Quaternions for Physics and Engineering: Orientation, Notation, and Conventions

Quaternions are widely used to represent three-dimensional orientation in spacecraft attitude determination and control, inertial navigation, robotics, multibody dynamics, computer graphics, and rigid-body simulation. Their practical advantages are substantial: a unit quaternion represents an arbitrary proper orientation with only four parameters, avoids the coordinate singularities of Euler Angle descriptions, composes frame transformations efficiently, and can be propagated directly from angular velocity.

The difficulty is not usually the quaternion algebra itself. The difficulty is that several independent convention choices occur in the literature and in software. Two sources can display nearly identical equations while assigning opposite meanings to them. A reliable engineering treatment must therefore state its conventions before using quaternion formulas.

This entry establishes the convention used by the PhysicsLibrary quaternion series. The collection uses Hamilton multiplication, scalar first display notation, right handed physical frames, passive frame to frame coordinate transformations as the primary attitude interpretation, and intrinsic moving axis sequences when an orientation is specified by successive named rotations.

The algebraic details of quaternion multiplication, conjugation, norm, inverse, axis angle construction, coordinate transformation, direction cosine matrices, Euler angles, and quaternion kinematics are developed in subsequent entries.

1 The convention choices that must not be conflated

A quaternion implementation normally makes several logically separate choices:

1. the quaternion multiplication law;
2. the ordering used to display or store the four components;
3. the handedness of the physical coordinate frames;
4. the sign convention for positive physical or frame rotation;
5. whether the operation is interpreted as an active vector rotation or a passive coordinate transformation;
6. the direction of a frame to frame coordinate transformation; and
7. for a sequence of rotations, whether the axes are intrinsic (moving axis) or extrinsic (fixed-axis).

These choices are related in applications, but they are not identical. In particular, scalar first versus scalar last storage does not determine the quaternion multiplication law, and active versus passive does not by itself determine whether Hamilton or flipped multiplication is being used.

Likewise, the words “yaw–pitch–roll” or “3–2–1” are not complete convention declarations unless the sequence is identified as intrinsic or extrinsic and the frame-map direction is stated.

For this reason, identifying a convention only by a label such as “JPL,” “Hamilton,” “engineering,” “aerospace,” “right handed,” or “left handed” is unsafe unless the defining equations are also given.

2 PhysicsLibrary convention contract

Unless an article explicitly states otherwise, the PhysicsLibrary quaternion series uses the following convention.

The corresponding passive direction cosine matrix uses the same frame labels:

(1)

Thus the quaternion and matrix representations have the same map direction,

(2)

and the same frame chain order,

(3)

This alignment is deliberate. It preserves Hamilton’s original algebra while making quaternion, direction cosine-matrix, and Euler angle frame notation behave consistently in mechanics, navigation, robotics, and attitude estimation.

3 Quaternion algebra versus component storage

A quaternion is an element of a four-dimensional real algebra,

(4)

The symbols i,j,k satisfy

(5)

Consequently,

(6)

and similarly for cyclic permutations. The product is therefore not commutative.

The same quaternion may be stored in several array layouts. For example,

(7)

may be stored scalar first as

(8)

or scalar last as

(9)

The algebra has not changed. Only the memory or display layout has changed.

This distinction is critical when moving equations into software. Reordering an array is a permutation of components; changing the quaternion multiplication law changes the signs and order in the algebra itself.

4 Hamilton and flipped quaternion multiplication

Let

(10)

where p and q denote the three component vector parts. Hamilton multiplication is

(11)

The sign of the cross product term encodes the familiar Hamilton rule ij = k.

A second multiplication is common in parts of the spacecraft literature. It may be defined by reversing the Hamilton factors,

(12)

where the product on the right is Hamilton multiplication. This gives

(13)

Hence a rapid diagnostic is

(14)

whereas

(15)

Sommer et al. call the latter the flipped or Shuster multiplication and explain how it arose in spacecraft attitude work. Shuster’s historical discussion emphasizes that the alternative product was introduced to recover a convenient written composition order for a particular passive attitude matrix assignment.

PhysicsLibrary uses Hamilton multiplication and does not redefine the product to obtain passive frame transformations. Instead, the quaternion itself is defined in the same coordinate-map direction as the passive matrix. A positive rotation of the frame therefore uses the conjugate of the familiar positive active rotor.

A note on “right handed” and “left handed” quaternions

Some books, papers, software documentation, and engineering discussions refer to the two multiplication/sign conventions as right handed and left handed quaternion conventions. This terminology can be useful as an informal mnemonic because changing the sign of the vector cross product term reverses the orientation of the multiplication table. It is also highly overloaded.

“Right-handed” may instead refer to the handedness of the coordinate basis, the sign convention for positive rotation, the direction associated with a right handed screw, or the orientation chosen for a vector product. Historical quaternion literature itself contains incompatible uses of the term. Joly, for example, explicitly warned in 1905 that his right handed screw convention for positive rotation was opposite to Hamilton’s convention, even though Hamilton also called his own convention right handed.

Accordingly, PhysicsLibrary does not use “right handed quaternion” or “left handed quaternion” as the primary definition of an algebra. We state the unambiguous test instead:

(16)

Coordinate-frame handedness, positive frame rotation, and frame-map direction are stated separately.

5 Passive coordinate transformation and active vector rotation

This distinction is one of the most common sources of quaternion sign errors. PhysicsLibrary takes the passive frame transformation as the canonical attitude operation.

5.1 Passive viewpoint: the PhysicsLibrary default

A passive transformation leaves the physical vector fixed and changes the basis used to describe it.

Let frame B be obtained by rotating the axes of frame A through +90^∘ about +z. The positive frame rotation is represented by

(17)

The passive coordinate map is

(18)

For a fixed physical vector whose A-coordinates are

(19)

the B-coordinates are

(20)

Nothing physical has rotated in this calculation. The minus sign appears because the new B axes themselves have rotated positively relative to A.

5.2 Active viewpoint: the inverse geometric operation

An active rotation changes a physical vector while the reference frame remains fixed. For the same positive geometric angle, define the active rotor

(21)

It is the conjugate of the PhysicsLibrary passive frame quaternion:

(22)

when B is obtained from A by the corresponding positive frame rotation.

The active sandwich is

(23)

For +90^∘ about +z,

(24)

so

(25)

Thus the passive coordinate transformation gives

(26)

for a fixed physical vector, while the corresponding active operation gives

(27)

in a fixed basis.

5.3 Why conjugate quaternion values appear

The active and passive descriptions encode the same relative geometry but answer inverse questions. For a positive frame rotation, the PhysicsLibrary passive quaternion has the negative axis-angle vector part,

(28)

while the positive active rotor has the positive vector part,

(29)

For unit quaternions these are conjugates and inverses:

(30)

A statement such as “this quaternion is passive” remains incomplete unless the source also states the coordinate-map direction. PhysicsLibrary uses explicit frame superscripts and subscripts for that purpose.

6 Frame direction, inverse maps, and composition

Under the PhysicsLibrary convention,

(31)

maps coordinate components from frame A into frame B:

(32)

The inverse map is represented by the conjugate:

(33)

Indeed,

(34)

Now let ^Bq _A map A → B, and let ^Cq _B map B → C. Starting from a vector expressed in A,

(35)

then

(36)

Substitution gives

(37)

Therefore

(38)

The intermediate frame B cancels visually, just as it does in tensor and coordinate-transformation notation.

The passive direction cosine matrices obey the identical chain:

(39)

This same-order correspondence is one of the principal reasons for adopting ^Bq _A as the canonical PhysicsLibrary attitude quaternion.

7 Intrinsic rotation sequences

A unit quaternion by itself represents a relative orientation; the words intrinsic and extrinsic become important when that orientation is constructed from a sequence of named elementary rotations.

PhysicsLibrary uses intrinsic sequences by default. In an intrinsic i–j–k sequence,

1. the first rotation is about axis i of the original frame;
2. the second rotation is about axis j of the first intermediate frame;
3. the third rotation is about axis k of the second intermediate frame.

Let the first, second, and third positive frame rotations be α,β,γ. Define passive elementary quaternions

(40)

and similarly for q_j^P (β) and q _k^P (γ).

Successive passive frame chaining gives

(41)

The corresponding direction cosine matrix is

(42)

Thus the quaternion and matrix products have exactly the same factor order.

For the common intrinsic 3–2–1 yaw–pitch–roll sequence,

(43)

and therefore

(44)

matching

(45)

This is the convention shared by the cleaned-up PhysicsLibrary quaternion and Euler angle series.

8 Why the flipped multiplication appeared in spacecraft work

The convention disagreement is not merely a matter of taste. It arose from a specific composition problem.

Suppose a source stores a quaternion with the numerical sign convention of a positive active rotor,

(46)

but assigns to those same components a passive world to body attitude matrix by transposing the active matrix. For that assignment, Hamilton multiplication can produce an anti-homomorphic matrix relationship of the form

(47)

The quaternion factor order is then reversed relative to the corresponding passive matrix product.

Shuster advocated a flipped multiplication,

(48)

so that the written quaternion composition could be made to follow the desired matrix order for that convention.

Sommer et al. later emphasized that changing Hamilton’s multiplication is not necessary. One may instead retain Hamilton multiplication and make the quaternion to matrix assignment consistent with the chosen frame-map direction.

PhysicsLibrary follows that approach. The canonical passive quaternion is the coordinate map itself:

(49)

Its sandwich matrix is defined directly as

(50)

and Hamilton multiplication is then homomorphic with passive matrix composition:

(51)

when p and q are interpreted as successive passive coordinate maps in the matching frame chain.

The practical lesson is more important than the historical dispute:

Never infer a quaternion convention from component order or from the word “aerospace.” Test the multiplication law, determine the frame-map direction, inspect the axis-angle sign, and verify one known coordinate transformation.

9 A minimum convention declaration for engineering work

Any paper, simulation, flight software interface, or PhysicsLibrary entry that uses quaternions should make enough information available to reconstruct the following checklist:

1. Are the physical coordinate frames right handed or left handed?
2. What is the positive sense of frame rotation?
3. Is ij = +k or −k?
4. Is the scalar component displayed or stored first or last?
5. What does the frame notation ^Bq _A mean?
6. Does the quaternion map A coordinates to B coordinates, or the reverse?
7. Is the equation describing an active vector rotation or a passive coordinate change?
8. What sign is used in the positive axis-angle quaternion?
9. What is the quaternion to matrix relationship?
10. In a frame chain product q₂q₁, which coordinate transformation occurs first?
11. If Euler or other named rotations are used, are the axes intrinsic or extrinsic?

If these questions are answered, most apparent quaternion disagreements become simple conversion problems.

10 Worked example 1: storage order is not multiplication convention

Suppose a software library documents a quaternion as

(52)

and also states

(53)

The first statement says only that the array is scalar last. The second statement identifies the multiplication law as Hamilton multiplication. Therefore this library is scalar last Hamilton, not automatically a flipped or JPL multiplication implementation.

Now consider a positive 45^∘ rotation of frame B relative to frame A about +z. Under the PhysicsLibrary passive convention,

(54)

PhysicsLibrary displays this scalar first as

(55)

whereas the hypothetical scalar last library stores the same Hamilton quaternion as

(56)

Only the component layout has changed.

11 Worked example 2: passive and active signs

Consider again a +90^∘ rotation about +z.

For the PhysicsLibrary passive frame transformation,

(57)

and a fixed physical vector transforms in coordinates as

(58)

The corresponding positive active rotor is the conjugate,

(59)

and it sends

(60)

in a fixed frame.

The opposite signs are not contradictory. One operation rotates the basis; the other rotates the physical vector.

12 Worked example 3: detecting the multiplication convention

An unfamiliar source gives the basis products

(61)

The vector cross product term has the opposite sign from the Hamilton product. Therefore ⋆ is the flipped multiplication relative to the PhysicsLibrary/Hamilton convention.

Before importing any coordinate-transformation, DCM, Euler angle, or kinematic equation from that source, one must either convert the multiplication convention or consistently convert the associated axis-angle sign, quaternion to matrix relationship, frame-map direction, and composition rule as a set.

13 Worked example 4: intrinsic 3–2–1 composition

Suppose frame B is obtained from frame A by the intrinsic sequence

(62)

The passive elementary frame quaternions are

(63)

(64)

and

(65)

The coordinate maps chain as

(66)

so

(67)

The factor order is identical to the passive intrinsic DCM,

(68)

14 Common pitfalls

1. Assuming scalar last means JPL. It does not. Storage order and multiplication are independent choices.
2. Using the positive active axis-angle quaternion as a positive passive frame quaternion. Under the PhysicsLibrary convention a positive frame rotation has a negative quaternion vector part.
3. Calling a formula active or passive without naming the frames. A quaternion value alone does not determine the interpretation.
4. Mixing qvq^∗ and q^∗vq from different sources. For a unit quaternion these represent inverse transformations unless the quaternion itself has also been conjugated or redefined.
5. Reversing frame labels without conjugating the quaternion. Under the PhysicsLibrary convention,
   
6. Copying a quaternion-to-DCM equation without its convention. A transpose may represent the same physical orientation under the opposite mapping direction.
7. Assuming “right handed quaternion” is unambiguous. The phrase has been used for different algebraic, coordinate, and screw sense conventions.
8. Changing the attitude-map convention but not kinematics. Quaternion propagation equations depend on multiplication convention, frame-map direction, the chosen attitude state, and the frame in which angular velocity is resolved.
9. Calling a 3–2–1 sequence complete without saying intrinsic or extrinsic. PhysicsLibrary uses intrinsic moving axis sequences by default.
10. Forgetting the double representation. The unit quaternions q and −q represent the same proper frame transformation.

15 Exercises

1. A library stores quaternions as [q_x,q_y,q_z,q_w] and reports ij = k. Identify the component order and the multiplication convention.
2. A second library stores [q_w,q_x,q_y,q_z] but reports i ⋆ j = −k. Is it Hamilton multiplication? Does scalar first storage change your answer?
3. Frame B is obtained from frame A by a positive 90^∘ rotation about +z. Write ^Bq _A under the PhysicsLibrary passive convention and use it to determine the B-coordinates of a fixed vector whose A-coordinates are x.
4. For the situation in Exercise 3, write the corresponding positive active rotor. What does it do to x in a fixed frame?
5. Show directly that replacing ^Bq _A by −^Bq _A in the passive coordinate law does not change ^Bv.
6. Suppose ^Bq _A maps A coordinates to B coordinates and ^Cq _B maps B coordinates to C coordinates. Derive the quaternion that maps A coordinates directly to C coordinates.
7. Write the passive intrinsic 3–2–1 yaw–pitch–roll quaternion product using ψ for yaw, 𝜃 for pitch, and ϕ for roll.

16 Solutions

1. Scalar-last Hamilton

The array order [q_x,q_y,q_z,q_w] is scalar last. The defining product

(69)

identifies Hamilton multiplication. The two facts are independent.

2. Scalar-first flipped multiplication

The array is scalar first. Since

(70)

the multiplication is the flipped convention relative to Hamilton multiplication. Scalar-first storage does not alter this conclusion.

3. Positive passive 90^∘ frame rotation

For a positive rotation of frame B relative to A about +z,

(71)

Applying

(72)

to ^Av = i gives

(73)

Thus

(74)

4. Corresponding active rotor

The positive active rotor is the conjugate:

(75)

Therefore

(76)

It rotates the physical vector from +x to +y in a fixed frame.

5. The sign of a unit quaternion does not change the passive map

Using (−q)^∗ = −q^∗,

(77)

Thus q and −q generate the same passive coordinate transformation.

6. Frame-chain composition

Apply the A-to-B map first and the B-to-C map second:

(78)

Associativity gives

(79)

Therefore

(80)

7. Passive intrinsic 3–2–1 yaw–pitch–roll

The first rotation is yaw about axis 3, the second is pitch about the new axis 2, and the third is roll about the newest axis 1. The passive elementary quaternions are

(81)

(82)

and

(83)

Frame chaining gives

(84)

This has the same factor order as

(85)

17 Relation to later PhysicsLibrary entries

This entry deliberately stops before developing the quaternion algebra and application formulas in full. The subsequent sequence derives:

1. quaternion definition and elementary algebra;
2. scalar–vector form of the Hamilton product;
3. conjugate, norm, and inverse;
4. passive axis angle and unit quaternions;
5. passive coordinate transformations with quaternions;
6. composition of finite frame transformations;
7. quaternion DCM conversion;
8. passive intrinsic quaternion Euler angle conversion;
9. quaternion kinematics and angular velocity;
10. relative attitude and error quaternions;
11. numerical propagation and normalization;
12. a consolidated engineering exercise set with complete solutions.

The algebra-only articles require little or no convention change. The orientation, Euler angle, DCM, kinematics, relative attitude, and IMU propagation articles are convention sensitive and must use the passive frame map declared here.

18 Sources and convention notes

The present article is an original synthesis written for PhysicsLibrary. Its convention discussion is cross-checked against modern engineering literature, historical quaternion sources, and openly licensed instructional material.

Sommer et al. is especially useful for separating Hamilton multiplication from its flipped spacecraft variant and for identifying the interaction between quaternion multiplication and passive attitude matrix assignment. Shuster provides the historical spacecraft rationale for the alternative multiplication ordering. Moore provides a modern engineering presentation of reference-frame orientation and direction cosine matrices. Henderson provides a classic aerospace treatment connecting Euler angles, quaternions, and transformation matrices. Joly provides a useful historical warning that even the term “right handed” has been used with conflicting rotation conventions.

References

License

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