Question

In SLAM implementations, how does an Extended Kalman Filter (EKF) address the linearization error that occurs when predicting non-linear vehicle motion models?

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The Extended Kalman Filter addresses non-linear motion by using a technique called first-order Taylor series expansion to perform linearization, which approximates a complex curve as a straight line at a specific point. Because vehicle motion involves trigonometric functions like sines and cosines that are non-linear, the filter calculates the Jacobian matrix, a collection of partial derivatives that represent the slope of the motion model at the current estimated position. By using this Jacobian to project the uncertainty—how much we doubt our current position—forward in time, the filter treats the non-linear curve as a locally linear movement. Think of this like trying to map the curvature of the Earth: while the planet is a sphere, if you zoom in closely enough, the ground beneath your feet looks like a flat, straight line, allowing you to use simple geometry for navigation. This process allows the filter to maintain a Gaussian distribution, a bell-shaped curve representing probability, even though the actual motion path is curved. However, because this is only an approximation, the filter remains vulnerable to errors if the motion model is highly non-linear or if the initial uncertainty is too large, as the straight-line approximation will drift further away from the actual path the further the vehicle travels.

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In SLAM implementations, how does an Extended Kalman Filter (EKF) address the linearization error that occurs when predicting non-linear vehicle motion models?

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