The Kalman Filter by Hand: Noisy Measurements, Predict & Update Steps, Kalman Gain, Constant-Velocity Tracking, Sensor Fusion & Tuning

What you'll build

What you will be able to do

Each lab ends with a concrete result that you build yourself and the checker confirms, cell by cell.

• Turn ten noisy level readings into a running average that is updated one reading at a time, check that it equals the plain average, and compare its error with the error of a single reading.
• Measure the noise of two sensors as a variance, weight their live readings by inverse variance, combine them into one estimate and show that the combined variance is smaller than either sensor's.
• Run a one-dimensional Kalman filter on twelve tank readings: predict the state and its variance, compute the Kalman gain, update the estimate and the variance, and watch the gain settle.
• Compute the steady gain for four process-noise settings, measure how far a cautious and a nimble filter lag behind a sudden level jump and chart both against the readings.
• Run one predict–update step for a toy cart with two states, position and velocity, write the 2×2 covariance prediction out as four cells, compute the velocity gain and correct the velocity from a position-only reading.
• Fuse a noisy and a precise level sensor in one filter step by two sequential updates, then reach the same estimate and variance in one go with inverse-variance weights.
• Handle a sensor that sometimes sends nothing and once sends nonsense: compute the innovation variance, decide per step whether to use, skip or reject a reading with a gate of k·√S, and set the gain to zero when the update is skipped.
• Score five estimators of a toy cart's position by RMSE – a constant-velocity Kalman filter, an alpha–beta filter, two level filters with different Q/R and a moving average – compute the improvement and chart the best filter against the readings.