yes.

There is no formula for choosing R.

From the experience of other successful choices...

Without additional information, it is not possible to provide an answer. The two previous replies assume that the uncertainty is a 1-sigma number. There is no basis to believe that is true. You need to find out how the vendor defines that spec.

Nabil Shaukat

J. Dzielski what happens if the vendor specifies this value of measurement uncertainty at 2 sigma? Thank you

The only other typical way of specifying this is as an upper bound, in which case you treat the distribution as uniform and calculate the variance accordingly. I would also advise a sanity check on what you're trying to accomplish. By saying you want to implement a KF, you're implicitly saying the 0.015mm spec is not good enough. If it is good enough, why bother? I don't know your whole system or what you're trying to do, but in problems where nonlinearities are present, KF's typically produce worse results than just fitting a polynomial or spline the the past couple of measurements. If you don't REALLY know all the statistical parameters of the noise sources, the KF could be a waste of time.

If the matrix Rk is the covariance matrix of the measurement noise n_{k} in the output equation:

y_{k}=Cx_{k}+n_{k}

Then, you can try to estimate the noise covariance matrix from the data. You can obtain an initial estimate, and then you can recursively update the covariance matrix estimate. Note that the standard assumption when deriving the Kalman filter is that the noise n_{k} is white noise. You can test this assumption by performing a white-noise hypothesis test.

Note the the process n_{k} mentioned above is uncorrelated. It is NOT white. The underlying process may be assumed to be white, but the observation sequence n_k is not because E[n_k n_k] is finite.

From a physical modeling perspective, it's a bad idea to think of the observation noise process as white. Think for a minute about what a white noise process is. A process that is bounded on a finite time interval has finite power so it can't be white. Saying the noise process n(t) is white implies that the observation process y(t)=Cx(t) + n(t) is also white. For the physical problem you described, your measurement error with always be very small. It's not white. Anytime you introduce a white noise process into a physical model, you should look to see where the process is integrated before appearing in a measurement. If it's not integrated, don't model that process as white.

J. Dzielski in the actual textbook I refer to they say that, given the measurement model Z_{k} = H_{k}X_{k} + V_{k}, V_{k} is a zero mean white noise sequence of covariance R_{k} (as @Aleksandar Haber said) and that is statistically independent of both process noise and initial state. I recap that I want to simulate the effect of noise adding it to the simulated measured data which I DON'T generate using the above equation. Maybe I'm confusing the meaning of Z. The question was if I can translate the device accuracy specified at 2 sigma as measurement noise and add it to the simulated values of measurements. Thank you

I would assume 0.015 is 1-sigma or assuming the error is uniformly distributed sigma^2 = (2*0.015)^2/12. I am assuming that V_{k} is a measurement of a process V(t) at an instant of time. It may be that V(t) is white, and if your book says that, that's fine. If your book says V_{k} is white, that's wrong.

Michel Crimeni, those parameters are knobs you turn to tune the system so you should experiment with a range of values. But that's probably a good assumption to start with.

Best!!

AN

My suggestion is to revise the derivation of the Kalman filter equations in order to better understand what the matrices Q and R are. They are covariances of the process noise (disturbances affecting the state equation) and measurement noise. J. Dzielski : A pure Kalman filter is derived under the assumption that the process noise and the measurement noise are white, zero-mean, uncorrelated, and have known covariance matrices. A nice summary of the derivation of the Kalman filter equation is given in

Simon, D. (2006). Optimal state estimation: Kalman, H infinity, and nonlinear approaches. John Wiley & Sons.

See Chapter 5.1, page 124.

If the standard assumptions are not met (also the linearity assumption), which is often the case, then you should look into extensions and generalizations of the Kalman filter (see Chapters 7, 13-15 in the book). There are generalizations for the colored measurement noise and for nonlinear cases.

Final note: I do not think it is justified to randomly choose the covariance matrices. Maybe they can be seen as tunning parameters, however, if you want your filter to have a minimum (co)variance property then you should try to estimate these covariance matrices from the data, or look into adaptive filters that dynamically update covariance matrices.

A good reference to adaptive filtering is:

Sayed, Ali H. Adaptive filters. John Wiley & Sons, 2011.

For those of you who follow this question, here is a high-quality paper that explains how to estimate the covariance matrices from data:

The Autocovariance Least-Squares Method for Estimating Covariances: Application to Model-Based Control of Chemical Reactors Brian J. Odelson, Alexander Lutz, and James B. Rawlings

https://minds.wisconsin.edu/bitstream/handle/1793/10890/file_1.pdf?sequence=1

As I mentioned previously, it is not a justified practice to simply assume these matrices. Instead, they should be estimated from the data.

If you do not change or estimate the measurement noise matrix (according to the measurement geometry or any other effect) in each Kalman filter iteration then R will remain constant.

Assuming the measurement modelling (the relation between the state and the measurements) is error free you and assuming the laser device noise is zero mean Gaussian with std=1.5e-2 m, you can take R=std^2*eye(m) where m is the dimension of the measurement vector at each time step. If you are not sure about the measurement model is good, then you can take slightly higher R.

Here is a recent paper I wrote that explains and uses a variant of the autocovariance method for estimating and tuning covariance matrices of the Kalman filter.