Euler Angles

1.2 Euler Angles

Euler angles describe the orientation of a body–fixed frame B = {b₁, b₂, b₃} with respect to a reference (often inertial) frame I = {i₁, i₂, i₃} using three rotations applied in a specific order.

A label such as 3–2–1 means: rotate about axis 3, then axis 2, then axis 1 of the current (moving) frame. In aerospace, two sequences appear again and again:

• 3–2–1: yaw–pitch–roll for aircraft and spacecraft attitude.
• 3–1–3: node–inclination–argument of periapsis for orbital geometry.

1.2.1 Rotation Sequences

Intrinsic vs extrinsic rotations

• Intrinsic sequence (our convention): each rotation is about a body axis after it has already moved. After the first rotation, the axes rotate with the body; the second angle is measured about this new axis, and so on.
• Extrinsic sequence: each rotation is about the original reference axes (fixed in space). A given intrinsic 3–2–1 sequence is equivalent to an extrinsic 1–2–3 sequence applied in reverse order.

(a) 3–2–1 sequence (yaw–pitch–roll)

• Yaw ψ: rotate about axis b₃.
• Pitch θ: rotate about the new axis b₂.
• Roll φ: rotate about the new axis b₁.

This is the standard heading–pitch–bank description of an aircraft or spacecraft relative to a local navigation or inertial frame.

(b) 3–1–3 sequence (orbital angles)

• Rotate about axis 3 by Ω – aligns the reference direction with the ascending node.
• Rotate about the new axis 1 by i – tilts the orbit plane by the inclination.
• Rotate about the new axis 3 by ω – aligns the periapsis direction, giving the argument of periapsis.

The three angles (Ω, i, ω) act like an “attitude” for the orbital plane and periapsis line with respect to the inertial frame.

1.2.2 Euler Angle Geometry

A. 3–2–1 (Yaw–Pitch–Roll) Geometry

• Yaw ψ – turns the body left/right in the horizontal plane (heading).
• Pitch θ – nose-up / nose-down rotation relative to the horizontal.
• Roll φ – rotation about the longitudinal axis (bank).

DCM from 3–2–1 Euler angles

Let C^B_I map components from frame I to frame B. For a 3–2–1 intrinsic sequence,

C^B_I = R₁(φ) R₂(θ) R₃(ψ),

where R_k(·) is the basic rotation about axis k. Expanding gives

Cᵇᵢ =
⎡ cθ cψ              cθ sψ              −sθ          ⎤
⎢ sφ sθ cψ − cφ sψ   sφ sθ sψ + cφ cψ   sφ cθ        ⎥
⎣ cφ sθ cψ + sφ sψ   cφ sθ sψ − sφ cψ   cφ cθ        ⎦
      

with c· = cos(·) and s· = sin(·).

Recovering 3–2–1 angles from a DCM

If we write C^B_I = [c_ij], then for the 3–2–1 set

θ = −arcsin(c₁₃)
ψ = atan2(c₁₂, c₁₁)
φ = atan2(c₂₃, c₃₃)
      

These relations are valid as long as θ ≠ ± 90° (i.e. away from the singularity).

B. 3–1–3 Geometry (Orbital Angles)

For the 3–1–3 sequence with angles (Ω, i, ω):

• Ω: rotation about inertial axis 3 – swings the x-axis onto the line of nodes.
• i: rotation about the new axis 1 – tilts the orbital plane by the inclination.
• ω: rotation about the new axis 3 – sets the direction of periapsis within the orbital plane.

The corresponding DCM is

C^B_I = R₃(ω) R₁(i) R₃(Ω).

From this DCM, the inverse relations are

Ω = atan2(C₃₁, −C₃₂)
i = arccos(C₃₃)
ω = atan2(C₁₃, C₂₃)
      

again assuming i ≠ 0°, 180° (so that we are not at the singular attitude).

1.2.3 Singularities & Gimbal Lock

Loss of a degree of freedom

• Every Euler set has attitudes where the three angles are not unique.
• At these attitudes, two effective rotation axes line up, so the three-axis chain behaves like only two independent axes.

This loss of one degree of freedom is the Euler singularity or gimbal lock.

3–2–1 singularity

For the 3–2–1 sequence, θ = ± 90° causes the yaw and roll axes to line up. Only the combination (ψ + φ) is meaningful; ψ and φ individually become ambiguous.

3–1–3 singularity

For the 3–1–3 sequence, i = 0° or i = 180° makes the first and third rotation axes coincident, so Ω and ω can change while the physical attitude remains the same.

In general:

• For symmetric sequences (like 3–1–3), the singularity occurs at θ₂ = 0° or 180°.
• For asymmetric sequences (like 3–2–1), it occurs at θ₂ = ± 90°.

In all cases the second angle is the one that drives the singularity.

1.2.4 Combining Successive Euler Rotations

Suppose a rigid body first undergoes a rotation with Euler angles θ = {θ₁, θ₂, θ₃}, then a second rotation with the same type of sequence φ = {φ₁, φ₂, φ₃}.

The overall attitude relative to the original frame can be written as

C(final) = C(φ) C(θ) = C(ψ),
      

where ψ = {ψ₁, ψ₂, ψ₃} is an equivalent single set of Euler angles.

In principle one can solve for ψ from the matrix product, but:

• For general (especially asymmetric) sequences such as 3–2–1, the algebra becomes long and yields no neat closed form.
• For symmetric sets like 3–1–3, spherical-geometry relations give somewhat simpler expressions, but they are still not convenient for computation.

Takeaway: Euler angles are excellent for understanding rotations, but clumsy for chaining many rotations. In practice, attitude calculations are done with quaternions or DCMs, and Euler angles are used only for inputs and outputs.

1.2.5 Euler Angle Kinematic Differential Equations

In dynamics we often know the body angular velocity vector

ω = [ ω₁  ω₂  ω₃ ]ᵀ  (in the body frame)
      

and we want the time evolution of the Euler angles.

For the 3–2–1 set, the relation between body rates (p, q, r) and the Euler-angle rates (˙φ, ˙θ, ˙ψ) is

⎡ p ⎤   ⎡ 1   0            −sinθ          ⎤ ⎡ φ̇ ⎤
⎢ q ⎥ = ⎢ 0   cosφ         sinφ cosθ      ⎥ ⎢ θ̇ ⎥
⎣ r ⎦   ⎣ 0  −sinφ         cosφ cosθ      ⎦ ⎣ ψ̇ ⎦
      

This comes from writing ω as a combination of the Euler-rate vectors and also directly in body components (p, q, r), then equating the two expressions.

The coefficient matrix becomes singular at the same attitudes as before:

• 3–2–1: θ = ± 90°.
• 3–1–3: i = 0° or 180°.

At these points it is impossible to invert the equations to obtain Euler angle rates from (p, q, r).

1.2.6 Practical Remarks on Euler Angles

Advantages

• Only three parameters – compact and easy to visualise.
• Direct link to pilot-style concepts (yaw, pitch, roll) and to orbital elements (RAAN, inclination, argument of periapsis).

Limitations

• Every Euler set has at least one singular orientation.
• Any attitude is at most 90° away from some singularity.
• Kinematic differential equations are nonlinear and contain many trigonometric functions.
• Symmetric sets like 3–1–3 place the small-angle regime close to a singular attitude, which is awkward for describing small deviations.

Because of these issues, modern attitude estimation and control usually:

• use quaternions or DCMs internally for propagation and control,
• convert to Euler angles only for display or user-level commands.