Extended Kalman Filter - Determining the state vector

Question

I am trying to make sense of a paper that defines a EKF approach to estimating the heading of a device.

The paper says the following:

The state vector $x$ for the filter is composed of the rotation quaternion $(q = [q_1, q_2, q_3, q_4])$, augmented by the tri-axial gyroscope scale $(s = [s_x,s_y,s_z])$ and bias factors $(b = [b_x,b_y,b_z])$ and external magnetic disturbance vector $(d = [d_x,d_y,d_z])$, that is, $x=[q,s,d,b]$.

Does this mean that the state vector is 13 dimensional vector? (4 from quaternions, 3 from gyro scale, 3 from the bias, and the 3 from the discturbance.

Apologies if this is a noob question, its a bit confusing the way it is written and the way it is represented at the end as $x=[q,s,d,b]$

Answer 1

Yes, the state vector is effectively 13-dimensional. This way of presenting the state vector allows for more concise language when describing state and changes in state. E.g. if only one part of the state vector changes you can say something like

• "The new state vector is $x'=[q,s,d,b']$ where $b'=$blah blah"

instead of

• "The new state vector is $x'=[q_1,q_2,q_3,q_4,s_x,s_y,s_z,d_x,d_y,d_z,b_x',b_y',b_z']$ where $[b_x',b_y',b_z']=$blah blah"

It's a little like using composite data structures in a programming language.