A sensor gives you a noisy dot every tick — the true position is somewhere in the cloud, never exactly where you measured. A Kalman filter fuses two beliefs at each step: where its motion model predicted the object would be, and where the new measurement says it is. It weighs them by how much it trusts each, and the result tracks the truth far better than the raw measurements ever could.
true path — the hidden ground truthmeasurements — noisy sensor dotsestimate — the filtered trackcovariance — 2σ uncertainty ellipseEach tick runs two steps. Predict: push the estimate forward with a constant-velocity model, x ← Fx, and let the uncertainty grow, P ← FPFᵀ + Q — with no new data the ellipse swells. Update: a measurement z arrives; the filter forms the Kalman gain K, which is large when the estimate is shakier than the sensor and small when it's more confident, then nudges the estimate toward the measurement by K and shrinks the ellipse. Turn measurement noise up and the filter leans on its model, smoothing hard but lagging; turn it down and it snaps to every dot. That single trade-off — model versus measurement — is the whole idea, and it's what fuses GPS with an accelerometer in your phone, tracks aircraft on radar, and keeps a robot localized as it drives.