Library / Flight dynamics and control / FLT-01
Flight dynamics and control
Euler angle rates from body rates
Rate of change of the heading (yaw) angle from the body angular rates, for the yaw–pitch–roll sequence.
- Ref
- FLT-01
- Rev
- v1
- Field
- Flight dynamics and control
- Result
- ψ̇ [°/s]
φ, θ and ψ are the roll, pitch and yaw Euler angles in the yaw–pitch–roll (3–2–1) sequence, and p, q and rb are the body roll, pitch and yaw rates a gyro measures. The three Euler angle rates are φ̇ = p + (q sin φ + rb cos φ) tan θ, θ̇ = q cos φ − rb sin φ and ψ̇ = (q sin φ + rb cos φ)/cos θ. The heading rate ψ̇ is the output; the other two are shown among the derived results. The relations are singular at pitch ±90°.
Rate of change of the heading (yaw) angle from the body angular rates, for the yaw–pitch–roll sequence. The roll rate is φ̇ = p + (q sin φ + rb cos φ) tan θ. The relations become singular at pitch ±90°.
Calculator
- Roll angle rate, φ̇ = p + (q sin φ + rb cos φ) tan θ0.1633 °/s
- Pitch angle rate, θ̇ = q cos φ − rb sin φ1.232 °/s
- Factor 1 / cos θ1.004
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- In a 30° bank the pitch rate (2 °/s) and the yaw rate (1 °/s) both feed the heading, giving ψ̇ = 1.873 °/s. At wings level (φ = θ = 0) the Euler rates equal the body rates, so a pitch rate q of 0.10 rad/s changes the pitch angle at 5.73 °/s.
- The 1/cos θ factor is 1.004 at 5° pitch but 5.76 at 80° and unbounded at 90°. That singularity is why attitude is usually propagated with quaternions and converted to Euler angles only for display.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| rb Body yaw rate | −10 to 10 °/s | −7.69 | 9.697 | 17.39 | +0.4 °/s | +0.348 °/s |
| q Body pitch rate | −10 to 10 °/s | −4.15 | 5.888 | 10.04 | +0.4 °/s | +0.201 °/s |
| φ Roll angle | −60 to 60 ° | −1.237 | 2.241 | 3.477 | +2.4 ° | +0.0501 °/s |
| θ Pitch angle (|θ| < 90°) | −30 to 30 ° | 1.866 | 2.155 | 0.2887 | +1.2 ° | +0.00385 °/s |
| p Body roll rate | −20 to 20 °/s | 1.873 | 1.873 | 0 | +0.8 °/s | 0 °/s |
Sweep of Roll angle
| Roll angle (°) | Heading rate (°/s) |
|---|---|
| −60 | −1.237 |
| −45 | −0.7098 |
| −30 | −0.1345 |
| −15 | 0.45 |
| 0 | 1.004 |
| 15 | 1.489 |
| 30 | 1.873 |
| 45 | 2.129 |
| 60 | 2.241 |
Sweep one variable
Variables and how each one contributes
Heading rate
What the equation gives you. Everything else feeds it.
Roll angle
The roll angle in degrees. Positive is right wing down. In a bank the pitch and yaw rates both feed the heading rate in proportions set by sin φ and cos φ. Raising it by 10° (to 40°) moves the result by +9.95 %, from 1.873 to 2.059 °/s.
Pitch angle (|θ| < 90°)
The pitch angle in degrees. The formula fails near ±90°. The 1/cos θ factor grows with pitch, so the heading rate rises for a given body rate, slowly at first and steeply near 90°. Raising it by 10° (to 15°) moves the result by +3.13 %, from 1.873 to 1.932 °/s.
Body roll rate
The body roll rate in °/s. No effect on the heading rate. It enters the roll angle rate: φ̇ = p + (q sin φ + rb cos φ) tan θ = 0.1633 °/s at the defaults.
Body pitch rate
The body pitch rate in °/s. Enters through sin φ: at a larger roll angle more of the pitch rate turns the heading. Raising it by 10 % (to 2.2) moves the result by +5.36 %, from 1.873 to 1.974 °/s.
Body yaw rate
The body yaw rate in °/s. Enters through cos φ: with the wings level it is the heading rate itself (divided by cos θ). Raising it by 10 % (to 1.1) moves the result by +4.64 %, from 1.873 to 1.96 °/s.
Worked example with the values above
- Resolve the rates: q sin φ + rb cos φ = 2 × 0.5 + 1 × 0.866 = 1.866 °/s.
- Divide by cos θ: cos 5° = 0.9962.
- ψ̇ = 1.866 / 0.9962 = 1.873 °/s.
Assumptions and limits
- The yaw–pitch–roll (3–2–1) Euler sequence with body x forward, y right and z down, and rates in consistent units.
- Pitch angle away from ±90°. Rates are measured in the body frame by rate gyros.
Common mistakes
- Assuming angle rates equal the gyro rates when the aircraft is banked or pitched. They are equal only at φ = θ = 0, or approximately for small angles.
- Mixing the body axes with a different convention (z up, or x east). Check the axes before using the equations.
Statistics and uncertainty
A constant rate-gyro bias integrates into attitude error: a bias of 0.1 °/s in the yaw channel with the wings level gives a heading drift of 6° per minute, and the bias passes through the same trigonometric factors as the true rate. Errors in φ and θ affect the heading rate through sin φ, cos φ and 1/cos θ, which are insensitive near level flight and sensitive near the singularity.
Where it is used
- Propagating aircraft attitude from rate-gyro measurements.
- Understanding why body rates and Euler-angle rates differ in a bank.
Maths you need
- Euler angles: roll, pitch and yaw in a fixed rotation sequence
- Body-frame rates: what rate gyros measure