1
$\begingroup$

I am currently reading lots of material on the Kalman filter (in order to do some experiments), and there is something that I don't get, and I can't get a clear understanding.

I'll stick to the Wikipedia page notation, $\mathbf{F}_k$ is the state transition model and defines how the state goes from step $k$ to step $k+1$.

As I read it, there is a subscript, which means to me that it can change over time (and that makes sense).

However, the "update" step does not describe how I get the new estimation of $\mathbf{F}$.

Example: Consider a robot moving (more or less) randomly in a 2D space. At first, say I have a 45° line motion. If speed is constant, my state is described by $\left[ x \: y \: vx \: vy \right]$, and I'll have $$ \begin{bmatrix} x_{k+1} \\ y_{k+1} \\ vx_{k+1} \\ vy_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_k \\ y_k \\ vx_k \\ vy_k \end{bmatrix} $$

But when the robot changes directions, then that state matrix is not valid anymore!

How do I get the updated $\mathbf{F}$ matrix? What is the point I missed?

Edit to clarify: I have some sample working code that is able to provide a good tracking, and it keeps the same state-mode matrix all the way. But I just don't understand how that can work.

Edit2: forgot to mention, but the estimation of the covariance matrix $\mathbf{P}$ at a given step also relies on $\mathbf{F}$: $$ \mathbf{P} = \mathbf{F} \: \mathbf{P} \: \mathbf{F}^{-1} + \mathbf{Q} $$ How can that possibly work if the motion (expressed by the state-mode matrix $\mathbf{F}$) is incorrect? I really must have something wrong here...

$\endgroup$
2
  • $\begingroup$ The state transition matrix is given exogeneously; it is an input to the Kalman filter. It is not "estimated" or "updated" by the Kalman filter. I don't know anything about robot movement, it may require a different perspective than the one you've described here to use the Kalman filter for that. $\endgroup$
    – Chris Haug
    Nov 21, 2021 at 18:48
  • $\begingroup$ Thanks for your comment. This is a mystery for me: how can the filter provide a good estimate (and it does!) with an incorrect state update model. $\endgroup$
    – kebs
    Nov 21, 2021 at 19:00

2 Answers 2

1
$\begingroup$

Answers:

  1. "which means to me that it can change over time". Yes, the state-space coefficients can indeed depend on time. However, to be more precise, it is the time interval $\Delta t_k$ the coefficient usually depends on instead of time $t$. For example, if you have uniform $\Delta t_k$ for all $k$'s, then $F$ is independent of $k$.

  2. "However, the "update" step does not describe how I get the new estimation of $F$". The Kalman update step does not estimate $F$.

  3. "But when the robot changes directions, then that state matrix is not valid anymore! How do I get the updated $F$ matrix? What is the point I missed?". The model you are showing is a Wiener velocity model which describes any target motion in terms of position and velocity. When the robot changes its direction, the model is still a valid model, simply because you have forgotten the noise term. Denote $z = [x\,y\,v_x\,v_y]$. The complete model is $z_k = F \, z_{k-1} + q_{k-1}$, where $q_{k-1} \sim N(0, Q)$ is a Gaussian r.v. with covariance $Q$ depending only on $\Delta t_k$. This noise term $q$ accounts for the permutation/uncertainty of target's position and velocity. Without $q$, your model target will go along a straight line, and of course the direction will not change. With $q$, your model target will have a random trajectory with direction changes. You can try simulate the trajectory and see.

Think of the dynamic model as a prior on your target. The Kalman filter update step's job is to "correct" the prediction from the prior model by using the likelihood model (i.e., your measurement model) and measurement/data.

  1. "How do I get the updated $F$ matrix? What is the point I missed". If you really want to get more precise $F$ (and $Q$), you may parameterise $F$ ($Q$) with some parameters, say, $\theta$, and get $F(\theta)$ (and $Q(\theta)$). Then you can estiamte $\theta$ by performing maximum likelihood estimation.
$\endgroup$
1
  • $\begingroup$ Thanks for that detailed answer! $\endgroup$
    – kebs
    Nov 22, 2021 at 20:23
1
$\begingroup$

$F_k$ can change over time, but doesn't need to. The wikipedia article is suggesting that $F_k$ can change, perhaps as part of your filter's design (i.e. you may choose how it evolves in your design). But often, $F_0 = F_1 = ... F_k$.

So you are doing nothing wrong, just you are seeing a generalized syntax.

$\endgroup$
5
  • $\begingroup$ Thanks for you answer. As I said in comment above, I don't understand how the filter can provide a good estimate (and it -usually-does) with an incorrect state update model!! $\endgroup$
    – kebs
    Nov 21, 2021 at 19:02
  • $\begingroup$ Why do you think the state matrix isn't valid anymore? $\endgroup$ Nov 21, 2021 at 19:40
  • $\begingroup$ Well, if the object changes directions (say, 180° turn), then we go from $x_{k+1} = x_k + \Delta t. vx$ to $x_{k+1} = x_k - \Delta t. vx$. $\endgroup$
    – kebs
    Nov 21, 2021 at 20:09
  • $\begingroup$ Arg... I think I am starting to understand were my assumptions are bad... Let me check. $\endgroup$
    – kebs
    Nov 21, 2021 at 20:11
  • $\begingroup$ Please fill us in on your solution - it will help others! $\endgroup$ Nov 22, 2021 at 2:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.