The big idea: push points, not slopes

Here is the problem: you have a Gaussian guess (mean and covariance) and a nonlinear function. What is the mean and covariance of the output?

• The EKF draws a tangent line at the mean and pushes the Gaussian through the line.
• The particle filter pushes thousands of random samples through the function.
• The UKF pushes a small, deterministic set of points, \(2n+1\) for an \(n\)-dimensional state, chosen so that their mean and covariance exactly match the Gaussian. Then it measures the mean and covariance of the transformed points.

Jeffrey Uhlmann’s insight was: it is easier to approximate a probability distribution than an arbitrary nonlinear function.

For a 3-state rover you push only 7 points. The result captures the mean and covariance correctly to second order (the EKF is only first-order accurate for the mean), and you never write a derivative.

Picture a pendulum whose angle you know roughly, and a sensor that sees the bob’s horizontal position \(L\sin\theta\). Near the bottom, \(\sin\) is almost straight and both filters agree. Near 90°, \(\sin\) is flat on top, and the EKF’s tangent line badly misrepresents the spread, while the sigma points “feel” the curve.