Question

Why must an Extended Kalman Filter (EKF) use a Jacobian matrix to linearize the state transition and measurement models?

Accepted Answer

The Extended Kalman Filter must use a Jacobian matrix because the standard Kalman Filter only operates on linear systems where the relationship between states is represented by a straight line or constant matrix. In many real-world applications, state transition models and measurement models are nonlinear, meaning the output changes in curved or complex ways relative to the input. A Jacobian matrix is a mathematical tool that calculates the partial derivatives of these nonlinear functions, providing a linear approximation of the system at a specific point. By using the Jacobian, the Extended Kalman Filter creates a local linear approximation that mimics a tangent line touching the curve of the nonlinear function. This allows the filter to apply standard Kalman Filter equations to systems that would otherwise be mathematically incompatible. For example, if a robot moves in a circular path, the position coordinates involve trigonometric functions like sine and cosine; the Jacobian allows the filter to treat these circular movements as locally linear segments so the filter can update its position estimate effectively. Without this linearization, the standard Kalman Filter equations would fail because they cannot process the nonlinear nature of the underlying math.

Home → All Courses → Engineering and Technology Courses → Robotics and Autonomous Systems → Flashcard

Why must an Extended Kalman Filter (EKF) use a Jacobian matrix to linearize the state transition and measurement models?