Extended Kalman Filter

Compute the Jacobian by taking partial derivatives: F[i,j] = ∂f_i/∂x_j evaluated at the current estimate. For complex functions, use automatic differentiation tools (e.g., PyTorch, JAX) to avoid errors. Always verify numerically by comparing against finite difference approximations: (f(x+ε)-f(x))/ε.

The EKF linearizes the nonlinear function using a Taylor expansion (Jacobian). The UKF instead samples a set of deterministically chosen points (sigma points) around the estimate and propagates these through the nonlinear function, capturing higher-order nonlinearity without explicit Jacobians. UKF is often more accurate for strongly nonlinear systems but slightly more computationally expensive.

Monitor the normalized innovation (measurement residual divided by predicted measurement covariance). For a well-tuned filter, these innovations should be white (mean zero, constant variance). If innovations are consistently biased or have increasing variance, the filter is diverging. Also check that predicted covariance decreases over time; if it increases, the model uncertainty Q is too large.

Standard EKF does not handle time delays in measurements. To use EKF with delayed measurements, augment the state vector to include past states, or use a delayed Kalman Filter variant that stores and processes out-of-order measurements. Another approach is to increase measurement noise covariance R to account for the delay uncertainty.