Euler Angles: 3-2-1 Yaw Pitch Roll Examples, Exercises, and Solutions

This entry is the self study companion to Euler Angles: 3-2-1 yaw pitch roll.

The problems develop the intrinsic 3-2-1 sequence from its moving-frame geometry through the passive direction cosine matrix, aerospace sign interpretation, inverse extraction, singularity, quaternion representation, angular velocity relation, and numerical implementation.

All exercises are stated first. Complete worked solutions follow afterward.

1 Convention summary

PhysicsLibrary uses the passive coordinate map

(1)

For intrinsic 3-2-1 yaw pitch roll,

(2)

and

(3)

For a conventional aircraft body frame,

The common principal branch is

(4)

2 Visual reference

3 Exercises

1. Map direction and chronological order.

   State what

   

   means physically. Then state the chronological intrinsic 3-2-1 frame rotations associated with (ψ,𝜃,ϕ).

2. Intermediate-frame construction.

   Introduce A₀,A₁,A₂,A₃ for intrinsic 3-2-1 and write the three frame-labeled passive coordinate maps. Show how they compose to form ^BC _A.

3. Aerospace positive-angle interpretation.

   For a Forward-Right-Down body frame, explain the physical sense of positive ψ, positive 𝜃, and positive ϕ.

4. Elementary passive matrices and single-axis checks.

   Write C₁(ϕ), C₂(𝜃), and C₃(ψ). Then state the three single-angle reductions of the complete 3-2-1 DCM.

5. Derive the full passive 3-2-1 DCM.

   Starting from

   

   multiply the matrices and obtain all nine elements.

6. Interpret rows and columns.

   For

   

   state the geometric meaning of row i and column j. What does the first row represent in the aerospace specialization?

7. Recover yaw and pitch from the forward axis.

   A body forward axis expressed in North-East-Down coordinates is

   

   Assuming the principal branch, find yaw and pitch. Can roll be obtained from this vector alone?

8. Principal ranges.

   State the common principal ranges for ψ, 𝜃, and ϕ. Explain why cos 𝜃 ≥ 0 on this branch.

9. Inverse DCM extraction.

   For a nonsingular passive DCM

   

   write the principal extraction formulas for 𝜃, ϕ, and ψ.

10. Alternative pitch extraction.

    Show why

    

    is equivalent to the principal arcsine formula away from singularity.

11. Numerical DCM.

    Compute the passive 3-2-1 DCM for

    
12. Numerical round trip and DCM checks.

    Using the matrix from Exercise 11:

    1. recover (ψ,𝜃,ϕ);
    2. state the orthogonality check;
    3. state the determinant check;
    4. write the reverse coordinate map.
13. Locate and interpret the singularity.

    State the 3-2-1 singularity condition. Explain geometrically what happens to z_A and x₂ at 𝜃 = +90^∘ and 𝜃 = −90^∘.

14. Exact coupling at positive ninety degrees.

    Set

    

    in the full 3-2-1 DCM and show that only one combination of yaw and roll remains observable. Identify that combination.

15. Exact coupling at negative ninety degrees.

    Repeat Exercise 14 for

    

    Which combination of yaw and roll remains observable?

16. A singular-case software convention.

    At 𝜃 = +90^∘, suppose the original Euler angles are

    

    The implementation chooses ϕ = 0 at singularity. What yaw should it return so that the same orientation is represented?

17. Equivalent extrinsic description.

    Write the extrinsic sequence and chronological angle order equivalent to intrinsic

    
18. Active versus passive.

    If C = ^BC _A is the PhysicsLibrary passive 3-2-1 matrix, what active rotation matrix represents the corresponding physical rotation? Explain why transposed formulas often appear in references.

19. Passive quaternion composition.

    Write the passive Hamilton quaternion product corresponding to intrinsic 3-2-1. Then write its four scalar-first components in terms of half-angle sines and cosines.

20. Numerical quaternion consistency.

    For

    

    compute the passive quaternion and state the DCM consistency relation that should hold.

21. Euler rates to body angular velocity.

    At

    

    let

    

    Compute the body angular velocity components (p,q,r).

22. Body angular velocity to Euler rates.

    At the same attitude, suppose

    

    Recover (ϕ,𝜃,ψ). Why does this inverse relation become singular at 𝜃 = ±90^∘?

23. Small-angle approximation.

    Derive the first-order passive DCM for small (ϕ,𝜃,ψ) and express it using the cross-product matrix of

    
24. Exact relative attitude versus Euler subtraction.

    Two body attitudes relative to the same reference frame are

    

    and

    

    First form the naive Euler-coordinate difference. Then describe the correct DCM operation for the exact relative attitude. The exact relative 3-2-1 principal angles are approximately

    

    Explain what this comparison demonstrates.

25. Convention and implementation audit.

    A source says only “use yaw pitch roll.” List at least eight convention or implementation questions that should be answered before directly copying its formulas into a PhysicsLibrary implementation.

4 Solutions

Solution 1: map direction and chronological order

The equation

means that the physical vector is unchanged while its coordinates are re-expressed from frame A into frame B.

The chronological intrinsic rotations are

(5)

Thus the first axis belongs to the initial frame, while the second and third axes belong to successive intermediate frames.

Solution 2: intermediate-frame construction

Let

The three passive frame maps are

(6)

(7)

and

(8)

Matching adjacent frame labels gives

(9)

Solution 3: aerospace positive-angle interpretation

For Forward-Right-Down axes:

• positive yaw rotates the current x axis toward the current y axis;
• positive pitch rotates the current x axis toward the negative current z direction, corresponding to nose up;
• positive roll rotates the current y axis toward the current z axis, corresponding to right wing down.

These are physical frame rotations following the right hand rule.

Solution 4: elementary passive matrices and single-axis checks

The passive elementary matrices are

(10)

(11)

and

(12)

Therefore

(13)

(14)

and

(15)

Solution 5: derive the full passive 3-2-1 DCM

Use

with analogous notation for 𝜃 and ψ.

First,

(16)

Premultiplication by C₁(ϕ) gives

(17)

Solution 6: interpret rows and columns

For

its entries satisfy

(18)

Thus row i is the body basis vector e_i^B expressed in frame A, and column j is the reference basis vector e_j^A expressed in frame B.

The first row is the body forward axis expressed in the reference frame:

(19)

Solution 7: recover yaw and pitch from the forward axis

The horizontal components are equal and positive, so

(20)

The Down component gives

hence

(21)

Roll cannot be determined from the forward axis alone because the final roll is about that same forward axis and therefore does not change its direction.

Solution 8: principal ranges

A common principal branch is

(22)

(23)

and

(24)

Because 𝜃 lies between minus and plus ninety degrees,

(25)

Solution 9: inverse DCM extraction

Away from the singularity,

(26)

(27)

and

(28)

The two-argument arctangent is required to preserve the correct quadrant.

Solution 10: alternative pitch extraction

From the first row of the DCM,

Therefore

(29)

On the principal branch, c_𝜃 ≥ 0, so

(30)

Since

the atan2 expression uses (sin 𝜃, cos 𝜃) and returns the same principal pitch.

Solution 11: numerical DCM

For

(31)

Solution 12: numerical round trip and DCM checks

The extraction equations give

and

Thus

(32)

A valid DCM must satisfy

(33)

and

(34)

The reverse map is

(35)

Solution 13: locate and interpret the singularity

The singularity condition is

(36)

so

(37)

At 𝜃 = +90^∘, the third rotation axis x ₂ is anti-aligned with the first rotation axis z_A:

(38)

At 𝜃 = −90^∘, they are aligned:

(39)

The physical orientation remains valid; only the Euler coordinate chart loses local uniqueness.

Solution 14: exact coupling at positive ninety degrees

Setting

in the full DCM gives

(40)

Only

(41)

appears. Therefore yaw and roll cannot be recovered independently.

Solution 15: exact coupling at negative ninety degrees

At

(42)

Only

(43)

is observable.

Solution 16: a singular-case software convention

At positive ninety degrees,

For the original angles,

(44)

If software imposes

then it must satisfy

Therefore it should return

(45)

This new pair represents the same singular orientation because the observable combination remains ϕ − ψ = −30^∘.

Solution 17: equivalent extrinsic description

Reverse the axis order and angle association:

(46)

Both descriptions produce the same final orientation and the same passive DCM.

Solution 18: active versus passive

The corresponding active rotation matrix is

(47)

Thus an active-vector reference can legitimately display the transpose of the PhysicsLibrary passive coordinate map. Apparent sign or factor-order conflicts should be investigated at the transformation-semantics level before being called errors.

Solution 19: passive quaternion composition

For a positive passive frame rotation,

Therefore

(48)

Define

with analogous notation for 𝜃 and ψ.

Using Hamilton multiplication and scalar-first storage,

(49)

Solution 20: numerical quaternion consistency

For

the passive quaternion is approximately

(50)

It should have unit norm to numerical precision and satisfy

(51)

This is one of the strongest convention checks available because it compares two independent attitude representations.

Solution 21: Euler rates to body angular velocity

For intrinsic 3-2-1,

(52)

At ϕ = 10^∘, 𝜃 = 20^∘, and the stated Euler rates,

(53)

(54)

and

(55)

Thus (p,q,r) are not simply (ϕ,𝜃,ψ).

Solution 22: body angular velocity to Euler rates

Away from the singularity,

(56)

Substitution gives approximately

(57)

The inverse contains tan 𝜃 and 1∕ cos 𝜃. It therefore becomes singular when

(58)

This is the same geometric singularity found in the Euler coordinate chart.

Solution 23: small-angle approximation

For small angles,

Keeping only first-order terms gives

(59)

Define

Then

(60)

This local relationship does not make finite yaw pitch roll triples ordinary vectors.

Solution 24: exact relative attitude versus Euler subtraction

The naive coordinate difference is

(61)

Let

The exact passive relative map from frame B₁ coordinates to frame B₂ coordinates is

(62)

For the stated attitudes,

(63)

Extracting intrinsic 3-2-1 principal angles gives approximately

(64)

This differs from (5^∘, 3^∘, 3^∘) because Euler coordinates are nonlinear sequence-dependent coordinates. Exact finite relative attitude must be formed by attitude composition rather than coordinate subtraction.

Solution 25: convention and implementation audit

Before copying a yaw pitch roll formula, determine at least:

1. whether the sequence is intrinsic or extrinsic;
2. whether the transformation is active or passive;
3. the coordinate-map direction;
4. whether vectors are represented as columns or rows;
5. the positive-angle convention;
6. whether 3-2-1 denotes chronological rotations or written matrix factor order;
7. the reference-frame axis directions, such as North-East-Down or East-North-Up;
8. the body-axis convention, such as Forward-Right-Down;
9. the principal Euler Angle ranges;
10. the singular-case extraction policy;
11. the quaternion multiplication and storage conventions if quaternions are also used;
12. whether body angular velocity or Euler angle rates are being supplied by the software interface.

A robust implementation should also verify zero-angle and single-axis cases, orthogonality, determinant +1, transpose reversal, DCM-to-Euler round trips, quaternion agreement, and expected behavior near the pitch singularity.

5 Compact review

The passive intrinsic 3-2-1 map is

(65)

Away from singularity,

(66)

(67)

and

(68)

The singularity is

(69)

with outer-angle coupling

(70)

and

(71)

The passive quaternion is

(72)

Body angular velocity and Euler coordinate rates are related by a state-dependent matrix and are not generally equal component by component.

6 Sources and exercise provenance

The exercises and worked solutions in this companion are newly written for PhysicsLibrary to reinforce the convention and derivations developed in Euler angles: 3-2-1 yaw pitch roll.

References

License

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