Euler Angles: Direction Cosine Matrix to Euler Angles Examples, Exercises, and Solutions

This entry is the self study companion to Euler Angles: direction cosine matrix to Euler angles.

The exercises develop the inverse attitude problem from basic convention recognition through numerical extraction, branch selection, singularity handling, DCM validation, reconstruction checks, and software-oriented diagnostics. The same passive intrinsic convention used throughout the PhysicsLibrary Euler series is retained.

All exercises appear first. Complete worked solutions follow afterward.

1 Convention summary

PhysicsLibrary uses

(1)

For an intrinsic i-j-k Euler sequence,

(2)

For Tait Bryan sequences the standard middle-angle branch is

(3)

while for proper Euler sequences it is

(4)

The outer angles are commonly normalized to

(5)

2 Visual reference

3 Exercises

1. What information is missing from “DCM to Euler angles”?

   Explain why a direction cosine matrix does not determine a unique Euler triple until additional choices are specified. List at least four such choices.

2. Passive map direction.

   A vector has coordinates

   

   and the passive DCM from A coordinates to B coordinates is

   

   Compute ^Bv. What physical object has changed: the vector itself or only its coordinates?

3. Is the input a proper rotation matrix?

   For

   

   compute the dot product of the second and third rows and their squared norms. Is M exactly orthogonal? Should Euler extraction silently assume that it is?

4. Why determinant matters.

   Consider

   

   Show that RR^T = I but det R = −1. Why is R not a valid attitude DCM?

5. Why use atan2?

   Suppose a pair of DCM elements is proportional to

   

   Compute λ using atan2 and compare with the result from arctan(y∕x). Explain the quadrant problem.

6. Derive the 3-2-1 middle-angle formula.

   For intrinsic passive 3-2-1,

   

   Using

   

   derive the principal-branch formula

   
7. Derive 3-2-1 yaw and roll extraction.

   Away from the 3-2-1 singularity, derive formulas for ψ and ϕ from the DCM elements.

8. Numerical 3-2-1 extraction.

   For

   

   recover the principal intrinsic 3-2-1 angles (ψ,𝜃,ϕ).

9. Reconstruction check for 3-2-1.

   Using the angles found in the previous exercise, reconstruct

   

   State what a small reconstruction residual tells you and name three errors that a large residual could reveal.

10. Active/passive transpose trap.

    The same numerical orientation relationship is represented actively by

    

    Suppose a software library returns the active matrix but the PhysicsLibrary 3-2-1 extraction formulas are applied to it as though it were passive. Why should the resulting Euler angles not be trusted even if they look plausible?

11. Derive the 3-1-3 middle-angle formula.

    For intrinsic passive 3-1-3,

    

    use

    

    to derive a robust principal-branch formula for β.

12. Derive 3-1-3 outer-angle extraction.

    Away from sin β = 0, derive formulas for α and γ from the 3-1-3 DCM.

13. Numerical 3-1-3 extraction.

    For

    

    recover the principal intrinsic 3-1-3 triple (α,β,γ).

14. Tait Bryan principal branch.

    Why does the standard Tait Bryan middle-angle range

    

    make

    

    rather than | cos β| ambiguous inside the extraction formulas? Where are the singular endpoints?

15. Proper Euler principal branch.

    Why does

    

    make

    

    on the principal branch? Where are the two singular endpoints?

16. 3-2-1 singularity at positive ninety degrees.

    A 3-2-1 orientation is generated by

    

    At this singularity the software policy sets ϕ = 0 and uses

    

    What yaw does the policy report? Why is it different from the original 50^∘ yet still represents the same physical orientation when combined with 𝜃 = 90^∘ and ϕ = 0?

17. 3-2-1 singularity at negative ninety degrees.

    Repeat the previous exercise for

    

    What yaw is reported under the same ϕ = 0 policy?

18. 3-1-3 singularity at beta equals zero.

    For intrinsic 3-1-3, set β = 0 and show that the DCM depends only on α + γ. If a singular policy sets γ = 0, what should be reported for α?

19. 3-1-3 singularity at beta equals pi.

    Set β = π and show that only α−γ is observable. Under the same γ = 0 policy, derive the reported α.

20. Normalize, do not blindly modulo every angle.

    Explain the difference between

    

    and

    

    Why should the middle angle not simply be treated as another freely wrapped 2π-periodic coordinate?

21. Alternate 3-2-1 branch.

    Show that the two intrinsic 3-2-1 triples

    

    and

    

    produce the same DCM, even though only the first lies on the usual Tait Bryan principal branch.

22. A non-3-2-1 Tait Bryan example.

    For intrinsic 1-3-2,

    

    and the DCM is

    

    Using the sequence-specific formulas

    
    

    and

    

    recover the principal angles.

23. A non-3-1-3 proper Euler example.

    For intrinsic 2-3-2,

    

    and

    

    Use

    
    

    and

    

    to recover the principal angles.

24. Near-singular conditioning.

    For a Tait Bryan sequence, the outer-angle extraction becomes poorly conditioned as | cos β| approaches zero. Compare

    

    with

    

    by computing | cos β| for each. Approximately how many times smaller is the common outer-angle scale factor at 89.9^∘?

25. relative attitude and final implementation audit.

    Frame B₁ has intrinsic 3-2-1 angles

    

    and frame B₂ has

    

    both relative to frame A.

    Explain why simple componentwise subtraction is not the exact finite relative attitude. Form the correct expression

    

    and state the approximate principal 3-2-1 relative angles obtained by extracting from this matrix.

    Finally, list at least six checks that should appear in a production DCM-to-Euler implementation.

4 Solutions

1. What information is missing from “DCM to Euler angles”?

   A DCM stores the orientation itself, whereas Euler angles are coordinates of that orientation in a chosen chart. The inverse map is therefore not unique until the chart is specified.

   At minimum one must specify:

   • the Euler sequence, such as 3-2-1 or 3-1-3;
   • intrinsic versus extrinsic interpretation;
   • active versus passive matrix convention;
   • DCM map direction, such as ^BC _A versus ^AC _B;
   • principal angle ranges;
   • a singularity / gauge policy;
   • an angle-wrapping convention for the outer angles.

   Thus the meaningful inverse operation is

   
2. Passive map direction.

   Apply the passive coordinate map:

   (6)

   Hence

   (7)

   The physical vector has not changed. Only the coordinates used to describe that same vector have changed from frame A to frame B.

3. Is the input a proper rotation matrix?

   Let the second and third rows be

   

   Their dot product is

   (8)

   Their squared norms are

   (9)

   and the same for r₃.

   Thus the rows are mutually perpendicular but not exactly unit length. Therefore

   

   exactly.

   Euler extraction should not silently assume arbitrary input matrices are perfect rotations. A production implementation should test orthogonality and determinant and, when appropriate, repair a noisy matrix before extraction.

4. Why determinant matters.

   Because R is diagonal,

   (10)

   However,

   (11)

   A proper three-dimensional rotation must lie in SO(3) and therefore satisfy

   (12)

   The matrix R is an improper orthogonal transformation containing a reflection. It is not a physically valid attitude DCM.

5. Why use atan2?

   Take

   

   Then

   (13)

   By contrast,

   (14)

   The tangent is the same for angles differing by 180^∘, so arctan(y∕x) cannot determine the quadrant. The two-argument atan2 preserves the signs of both inputs and returns the correct quadrant.

6. Derive the 3-2-1 middle-angle formula.

   From

   

   we obtain

   (15)

   On the principal Tait Bryan branch,

   

   so

   

   Therefore

   (16)

   Also,

   

   Hence

   (17)

7. Derive 3-2-1 yaw and roll extraction.

   Away from the singularity, cos 𝜃≠0.

   For yaw,

   

   The common positive factor c_𝜃 cancels in the two-argument inverse, giving

   (18)

   For roll,

   

   so

   (19)

   Thus the complete nonsingular principal extraction is

   (20)

   (21)

   (22)

8. Numerical 3-2-1 extraction.

   Yaw:

   (23)

   Pitch:

   (24)

   Roll:

   (25)

   Therefore

   (26)

9. Reconstruction check for 3-2-1.

   The reconstructed matrix is

   (27)

   Numerically,

   (28)

   Thus C matches the supplied C to the shown precision.

   A small reconstruction residual strongly suggests that the extraction formula, sequence, signs, branch, and matrix convention are mutually consistent.

   A large residual can reveal, among other things:

   • the wrong Euler sequence;
   • active/passive transpose confusion;
   • wrong map direction;
   • a sign error in an elementary rotation matrix;
   • swapped first and third angles;
   • an invalid input DCM.
10. Active/passive transpose trap.

    The PhysicsLibrary formulas assume the passive map

    

    For the same orientation geometry, the corresponding active matrix is

    (29)

    Transposition changes the matrix elements used by the sequence-specific inverse formulas. Applying passive extraction to the active matrix is therefore solving a different inverse problem.

    The returned numbers may still look smooth and physically plausible because the transpose is itself a valid rotation matrix. Plausibility is not a convention check. Reconstruction with the intended passive product is the appropriate diagnostic.

11. Derive the 3-1-3 middle-angle formula.

    From

    

    we have

    (30)

    On the proper Euler principal branch

    

    so

    

    Therefore

    (31)

    Since

    

    the robust middle-angle formula is

    (32)

12. Derive 3-1-3 outer-angle extraction.

    Away from sin β = 0,

    

    Hence

    (33)

    Also,

    

    so

    (34)

13. Numerical 3-1-3 extraction.

    First,

    (35)

    Then,

    (36)

    Finally,

    (37)

    Therefore

    (38)

14. Tait Bryan principal branch.

    For

    

    the cosine is nonnegative:

    (39)

    Therefore

    (40)

    throughout the principal branch.

    The singular endpoints are

    (41)

    where cos β = 0 and the two outer rotations lose independent observability.

15. Proper Euler principal branch.

    For

    

    we have

    (42)

    Thus

    (43)

    The singular endpoints are

    (44)

    At both endpoints the first and third physical rotation axes become collinear.

16. 3-2-1 singularity at positive ninety degrees.

    For

    

    the singular DCM contains only the coupled outer-angle combination

    

    Indeed the relevant matrix elements are

    

    The chosen gauge gives

    (45)

    Therefore the software may report

    (46)

    This differs from the original coordinate triple but represents the same DCM. At the singularity, ψ and ϕ are not independently observable.

17. 3-2-1 singularity at negative ninety degrees.

    At

    

    the observable combination for this convention is

    

    With

    

    the coupled value is

    

    The DCM gives approximately

    

    Hence

    (47)

    Under the ϕ = 0 gauge the reported triple is therefore

    (48)

18. 3-1-3 singularity at beta equals zero.

    At β = 0,

    (49)

    Thus only

    (50)

    is observable.

    The matrix has

    

    Therefore

    (51)

    If the policy chooses γ = 0, then it reports

    (52)

19. 3-1-3 singularity at beta equals pi.

    At β = π, the full matrix depends on the outer angles only through

    (53)

    The upper-left pair becomes

    

    Hence

    (54)

    With the policy γ = 0,

    (55)

    The numerical formula happens to look the same as at β = 0, but the underlying observable combination is different: sum at one endpoint, difference at the other.

20. Normalize, do not blindly modulo every angle.

    An outer Euler Angle is ordinarily periodic modulo 2π, so a value such as

    

    may be wrapped to

    

    The middle Euler angle is different because the selected Euler chart imposes a restricted principal interval.

    For Tait Bryan sequences,

    

    while for proper Euler sequences,

    

    Moving the middle angle outside those ranges generally requires a coordinated change in the two outer angles to preserve the same orientation.

    Therefore a robust implementation should normalize the Euler triple to the selected principal branch, not apply an independent modulo-2π operation to every coordinate.

21. Alternate 3-2-1 branch.

    The first triple is

    

    The alternate Tait Bryan branch rule is

    (56)

    Applying it gives

    

    Substituting either triple into

    (57)

    produces the same matrix.

    The first triple lies on the usual principal branch because

    

    The second does not. DCM-to-Euler extraction therefore requires a branch choice even away from singularity.

22. A non-3-2-1 Tait Bryan example.

    For 1-3-2,

    (58)

    The middle angle is

    (59)

    The third angle is

    (60)

    Therefore

    (61)

    This example demonstrates why the 3-2-1 matrix-element pattern cannot simply be reused for another Tait Bryan sequence.

23. A non-3-1-3 proper Euler example.

    For 2-3-2,

    (62)

    The middle angle is

    (63)

    The third angle is

    (64)

    Thus

    (65)

    Again, the DCM elements used by the inverse map are sequence specific.

24. Near-singular conditioning.

    For

    

    (66)

    For

    

    (67)

    The ratio is

    (68)

    Thus the common outer-angle scale factor is about 286 times smaller at 89.9^∘ than at 60^∘.

    This explains why small perturbations in DCM elements can produce large changes in the recovered outer angles near a Tait Bryan singularity even before the singularity is reached exactly.

25. Relative attitude and final implementation audit.

    The two Euler triples are coordinates of two separate rotations. Euler coordinates do not obey finite vector subtraction.

    The exact passive relative map is

    (69)

    Using

    

    and

    

    one obtains approximately

    (70)

    Extracting intrinsic passive 3-2-1 angles gives approximately

    (71)

    (72)

    and

    (73)

    Simple componentwise subtraction would have produced

    

    which is close in two components for this particular example but is not the exact finite relative orientation.

    A production DCM-to-Euler implementation should include checks such as:

    • explicit passive/active convention identification;
    • explicit DCM map direction;
    • explicit Euler sequence selection;
    • orthogonality residual check;
    • determinant check;
    • appropriate DCM repair or rejection policy;
    • sequence-specific singularity detection;
    • documented singular gauge policy;
    • principal-branch normalization;
    • outer-angle wrapping convention;
    • reconstruction of C;
    • reconstruction residual test;
    • tests near and exactly at singular configurations;
    • regression tests for multiple Euler sequences.

5 Further study

The parent article Euler angles: direction cosine matrix to Euler angles contains the full extraction tables for all twelve standard intrinsic Euler sequences and develops the numerical workflow in detail.

The next article in the Euler series develops singularities, gimbal lock, and nonuniqueness as geometric properties of Euler coordinate charts rather than failures of the underlying physical attitude.

For numerical state propagation, a DCM or quaternion is generally preferable to an Euler triple. Euler angles are especially useful as human-readable derived outputs when the selected chart remains well conditioned.

6 References and further reading

References

License

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