P7-T3 — Track State and Motion Estimation

Learning objectives

After this lesson, you should be able to:

• Define a track state and write the constant-velocity prediction as a matrix equation.
• Explain when constant-velocity and constant-acceleration models are appropriate.
• Apply the alpha-beta update to a position measurement.
• Compute the prediction error that builds up after a manoeuvre and the effect of the filter gain on noise and lag.

The idea: remember how the object was moving

A pedestrian who was walking to the right a moment ago is probably still walking to the right. Using that expectation is what prediction means: the tracker carries, beside each object’s position, a compact summary of its recent motion and uses it to say where the object should be in the next frame.

What a track stores

A track holds a state: the numbers that, together with a model, are enough to predict the future. For an object on the ground plane the usual state is position and velocity, x = (p_x, p_y, v_x, v_y). Around the state a track keeps its identity, its age and hit count (Lesson P7-T5), its class, its size, and its uncertainty (a covariance matrix, Lesson P7-T4). The state is not measured: only the position is observed. The velocity is inferred from how the position changes, which is why it needs several frames to settle.

Motion models

1. Constant velocity. The object keeps its velocity and any change is treated as small random noise. It suits pedestrians, vehicles on straight roads and objects in short intervals, and is the default of most trackers.
2. Constant acceleration. The state includes acceleration, so the model can follow speeding up and braking. It needs more frames to settle and amplifies noise, so it is used when accelerations are large and persistent.
3. Constant turn rate. The speed and the turn rate are held constant, which follows curved paths such as a car on a bend.
4. Several models at once. When the object may do any of these, trackers run several filters and weigh them by how well each fits the data (interacting multiple models). It costs more and handles manoeuvres better.

All models are wrong in some situations. The question is whether they are wrong by a margin that the tracker’s noise settings and gate can absorb.

The alpha-beta filter

The simplest estimator that uses a motion model is the alpha-beta filter. It predicts with constant velocity, compares the prediction with the new measurement, and corrects the position by a fraction α of the difference and the velocity by a fraction related to β. Its two numbers play the role the Kalman filter’s gain plays in the next lesson, but they are set by hand and do not change.

The mathematics: state-space models and the alpha-beta filter

For one coordinate, with state x_k = (p_k, v_k)ᵀ and time step Δt, the constant-velocity model is

F

the transition matrix: the new position is the old position plus velocity times Δt; the velocity is unchanged.

w_k

process noise: the part of the motion the model does not explain (random accelerations). Its size is the model’s admission of ignorance.

H

the observation matrix: only the position is measured.

n_k

measurement noise with variance r.

Without noise, the prediction after k steps is p + k Δt v. For a 2D object the same model is applied to each coordinate, or written with a 4 × 4 matrix.

A constant-acceleration model adds a third state a: p ← p + Δt v + ½ Δt² a, v ← v + Δt a.

The alpha-beta filter predicts, then corrects with the residual (the measurement minus the prediction):

p_pred

the predicted position.

r

the residual: how far the measurement is from the prediction. A large residual means a new motion or a bad measurement.

α

position gain between 0 and 1: α = 1 copies the measurement, α = 0 ignores it.

β

velocity gain: how strongly a residual changes the velocity. A common choice is β = α² / (2 − α), which ties the two gains together so that one number sets the whole filter.

The gains set a trade-off. A small α averages many measurements, so the noise in the estimate is low; but when the object changes its motion, the estimate lags behind it. A large α follows quickly and passes the noise through. The two goals, low noise and low lag, pull α in opposite directions (Lab P7-L3, stage 1).

Prediction error after a manoeuvre. If the object changes velocity by Δv and the model keeps the old one, the prediction error after k frames is

The error grows linearly. The gate (Lesson P7-T2) must be wide enough to include it, or the tracker loses the object.

Worked examples: one update, one manoeuvre

One alpha-beta update. A track has position p = 10 and velocity v = 2 units per frame; Δt = 1. The next measurement is z = 13.5. Take α = 0.5 and β = α² / (2 − α) = 0.25 / 1.5 = 0.167.

The measurement said the object was 1.5 ahead of where the model expected. The filter moved the position by half of that and raised the velocity a little. Repeated over frames, a consistent residual pushes the velocity toward the real one.

A manoeuvre. An object moves at 3 units per frame until frame 6, where it has position 18, then slows to 1 unit per frame. A prediction made at frame 6 with the old velocity gives 18 + 3k after k frames; the object is at 18 + k. The error is 2k: 6 units after three frames and 12 after six. If the gate is 8 units, the track is lost after four frames with no detection to correct it. With measurements arriving each frame, the filter corrects the velocity as the residuals grow: with α = 0.5 the first residual after the change is about 2, so the velocity falls by about 0.33 on that frame (β = 0.167 times the residual), and the filter needs several frames to get down to 1. During those frames the estimate lags the object. A higher gain recovers faster but passes more noise.

The cost of that choice. For measurements with σ = 3, Lab P7-L3 shows an RMS error between about 2.1 and 2.5, below the measurement noise of 3, for α between 0.5 and 0.7, and about 3.4 at α = 0.3, where the lag after the manoeuvre dominates. The best gain depends on both the noise and how often the target manoeuvres.

Engineering implications and limitations

Choose the motion model for the object, not for the tracker. Pedestrians turn on a dime; a truck cannot. Per-class models, or per-class process noise, give better gates and better prediction.

Initialise the velocity honestly. A new track has a position but no velocity. Give it zero velocity with a large velocity uncertainty, or estimate it from the first two detections and mark it unconfirmed; do not claim a speed you have not measured.

Work in a consistent frame. Track in a world or ground frame, not in image pixels, when the camera or robot moves, or the robot’s own motion will appear as target motion (Lesson P1-T4). Compensate ego-motion before the prediction.

Use real time stamps. The prediction step needs the actual Δt between measurements; a fixed Δt with irregular frames makes the filter see accelerations that are not there.

Treat a growing residual as information. A run of large residuals in one direction means the model is wrong (a manoeuvre). Raising the process noise temporarily, or switching model, helps the filter recover.

Do not trust the velocity too early. The velocity of a track with fewer than a handful of updates is uncertain; consumers such as planners should see the uncertainty or a confirmed flag.

Common misconceptions

Engineering takeaways

1. A track’s state holds position and velocity (and sometimes acceleration or turn rate); only position is observed.
2. Constant velocity is the default model: x_k+1 = F x_k with F = [[1, Δt], [0, 1]].
3. The alpha-beta filter predicts, forms the residual and corrects position by α·r and velocity by (β/Δt)·r; with one update 10, 2, z = 13.5 gives p = 12.75 and v = 2.25 at α = 0.5.
4. After a velocity change Δv the prediction error grows as k·Δv·Δt: 2 per frame in the example, so a gate of 8 is exceeded after four frames.
5. The gain trades noise against lag; choose it from the noise level and the manoeuvres expected.

References and further reading

Textbooks and the classic paper for the alpha-beta relation. The numerical examples are constructed for teaching; the RMS figures quoted from Lab P7-L3 are from its simulator.

• Bar-Shalom, Y., Li, X.-R. and Kirubarajan, T. (2001). Estimation with Applications to Tracking and Navigation. Wiley: motion models, alpha-beta filters and interacting multiple models.
• Blackman, S. and Popoli, R. (1999). Design and Analysis of Modern Tracking Systems. Artech House.
• Benedict, T. R. and Bordner, G. W. (1962). Synthesis of an Optimal Set of Radar Track-While-Scan Smoothing Equations. IRE Transactions on Automatic Control: the alpha-beta relation β = α²/(2 − α).
• Thrun, S., Burgard, W. and Fox, D. (2005). Probabilistic Robotics. MIT Press.

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

Next: the Kalman filter, which sets the gain from the stated uncertainties instead of leaving it as a hand-tuned number.

Related content

Nothing below is required to finish this lesson.

• Before
  P7-T2: Object Association and Identity

  Associate detections to tracks with a distance, gating and an assignment.

• Lab
  P7-L3: Investigate How Noisy Detections Affect Track Quality

  Add a motion model and a filter and measure track error against noise.