Imagine a vehicle moving between GPS fixes: the IMU provides frequent motion measurements, while GPS periodically offers a position observation. The navigation system has to combine them without treating either as exact. A scalar Kalman update makes the central idea visible before adding the full position-and-velocity state used in a real system.
Predict Between Position Fixes
Between GPS observations, an inertial navigation system propagates its state using motion measurements and a model of how the vehicle moves. In a simple one-dimensional position example, suppose that prediction is 10 meters with variance 4 square meters. The variance describes uncertainty in the estimate; it is not a statement that the vehicle is literally spread across a four-meter region.
Blend The New Measurement
Now let a GPS observation report 12 meters with variance 1 square meter. For a scalar position measurement, the Kalman gain is prior variance divided by the sum of prior and measurement variances, so here it is 4 / (4 + 1) = 0.8. The updated estimate is 10 + 0.8 × (12 – 10) = 11.6 meters: it moves toward GPS because this example assigns GPS the smaller variance.
Know What The Numbers Assume
Under the scalar model, the updated variance is (1 – gain) times the prior variance, which gives 0.8 square meters here. This calculation assumes the stated uncertainties are meaningful and that the errors are independent and adequately modeled; real navigation systems also track multiple axes, velocity, sensor biases, and coordinate frames. A GPS fix affected by multipath may be a poor measurement despite a small reported uncertainty, so the filter’s result still depends on sensor quality and modeling choices.
Keep Time And Frames Aligned
Before trusting a sensor-fusion result, check that the IMU and GPS observations refer to compatible timestamps and coordinate frames. A delayed GPS measurement compared against a current prediction can look like a position error even when both sensors are behaving as expected. The practical lesson is simple: the update equation combines estimates, but the measurement model determines what those estimates mean.