The Extended Kalman Filter (EKF) is a mathematical algorithm that serves as the de facto standard for state estimation in navigation systems, including GNSS receivers, inertial navigation systems, and sensor fusion applications. Building upon the optimal linear Kalman filter, the EKF extends its capabilities to handle the nonlinear system dynamics and measurement relationships inherent in real-world navigation problems.
At its core, the Kalman filter family of algorithms provides a statistically optimal method for combining uncertain measurements with predictions from a dynamic system model. In navigation applications, the filter maintains estimates of the system state, typically including position, velocity, and attitude, along with the statistical uncertainty of these estimates expressed as a covariance matrix. With each new sensor measurement, the filter updates its state estimate by optimally weighting the new information against its prior prediction based on their respective uncertainties.
The Extended Kalman Filter handles nonlinearity by linearizing the system dynamics and measurement equations around the current state estimate at each time step. This linearization uses first-order Taylor series approximations (Jacobian matrices) to convert nonlinear relationships into locally linear forms compatible with the standard Kalman filter equations. While this approximation introduces some theoretical limitations compared to truly optimal nonlinear estimation, the EKF provides an excellent practical balance of accuracy and computational efficiency for most navigation applications.
Modern GNSS/INS integration systems rely heavily on EKF implementations to fuse satellite positioning data with inertial measurements from accelerometers and gyroscopes. The filter continuously estimates not only position and velocity but also sensor errors including IMU biases, GNSS clock offsets, and atmospheric delays. By jointly estimating these error states alongside navigation parameters, the EKF can partially compensate for systematic sensor errors and provide more accurate, robust positioning than any single sensor alone.
