternion is more general than an attitude quaternion.}
The definition article develops the algebra; the axis-angle article introduces
unit quaternions used for spatial orientation.
\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 unchanged while its coordinates are transformed between
reference frames.
\item \textbf{Intrinsic describes the sequence construction, not the quaternion
algebra.}
When an orientation is specified by yaw, pitch, roll, or another named
three-angle sequence, each successive axis belongs to the current moving frame.
\item \textbf{Quaternion and DCM frame chains are deliberately aligned.}
The definitions
${}^{B}q_A$ and ${}^{B}C_A$ both map $A$ coordinates to $B$ coordinates, so
their composition equations have the same written order.
\item \textbf{A positive frame rotation has the opposite quaternion vector-part
sign from the corresponding positive active vector rotor.}
The two are conjugates of one another.
\item \textbf{The unit quaternion and direction-cosine matrix carry the same
orientation information.}
The map is two-to-one because $q$ and $-q$ produce the same matrix.
\item \textbf{Euler angles are useful local coordinates, not a globally
nonsingular orientation representation.}
Quaternions are preferable for propagation and estimation, but the two
representations must use the same frame convention when they are converted.
\item \textbf{A full quaternion state and a three-component attitude error are
complementary.}
The nominal state can remain global while its estimator error is locally
linearized, provided the relative frame map and error sign are explicitly
declared.
\item \textbf{IMU propagation is frame-transformation composition.}
Gyro samples produce small passive body-frame increments whose multiplication
order, sign, bias correction, and noncommutativity matter in real strapdown
systems.
\end{enumerate}
\section{Where deeper mathematics fits}
The present collection intentionally reaches engineering applications before
pursuing the deeper algebra. A later mathematical branch can build from the
\PMlinkname{quaternion}{Quaternion} and
\PMlinkname{axis-angle representation and unit quaternion}
{AxisAngleRepresentationAndUnitQuaternion}
articles toward 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 topics 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 collection is designed so that each article has a
clear linking role. \PMlinkname{Quaternion}{Quaternion} is the canonical
definition of the mathematical object, while
\PMlinkname{Quaternions for Physics and Engineering: Orientation, Notation, and Conventions}
{NotationQuaternionsForPhysicsAndEngineeringOrientation}
is the convention reference.
The algebra articles establish multiplication, conjugation, norm, and inverse.
The orientation articles develop passive axis-angle frame transformations,
coordinate transformation, composition, direction-cosine matrices, and
intrinsic Euler angles. The later engineering articles carry those ideas into
angular-rate kinematics, relative attitude and estimation errors, and sampled
IMU state propagation.
The collection-level attitude convention is summarized by
\begin{equation}
{}^{B}v
=
{}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*
=
{}^{B}C_A\,{}^{A}v.
\end{equation}
For a positive right-handed frame rotation,
\begin{equation}
{}^{B}q_A=\cos\frac{\theta}{2}-\widehat{\mathbf u}\sin\frac{\theta}{2}.
\end{equation}
For frame chaining,
\begin{equation}
{}^{C}q_A
=
{}^{C}q_B\,{}^{B}q_A.
\end{equation}
These three equations define the passive quaternion convention used by new and
revised PhysicsLibrary attitude articles.
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 its multiplication order from frame labels, test it against simple
coordinate transformations, and use it 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.