Strapdown Inertial Navigation Examples: Comparing DCM, Euler-Angle, Quaternion, and Rotation-Vector Propagation

This companion to INS07 uses the same gyroscope data in several attitude representations. The purpose is not to declare one representation universally superior. The goal is to show that a direction cosine matrix, Euler Angles, a quaternion, and a rotation vector describe the same physical orientation when their conventions and numerical propagation are handled consistently.

Throughout the article, the gyro rate has already been corrected for navigation-frame rotation unless a problem explicitly says otherwise. Thus the common propagation input is

(1)

For a short interval Δt over which the rate is approximately constant, define

(2)

The DCM update is

(3)

the finite quaternion increment is

(4)

and

(5)

For 3-2-1 roll-pitch-yaw angles,

(6)

Finally, a finite rotation vector is

(7)

1 Exercises

Exercise 1: One-axis yaw represented four ways

A level vehicle begins aligned with NED and rotates at the constant body-relative rate

(8)

for 4 s.

1. Compute the final DCM C_bⁿ.
2. Compute the final 3-2-1 Euler angles.
3. Compute the scalar-first Hamilton quaternion.
4. Compute the finite rotation vector.
5. Verify that the quaternion and rotation vector reproduce the DCM.

Exercise 2: One finite three-axis gyro increment

An initially aligned vehicle has

(9)

for Δt = 0.25 s.

1. Form Δ𝜃 and its magnitude α.
2. Use Rodrigues’ formula to compute the exact DCM increment.
3. Compute the equivalent quaternion increment.
4. Extract the equivalent 3-2-1 Euler angles from the resulting DCM.
5. Show that all representations correspond to the same finite rotation.

Exercise 3: Euler-angle rate propagation at a nonzero attitude

The initial attitude is

(10)

and the body-relative rate is

(11)

For Δt = 0.02 s:

1. Compute dϕ∕dt, d𝜃∕dt, and dψ∕dt.
2. Use one forward Euler step on the Euler angles.
3. Independently propagate the initial DCM using the exact exponential increment.
4. Extract Euler angles from that exact DCM and compare the two results.

Exercise 4: Euler-angle singularity with modest gyro rates

At

(12)

suppose

(13)

Compute the three Euler-angle rates. Explain why very large roll and yaw rates do not imply a physically enormous angular velocity. State how DCM, quaternion, and rotation-vector propagation behave for the same physical rate.

Exercise 5: DCM forward Euler versus the exponential map

Reuse the increment from Exercise 2. Beginning from C₀ = I, compare

(14)

with

(15)

Compute C_FE^T C _FE − I, its Frobenius norm, and det C_FE. Explain which invariants the exact update preserves automatically.

Exercise 6: Quaternion forward Euler versus a finite unit quaternion

Again use Exercise 2, beginning from

(16)

A first-order step gives

(17)

1. Compute ∥q_FE∥.
2. Normalize it.
3. Compare the normalized approximation with the exact finite quaternion.
4. Compute the attitude-angle difference between the normalized first-order quaternion and the exact quaternion.

Exercise 7: Two gyro increments that do not commute

Starting from identity attitude, apply

(18)

Compare

(19)

with the naive single-vector approximation

(20)

Find the attitude difference and determine the equivalent rotation vector of the sequential result.

Exercise 8: A stationary Earth-fixed vehicle

A level vehicle is stationary at latitude 45^∘ and aligned with NED. Ignore transport rate. The ideal gyro therefore measures Earth rate,

(21)

1. Compute the gyro measurement in rad/s and degrees per hour.
2. Compute ω_nb^b after navigation-frame-rate subtraction.
3. State the one-hour attitude change predicted by DCM, Euler-angle, quaternion, and rotation-vector propagation after the correction.
4. Determine the erroneous one-hour rotation magnitude if the raw gyro were integrated without Earth-rate subtraction.

Exercise 9: Finite increment from a nonidentity attitude

The initial attitude is

(22)

A finite gyro increment is

(23)

Propagate the initial attitude once using both DCM multiplication and quaternion multiplication. Convert both final states to Euler angles and verify agreement.

Exercise 10: Four-sample attitude mechanization comparison

An initially aligned vehicle receives four consecutive corrected gyro samples. Each sample lasts 0.05 s:

Propagate the complete sequence using:

1. exact DCM increments;
2. exact incremental quaternions;
3. numerical integration of the 3-2-1 Euler kinematic equations;
4. numerical integration of the Bortz rotation-vector equation.

Compare the final attitude from all four methods.

2 Worked solutions

Solution 1: One-axis yaw represented four ways

The angular displacement is

(24)

For a positive body z rotation from an initially aligned frame,

(25)

The 3-2-1 angles are simply

(26)

The unit rotation axis is u = [0, 0, 1]^T , so

(27)

The rotation vector is

(28)

Substitution of the quaternion into the quaternion-to-DCM formula gives the same matrix above. Rodrigues’ formula with σ does the same. This is a useful unit test for sign, quaternion ordering, and passive DCM convention.

Solution 2: One finite three-axis gyro increment

The finite increment is

(29)

Its magnitude is

(30)

Rodrigues’ formula gives

(31)

The exact incremental quaternion is

(32)

The equivalent Euler angles extracted from δC are

(33)

Notice that these Euler angles are not simply the three components of Δ𝜃 converted to degrees. The rotation-vector components describe one axis-angle rotation, while the Euler angles describe a sequence of three rotations about different intermediate axes.

Solution 3: Euler-angle rate propagation at a nonzero attitude

With

(34)

and

(35)

the Euler-rate map gives

(36)

In degrees per second,

(37)

One forward Euler step over 0.02 s predicts

(38)

Now propagate the DCM exactly:

(39)

Extracting Euler angles gives

(40)

The differences are only about

(41)

because the time step is short. The example shows two distinct ideas: the Euler rate is not the gyro vector, and a numerical Euler-angle integrator still has discretization error.

Solution 4: Euler-angle singularity with modest gyro rates

At 𝜃 = 89^∘,

(42)

Substitution gives

(43)

In degrees per second,

(44)

Yet the physical body rate has magnitude only

(45)

The large Euler rates are caused by the coordinate singularity. DCM, quaternion, and rotation-vector descriptions remain finite for the same physical motion. Their numerical difficulties are different, but they do not have the 3-2-1 gimbal-lock singularity at this attitude.

Solution 5: DCM forward Euler versus the exponential map

For Exercise 2,

(46)

Thus

(47)

Its orthogonality defect is

(48)

The Frobenius norm is

(49)

and

(50)

The exact matrix exponential lies on SO(3), so in exact arithmetic

(51)

This is one reason finite rotation updates are preferable to repeatedly taking first-order DCM steps.

Solution 6: Quaternion forward Euler versus a finite unit quaternion

The first-order quaternion is

(52)

Its norm is

(53)

Thus a direct first-order step leaves the unit sphere. After normalization,

(54)

The exact finite quaternion from Exercise 2 is

(55)

For two unit quaternions describing nearby attitudes, the relative rotation angle may be obtained from

(56)

Here,

(57)

Normalization repairs the quaternion norm, but it does not make a first-order integration step mathematically identical to the exact finite rotation.

Solution 7: Two gyro increments that do not commute

Sequential multiplication gives

(58)

The naive sum gives

(59)

The relative attitude angle is approximately

(60)

The equivalent rotation vector of the sequential result is

(61)

The small z component is the key result. It was not present in either individual increment and is generated by the noncommutative composition. The leading cross term is consistent with the Baker-Campbell-Hausdorff structure and motivates the coning corrections studied later in the series [4, 5].

Solution 8: A stationary Earth-fixed vehicle

At 45^∘ latitude,

(62)

In degrees per hour this is

(63)

Because the vehicle is stationary relative to NED,

(64)

and therefore

(65)

All four correctly driven attitude representations therefore predict zero Earth-relative attitude change over one hour:

(66)

If the raw gyro is integrated as though it were body-relative-to-NED rate, the one-hour rotation magnitude is

(67)

This is a powerful static test for a local-level attitude mechanization.

Solution 9: Finite increment from a nonidentity attitude

Construct the initial DCM from

(68)

The initial scalar-first quaternion consistent with that DCM is approximately

(69)

From

(70)

form δC = exp([Δ𝜃]_×) and the equivalent δq. Then

(71)

Quaternion propagation gives

(72)

Converting either C₁ or q₁ to Euler angles gives

(73)

The maximum elementwise difference between the DCM formed from q₁ and the directly propagated DCM is at floating-point roundoff in this calculation. The order of multiplication is important. Because the increment is resolved in body coordinates, it multiplies the body-to-navigation DCM on the right.

Solution 10: Four-sample attitude mechanization comparison

For each sample, form

(74)

Sequential exact DCM propagation gives

(75)

Sequential finite quaternion propagation gives

(76)

The equivalent Euler angles are

(77)

Numerically integrating the Euler kinematic equations with a fourth-order method over the same four piecewise-constant samples gives the same values to the shown precision.

Integrating the Bortz equation gives the final total rotation vector

(78)

Applying Rodrigues’ formula to this σ_f reproduces the DCM above to numerical roundoff.

This exercise is the main lesson of INS07E1. The four state descriptions are not competing physical models. They are coordinate systems and numerical mechanisms for propagating one underlying orientation. When the rate convention, frame convention, multiplication order, and numerical accuracy are consistent, they agree.

3 Cross-method checks for software

The worked examples suggest a compact set of tests for an attitude library.

1. One-axis sign test. A positive z rate from identity should produce the expected positive yaw DCM and quaternion.
2. DCM invariant. Verify

3. Quaternion invariant. Verify

4. Representation agreement. Convert the DCM, quaternion, and rotation vector to one common DCM and compare matrix elements.
5. Euler warning test. Near 𝜃 = ±90^∘, do not interpret large Euler rates as large physical angular velocity.
6. Stationary-Earth test. A level Earth-fixed vehicle must have zero NED-relative attitude rate after Earth-rate subtraction.
7. Order test. Swapping two finite cross-axis increments should change the final attitude.

4 Summary

The DCM, Euler-angle, quaternion, and rotation-vector descriptions all propagate the same physical attitude, but they expose different numerical properties.

The DCM is geometrically direct and redundant. Exact exponential updates preserve orthogonality.

Euler angles are intuitive and minimal, but their rate equations depend on the current attitude and become singular at gimbal lock.

Quaternions are compact and nonsingular for attitude propagation, but they require a unit-norm constraint and an explicitly stated multiplication convention.

Rotation vectors provide a natural finite-increment description and connect directly to the exponential map, Rodrigues’ formula, and higher-order strapdown integration algorithms.

For implementation, the most reliable strategy is to keep one primary nonsingular propagation state, usually a quaternion or DCM, and derive Euler angles for display. Independent DCM, quaternion, and rotation-vector implementations are also valuable during development because agreement among them provides a strong numerical consistency check.

References