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In this video, we're going to look at the measurement innovation calculations and how that applies

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for the extended coming Filton.

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So the common field of cricks for the state estimation errors by fitting back a weighted term based

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on the observation of measurement errors.

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So the measurement error is pretty much the difference between the predicted measurement from the process

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model and the actual observed measurement from the actual sensor.

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So the first step in the update process is to calculate the measurement innovation, which is this measurement

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error.

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And we also want to work out the measurement error covariance.

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So this is going to be how much error we think is inside this innovation measurement.

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So we'll start by looking at the measurement innovation, and this is for the state, we can predict

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the current measurement.

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So this is our said K. estimate.

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So this is going to be our predicted measurement four times a week based on the current Empire State,

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which is going to be ex hat for the Tompsett.

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K. And this is going to be a AIPA because we have the negative.

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So this is going to be using all the information up to this time step by not including this timestep.

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And we can use the nonlinear measurement function hech so we have our predictive measurement is going

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to be the measurement model evaluated at our current best estimate and with zero measurement noise.

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We will also define the innovation, make the music to be the difference between the predicted measurement,

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al-Said said hat and the true measurement value said, sorry, this is what we're going to be getting

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from the sensor.

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So the innovation is just going to be simply defined as here.

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So the innovation vector is going to be the difference between the true measurement and what our predicted

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measurement is.

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And again, I'll predict the measurement is just going to take a current based estimate of the state,

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put it into our measurement model to calculate a predictive measurement, and we just going to get the

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difference between the two.

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So you can see already the difference between the linear carbon filter and the extended carbon filter

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in the linear algebra, we use the linear model here, while in the extent that we're using the full

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nonlinear model here to calculate the innovation.

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So the calculation of the measurement innovation for the state is fairly straightforward, so now we

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need to work out how we can calculate the covariance of this measurement innovation.

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So how do we work out the covariance matrix for the innovation?

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So the innovation covariance is the matrix and we've seen is previously defined using this equation

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here.

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So so the covariance matrix is for the innovation is simply the expectation operator on the innovation

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squared.

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So this is basically innovation is a measurement error.

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So this is the measurement error squared.

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So this gives us the covariance matrix for the innovation.

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And we can calculate this term by substituting in our equation for our innovation, and this gives us

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this equation here.

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So this equation is very similar to the linear algebra, is it, of using our measurement model metric

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D.H we're now using the Jacobins instead.

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So the Jacobin here is a Jacobean of the measurement model with respect to the state and the describing

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of here is a Jacobean of the measurement model with respect to the noise.

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So this is very similar to the linear algebra set of the matrixes and metric we now use that you Caribbeans

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and again these definition of the describing list down here.

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So we have the describing or the measurement model with respect to the state vector X, and we're going

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to evaluate this at a current best estimate.

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And the same thing for our measurement noise Jacobean.

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We have the we have the partial derivatives of the measurement model with respect to the noise.

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And again, we're evaluating it about a current state estimate.

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So we can take a direct comparison between the two different types of the common filters that we've

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covered, so for the linnear common filter, we calculate the innovation vector by basically the difference

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between the current measurement and the measurement model.

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And out of that, we get the innovation now for the extend the current filter instead of just using

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the linear model, which we now have access to the nonlinear model.

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So we basically subtract off the nonlinear model from our current measurement, so we get our nonlinear

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innovation here.

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So they're very similar, except now we're using the nonlinear measurement model compared to the linear

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measurement model.

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So that's for the innovation.

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We can also have a look at the calculation of the innovation covariance.

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The linear comma here, as we've seen before, just uses the measurement model metrics again to work

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out the innovation covariance.

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So basically it uses the covariance at the current time step multiplies it using the uncertainty, transformation

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for the linear measurement model and adds on the additional noise from the sensor.

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And this is where we get our innovation covariance.

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The extend coming filter is pretty much exactly the same, except now we're using the linear approximation

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by using the Jacobean of the measurement models instead of the matrices here.

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And again, we can add on the noise covariance at the end if we just have additive noise.

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So you can see comparing the two, they're very similar, we're just using a linear approximation instead

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of using the linear model and we have to use a linear approximation because we have a nonlinear model.

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So in summary, when we calculate the measurement innovation, we can calculate the nonlinear measurement

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innovation by taking the difference between the measurement from the sensor, minus the estimated measurement

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from the nonlinear measurement model, we can calculate the covariance inside the innovation, the metrics

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by just using the linear approximation, which is the which is just using the Jacobean matrices, which

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are the partial derivatives of the measurement model with respect to the state vector or the measurement

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model with respect to the noise.

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So the calculation of the innovation and the innovation covariance should be fairly straightforward

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to what we've already done, it should follow pretty much similar steps as linear common filter is that

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now we're going to be using the nonlinear model and we're going to use a linear approximation of the

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nonlinear model.