Uncertainty & State Estimation
Covariances as shapes, filters as products of Gaussians, and least squares as the thing underneath both.
Estimation is where robotics stops being geometry and starts being inference. The mathematics is standard and the intuition usually is not: people can derive the Kalman gain and still not predict which way an estimate will move when a sensor is precise in one direction and vague in another.
This topic builds the geometric picture first — a covariance is an ellipse, a measurement is another ellipse, fusion is their overlap — and then shows that filtering and nonlinear least squares are two views of the same object.
- 01 A covariance is a shape Foundation 8′ Eigenvectors are the directions; square-rooted eigenvalues are the extents. Correlation is a tilt. Repairs Reading a covariance as "the error in x and the error in y", which discards the off-diagonal information that is usually the interesting part.
- 02 Fusing measurements Core 1 interactive 11′ The Kalman gain is a precision-weighted average, and it is visible in the geometry. Repairs Knowing the Kalman equations but not being able to predict, before computing, where the posterior will sit.
- 03 Nonlinear least squares Advanced 11′ Gauss–Newton, damping, and robust kernels — three answers to "the linearisation was optimistic". Repairs Using an optimiser as a black box and having no model of why it diverged, stalled, or was dragged off by one bad correspondence.