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Position error from accelerometer bias
Position error after double integration of an uncorrected constant accelerometer bias.
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- Result
- δp [m]
An accelerometer bias ba that is not removed looks like a real constant acceleration. Integrating once gives a velocity error bat, and integrating again gives a position error ½bat². Because of the square, the error grows faster and faster, which is why inertial navigation cannot run unaided for long without help from GNSS, a map or another sensor.
Position error after double integration of an uncorrected constant accelerometer bias. It grows with the square of time, which is why inertial navigation drifts without outside aiding.
Calculator
- Velocity error, bat0.6 m/s
- Bias in milli-g1.019 mg
- Time to reach a 100 m error, √(200/ba)141.4 s
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.
- A bias of 0.01 m/s² (1.02 mg) gives 18 m after 60 s, 72 m after twice as long, and 1,800 m after ten minutes.
- Tilt acts like bias: an attitude error of 0.1° rotates gravity into the horizontal and creates an apparent acceleration of about 0.0171 m/s², which is larger than the bias here. In practice tilt error is often the bigger part of the drift.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| t Time of integration | 10 to 600 s | 0.5 | 1,800 | 1,800 | +10% | +21 % |
| ba Uncorrected accelerometer bias | 0.001 to 0.1 m/s² | 1.8 | 180 | 178.2 | +10% | +10 % |
Sweep of Time of integration
| Time of integration (s) | Position error (m) |
|---|---|
| 10 | 0.5 |
| 83.75 | 35.07 |
| 157.5 | 124 |
| 231.3 | 267.4 |
| 305 | 465.1 |
| 378.8 | 717.3 |
| 452.5 | 1,024 |
| 526.3 | 1,385 |
| 600 | 1,800 |
Sweep one variable
Variables and how each one contributes
Position error
What the equation gives you. Everything else feeds it.
Uncorrected accelerometer bias
The uncorrected constant accelerometer bias in m/s². One milli-g (1 mg) is 0.00981 m/s². Directly proportional: double the bias, double the position error. Raising it by 10 % (to 0.011) moves the result by +10 %, from 18 to 19.8 m.
Time of integration
The time over which the bias is integrated, in s. The error grows with the square of time, so a 10 % longer run gives about 21 % more error. Raising it by 10 % (to 66) moves the result by +21 %, from 18 to 21.78 m.
Worked example with the values above
- Time squared: t² = 60² = 3,600 s².
- Half the bias times that: ½ × 0.01 × 3,600.
- δp = 18 m, together with a velocity error of 0.6 m/s.
Assumptions and limits
- A constant, uncorrected accelerometer bias on one axis, with zero initial velocity and position error.
- A simple integration model: no aiding, no attitude error, and no scale-factor error.
Common mistakes
- Expecting the drift to grow in proportion to time. It grows with the square of time, so a short test hides the problem.
- Forgetting that tilt and gyro errors turn gravity into an apparent acceleration, which adds to the drift beyond the accelerometer bias.
Statistics and uncertainty
Because position error is proportional to bat², a 10 % error in the time (or run length) is a 21 % error in the result, and a 10 % error in the bias is a 10 % error. A bias that is estimated and removed to within 10 % leaves only 10 % of the drift, which is how aiding sensors keep the drift small: 1.8 m instead of 18 m here.
Where it is used
- Judging how long an inertial sensor can navigate unaided.
- Specifying the grade of sensor needed for a given drift budget.
Maths you need
- Integration: velocity is the integral of acceleration, position of velocity
- Milli-g: 1 mg = 0.00981 m/s²