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I have been reading about kalman filters for IMU and I am confused on the error state formulation.

Reading this paper, starting on page 63 and going onto 64 the paper addresses the reset of the error state.

Error Reset

https://arxiv.org/pdf/1711.02508.pdf

My confusion is what is meant by line 284? 285 just looks like it sets the error state to zero and 286 is applying a linear transformation to the error state covariance, but I don't understand what function is really being used in 284 for the quaternion.

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My confusion is what is meant by line 284?

In section 4 of the paper they define the $\oplus$. Then they define $\ominus$ as being the inverse: i.e., if $a = b \oplus c$, then $c = a \ominus b$.

They left out the time index, and you need to peer closely to see that they'e distinguishing $\delta \mathbf x$ (the error state) from $\hat {\delta \mathbf x}$ (the estimate of the error state, with a hat).

Adding indexes, that's $$\delta \mathbf x_k \leftarrow g(\delta \mathbf x_{k-1}) = \delta \mathbf x_{k-1} - \hat {\delta \mathbf x}_{k-1} \tag {284-update}$$

285 just looks like it sets the error state to zero

(285) sets the estimated error state to zero -- again, note the hat. The actual error state is, formally, unknown, being the difference between the actual state of the system (whatever that is) and your estimate of the actual state.

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  • $\begingroup$ I think I am following but in this case quaternions are used with multiplication. I am confused on the G matrix still, I understand it's the jacobian but I still don't really follow. I understand it gets into some niche lie theory. $\endgroup$ Commented Apr 8, 2023 at 23:16
  • $\begingroup$ When you read section 4 of the paper -- the one that addresses your confusion -- which bits didn't you understand? Perhaps you should edit your question with those bits, and how they confuse you. $\endgroup$ Commented Apr 9, 2023 at 4:09
  • $\begingroup$ Reviewing this basically, you need to recenter the error state system. You apply another linearization to update the covariance matrix for the reset operation. Simply setting the error vector to zero doesn't update the covariance matrix. $\endgroup$ Commented Jan 15, 2024 at 7:42

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