Relative Attitude and Error Quaternions

An attitude quaternion describes the orientation of one frame relative to another. In estimation and control, however, the quantity of immediate interest is often not the attitude itself but the difference between two attitudes.

Examples include:

• the difference between a commanded spacecraft attitude and the measured attitude;
• the difference between the true attitude and a filter estimate;
• the relative orientation between two vehicles;
• the correction that must be injected into a nominal quaternion state;
• the small attitude error used in a Kalman filter or feedback controller.

Attitude error is not obtained by subtracting quaternion components. Because three dimensional orientation is a noncommutative rotation group, the exact relative attitude is formed by quaternion multiplication with an inverse.

PhysicsLibrary uses passive frame quaternions. Let

(1)

denote the current or actual attitude, mapping reference frame I coordinates into actual body frame B coordinates.

Let

(2)

denote the desired attitude, mapping the same reference frame coordinates into desired frame D coordinates.

Two multiplicative error definitions are useful:

(3)

and

(4)

They are called the left and right errors because they reconstruct the desired attitude as

(5)

or

(6)

Both represent the same physical relative orientation, but their vector parts are resolved on different sides of the attitude transformation.

1 Convention declaration

PhysicsLibrary uses Hamilton multiplication,

with reversed products changing sign.

Quaternion components are displayed scalar first:

(7)

The canonical passive coordinate map is

(8)

For a positive frame rotation through angle 𝜃 about unit axis u,

(9)

This negative vector sign is essential when converting an error quaternion to a physical small frame rotation vector.

2 Why subtraction is not an attitude error

A quaternion is not an ordinary vector in orientation space.

The component difference

is a four dimensional Euclidean difference between two parameter vectors. It is not itself a unit quaternion and does not directly represent the finite rotation that carries one attitude into the other.

The exact relative transformation must instead satisfy a composition equation.

For a left error,

Right multiply by q^∗:

Therefore

(10)

For a right error,

Left multiply by q^∗:

Therefore

(11)

3 Geometric interpretation of the left error

Because

and

the conjugate of the current attitude is

Therefore the left error has the explicit frame chain

Hence

(12)

The left error is the direct passive coordinate transformation from the current actual frame into the desired frame.

It is therefore especially natural when the correction is applied on the body or output side:

4 Geometric interpretation of the right error

The right error is

It reconstructs the desired attitude by right multiplication:

The right error is the same relative rotation transported to the reference or input side of the current attitude transformation.

The two errors are related exactly by conjugation.

Starting from

multiply on the left by q and on the right by q^∗:

Thus

(13)

Equivalently,

(14)

This relation explains why left and right errors have the same scalar part and the same rotation magnitude but generally different vector components.

5 Relative attitude diagram

The following diagram summarizes the passive frame maps and the two multiplicative errors.

Figure. The actual passive attitude is q = ^Bq _I and the desired passive attitude is q_d = ^Dq _I. The left error is the direct actual to desired frame map δq_L = q_dq^∗ and is applied on the left. The right error δq_R = q^∗q _d is the conjugate transported version applied on the right. For a small positive frame error, the quaternion vector part has the opposite sign from the physical rotation vector.

6 Direction cosine matrix form

Let

and

be the passive DCMs.

Because

and

the left error matrix is

(15)

The right error matrix is

(16)

The exact conjugation relation becomes

(17)

Thus the quaternion and DCM error definitions are completely consistent.

7 Exact axis angle form of a passive error quaternion

Choose a unit error quaternion

Under the PhysicsLibrary passive convention,

(18)

For a principal representative with

the principal error angle is

(19)

If the vector part is nonzero, the corresponding positive frame error axis is

(20)

The minus sign is the passive counterpart of the usual active axis extraction formula.

8 Principal sign choice

The unit quaternions

and

represent the same physical relative attitude.

For local estimation and control, it is usually convenient to choose the representative with nonnegative scalar component:

(21)

This selects a principal error angle in the interval

(22)

If an initially computed error has

replace it by

before forming a local small error vector.

At exactly 180^∘,

and the sign choice is intrinsically ambiguous. The two quaternion signs are equally valid at that point.

9 Small attitude error approximation

For a small positive frame error vector

use

and

Then the passive error quaternion becomes

(23)

In scalar first array notation,

(24)

Therefore the physical small frame error vector is recovered from the quaternion vector part by

(25)

This is one of the most important sign changes from an active quaternion error model.

10 Left and right small error vectors

Let

denote the small frame error vector associated with δq_L, and let

denote the small frame error vector associated with δq_R.

The exact quaternion relation is

To first order,

Therefore

The quaternion sandwich transforms pure vector coordinates according to the current passive attitude:

Hence

(26)

Equivalently,

(27)

Thus the left and right small errors represent the same geometric misalignment in different coordinate resolutions.

11 Check: desired yaw differs by

Suppose the current passive attitude is a positive 10^∘ frame yaw about +z:

Suppose the desired attitude is a positive 25^∘ frame yaw:

Because both rotations use the same axis, they commute. The left error is

Therefore

(28)

This is the passive quaternion for a positive 15^∘ frame error about +z.

The exact physical error is therefore

(29)

12 Check: left and right errors need not be equal

Take the current attitude as a positive 90^∘ frame rotation about +x:

(30)

Take the desired attitude as a positive 90^∘ frame rotation about +y:

(31)

The current conjugate is

The left error is

Thus

(32)

The right error is

Thus

(33)

The two errors have the same scalar part and the same vector magnitude but different vector directions.

Finite attitude error is therefore convention sensitive even when the error angle itself is unchanged.

13 Same physical attitude with opposite quaternion signs

Suppose the stored current and desired quaternions satisfy

They represent exactly the same physical attitude.

The left error is

Thus

(34)

The quaternion −1 represents the identity orientation just as +1 does.

For a local error representation, change the sign to the principal representative

(35)

The physical attitude error is zero, not 360^∘.

14 Desired versus actual attitude in control

For control, let

be the measured or estimated actual passive attitude and let

be the commanded passive attitude.

A body side correction can be formed with

For a small error,

Therefore a controller that requires a physical positive frame correction vector should use

(36)

away from the 180^∘ sign ambiguity.

The sign factor selects the locally shortest quaternion representative.

A control law should not use the raw quaternion vector part without first accounting for the active or passive convention.

15 True versus estimated attitude in estimation

Let the true passive attitude be

and a nominal estimate be

A left multiplicative error is defined by

(37)

Therefore

(38)

A right multiplicative error is defined by

(39)

so

(40)

For a passive small error state,

(41)

The filter covariance, Jacobians, reset equation, and correction injection must all use the same left or right definition.

16 Error injection

Suppose an estimator has computed a correction quaternion δq.

For a left error model, update the nominal attitude by

(42)

For a right error model,

(43)

Mixing a right error covariance model with a left injection, or vice versa, changes the coordinate interpretation of the correction and is generally inconsistent.

17 Relative attitude between two physical bodies

The same algebra applies when neither attitude is merely a desired command.

Let two physical frames B₁ and B₂ have passive attitudes

and

The direct coordinate map from B₁ into B₂ is

(44)

This is exactly the same form as the left error quaternion.

Therefore relative attitude and attitude error are mathematically the same operation. The difference is the engineering interpretation assigned to the two endpoint frames.

18 Why Euler angle subtraction is generally not exact

Suppose the actual and desired attitudes are displayed as yaw, pitch, and roll triples.

The component difference

is generally not the exact finite attitude error vector.

Euler Angles are nonlinear coordinates associated with an ordered sequence of rotations. Finite rotations do not commute, so subtracting the three coordinates does not reproduce the group product

or

For sufficiently small errors and away from an Euler singularity, angle differences may provide a local approximation. The exact error should nevertheless be formed first with quaternions or DCMs.

19 Error near

For a principal passive error quaternion,

Therefore

corresponds to

Near this angle, the two equivalent representatives δq and −δq have nearly equal scalar magnitude and opposite vector parts.

Consequently a rule that always forces a positive scalar component can switch sign under very small perturbations near 180^∘.

The physical orientation remains continuous. The discontinuity belongs to the chosen quaternion representative.

This issue is especially important in controllers, optimizers, and time histories that assume quaternion component continuity.

20 Exact error angle without choosing an axis sign

If only the magnitude of the principal attitude error is needed, choose the quaternion sign so that

Then

(45)

An equivalent numerically robust form is

(46)

The second form behaves well for both small and large angles.

21 Small error norm

The first order passive error quaternion is

Its squared Euclidean norm is

Thus

(47)

The unit norm error introduced by the first order small error approximation is second order in the attitude error magnitude.

22 Implementation checks

A relative attitude implementation should pass the following checks.

1. Identity.

   If

   

   then both errors must equal +1 up to quaternion sign.

2. Opposite stored sign.

   If

   

   the physical error must still be zero.

3. Same axis rotations.

   For current yaw 10^∘ and desired yaw 25^∘, the passive error must represent a positive 15^∘ frame yaw and therefore have a negative k vector component.

4. Left and right reconstruction.

   Verify

   

   and

   
5. Conjugation relation.

   Verify

   
6. DCM consistency.

   Verify

   

   and

   
7. Small error sign.

   For a small positive frame rotation vector δ𝜃, verify

   

23 Common pitfalls

1. Subtracting quaternion components.

   The exact relative attitude is multiplicative, not additive.

2. Using the old active small error sign.

   For the PhysicsLibrary passive frame quaternion,

   
3. Calling an error simply “the quaternion difference.”

   The order must be stated. q_dq^∗ and q^∗q _d are generally different.

4. Mixing left error propagation with right error injection.

   The filter or controller must keep the chosen side consistent.

5. Ignoring the versus equivalence.

   An error quaternion near −1 can represent almost zero physical error.

6. Forcing sign continuity and principal angle simultaneously without a policy.

   Near 180^∘, the principal sign choice can switch discontinuously.

7. Interpreting the raw passive vector part as a positive physical correction vector.

   The small physical frame error is approximately −2δq_v.

8. Subtracting Euler Angle triples for a large finite error.

   Euler coordinates are sequence dependent nonlinear coordinates, not a global rotation vector.

9. Assuming left and right error vectors have the same coordinates.

   They represent the same geometric error but are transported through the current attitude.

24 Relationship to adjacent PhysicsLibrary entries

The preceding article, quaternion kinematics and angular velocity, derives the passive quaternion differential equations used to propagate the nominal attitude.

The present article defines exact relative attitude and multiplicative error quaternions and establishes the passive small error sign used in estimation and control.

A separate companion entry, Relative Attitude and Error Quaternions: Examples, Exercises, and Solutions, provides the Q12E self study problem bank.

The next main article, numerical quaternion propagation and IMU attitude state integration, combines quaternion kinematics, sampled gyro increments, bias correction, and small error concepts in a discrete navigation implementation.

25 Sources and convention notes

Relative attitude and multiplicative quaternion errors are standard tools in spacecraft attitude estimation, inertial navigation, robotics, and nonlinear state estimation. Their exact signs and multiplication sides depend on the chosen quaternion frame map.

Sommer and coauthors provide a modern discussion of quaternion convention management and passive frame transformations. Markley and Crassidis provide a broad spacecraft attitude treatment including quaternion errors and estimation. Shuster discusses historical spacecraft quaternion conventions. Solà provides a detailed engineering discussion of quaternion perturbations and error state Kalman filtering; its formulas should be translated carefully when the quaternion map differs from the PhysicsLibrary passive convention.

References

License

Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution ShareAlike 4.0 International license.