Error propagation

Until now we could pretend the world was a number. Part IV refuses. Every encoder tick, every pixel, every range is a random variable. The useful first model is a Gaussian: a mean you act on and a covariance you refuse to hide. Students who skip covariance write planners that thread gaps they cannot prove they are in.

Error propagation is the chain rule for uncertainty. If \(y=f(x)\) and \(x\) is uncertain, a linearization \(y \approx f(\hat{x}) + J(x-\hat{x})\) gives \(\Sigma_y \approx J\Sigma_x J^\top\). That formula is why odometry ellipses grow, especially in heading. A little yaw noise at the start of a long drive is a large lateral uncertainty at the end. Draw it. The studio Kalman band is the 1D cousin of this picture.

Independence is fragile. Two range readings may be independent at the sensor and become correlated once both are used to estimate the same wall. A shared calibration error correlates everything that used the calibration. When you add measurements, you must say whether their noises share ancestors. Blind diagonal covariances are how filters become overconfident — the worst failure, because the robot is sure.

An error ellipse (a level set of a 2D Gaussian) is a teaching gift. Long and skinny means you know one direction and not the other — typical when a single range to a long wall pins you laterally but not along the wall. Fat and round means you are lost-ish in both. If the ellipse does not shrink when you think you measured something, your \(h(x)\) is uninformative in that direction. That is a feature problem, not a “need more gain” problem.

Do a hallway thought experiment: start with a tight pose, drive 10 m with wheel noise, watch the ellipse grow, then take one lidar scan of a side wall and watch it flatten. That motion is the Kalman gain in public. Read the book chapter for the derivations this note is not permitted to lift.