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I am trying to implement the paper Adaptive Kalman Filter for INS/GPS and there are a couple of expressions where the paper says that it comes from the Standard Kalman Filter theory, but I can't see why.

In the Annex a. Equation A1 says $$ C_{v_k}^{-1}v = R^{-1}v $$ Being $ C_v $ the covariance of the innovation, $ v$ the innovation itself and $R$ the measurement model covariance. I just can't see why is that.

What I think that this only happens only two conditions. In the optimal case, $$ C_{v_k} = HP_{k|k}H^T + R $$ Being $H$ the measurement model matrix and $P_{k|k}$ the estimated covariance matrix. It seems that for the first equation to be true, the first term of this last equation should disappear.

So I am guessin that for it to be true, 1. We must be in Steady state, where $P_{k|k} = P_{k-1|k-1}$ 2. In addition $P_{k|k}$ must be zero!

Is there any other explanation that I am not seeing?

The second equation that I don't see is equation A4. $$ (C_{v_k} - R)C_{v_k}^{-1} = H P_{k|k} H^T R^{-1} $$

Again, if I just take $$ C_{v_k} = HP_{k|k}H^T + R $$ and multiply by $R^{-1}$, I get to a situation where apparently $C_{v_k} = R$ I have to do the same assumptions, then, right?

Lastly, I would like to know why in equation (23), you can discard the terms related with the covarince matrix. The equation looks like this:

$$ \hat Q = \frac{1}{N} \sum_{j=j0}^k qq^T + P_{k|k} - H P_{k-1|k-1} H^T $$

and then just consider

$$ \hat Q = \frac{1}{N} \sum_{j=j0}^k qq^T $$

In a setting where H is the identity, and we are in a steady state, I see that you can exaclty do that. If it is not the case, then the two terms con be very different, can't they?

And a final note is that I think that there is a mistake, in equatino (16). The derivative $\frac{dQ_{j-1}}{d\alpha_k}$ should be $\frac{dQ_{j}}{d\alpha_k}$, I believe. Am I wrong?

Thank you very much. I hope I can open a nice discussion about the topic, and be of interest for people looking at the same nice stuff.

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