Euler 213 Sequence

The Euler 213 sequence is an intrinsic three rotation description of frame orientation.

This entry uses the PhysicsLibrary passive intrinsic convention and replaces the older generic ϕ,𝜃,ψ sequence notation with

(1)

where α is the first rotation, β the second, and γ the third.

For this sequence, the chronological intrinsic axis order is

This is a Tait Bryan sequence.

1 Convention and frame chain

Let

The intrinsic frame construction is:

1. rotate frame A₀ through α about axis 2 of A₀ to form A₁;
2. rotate frame A₁ through β about axis 1 of A₁ to form A₂;
3. rotate frame A₂ through γ about axis 3 of A₂ to form A₃ = B.

For any fixed physical vector,

(2)

The intermediate passive coordinate maps are

(3)

(4)

and

(5)

Therefore

(6)

The rightmost factor acts first on a coordinate column.

2 Elementary passive matrices

The elementary matrices used in this article are

(7)

(8)

and

(9)

3 Expanded Euler 213 direction cosine matrix

Define

(10)

(11)

and

(12)

Carrying out the matrix multiplication gives

(13)

This is the canonical PhysicsLibrary passive intrinsic Euler 213 matrix.

4 Single angle checks

The matrix must reduce to the correct elementary transformation when two sequence angles are zero.

If

then

(14)

If

then

(15)

If

then

(16)

These reductions are useful sign and order checks.

5 Inverse extraction

Let

Away from the singularity, the principal angles may be extracted as

(17)

(18)

and

(19)

The two-argument function atan2(y,x) is required so that the correct quadrant is retained.

6 Principal ranges

A standard principal branch is

(20)

On this branch, cos β ≥ 0.

7 Coordinate singularity

The sequence becomes singular when

(21)

that is,

(22)

At this configuration the first and third physical rotation axes become aligned or anti-aligned. The orientation remains valid, but α and γ are no longer independently recoverable. Only one combination of the two outer angles remains observable.

Near this singularity, the nonsingular inverse formulas below should not be used without a numerical threshold and an explicitly chosen singularity policy.

8 Alternate Euler branch

Away from the singularity, one equivalent nonprincipal triple is

(23)

with the outer angles wrapped by integer multiples of 2π as needed.

This identity explains why inverse Euler extraction requires a declared principal branch.

9 Equivalent extrinsic description

The intrinsic Euler 213 sequence with angle list (α,β,γ) is equivalent to the extrinsic sequence

with chronological angle list

Thus

(24)

This equivalence changes the verbal axis construction, not the final orientation.

10 Passive quaternion equivalent

The migrated PhysicsLibrary quaternion series uses the same frame map direction.

For a positive frame rotation about axis n,

(25)

The Euler 213 sequence therefore has quaternion product

(26)

The quaternion and DCM represent the same passive map:

(27)

11 Numerical example

Take

(28)

Substitution into the Euler 213 product gives approximately

(29)

This matrix is orthogonal to numerical precision and has determinant +1.

Applying the nonsingular extraction formulas to this matrix recovers the principal triple

12 Verification properties

A correct implementation should satisfy

(30)

(31)

and

(32)

It should also agree with the corresponding passive quaternion:

(33)

13 Common mistakes

1. Using the legacy ϕ,𝜃,ψ symbols as generic first, second, and third angles while also interpreting them as roll, pitch, and yaw.
2. Treating the sequence digits as written matrix order rather than chronological intrinsic axis order.
3. Mixing the active elementary matrices with the passive PhysicsLibrary coordinate map.
4. Applying inverse formulas from another Euler sequence.
5. Ignoring the sequence singularity when extracting angles.
6. Comparing Euler triples directly rather than comparing their DCMs when alternate branches may be present.
7. Forgetting to reverse both axis order and angle association when translating to an extrinsic description.

14 Summary

For the intrinsic Euler 213 sequence,

(34)

The sequence belongs to the Tait Bryan family.

Its passive DCM, inverse extraction formulas, singularity, alternate branch, and passive quaternion product are all determined by that one declared sequence and convention.

15 References and further reading

Henderson provides a classic engineering tabulation of all twelve Euler sequences and transformation matrix relationships.

Diebel gives a compact unified treatment of Euler angles, DCMs, quaternions, and rotation vectors.

Moore develops orientation through reference frame basis relationships and is useful for interpreting the passive coordinate map.

References

License

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