In this video, we're going to look at the measurement innovation calculations and how that applies
for the extended coming Filton.
So the common field of cricks for the state estimation errors by fitting back a weighted term based
on the observation of measurement errors.
So the measurement error is pretty much the difference between the predicted measurement from the process
model and the actual observed measurement from the actual sensor.
So the first step in the update process is to calculate the measurement innovation, which is this measurement
error.
And we also want to work out the measurement error covariance.
So this is going to be how much error we think is inside this innovation measurement.
So we'll start by looking at the measurement innovation, and this is for the state, we can predict
the current measurement.
So this is our said K. estimate.
So this is going to be our predicted measurement four times a week based on the current Empire State,
which is going to be ex hat for the Tompsett.
K. And this is going to be a AIPA because we have the negative.
So this is going to be using all the information up to this time step by not including this timestep.
And we can use the nonlinear measurement function hech so we have our predictive measurement is going
to be the measurement model evaluated at our current best estimate and with zero measurement noise.
We will also define the innovation, make the music to be the difference between the predicted measurement,
al-Said said hat and the true measurement value said, sorry, this is what we're going to be getting
from the sensor.
So the innovation is just going to be simply defined as here.
So the innovation vector is going to be the difference between the true measurement and what our predicted
measurement is.
And again, I'll predict the measurement is just going to take a current based estimate of the state,
put it into our measurement model to calculate a predictive measurement, and we just going to get the
difference between the two.
So you can see already the difference between the linear carbon filter and the extended carbon filter
in the linear algebra, we use the linear model here, while in the extent that we're using the full
nonlinear model here to calculate the innovation.
So the calculation of the measurement innovation for the state is fairly straightforward, so now we
need to work out how we can calculate the covariance of this measurement innovation.
So how do we work out the covariance matrix for the innovation?
So the innovation covariance is the matrix and we've seen is previously defined using this equation
here.
So so the covariance matrix is for the innovation is simply the expectation operator on the innovation
squared.
So this is basically innovation is a measurement error.
So this is the measurement error squared.
So this gives us the covariance matrix for the innovation.
And we can calculate this term by substituting in our equation for our innovation, and this gives us
this equation here.
So this equation is very similar to the linear algebra, is it, of using our measurement model metric
D.H we're now using the Jacobins instead.
So the Jacobin here is a Jacobean of the measurement model with respect to the state and the describing
of here is a Jacobean of the measurement model with respect to the noise.
So this is very similar to the linear algebra set of the matrixes and metric we now use that you Caribbeans
and again these definition of the describing list down here.
So we have the describing or the measurement model with respect to the state vector X, and we're going
to evaluate this at a current best estimate.
And the same thing for our measurement noise Jacobean.
We have the we have the partial derivatives of the measurement model with respect to the noise.
And again, we're evaluating it about a current state estimate.
So we can take a direct comparison between the two different types of the common filters that we've
covered, so for the linnear common filter, we calculate the innovation vector by basically the difference
between the current measurement and the measurement model.
And out of that, we get the innovation now for the extend the current filter instead of just using
the linear model, which we now have access to the nonlinear model.
So we basically subtract off the nonlinear model from our current measurement, so we get our nonlinear
innovation here.
So they're very similar, except now we're using the nonlinear measurement model compared to the linear
measurement model.
So that's for the innovation.
We can also have a look at the calculation of the innovation covariance.
The linear comma here, as we've seen before, just uses the measurement model metrics again to work
out the innovation covariance.
So basically it uses the covariance at the current time step multiplies it using the uncertainty, transformation
for the linear measurement model and adds on the additional noise from the sensor.
And this is where we get our innovation covariance.
The extend coming filter is pretty much exactly the same, except now we're using the linear approximation
by using the Jacobean of the measurement models instead of the matrices here.
And again, we can add on the noise covariance at the end if we just have additive noise.
So you can see comparing the two, they're very similar, we're just using a linear approximation instead
of using the linear model and we have to use a linear approximation because we have a nonlinear model.
So in summary, when we calculate the measurement innovation, we can calculate the nonlinear measurement
innovation by taking the difference between the measurement from the sensor, minus the estimated measurement
from the nonlinear measurement model, we can calculate the covariance inside the innovation, the metrics
by just using the linear approximation, which is the which is just using the Jacobean matrices, which
are the partial derivatives of the measurement model with respect to the state vector or the measurement
model with respect to the noise.
So the calculation of the innovation and the innovation covariance should be fairly straightforward
to what we've already done, it should follow pretty much similar steps as linear common filter is that
now we're going to be using the nonlinear model and we're going to use a linear approximation of the
nonlinear model.