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: passive coordinate transformations are the primary attitude maps, and named rotation sequences are interpreted intrinsically, meaning that successive rotations are about axes of the current, already-rotated frame.

Article sequence

ID	Article
Q00	Quaternions for Physics and Engineering: Orientation, Notation, and Conventions
Q01	Quaternion Definition and Basic Algebra
Q01	Quaternion Definition and Basic Algebra: Examples, Exercises, and Solution
Q02	Quaternion Product in Scalar Vector Form
Q02E	Quaternion Product in Scalar Vector Form: Examples, Exercises, and Solutions
Q03	Quaternion Conjugate
Q03E	Quaternion Conjugate: Examples, Exercises, and Solutions
Q04	Quaternion Norm
Q04E	Quaternion Norm: Examples, Exercises, and Solutions
Q05	Quaternion Inverse
Q05E	Quaternion Inverse: Examples, Exercises, and Solutions
Q06	Axis Angle Representation and Unit Quaternion
Q06E	Axis Angle Representation and Unit Quaternion: Examples, Exercises, and Solutions
Q07	Rotating Vectors with Quaternions
Q07E	Rotating Vectors with Quaternions: Examples, Exercises, and Solutions
Q08	Composition of Rotations and Quaternion Order
Q08E	Composition of Rotations and Quaternion Order: Examples, Exercises, and Solutions
Q09	Quaternions and Direction Cosine Matrices
Q09E	Quaternions and Direction Cosine Matrices: Examples, Exercises, and Solutions
Q10	Quaternions and Euler Angles
Q10E	Quaternions and Euler Angles: Examples, Exercises, and Solutions
Q11	Quaternion Kinematics and Angular Velocity
Q11E	Quaternion Kinematics and Angular Velocity: Examples, Exercises, and Solutions
Q12	Relative Attitude and Error Quaternions
Q12E	Relative Attitude and Error Quaternions: Examples, Exercises, and Solutions
Q13	Numerical Quaternion Propagation and IMU Attitude State Integration
Q13E	Numerical Quaternion Propagation and IMU Attitude State Integration: Examples, Exercises, and Solutions
Q14	Quaternion Exercises for Physics and Engineering

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 ${}^{B}C_A$ :

$\displaystyle {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^* = {}^{B}C_A\,{}^{A}v.$

(1)

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

$\displaystyle {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}.$

(2)

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:

$\displaystyle {}^{C}q_A = {}^{C}q_B\,{}^{B}q_A, \qquad {}^{C}C_A = {}^{C}C_B\,{}^{B}C_A.$

(3)

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.

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:

$\displaystyle q_{\mathrm{active}} = ({}^{B}q_A)^*.$

(4)

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 $+\widehat{\mathbf z}$ :

$\displaystyle {}^{B}q_A = \frac{1-\mathbf k}{\sqrt2}, \qquad {}^{B}C_A = \begin{bmatrix} 0&1&0\ -1&0&0\ 0&0&1 \end{bmatrix}.$

(5)

A fixed physical vector with $A$ coordinates $[1,0,0]^T$ therefore has $B$ coordinates $[0,-1,0]^T$ .

How the articles fit together

The material naturally falls into three stages.

1. Conventions and algebra

Begin with Quaternions for Physics and Engineering: Orientation, Notation, and Conventions, then read quaternion definition and basic algebra, 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.

2. Frame transformations and attitude representations

Continue with Axis Angle Representation and Unit Quaternion, rotating vectors 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,

$\displaystyle {}^{B}q_A = \exp \left( -\frac{\theta}{2}\widehat{\mathbf u} \right).$

(6)

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

$\displaystyle {}^{B}C_A=C({}^{B}q_A), \qquad C(q^*)=C(q)^T, \qquad C(-q)=C(q).$

(7)

For intrinsic $3$ - $2$ - $1$ yaw, pitch, roll, with $\psi$ yaw, $\theta$ pitch, and $\phi$ roll,

$\displaystyle {}^{B}q_A = q_1^P(\phi) q_2^P(\theta) q_3^P(\psi),$

(8)

$\displaystyle {}^{B}C_A = C_1(\phi) C_2(\theta) C_3(\psi).$

(9)

Quaternion and DCM sequence products therefore have the same written factor order.

3. Kinematics, 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

$\displaystyle q={}^Bq_I$

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

$\displaystyle \dot q = -\frac12 \omega^Bq.$

(10)

If the same physical angular velocity is resolved in the inertial frame,

$\displaystyle \dot q = -\frac12 q\omega^I.$

(11)

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:

$\displaystyle \delta q_L = q_dq^* = {}^Dq_B, \qquad q_d=\delta q_Lq.$

(12)

A right multiplicative error may instead be defined by

$\displaystyle \delta q_R=q^*q_d, \qquad q_d=q\delta q_R, \qquad \delta q_L=q\delta q_Rq^*.$

(13)

For a small positive passive frame error,

$\displaystyle \delta q \approx \begin{bmatrix} 1\ -\frac12\delta\boldsymbol\theta \end{bmatrix}, \qquad \delta\boldsymbol\theta \approx -2\delta\mathbf q_v.$

(14)

For sampled body rate propagation, first correct the measured rate or delta angle for estimated gyroscope bias. The passive body increment

$\displaystyle \delta q_{B,k} = {}^{B_{k+1}}q_{B_k}$

left multiplies the current inertial to body state:

$\displaystyle q_{k+1} = \delta q_{B,k}q_k, \qquad \delta q_{B,k} = \exp \left( -\frac12 \Delta\boldsymbol\theta_k^B \right).$

(15)

For $\Delta\theta=\lVert\Delta\boldsymbol\theta^B\rVert\neq0$ ,

$\displaystyle \delta\mathbf q_B = \begin{bmatrix} \cos(\Delta\theta/2)\ -\dfrac{\sin(\Delta\theta/2)}{\Delta\theta} \Delta\boldsymbol\theta^B \end{bmatrix}.$

(16)

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.

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:

Axis Angle Representation and Unit Quaternion: Examples, Exercises, and Solutions;
Rotating Vectors with Quaternions: Examples, Exercises, and Solutions;
Composition of Rotations and Quaternion Order: Examples, Exercises, and Solutions;
Quaternions and Direction Cosine Matrices: Examples, Exercises, and Solutions;
Quaternions and Euler Angles: Examples, Exercises, and Solutions;
Quaternion Kinematics and Angular Velocity: Examples, Exercises, and Solutions;
Relative Attitude and Error Quaternions: Examples, Exercises, and Solutions;
Numerical Quaternion Propagation and IMU Attitude State Integration: Examples, Exercises, and Solutions.

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.

Core formula map

The main formulas of the engineering sequence can be summarized compactly as

$\displaystyle q$	$\displaystyle =q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k,$
$\displaystyle q^{-1}$	$\displaystyle =q^*/\lVert q\rVert^2,$
$\displaystyle {}^{B}q_A$	$\displaystyle =\cos(\theta/2)-\widehat{\mathbf u}\sin(\theta/2),$
$\displaystyle {}^{B}v$	$\displaystyle ={}^Bq_A\,{}^Av\,({}^{B}q_A)^*,$
$\displaystyle {}^{C}q_A$	$\displaystyle ={}^Cq_B\,{}^Bq_A,$
$\displaystyle {}^{B}C_A$	$\displaystyle =C({}^{B}q_A),$
$\displaystyle {}^{B}q_A$	$\displaystyle =q_1^P(\phi)q_2^P(\theta)q_3^P(\psi) \quad\hbox{for intrinsic $3$-$2$-$1$},$
$\displaystyle \dot q$	$\displaystyle =-\tfrac12\omega^Bq =-\tfrac12q\omega^I \quad\hbox{for $q={}^Bq_I$},$
$\displaystyle \delta q_L$	$\displaystyle =q_dq^,\qquad \delta q_R=q^q_d,$
$\displaystyle \delta q$	$\displaystyle \approx[1,-\tfrac12\delta\boldsymbol\theta^T]^T,$
$\displaystyle \delta q_{B,k}$	$\displaystyle =\exp(-\tfrac12\Delta\boldsymbol\theta_k^B),$
$\displaystyle q_{k+1}$	$\displaystyle =\delta q_{B,k}q_k.$

This map is a guide, not a substitute for the individual derivations.

Conceptual relationships

Several relationships recur throughout the series.

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.
Storage order is not multiplication convention. Scalar-first and scalar last arrays can represent the same Hamilton quaternion.
The PhysicsLibrary attitude quaternion is a passive coordinate map. The physical vector is held fixed while its components are transformed between reference frames.
Quaternion and DCM frame directions are deliberately aligned. Both ${}^{B}q_A$ and ${}^{B}C_A$ map coordinates into coordinates.
Intrinsic describes a sequence construction. For a named Euler sequence, successive elementary rotations are about axes of the moving frame.
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.
Euler Angles are local coordinates. They remain valuable for interpretation but have coordinate singularities; quaternions and DCMs provide globally nonsingular orientation propagation.
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.
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.

Series wide verification cases

Convention sensitive formulas should be checked against a common battery of simple cases:

identity orientation;
positive $90^\circ$ frame rotations about , , and ;
inverse transformation by quaternion conjugation and DCM transpose;
versus ;
two noncommuting successive rotations;
quaternion to DCM to quaternion round trip;
intrinsic -- quaternion and DCM agreement;
zero angular rate and constant single axis angular rate;
left versus right multiplicative attitude error;
the passive small error sign $\delta\boldsymbol\theta\approx-2\delta\mathbf q_v$ ;
body resolved IMU increments left multiplying the inertial to body attitude state.

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.

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.

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:

$\displaystyle {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*, \qquad {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}.$

(17)

$\displaystyle {}^{C}q_A = {}^{C}q_B\,{}^{B}q_A, \qquad q_{k+1} = \delta q_{B,k}q_k.$

(18)

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.

License

This article is an original synthesis prepared for PhysicsLibrary and intended for release under CC BY-SA 4.0.

"quaternion series overview and article guide" is owned by bloftin.
(view preamble)