Library / Tracking and estimation / TRK-03
Tracking and estimation
Alpha–beta tracker update
One update of a fixed-gain tracker for position and velocity.
- Ref
- TRK-03
- Rev
- v1
- Field
- Tracking and estimation
- Result
- p+ [m]
The alpha–beta tracker is a fixed-gain filter for position and velocity. It predicts the position forward with the last velocity, forms the innovation ν = z − p⁻ (the gap between the measurement and the prediction), then adds α times ν to the position and (β/Ts) times ν to the velocity. Larger gains follow manoeuvres faster but pass more noise. The updated position is the output; the velocity is shown among the derived results.
One update of a fixed-gain tracker for position and velocity. The innovation ν, the gap between the measurement and the predicted position, is shared between a position correction and a velocity correction.
Calculator
- Innovation, ν = z − p−1 m
- Updated velocity, v+ = v + (β/Ts)ν4 m/s
- Next predicted position, p+ + Ts v+11 m
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 measurement is 1 m beyond the prediction. The tracker accepts 60 % of it as a position correction, moving the position from 10 m to 10.6 m, and raises the velocity from 2 to 4 m/s.
- Halving the sample interval to 0.05 s doubles the velocity correction β/Ts to 4 per s, so the velocity would jump to 6 m/s for the same innovation. The gains must be set for the sample time in use.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z Measured position | 0 to 100 m | 4 | 64 | 60 | +10% | +6.23 % |
| p- Predicted position | 0 to 100 m | 6.6 | 46.6 | 40 | +10% | +3.77 % |
| α Position gain | 0.1 to 1 | 10.1 | 11 | 0.9 | +10% | +0.566 % |
| v Previous velocity estimate | −10 to 10 m/s | 10.6 | 10.6 | 0 | +0.4 m/s | 0 % |
| Ts Sample interval | 0.05 to 1 s | 10.6 | 10.6 | 0 | +10% | 0 % |
| β Velocity gain | 0.02 to 0.5 | 10.6 | 10.6 | 0 | +10% | 0 % |
Sweep of Position gain
| Position gain | Updated position (m) |
|---|---|
| 0.1 | 10.1 |
| 0.2125 | 10.21 |
| 0.325 | 10.32 |
| 0.4375 | 10.44 |
| 0.55 | 10.55 |
| 0.6625 | 10.66 |
| 0.775 | 10.78 |
| 0.8875 | 10.89 |
| 1 | 11 |
Sweep one variable
Variables and how each one contributes
Updated position
What the equation gives you. Everything else feeds it.
Predicted position
The predicted position p⁻ in m: the previous position moved forward by Ts times the previous velocity. Moves the updated position nearly one for one, since it is the starting point. Raising it by 10 % (to 11) moves the result by +3.77 %, from 10.6 to 11 m.
Previous velocity estimate
The previous velocity estimate in m/s. No effect on the updated position here, because the prediction p⁻ is entered directly. It carries into the updated velocity: v⁺ = v + (β/Ts)ν.
Measured position
The measured position in m. Moves the updated position by the fraction α of its change. Raising it by 10 % (to 12.1) moves the result by +6.23 %, from 10.6 to 11.26 m.
Sample interval
The sample interval in s. No effect on the updated position, which depends on α only. It scales the velocity correction β/Ts, which doubles when Ts is halved.
Position gain
The position gain α, between 0 and 1: the share of the innovation added to the position. Directly raises the share of the measurement that is accepted, so the estimate follows the data more closely and more noisily. Raising it by 10 % (to 0.66) moves the result by +0.566 %, from 10.6 to 10.66 m.
Velocity gain
The velocity gain β: with Ts it sets how much of the innovation changes the velocity estimate. No effect on the updated position. It sets the velocity: v⁺ = 4 m/s at the defaults.
Worked example with the values above
- Innovation: ν = z − p⁻ = 11 − 10 = 1 m.
- Position: p⁺ = p⁻ + αν = 10 + 0.6 × 1 = 10.6 m.
- Velocity: v⁺ = 2 + (0.2/0.1) × 1 = 4 m/s.
Assumptions and limits
- A near-constant velocity along a line, with a fixed sample interval and fixed gains.
- Position measurements only, and gains that give a stable filter for the sample interval.
Common mistakes
- Measuring the innovation against the previous position instead of the predicted position p⁻ = p + Tsv.
- Reusing gains after changing the sample interval. The stability and noise performance depend on the sample time.
Statistics and uncertainty
Measurement noise passes into the velocity estimate multiplied by β/Ts. With a position noise of 0.5 m, one update changes the velocity by 1 m/s of pure noise, which is large next to the velocity itself. This is why β is kept small, and why the noise gain is much larger for the velocity than for the position.
Where it is used
- Tracking a moving target from noisy position fixes with a cheap fixed-gain filter.
- Checking a tracker's first update by hand.
Maths you need
- Prediction and correction: predict forward, then correct with the measurement
- Innovation: measurement minus prediction