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18

Roughly speaking, they are the amount of noise in your system. Process noise is the noise in the process - if the system is a moving car on the interstate on cruise control, there will be slight variations in the speed due to bumps, hills, winds, and so on. Q tells how much variance and covariance there is. The diagonal of Q contains the variance of each ...


9

I'm the author of the textbook linked to above. This is a new account and thus I am not allowed to reply to that answer. Anyway, the Gaussian chapter covers the definition of a covariance matrix. In general though your Q matrix will be full, not a diagonal, because there is correlation between the state variables. For example, suppose you are tracking ...


8

If you can do it that way, it makes it straight forward to implement the Kalman filter. As an example - consider a constant velocity model in Cartesian coordinates, but the only thing you are measuring is the position (not velocity) and you are measuring the position in a Cartesian coordinate system. Then your $\mathbf{H}$ matrix will only pick off the ...


8

The Kalman filter is the optimal filter under various assumptions. You need to check whether those assumptions hold in your case: a) the model perfectly matches the real system, b) the entering noise is white and Gaussian and c) the covariances of the noise are exactly known. Without further detail I can't say whether your statement My experiments ...


7

I think the confusion comes from the authors not parameterizing things clearly. Furthermore, by switching to geometric algebra rather than quaternions, some additional confusion can be cleared up. The main difference between normal vector algebra and geometric algebra is that we can multiply vectors. So of $e_x$, $e_y$, and $e_z$ are our (orthonormal) ...


7

Kalman filters really aren't that special, and you seem to be missing the point of a Kalman filter. A Kalman filter is really just a generally time-varying, generally IIR, generally multi-input multi-output filter that's been designed using a specific procedure. Can we deem that traditional filters such as FIR and low-pass filter are designed to be used ...


7

We can build a non linear dynamic model in order to estimate the parameters of a sine signal. Let's model the signal as $ a \sin \left( \phi \right) $ where $ \phi $ is the instantaneous phase. So the model could be also written as $ a \sin \left( \omega t + \psi \right) $. Then the model can be: $$ {a}_{k} \sin \left( {\omega}_{k} {t}_{k} + \psi \right) = {...


6

The covariance matrix of a Kalman filter is a function of the $ Q $ and $ R $ matrices of the model. If you use a model where $ R $ and $ Q $ are time invariant or known in prior then the calculation of the covariance matrix $ P $ can be done offline and isn't a function of the measurements. In some cases, advanced implementations of Kalman Filter estimate ...


6

My answer is that if it's linear system you should use KF; if it's nonlinear system with weak nonlinearity you should use EKF, if the nonlinear system with high nonlinearity you may consider the well-known UKF. I draw a graph for this, hopefully, it's useful.


6

I won't add any equations, I will just add some intuition. I will also limit my self for Additive Gaussian White Noise. Now, in that case the Kalman filter can written as a Least Squares problem to solve. I'd say even more, the Kalman Filter is linear, if you have the samples up to certain time $ T $, you can write the Kalman filter as weighted sum of all ...


6

First and foremost I really recommend this great textbook project about Kalman filtering. You can find some words about setting the process noise here. There is also a pdf version of it. In every step the filter estimates a multivariate normal distribution with parameters $\mu = x$ (state vector) and $\sigma = P$ (covariance matrix). By looking the Kalman ...


6

So this is just the start of an answer. I'll have to keep updating it as I go. The first attempt is to say that the quantities you are interested in are the location of the center of the four LEDs, and the roll, pitch, and yaw (rotation angles) of the LEDs. That means your Kalman FIlter state will be: $$ \mathbf{x}_k = \left[x_k\ y_k\ \alpha_k\ \beta_k\ \...


6

Well, in continuous time, a sinusoid with a bias can be seen as the output of the linear system \begin{align*} \begin{bmatrix}\dot x_1\\\dot x_2\\\dot x_3\end{bmatrix} &= \begin{bmatrix}0 & 1 & 0\\-\omega^2&0&0\\0 &0 &0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\\ y &= \begin{bmatrix}1&0&1\end{bmatrix} \...


6

Adaptive Filters are called "Adaptive" when they can adapt to changes in data. In the filters you mentioned above, which are part of the Linear Filters family the property means their coefficients are changing over time. Linear Filters are basically weighing and summing the data. For instance, given no prior information on data you may want to have exact ...


6

The Kalman Filter is basically a framework to fuse 2 things: Measurement. Dynamic Model (Dynamic in the sense we can predict next value from a current value). In your case the model is composed of two things: Model which connects $ {T}_{ext} $ and $ {W}_{fuel} $ to $ {T}_{in} $. Measurement of $ {T}_{ext} $. Now, the reason you estimate $ {T}_{ext} $ is ...


5

In addition to Peter's answer, if you have a nonlinear system that is well-behaved in a sense of being only mildly nonlinear or at least exhibiting no discontinuities, special variants of the Kalman filter can still be applied. Extended Kalman Filter This filter linearizes the system at the current state of the system using a first order Taylor Series ...


5

Well, let's look at the two issues: 1) linearity and 2) Gaussianity. Linearity If you're imaging moving 3D objects (people) with a single camera, then you're working with a 2D projection of those 3D objects. That dimensionality reduction can cause non-linearities to appear. Take a 2D to 1D example: an object moving in a circle in 2D. The object is ...


5

In ideal world you'd have the correct model and use it. In your case, the model isn't perfect. Yet the steps you're suggesting are based on a knowledge you have about the process - which you should incorporate into your process equation using your dynamic model matrix: The classic and correct way given F matrix is built correctly according to your knowledge....


5

I think I have the solution. I'd be happy to hear others' thought. Defining $ F \left(r, v, a, {T}_{tth} \right) = r + v {T}_{tth} + \frac{a {{T}_{tth}}^{2}}{2} $ which is the implicit function which connects all variables. Since we're dealing with non linear function the variance is given by: $$ var \left( {T}_{tth} \right) = J P {J}^{T} $$ Where $ P $ ...


5

as the first answer (with the most votes) says, the kalman filter is better in Any case when signal is changing. Notice the problem statement These use the algorithm to estimate some constant voltage. How could using a Kalman filter for this be better than just keeping a running average? Are these examples just oversimplified use cases of the filter? using a ...


5

Kalman filter is the best linear estimator regardless of stationarity or Gaussianity. Also in the Gaussian case it does not require stationarity (unlike Wiener filter). In the linear Gaussian case Kalman filter is also a MMSE estimator or the conditional mean.


5

Is this so far a reasonable scenario / approach to the Kalman filter? Answer 1: Yes, your model looks reasonable. You're treating the acceleration as constant, however. If it will change in your experiment, you should include that in your system error matrix $Q$. How do I choose the initial uncertainty covariance matrix $P_0 \in \mathbb{R}^{4 \times 4}$ ...


5

First of all let us assure that a Kalman filter (estimator) does not only remove Gaussian noise, but can remove (with certain success) any other type of noise as long as it's designed accordingly. However, what lies at the heart of the standard Kalman filter is the linear estimator; and that linear estimator will be the optimum minimum mean square error ...


5

I will tell you something, even if it is differntiable, use Unscented Kalman Filter for any non linear case. This flavor of Kalman Filter, based on the Unscented Transform, is almost always superior to the Extended Kalman due to its properties. The main reason is it is able to better predict the mean and variance (Which all Kalman Filter needs) of the ...


4

I suggest this reference regarding the comparison between least-squares and Kalman filters : Fundamentals of Kalman Filtering: A Practical Approach by P. Zarchan & H. Mussof Especially Chapter 3 (Recursive Least-Squares Filtering) and Chapter 4 (Polynomial Kalman Filters). In Chapter 4, the authors show that the discrete (time) n-th order polynomial ...


4

A lot has been said already, allow me to add some comments: Kalman filters are an application of Bayesian probability theory, which means that "a priori information" or "prior uncertainty" can (and must) be specified. As I understand, this is not the case with traditional least-squares fitting. While observations (data) can be weighted with probabilities in ...


4

You are most likely not having a control vector $u$. Maybe you can somehow model it into the filter, but it is not required in your case and uncommon. Furthermore $\overset{.}{x}$ is not your gyro input but it is the Kalman filtered rotation rate calculated after using all your measurements. Usually, the main effects on a magnetometer are hard iron and soft ...


4

The Kalman Gain $K_k$ for a discrete, linear system is computed using the state covariance matrix $P_k$, measurement matrix $H_k$ and measurement noise covariance matrix $R_k$: $$ K_k = P_kH_k^T(H_kP_kH_k^T+R_k)^{-1} $$ If the measurement matrix and the measurement noise covariance matrix are constant and the state covariance matrix converges to a steady-...


4

Few notes: First you should multiply the noise by the Standard Deviation (Root of the Variance for zero mean noise). You can do that by multiplying the Lower Cholesky Decomposition of matrix by a column vector of Gaussian noise. Yet since you assume no correlation, you can do that by independent multiplication. Something like: vZ = vZ + [sigma_range; ...


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