The Unscented Kalman Filter (UKF)

When a Gaussian passes through a curved function, the result is no longer Gaussian. The EKF approximates the curve with a straight line. The UKF instead pushes a handful of carefully chosen sample points, called sigma points, through the real function and fits a Gaussian to where they land.

Range Range uncertainty σr Bearing uncertainty σθ

Sigma points

For an $n$-dimensional Gaussian with mean $\boldsymbol\mu$ and covariance $\Sigma$, the UKF picks $2n + 1$ points

$$\mathcal{X}_0 = \boldsymbol\mu \qquad \mathcal{X}_{\pm i} = \boldsymbol\mu \pm \left(\sqrt{(n + \lambda)\Sigma}\right)_i$$

passes each through the nonlinear function, and recombines them with weights to get the new mean and covariance. No Jacobians are needed.

What to notice

Here a reading in range and bearing is converted to x and y, a curved transform. The gray dots are the true spread, from thousands of samples. With small bearing uncertainty, all three agree. Raise it and the true spread bends into a banana. The EKF's ellipse, navy, stays centered at the curve's outer edge and misses the shift of the mean. The UKF's ellipse, red, moves its center inward with the true mean and covers the spread better, using only five points.