Library / Tracking and estimation / TRK-04
Tracking and estimation
Process noise for a constant-velocity model
Process-noise covariance of a position–velocity state driven by continuous white acceleration noise.
- Ref
- TRK-04
- Rev
- v1
- Field
- Tracking and estimation
- Result
- Q11 [m²]
For a position–velocity state driven by continuous white acceleration noise of intensity qa (in m²/s³ for position in metres), the process-noise covariance over one sample interval has the entries Q₁₁ = qaTs³/3, Q₁₂ = qaTs²/2 and Q₂₂ = qaTs. The position variance Q₁₁ is the output; the others are shown among the derived results. A piecewise-constant random acceleration gives a different matrix.
Process-noise covariance of a position–velocity state driven by continuous white acceleration noise. It tells the filter how much the target may manoeuvre between updates, and it is set by the noise intensity qa.
Calculator
- Cross term, Q₁₂ = qaTs²/20.25 m²/s
- Velocity variance, Q₂₂ = qaTs0.5 m²/s²
- Position noise, √Q₁₁0.4082 m
- Velocity noise, √Q₂₂0.7071 m/s
- Equivalent acceleration noise over a sample, √(qa/Ts)0.7071 m/s²
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.
- At the defaults the position can wander by about 0.408 m and the velocity by 0.707 m/s between samples. The correlation between the two errors is fixed at √3/2 ≈ 0.87, because the velocity error carries into the position.
- Q₁₁ grows with the cube of Ts: doubling the interval to 2 s gives 1.333 m², eight times as much. A target that is watched less often is much less predictable in position.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ts Sample interval | 0.1 to 5 s | 1.667e-4 | 20.83 | 20.83 | +10% | +33.1 % |
| qa Acceleration noise intensity | 0.01 to 10 m²/s³ | 0.003333 | 3.333 | 3.33 | +10% | +10 % |
Sweep of Sample interval
| Sample interval (s) | Position process variance (m²) |
|---|---|
| 0.1 | 1.667e-4 |
| 0.7125 | 0.06028 |
| 1.325 | 0.3877 |
| 1.938 | 1.212 |
| 2.55 | 2.764 |
| 3.163 | 5.272 |
| 3.775 | 8.966 |
| 4.388 | 14.08 |
| 5 | 20.83 |
Sweep one variable
Variables and how each one contributes
Position process variance
What the equation gives you. Everything else feeds it.
Acceleration noise intensity
The acceleration noise intensity in m²/s³: how strongly the target's acceleration varies, per second of time. Directly proportional: every entry of the matrix scales with qa. Raising it by 10 % (to 0.55) moves the result by +10 %, from 0.1667 to 0.1833 m².
Sample interval
The sample interval in s. Q₁₁ grows as the cube of the interval, so a slightly longer interval adds much more position uncertainty. Raising it by 10 % (to 1.1) moves the result by +33.1 %, from 0.1667 to 0.2218 m².
Worked example with the values above
- Position variance: Q₁₁ = qaTs³/3 = 0.5 × 1 / 3 = 0.1667 m².
- Cross term: Q₁₂ = qaTs²/2 = 0.5 × 1 / 2 = 0.25.
- Velocity variance: Q₂₂ = qaTs = 0.5 × 1 = 0.5.
Assumptions and limits
- Continuous white acceleration noise integrated over one sample interval, for a constant-velocity model.
- qa in m²/s³, with position in metres. A piecewise-constant random acceleration model would have a different Q.
Common mistakes
- Using Q = qaTs times the identity matrix. That ignores the correlation between the position and velocity errors.
- Mixing the white-noise-acceleration model with the piecewise-constant one. They give different matrices for the same nominal intensity.
Statistics and uncertainty
qa is rarely measured; it is chosen so that √(qa/Ts), here 0.707 m/s², is of the order of the acceleration the target can plausibly have between samples. Setting it too small makes the filter slow to follow a manoeuvre, too large makes it noisy, so it is usually tuned against recorded data.
Where it is used
- Setting the process-noise matrix of a Kalman tracker from the expected manoeuvring of the target.
- Seeing how a longer revisit time degrades the predicted position.
Maths you need
- Covariance matrix: variances on the diagonal, covariance off it
- Integration of white noise: the effect of random acceleration on velocity and position