Question concerning the extended kalman filter

In summary: The Jacobian is the matrix of partial derivatives of the components of h with respect to the components of x.In summary, the conversation discusses the use of the Kalman filter and extended Kalman filter in a vehicle's motion model. The question at hand concerns the calculation of the Jacobian matrix for the measurement function h. The conversation concludes with a clarification on the function h and its corresponding Jacobian matrix.
  • #1
Anonymous123
1
0
Good day,

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 matrix h. 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 remind Extended 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 matrix H for my measurement-function h. The slide points that very well out, in the correction step is H the jacobian matrix and h is 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 appreciate any help and apologize for my english, because I am not a native speaker :)
 
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  • #2
http://www.google.com/url?sa=t&rct=...sg=AFQjCNFjl1YZdkoZZiaJ7-NMEi3vDKnU5Q&cad=rja

Anonymous123 said:
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]

In the treatments of the extended Kalman filter that I can read (such as http://www.google.com/url?sa=t&rct=...sg=AFQjCNFjl1YZdkoZZiaJ7-NMEi3vDKnU5Q&cad=rja
), [itex] h [/itex] is a vector valued function, not a matrix. In you example it would be the function [itex] h(x,y,\varphi,v) = (x,y,\varphi,v) [/itex]. So the Jacobian of this function is not found by differentiating constants.
 

1. What is the extended Kalman filter and how does it work?

The extended Kalman filter (EKF) is a recursive algorithm used to estimate the state of a nonlinear system. It combines the concepts of the Kalman filter, which is used for linear systems, with the extended Taylor series to handle nonlinear systems. The EKF uses a prediction step and an update step to iteratively estimate the state of the system based on measurements and a mathematical model.

2. What types of systems can the extended Kalman filter be used for?

The EKF can be used for nonlinear systems, as long as the system can be described by an underlying mathematical model. This includes systems in various fields such as aerospace, robotics, and control systems. The EKF is particularly useful for systems with complex dynamics or measurements that are subject to noise.

3. What are the limitations of the extended Kalman filter?

One limitation of the EKF is that it relies on a linearized version of the system's dynamics, which may not accurately represent the true nonlinear behavior. This can lead to errors in the estimation process, especially for highly nonlinear systems. Additionally, the EKF assumes that the measurement and process noise are Gaussian and independent, which may not always be the case in real-world scenarios.

4. How is the performance of the extended Kalman filter evaluated?

The performance of the EKF can be evaluated by comparing the estimated state values to the true state values of the system. This can be done by analyzing the error between the two values and measuring metrics such as root mean square error (RMSE) or mean absolute error (MAE). Additionally, the stability and convergence of the EKF can be assessed by examining the covariance matrix and the Kalman gain.

5. Are there any alternatives to the extended Kalman filter for nonlinear systems?

Yes, there are several alternatives to the EKF for nonlinear systems. These include the unscented Kalman filter (UKF), which uses a sigma-point approach to handle the nonlinearities, and the particle filter, which uses a Monte Carlo sampling method. Each of these alternatives has its own strengths and limitations, and the choice of which filter to use depends on the specific characteristics of the system and the available measurements.

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