I am using a linear Kalman filter for estimating the position using multiple IMUs. My model is adding the accelerometer readings' double integration onto the previous positions. The state vector is the positions of the IMUs and the input vector is the IMU accelerometer measurements.
Also, I have some constraints in the positions and I am using those constraints in the correction step of the Kalman filter.
So in the tuning step, I set the R matrix zero so that the Kalman filter relies on the measurement only (so not use the aprior estimate at all). It works fine unless there is no input. However, if I introduce some acceleration to the system, even if I set the R=0 and Q very high, the Kalman filter still takes the contribution from the aprior estimate. The question comes at that point: what is the functionality to set R=0 if it does not disregard the model error?
Another approach to satisfy this behaviour comes from the Kalman gain calculation. As R goes to zero, the Kalman gain should approach to the inverse of the observation matrix (H or C) and so that the a posterior estimate approaches to the measurement. Since the H matrix is not a square and full rank matrix in my case, I cannot intuitively check where exactly the contribution of the aprior estimates take place even though I set the R=0. If anyone has any comments, I would really appreciate.
Yes, you are absolutely right that the Kalman gain will approach inverse of H matrix when R is zero.
Also, the update equation is given by,
x(k) = x(k-1) + K*(z(k) - H*x(k-1))
which can be written as,
x(k) = x(k-1) + H_inv * z(k) - x(k-1)
OR
x_posteriori = H_inv * z(k)
which is independent of apriori estimate.
There is definitely something else going on in your implementation. Check the equations with constraints. I would also suggest you to put breakpoints on each equation of Kalman filter and investigate.
Hope this helps.
Your assuming zero noise on the process, defeata the purpose of using the Kalman filter
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