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i read a lot about the kalman filter and the extended kalman filter, but some things are still not clear to me. E.g. I have one question concerning the jacobian matrix of the measurement matrixh. I want to point out my problem with a concrete example:

A vehicle is represented by the following state vector: [tex]x=\begin{pmatrix} x \\ y \\ \varphi \\ v \end{pmatrix}[/tex] (position, rotation and speed).

The equations of the motion model are the following ones:

[tex]

x_x = x_x + x_v \cdot sin(x_\varphi);

[/tex]

[tex]

x_y = x_y + x_v \cdot cos(x_\varphi);

[/tex]

x y phi and v can every second be measured with a failure.

To remindExtended Kalman Filter: http://www.embedded-world.eu/fileadmin/user_upload/pdf/batterie2011/Armbruster.pdf (Slide 9)

Question:As visible on the slide 9, I have to calculate the jacobian matrixHfor my measurement-functionh. The slide points that very well out, in the correction step isHthe jacobian matrix andhis my measurement function.

So if I want to consider all elements of the measurement vektor, h would be in my opinion the following matrix:

[tex]h = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} [/tex]

because

[tex]z_k = h*x_t = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} * \begin{pmatrix} x_{t,x} \\ x_{t,y} \\ x_{t,\varphi} \\ x_{t,v} \end{pmatrix} [/tex]

(x_t is the current measurement vector)

Therefore [tex]z_k = x_t[/tex]

But in that case the jacobian matrix [tex]J(h)=H[/tex] becomes

[tex]H = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} [/tex]

And this means, that the whole correction step of the EKF does not work, because the Kalman gain-matrix also becomes a zero matrix (See again slide 9, there are some matrix multiplications where H is used. When H contains just zeros the gain [tex]K_K[/tex] also becomes zero).

So to conclude:I think I haven't understood the meaning of h and how to calculate H. I appriciate any help and apologize for my english, because Im not a native speaker :)

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# Question concerning the extended kalman filter

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