Kalman filter

What is it?

Formulas

$$K = P^-H^{\mathsf T}\big(HP^-H^{\mathsf T} + R\big)^{-1}, \qquad \hat x = \hat x^- + K\big(z - H\hat x^-\big)$$

$$\hat x = \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}x_1 + \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2}x_2$$

the 1D intuition: average two estimates, trusting the less noisy one more

The mathematics behind it

• Variance★★★★★fundamental

  The Kalman gain weighs prediction and measurement by their variances.

• Continuous distributions★★★★★fundamental

  Gaussian noise assumptions make the Kalman filter optimal and closed-form.

• Matrices and linear maps★★★★★frequent

  State transition, observation and covariance updates are all matrix products.

• Total differential and linearization★★★★★frequent

  The extended Kalman filter linearizes nonlinear dynamics and sensors at the current estimate.

• Jacobian matrix★★★★★frequent

  The extended Kalman filter propagates covariances with the Jacobians of the dynamics and measurement models.

• Probability density function★★★★★frequent

  The Kalman filter propagates Gaussian densities of the state through predictions and measurements.