Trouble Understanding Error State Kalman Filter State Transition Matrix
Hello. I am currently implementing a program for a microcontroller to do orientation estimation. I do not have a controls systems background and have been reading up on several papers to understand error state kalman filters and quaternions, though my lack of differential equations knowledge and other subjects has left me stumped on understanding this filter.
Here is the main paper I have been using to understand: https://arxiv.org/pdf/1711.02508
The main question I have is how the state transition matrix is obtained. Here is the discrete transition matrix in the paper:
They define R as the rotation matrix from a quaternion:
What I don't understand is passing in the accelerometer values in equation 311; on the 3rd column and 3rd row, they pass in the measured accelerometer minus the accelerometer bias into this rotation matrix equation. Is this meant to be obtained from the roll and pitch values?... Also for the other R values in the equation, is that just assumed to be my orientation equation from the gyro? And is it appropriate to use accelerometer values in the state transition matrix if the goal is to just do orientation estimation?
A lot of my confusion comes from looking at different implementations and other papers of the same problem (orientation estimation, sometimes with position as well). The F matrices I see in these implementations look nothing like what I have seen on here, and I am not really grasping how this transition matrix very well. I've looked at several resources and am a bit overwhelmed by the differences.
Any help would be greatly appreciated, thank you ^^