quaternion series overview and article guide
QuaternionSeriesOverviewAndArticleGuide
2026-08-24 23:53:02
2026-08-24 23:53:02
Topic
changing to passive rotation convention
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 rotation and attitude
representations, 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.
\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 physical 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 active vector rotation by a unit quaternion,
\begin{equation}
\boxed{\mathbf v'=q\mathbf v q^*;}
\end{equation}
\item frame quaternion notation in which ${}^{A}q_B$ maps coordinates from
frame $B$ to frame $A$.
\end{enumerate}
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{quaternion conventions for physics and engineering}{QuaternionConventionsForPhysicsAndEngineering}.
\section{How the articles fit together}
The material naturally falls into three stages.
\begin{enumerate}
\item \textbf{Conventions and algebra.} Begin with
\PMlinkname{quaternion conventions for physics and engineering}{QuaternionConventionsForPhysicsAndEngineering},
then read \PMlinkname{quaternion}{Quaternion},
\PMlinkname{quaternion product}{QuaternionProduct},
\PMlinkname{quaternion conjugate}{QuaternionConjugate},
\PMlinkname{quaternion norm}{QuaternionNorm}, and
\PMlinkname{quaternion inverse}{QuaternionInverse}.
\item \textbf{Rotations and attitude representations.} Continue with
\PMlinkname{axis angle representation and unit quaternion}{AxisAngleRepresentationAndUnitQuaternion},
\PMlinkname{rotating vectors 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{Quaternion conventions for physics and engineering}{QuaternionConventionsForPhysicsAndEngineering}
should normally be read before applying quaternion rotation 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 rotation, 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.
\subsection*{Quaternion}
\PMlinkname{Quaternion}{Quaternion} is the canonical PhysicsLibrary definition
entry for the mathematical object. It defines
\begin{equation}
\boxed{
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.
When a later entry needs to define the mathematical object itself, this is the
natural link target. The conventions article instead supplies the interpretation
rules needed to compare formulas from different sources.
\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}
\boxed{
(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 rotation
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}
\boxed{
q^{-1}=\frac{q^*}{\|q\|^2}
}
\end{equation}
for every nonzero quaternion. For a unit quaternion,
\begin{equation}
\boxed{q^{-1}=q^*.}
\end{equation}
The definitions are kept separate so later articles can link precisely to the
concept they require.
\section{Quaternions as three-dimensional rotations}
\subsection*{Axis angle representation and unit quaternion}
\PMlinkname{Axis angle representation and unit quaternion}{AxisAngleRepresentationAndUnitQuaternion}
derives the fundamental rotation parameterization
\begin{equation}
\boxed{
q=\cos\frac{\theta}{2}
+\widehat{\mathbf u}\sin\frac{\theta}{2}.
}
\end{equation}
The article explains why the quaternion contains 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*{Rotating vectors with quaternions}
\PMlinkname{Rotating vectors with quaternions}{RotatingVectorsWithQuaternions}
derives
\begin{equation}
\boxed{\mathbf v'=q\mathbf v q^*}
\end{equation}
for unit $q$ and shows that the result is exactly Rodrigues' rotation formula:
\begin{equation}
\boxed{
\mathbf v'
=
\mathbf v\cos\theta
+(\widehat{\mathbf u}\times\mathbf v)\sin\theta
+\widehat{\mathbf u}(\widehat{\mathbf u}\cdot\mathbf v)(1-\cos\theta).
}
\end{equation}
It also gives vector only implementation formulas that avoid explicitly forming
pure vector quaternions.
Its companion is
\PMlinkname{rotating vectors with quaternions: examples, exercises, and solutions}{RotatingVectorsWithQuaternionsExamplesExercisesAndSolutions}.
\subsection*{Composition of rotations and quaternion order}
\PMlinkname{Composition of rotations and quaternion order}{CompositionOfRotationsAndQuaternionOrder}
shows why finite rotations do not commute. If $q_1$ acts first and $q_2$ acts
second under the active convention, then
\begin{equation}
\boxed{q_{\mathrm{net}}=q_2q_1.}
\end{equation}
The article also develops frame chains and intrinsic versus extrinsic rotation
language.
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 rotation matrix associated with a unit quaternion. In compact
form,
\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}
The article develops the component matrix, the interpretation of its columns,
the relation
\begin{equation}
R(q^*)=R(q)^T,
\end{equation}
and matrix to quaternion recovery. It also emphasizes that
\begin{equation}
R(-q)=R(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 3-2-1 yaw-pitch-roll sequence as the primary example. The declared
active convention is
\begin{equation}
R=R_z(\psi)R_y(\theta)R_x(\phi),
\end{equation}
with quaternion composition
\begin{equation}
\boxed{
q=q_z(\psi)q_y(\theta)q_x(\phi).
}
\end{equation}
The article gives forward and inverse conversion formulas and explains the
3-2-1 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 quaternion attitude to measured or modeled angular rate. For
$q={}^Iq_B$, body resolved angular velocity satisfies
\begin{equation}
\boxed{
\dot q=\frac12 q\omega^B,
}
\end{equation}
whereas inertial resolved angular velocity satisfies
\begin{equation}
\boxed{
\dot q=\frac12\omega^I q.
}
\end{equation}
The article derives both forms, explains why the multiplication side changes,
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 the rotation from an actual attitude to a desired or reference
attitude. For actual attitude $q$ and desired attitude $q_d$, two common
multiplicative error definitions are
\begin{equation}
\boxed{\delta q_I=q_dq^*}
\end{equation}
and
\begin{equation}
\boxed{\delta q_B=q^*q_d.}
\end{equation}
They describe the same physical relative rotation with the error expressed in
different coordinates. For a small local error,
\begin{equation}
\boxed{
\delta q
\approx
\begin{bmatrix}
1\\[1mm]
\dfrac12\delta\boldsymbol\theta
\end{bmatrix}.
}
\end{equation}
This is the bridge to multiplicative attitude filters and quaternion based
control.
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 kinematic equation into an executable sensor driven state
update. A typical body gyro propagation chain is
\begin{equation}
\boldsymbol\omega_m^B
\longrightarrow
\boldsymbol\omega_c^B
\longrightarrow
\Delta\boldsymbol\theta
\longrightarrow
\delta q
\longrightarrow
q_{k+1}=q_k\delta q.
\end{equation}
For one constant rate sample,
\begin{equation}
\boxed{
\delta q
=
\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
\approx
\begin{bmatrix}
1\\[1mm]
\dfrac12\Delta\boldsymbol\theta
\end{bmatrix}.
\end{equation}
The article distinguishes this propagation approximation from the local
error state approximation used in the relative attitude article. It also
covers bias correction, Euler and Runge Kutta propagation, exact exponential
updates, coning, sign continuity, and the use of the propagated quaternion to
rotate accelerometer specific force inside a strapdown navigation state.
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{quaternion conventions for physics and engineering}{QuaternionConventionsForPhysicsAndEngineering};
\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{rotating vectors with quaternions}{RotatingVectorsWithQuaternions};
\item \PMlinkname{quaternions and direction cosine matrices}{QuaternionsAndDirectionCosineMatrices}.
\end{enumerate}
This route establishes conventions, unit quaternions, vector rotation, and the
matrix connection without immediately requiring dynamics or estimation.
\subsection*{Spacecraft attitude, robotics, or rigid-body simulation}
After the rotation 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.
\subsection*{Inertial navigation and IMU propagation}
The most direct application path emphasizes
\PMlinkname{axis angle representation and unit quaternion}{AxisAngleRepresentationAndUnitQuaternion},
\PMlinkname{rotating vectors 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 rate equation, the relative
attitude article supplies the small multiplicative error representation, and
the numerical propagation article applies both ideas to sampled gyro
measurements and a propagated 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{rotating vectors 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{product},\\
q^{-1}&=q^*/\|q\|^2,
&&\text{inverse},\\
q&=\cos(\theta/2)+\widehat{\mathbf u}\sin(\theta/2),
&&\text{axis angle},\\
\mathbf v'&=q\mathbf vq^*,
&&\text{vector rotation},\\
q_{\rm net}&=q_2q_1,
&&\text{successive active rotations},\\
\mathbf v'&=R(q)\mathbf v,
&&\text{matrix representation},\\
q&=q_z(\psi)q_y(\theta)q_x(\phi),
&&\text{3-2-1 Euler sequence},\\
\dot q&=\tfrac12q\omega^B,
&&\text{body rate kinematics},\\
\delta q_B&=q^*q_d,
&&\text{body resolved attitude error},\\
\delta q&\approx[1,\tfrac12\delta\boldsymbol\theta^T]^T,
&&\text{small local error},\\
q_{k+1}&=q_k\delta q_k,
&&\text{body gyro 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.
\section{Conceptual relationships}
Several relationships recur throughout the collection:
\begin{enumerate}
\item \textbf{A quaternion is more general than a rotation quaternion.}
The definition article develops the algebra; the axis angle article introduces
the unit quaternions used for spatial rotation.
\item \textbf{Storage order is not multiplication convention.} Scalar-first
and scalar-last arrays can represent the same Hamilton quaternion.
\item \textbf{Active and passive descriptions must be separated.} A physical
vector rotation and a coordinate transformation can describe the same geometry
with inverse operations.
\item \textbf{Quaternion order follows physical composition.} Finite rotations
are noncommutative, so factor order must be derived from the declared mapping.
\item \textbf{The unit quaternion and rotation 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 coordinates, not a globally nonsingular
orientation representation.} Quaternions are preferable for propagation and
estimation.
\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.
\item \textbf{IMU propagation is finite rotation composition.} Gyro samples
produce small quaternion increments whose multiplication order, 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{quaternion conventions for physics and engineering}{QuaternionConventionsForPhysicsAndEngineering}
is the convention reference. The algebra articles establish multiplication,
conjugation, norm, and inverse. The rotation articles develop axis angle,
vector rotation, composition, direction cosine matrices, and Euler angles. The
later engineering articles carry those ideas into angular rate kinematics,
relative attitude and estimation errors, and sampled IMU state propagation.
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, test it against simple rotations, 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.