control//state estimation//sensor fusion//multi-rate fusion

Multi-rate fusion is the way a filter combines sensors that report at different, unrelated rates: the fastest sensor drives the prediction at its own rate, and each slower sensor is a correction applied whenever one of its readings arrives. It is the normal case on any vehicle: a drone's inertial unit samples at hundreds of hertz or more, its barometer at tens of hertz, its GPS at 5 or 10 Hz, its magnetometer at its own pace, and nothing aligns them.

Multi-rate fusion is the way a filter combines sensors that report at different, unrelated rates: the fastest sensor drives the prediction at its own rate, and each slower sensor is a correction applied whenever one of its readings arrives. It is the normal case on any vehicle: a drone's inertial unit samples at hundreds of hertz or more, its barometer at tens of hertz, its GPS at 5 or 10 Hz, its magnetometer at its own pace, and nothing aligns them.

The predict-update cycle takes this in its stride. The inertial readings feed the prediction at every tick (in inertial navigation they usually enter as known inputs to the model, rather than as measurements), and each slow sensor brings its own measurement model, which part of the state it sees, and its own noise RRR (measurement noise). When a reading arrives it is fused; when none does, the filter only predicts, and its covariance grows until the next correction deflates it. A GPS outage is just a long run of predictions, with the uncertainty saying honestly how lost the filter is getting.

Independent sensors can be fused one after the other in any order, each with its own small update, which is cheaper than stacking them into one large correction and gives the same result (correlated measurements). Each can carry its own validity test (innovation gating), so a bad magnetometer reading is refused without touching the GPS.

Resampling everything to one common rate, by interpolating the GPS up to the inertial rate, is the tempting shortcut and a bad one: it invents readings, makes neighbouring samples share their errors and hides the latency each sensor really has.

The prediction at the fastest rate dominates the cost. A 24-state EKF costs tens of thousands of operations per prediction, little for a Cortex-M7 at a few hundred megahertz running it at hundreds of hertz in float32; the budget that breaks first is the timing (timestamps, latencies, synchronization), not the arithmetic (delayed measurements).