In the previous video, we've gone through the calculation steps for the measurement innovation, for
the extend the common filter and we've shown the and we've shown the equations for the innovation and
the innovation covariance.
So now we actually want to go through the derivation of how they get these equations so we can actually
prove that they're still valid.
And it is valid to use the Jacobean or the linear approximation of the nonlinear models inside our calculation
steps.
So we have seen that the innovation vector and the covariance matrix are calculated using these two
equations here.
So the first equation is for the innovation.
The second equation is for the innovation covariance.
But where do these equations actually come from?
How can we actually know that these are the correct equations?
They seem pretty straightforward if we were to use a linear comma filter and then apply the linear approximations,
but how do we know that that is actually a valid how do we actually know that mathematically?
We are valid to do that and we're not just using our best guess.
And again, we want to start with the Taylor series expansion, so this is how we take the linear approximation
of the non-linear model so we can take the Taylor series expansion of the measurement model.
So we have the measurement model said is equal to each of these.
So with the state vector, our noise vector is a measurement model and that is our actual measurement
vector.
And again, we want to take the Taylor series expansion of this measurement model.
We want to take it about our state vector X.
So basically we want to take the expansion about a best estimate X hat and we want to take it with the
expansion using zero using zero measurement noise.
So looking at this equation here, we take the Taylor series approximation, which gives us this equation
here.
So we have our measurement model evaluated at our best estimate, and then we have the Jacobin multiplied
by our preparation between the true state and our estimate of state, plus the Jacobean before the measurement
noise.
And again, we're going to have a lot of these higher order terms as we include more of these higher
order terms.
We get a more accurate Taylor series expansion estimate.
However, we just want to limit ourselves to the first term's early, so always higher order terms and
gets canceled out.
So when we do this, we get left with an estimate of our measurement model, which is just going to
be the Taylor series expansion, and it's going to be this equation set here.
So this is what we're going to based our equations for the extended coming field run.
So this is going to be the key identity that we're going to use.
So this is how we can turn the linear filter and make it applicable for non-linear systems, we use
a first order approximation of the nonlinear system.
So let's start off by looking at the extended feel, the innovation step for the innovation or the actual
state.
So using the conditional expectation of the predictive measurement which is shown here, so this is
saying the expected operator on the true measurement, given all the information up to came on, this
one is going to be our predictive measurement.
Now, on the previous slide, we've calculated what a first order approximation of the measurement is
going to be so we can take our approximation for a set and substitute it into this equation here to
calculate an approximation for our predictive measurement.
And of course, it looks like this.
So this is straightforward, we're just substituting indirectly for al-Said K from the equation, from
the previous slide.
So this is the Taylor series expansion.
Now having a look at this equation, there's a few things we can see now since this whole equation is
is inside the expectation operator, we can get used the expected value for anything that is stochastic.
Now, we know from our definition that the expected value of a best estimate state, given all the information
up to came on this one, is just going to be the best estimate of XQ.
So basically the difference between these two parameters here should pretty much be zero.
So if we take the expectation overall that we can just cancel this turn out here, we also know that
the expected value of the noise should be zero as well, because the noise should have zero.
I mean, which means these three terms cancel that.
And we're just left with the expectation of our measurement model and we just get left with the estimated
value of our measurement model evaluated at our best estimate.
Now we know the expected value of a constant is just the constant.
So we can pull it right out.
I mean, get this term here.
So the predictive measurement is just going to be out.
Measurement model evaluated at our best estimate with zero measurement noise.
Now, this is pretty much straightforward and you would have thought that we don't need to go through
the steps to work this out.
It should be self-evident.
But just going through the steps here, we've proved ourself correct.
So we are good to just say they are predictive measurement in the extended filter is just going to be
the measurement model evaluated at our best estimate.
So letting on from this, we can say that the innovation is simply the difference between a true and
the predictive measurement error which is given by this equation here.
So this is how we come up with our innovation equation.
So that has been looking at the innovation itself now, he wanted to do the same thing, the same steps,
but for the calculation of the innovation covariance.
And again, we start off with the definition.
We know the innovation covariance is defined as the matrix, which is just the expected value of the
innovation squared or the measurement error squared.
So the first step is that we can substitute in our approximation for said K. into the innovation victim
victime to give this equation here.
So we've taken our equation for said K. And we subjected it into this equation from here and we end
up with this term here.
So the first set of terms here is a Taylor series expansion, and then we're subtracting it off from
our predict for our predicted measurement.
So we're going to get minus one measurement model, which is minus our measurement model over here.
And again, looking at this equation, it should be fairly straightforward that these two terms here
are the same.
So these cancel out and we get left with this equation here.
So the innovation is pretty much the Jacobean of the probation and state plus the describing of the
noise.
So now that we've worked out the approximation for innovation, we can take our approximation of the
innovation and into this expectation of right up here and we get this equation here.
So this is the covariance of the innovation.
And it should be fairly straightforward to see how we've got into this equation.
So when you substitute this into this equation here, take the expectation of right up.
We're left with this definition down here.
So this is prove that we can use the Caribbean as a first order approximation of the nonlinear system.
So this direction has shown us that we haven't just taken the linear, common field of equations and
randomly, randomly substituted in for the matrixes.
Mathematically, it makes mathematically it is valid to use the describing of the system to come up
with these equations here.
So in summary, when we look at the measurement innovation, we can calculate the measurement innovation
using this equation.
So the innovation is just the difference between the true sense and measurement, minus the predicted
since measurement based on the measurement model evaluated at the best estimate and the measurement
innovation, covariance, again, is just going to be this covariance transformation where the transformation
matrix is for the president's model and for the measurement model.
I'm just going to be the describing of the measurement model with respect to X the vector and the describing
of the measurement model, respective noise for the measurement noise.