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Pitch from a stationary accelerometer
Pitch and roll of a motionless body from the specific force its three accelerometers measure (x forward, y right, z down).
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In body axes with x forward, y right and z down, a motionless accelerometer measures the specific force f = −C g, the reaction to gravity. Level, that reads fz = −g. Tilted by pitch θ and roll φ it reads fx = g sin θ, fy = −g sin φ cos θ and fz = −g cos φ cos θ, which can be inverted for the angles. The pitch is the output, and the roll is shown among the derived results. Heading cannot be found this way.
Pitch and roll of a motionless body from the specific force its three accelerometers measure (x forward, y right, z down). At rest an accelerometer senses only the reaction to gravity, f = -g in the z-axis when level.
Calculator
- Roll angle, atan2(−fy, −fz)10 °
- Magnitude of the specific force (should be about 9.81)9.81 m/s²
- Departure from g−0.002021 %
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.
- The default readings correspond to 5° nose up and 10° right wing down. Their magnitude is 9.81 m/s², consistent with gravity, so the body is at rest and the calculation is meaningful.
- Rotating about the vertical axis leaves all three readings unchanged, so heading cannot be found from accelerometers. A magnetometer, gyro-compass or GNSS is needed for that.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fx Forward specific force | −3 to 3 m/s² | −17.07 | 17.07 | 34.13 | +0.12 m/s² | +0.697 ° |
| fz Downward specific force | −12 to −8 m/s² | 4.035 | 5.969 | 1.933 | +0.08 m/s² | +0.0404 ° |
| fy Rightward specific force | −3 to 3 m/s² | 4.848 | 5.077 | 0.2289 | +0.12 m/s² | +0.0103 ° |
Sweep of Forward specific force
| Forward specific force (m/s²) | Pitch angle (°) |
|---|---|
| −3 | −17.07 |
| −2.25 | −12.97 |
| −1.5 | −8.726 |
| −0.75 | −4.389 |
| 0 | 0 |
| 0.75 | 4.389 |
| 1.5 | 8.726 |
| 2.25 | 12.97 |
| 3 | 17.07 |
Sweep one variable
Variables and how each one contributes
Pitch angle
What the equation gives you. Everything else feeds it.
Forward specific force
The accelerometer reading along the forward axis in m/s². A nose-up pitch makes it positive. Larger forward reading means a larger nose-up pitch. Raising it by 0.5 m/s² (to 1.355 m/s²) moves the result by +57.9 %, from 5 to 7.894 °.
Rightward specific force
The accelerometer reading along the right-hand axis in m/s². A right-wing-down roll makes it negative. Changes the roll first; it affects the pitch only through the denominator, slightly. Raising it by 0.5 m/s² (to −1.197 m/s²) moves the result by +0.762 %, from 5 to 5.038 °.
Downward specific force
The reading along the downward axis in m/s². Level it is about −9.81. A less negative reading means a larger tilt overall, so the pitch rises a little. Raising it by 0.5 m/s² (to −9.124 m/s²) moves the result by +5.27 %, from 5 to 5.264 °.
Worked example with the values above
- Horizontal-plane part: √(fy² + fz²) = √(2.88 + 92.62) = 9.772 m/s².
- Pitch: θ = atan2(0.855, 9.772) = 5°.
- Roll: φ = atan2(1.697, 9.624) = 10°.
Assumptions and limits
- The body is at rest or moving at a constant velocity, so the accelerometers sense only gravity.
- Axes are body x forward, y right, z down, with the readings calibrated for bias and scale factor.
Common mistakes
- Using it while the vehicle accelerates or vibrates. Real acceleration appears as false tilt.
- Getting the sign wrong. In these axes a level accelerometer reads −g in z, not +g.
Statistics and uncertainty
A bias of 0.05 m/s² on an accelerometer tilts the apparent vertical by about 0.29°, so cheap sensors limit the tilt accuracy to a fraction of a degree. Vibration and any real acceleration add to the readings and are not averaged out unless the data are filtered over a long enough time.
Where it is used
- Initial alignment of an inertial unit and tilt sensing in vehicles and platforms.
- Checking whether an accelerometer's readings are consistent with gravity.
Maths you need
- Specific force: the acceleration an accelerometer measures, which is gravity-free
- Body axes: x forward, y right, z down