I am trying to understand the EKF theory. Can the state transition function depend on variables that are not part of the state space? For example, the state propagation below depends on the quaternions that keep changing. If I get the quaternions from a very dependable source and I dont want to filter them, can I take them out of the state space? In that case, when calculating the Jacobian I will treat the quaternions as constants even though its a dynamic value from some external sensor. What are the implications of this approach? $$ \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}_{k+1} = \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}_{k} + \begin{bmatrix} Rotation Matrix\\ using\\ Quaternions \end{bmatrix} * \begin{bmatrix} u \\ v \\ w \\ \end{bmatrix}_{k} $$

The state space I am using is $$ \begin{bmatrix} x& y& z& u& v& w \end{bmatrix}^T $$ where x y z are the position coordinates and u v w are the linear velocities.


1 Answer 1


if the quaternion is correct, this approach will work.

  • $\begingroup$ It would be great if you could point me to some examples using similar approach. I have come across simple examples and all of them use state space variables that depend linearly or non-linearly on previous state. $\endgroup$
    – Vinmean
    Nov 15, 2016 at 20:11
  • $\begingroup$ i don't have any examples for you but I have done that sort of thing plenty of times with my own work. I assume the IMU/autopilot is doing a good job estimating attitude and just take it as truth. $\endgroup$
    – holmeski
    Nov 15, 2016 at 20:25
  • 1
    $\begingroup$ look into unscented kalman filtering. it is much easier and is more robust to large differences between initial estimated state and true initial state. But most importantly, jacobians do not need to be calculated. $\endgroup$
    – holmeski
    Nov 15, 2016 at 20:28

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