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advanced24 min studyLesson 153 in Physics

Euler angles and coordinate singularities

Use a declared Z-X-Z orientation chart and identify when angle coordinates fail despite smooth physical motion.

Before you begin
On this page

The question for today

What you’ll understand

  • Interpret active Z-X-Z Euler angles with an explicit matrix order.
  • Convert Euler-angle rates into body angular velocity.
  • Recognize a coordinate singularity without inventing a physical loss of freedom.

Three parameters can describe an orientation locally, but three angles cannot provide one smooth, unique coordinate chart for every possible orientation. Euler angles make rotations interpretable as precession, tilt, and spin. They also expose a useful distinction: a coordinate system can become singular while the body itself continues moving smoothly with a finite angular velocity.

Fix the rotation order before naming the angles

Define the body-to-space orientation as a product of active right-handed rotations: a rotation by phi about z, then a tilted intermediate x rotation by theta, then a final body z rotation by psi in the intrinsic interpretation. Matrix factors act on column components from right to left. Equivalently, the product written below directly defines the convention, removing any ambiguity about verbal rotation sequences.

The third body axis is the third column of the resulting matrix. Choose theta between zero and pi; its value is the polar angle from the fixed space z axis. Away from the poles, the axis azimuth measured from space positive x is phi minus ninety degrees for this x-tilt convention. Psi rotates the first two body axes around that third axis. A source using passive basis-component matrices can display a reversed product or transposed matrix for the same orientation.

R(ϕ,θ,ψ)=Rz(ϕ)Rx(θ)Rz(ψ),rs=RrbR(\phi,\theta,\psi)=R_z(\phi)R_x(\theta)R_z(\psi),\qquad\mathbf r_s=R\mathbf r_b
Each elemental matrix is an active right-handed rotation. The body-to-space map is the transpose of a source matrix that instead stores body basis vectors as rows.↔ Scroll wide equations horizontally.

Angle rates are not angular-velocity components

Each Euler-angle rate contributes an instantaneous rotation about a different axis. Those axes are not generally mutually perpendicular, so simply identifying the three rates with the three body angular-velocity components fails. Resolve each contribution onto the final body axes or evaluate the antisymmetric matrix formed from the orientation derivative. Both routes give the same components when the convention is consistent.

The third body component includes both the final spin rate and the projection of precession onto the body third axis. This extra contribution matters for spinning tops: the rate of a painted mark relative to an Euler coordinate can differ from the actual angular velocity along the symmetry axis. Use radians per second for all rates and check a limiting orientation before substituting numbers.

ω1=ϕ˙sin⁡θsin⁡ψ+θ˙cos⁡ψ,ω2=ϕ˙sin⁡θcos⁡ψ−θ˙sin⁡ψ,ω3=ψ˙+ϕ˙cos⁡θ\omega_1=\dot\phi\sin\theta\sin\psi+\dot\theta\cos\psi,\quad\omega_2=\dot\phi\sin\theta\cos\psi-\dot\theta\sin\psi,\quad\omega_3=\dot\psi+\dot\phi\cos\theta
These are body components for the declared active Z-X-Z body-to-space convention. They should not be transplanted into a different axis order.↔ Scroll wide equations horizontally.

A singular chart is not a stalled object

At zero tilt, both z rotations share the same axis. Orientation depends only on their sum, so increasing phi while decreasing psi by the same amount changes no physical orientation. The mapping from angle rates to angular velocity then has a nontrivial null direction. Near this configuration, inversion can require large and nearly canceling rates, making the coordinates numerically fragile.

Rotation matrices or normalized quaternions can carry the physical orientation through this chart boundary, although they introduce constraints or representation redundancy of their own. In the vector lab, first predict how the projections of a unit direction change between zero and ninety degrees, then check. The lab illustrates projected directions only; it does not implement Euler angles, gimbal hardware, or three-dimensional coordinate singularities.

Work it through

Two angle descriptions, one orientation

Set theta to zero. Compare Euler coordinates phi = 20 degrees, psi = 30 degrees with phi = 40 degrees, psi = 10 degrees. Then find omega_3 when phi-dot is 2 and psi-dot is -1 radians per second at zero tilt.

  1. At zero tilt, the middle rotation is the identity. The two z rotations combine by adding their angles, giving fifty degrees for the first description.

  2. The second description also gives fifty degrees. Although both coordinate pairs differ, the resulting matrix and every body-axis direction agree.

  3. With zero tilt and zero tilt rate, only the sum of the two z rates contributes. The third body component is two minus one, or one radian per second.

The result

The repeated description demonstrates coordinate redundancy at the pole. It does not mean that the object loses a rotational degree of freedom in physical space.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Projected direction versus orientation chart

Predict unit-vector components at 0 and 90 degrees, then inspect them with the angle control. Compare a direction described by components with an angle description.

Projected direction versus orientation chartPlayback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Current readings: x component: 1.393; y component: 0.975; Magnitude: 1.7. All parameters are adjustable with the labeled controls below.−7.46−3.7303.737.46−3.5−1.7501.753.5xyvvₓ = 1.39
Vectorx / y projectionsTrace marker
x component
1.393
y component
0.975
Magnitude
1.7
t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe diagram shows one planar direction. Euler-angle singularities require the three-dimensional analysis in the lesson; they are not simulated here.

Pause & reason

Check your understanding

At theta = 0 in the declared Z-X-Z convention, what does increasing phi by delta while decreasing psi by delta do?

Your turn

Try it yourself

For theta = 90 degrees, psi = 0, phi-dot = 3, theta-dot = 2, and psi-dot = 4 radians per second, calculate the three body angular-velocity components.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

At these angles, sine theta and cosine psi equal one, while cosine theta and sine psi equal zero.

Compare with the solution

Substitution gives omega_1 = 2, omega_2 = 3, and omega_3 = 4 radians per second. Their numerical order differs from the Euler-rate order because the instantaneous rotation axes must be resolved onto the final body axes. This regular orientation has no zero-tilt degeneracy.

Keep the idea

Take it with you

  • Euler angles require an axis order, an active or passive convention, and a map direction.
  • Euler-coordinate rates combine through projections to form angular velocity.
  • A coordinate singularity indicates a failing chart, not a failure of rigid-body motion.

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