How does robot_pose_ekf work?

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From what I can see from the code, the process model is defined in this file. The question you referenced asked for the transition and measurement matrices, but because this odometry process model is nonlinear, this matrix is not constant. Instead, it is generated by linearizing the process model about the current estimate. Check out the wikipedia page for the EKF for more on this.

You can see the computation of the expected value and the partial derivatives with respect to the process inputs (or if you like probabalistic terms more than systems terms, the conditioning variables). Anyone who is using robot_pose_ekf should make sure this process model is consistent with his or her robot's dynamics.

For the sensor data, these are all linear models, so you can immediately see their definition like the odom measurement model defined here. Similarly, anyone using this should make sure these measurement models are appropriate.

The interfaces that all these things use are defined by the BFL (Bayesian Filtering Library) API, specifically, interfaces for representing an analytic system model with gaussian uncertainty and an analytic conditional gaussian along with the linear model representations.