Alpha Beta Filter

Alpha-Beta Filter (gnc-autonomy/estimation-filtering/alpha-beta-filter)

Use when the task is an alpha-beta tracking filter for a constant-velocity target: predicted position and velocity, the alpha-beta gain update from a noisy position measurement, and the steady-state gain selection from the smoothing factor and the maneuverability index.

Domain quick reference

• Target model: constant velocity between samples. State is (position x, velocity v); each sample interval dt in seconds.
• Predict step: x_pred = x + dt * v and v_pred = v; the velocity persists and the position advances by dt * v.
• Residual: r = z - x_pred, the difference between the noisy position measurement z and the predicted position.
• Update step: x_new = x_pred + alpha * r and v_new = v_pred + (beta / dt) * r. alpha weights the position correction, beta weights the velocity correction; both gains are dimensionless constants.
• Gain ranges: stable for 0 <= alpha < 2 and 0 <= beta < 4 - 2*alpha. With alpha = 1 and beta = 1 the updated position equals the raw measurement, so the filter tracks the measurement exactly.
• Steady-state gains from the smoothing factor (Benedict-Bordner, critical damping): beta = alpha^2 / (2 - alpha) for alpha in (0, 2). Smaller alpha smooths more and lags a maneuvering target; larger alpha tracks faster and passes more measurement noise.
• Steady-state gains from the maneuverability index (Kalata tracking index): lambda = sigma_w * dt^2 / sigma_v, formed from the target process noise sigma_w, the sample interval dt, and the measurement noise sigma_v. The critical-damping gains follow the radical closed forms in the logic module; lambda -> 0 gives gains (0, 0) and lambda -> infinity gives gains tending to (1, 2).
• Tracking error metrics: root-mean-square error and maximum absolute error between the true and estimated position sequences.
• Units: position in the tracked unit (meters), velocity in unit per second (m/s), dt in seconds, gains dimensionless.
• ARP4754A (reference-only) frames development assurance for aircraft systems; the alpha-beta filter is common tracking-filter knowledge (Kalata, Benedict and Bordner).

Workflow

1. Set the sample interval dt and the target model: constant velocity between measurements.
2. Choose the gains. Either pick the smoothing factor alpha and derive beta with steady_state_gains(alpha), or form the maneuverability index lambda = sigma_w * dt^2 / sigma_v and derive both gains with gains_from_tracking_index(lambda).
3. Predict with predict(x, v, dt); confirm the position advanced by dt * v and the velocity held.
4. Compute the residual with residual(z, x_pred) and apply the update with update(x_pred, v_pred, z, dt, alpha, beta), which returns the residual, updated position, and updated velocity.
5. Run the full cycle with step(x, v, z, dt, alpha, beta), or filter a whole measurement batch with run_tracker(measurements, dt, alpha, beta, x0, v0) and inspect the position, velocity, residual, and predicted-position trajectories.
6. Assess the result with tracking_errors(true_positions, estimates); compare the RMSE and maximum error against the measurement-noise level.
7. For stateful use, keep a TrackFilter instance and call its step(z) method per measurement; the filter holds x, v, and the last residual.

Pitfalls

• Using gains outside the stable region; alpha >= 2 or beta >= 4 - 2*alpha makes the filter diverge, so the module raises ValueError.
• Confusing the smoothing factor with the maneuverability index; alpha is a dimensionless gain in (0, 2), lambda = sigma_w * dt^2 / sigma_v is a noise ratio, and the two connect through the steady-state gain formulas.
• Feeding a non-positive sample interval dt; the module raises ValueError, and a zero dt makes the velocity correction blow up.
• Forgetting the dt factor in the velocity update; the residual is divided by dt so the velocity unit stays unit per second.
• Expecting the filtered position to equal the measurement; with alpha < 1 the filter lags the measurement by design and trades lag against noise smoothing.
• Mixing units in one state; positions and velocities must share one unit set (meters and meters per second).

Behavior contract (gate 3)

The predict, residual, update, batch tracker, steady-state gain selection, tracking error metrics, and the stateful TrackFilter are exercised by the gate 3 contract test: scripts/test_alpha_beta_filter.py against scripts/alpha_beta_filter_logic.py (stdlib unittest, offline). Run: python3 scripts/test_alpha_beta_filter.py

Compliance

• ARP4754A is proprietary (SAE); name and paraphrase only per standards-map.yaml, reference-only: true.
• compliance: STANDARDS-REF, gated: false.