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We are now going to have a look at how we can use the measurement innovation inside the update step

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for the antenna to come and filter.

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So the APRA state estimate, this expert with the bar at the top can be updated with the current measurement

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innovation.

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Allow me to form the A posterior, I estimate.

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So this hat with the plus using the following equations.

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So these are the common of update equations, so the top equation is just the normal recursive predictor,

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correct characterful.

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It takes the current state estimate, it adds on a scaling gain based on the innovation to update the

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state.

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However, for the answer coming up, we're going to calculate the gain slightly differently.

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We're going to still use the inverse of the measurement innovation matrix.

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However, now we're going to calculate this covariance matrix.

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This is the cross covariance between the measurement and the current state.

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And then for recap, the pariah state four times HepC, our expat bar for TimeStep K is the expected

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value of the true state, using all the measurements up to the previous time, stepped up to K minus

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one while the posteriorly state is this Alleycat plus.

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At times Rapkay is the expected value, is the expected value of the true state using all the information

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up to the current timestep.

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Now the April state error covariance matrix Alpay minus can also be updated with the same information

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to form the posteriorly state estimate covariance matrix.

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Albay plus using the following equations.

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So we have the previous covariance before the update.

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We want to subtract off Alcaide times as times k transpose.

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So this is how much uncertainty we can reduce the estimates by to form the new covariance matrix.

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And again, for recap purposes, the APRA covariance will timestep k is just the covariance of the expected,

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just the error covariance between the true state and estimated state using all information up to the

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previous timestep so up to K minus one.

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While the posteriorly covariance outputs is the expected error between the true state and estimated

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dates for when we use all the information or all the measurements up to the current TimeStep K so we

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can compare the process between the extended comma filter and the unseen incoming filter for the update

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step for the state.

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So as you would have previously seen for the extended comma filter, we use the same predictive corrective

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form where we multiply the innovation by again Matrix K to get how much we need to update the current

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state to get the best estimate for the current information.

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And then we also use our Jacobean of the linear approximation of the measurement model to update our

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gain matrix.

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K Now for the unscented common filter.

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The first equation operates in the very same way because it's still a pretty to correct filter how to

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calculate the gain matrix.

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We're using this crosscourt variance tampoe, which is across covariance between the state and the measurement.

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However, we're still using our inverse over here.

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So you can think of this term up here is being how much I change measurement will affect the state multiplied

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by the uncertainty in the state.

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So this is pretty much transforming the uncertainty into the measurement form down here.

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The same thing.

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This is how much uncertainty there is between the state and the measurement divided by how much error

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is inside the measurement.

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So this here is basically the same thing, is basically a ratio of how much uncertainty there is in

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the measurement compared to how much uncertainty there is in the current state.

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Now for the update step for the covariance inside the extent we can use this equation here.

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So, again, this is using the Caribbean approximation of the nonlinear model, however, for the extended

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that we're going to use this approximation here.

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So this equation here can actually be used inside the external and filter as well.

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However, inside the extended common filter, since we have this Jakobsson approximation of the system,

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it is possible to simplify this equation down to this equation when we're using the Jacobean.

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However, since we don't have a Jakobsson in the unscented common filter, we have to keep it at this

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more higher level form as shown here.

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So in summary for the unscented coming through the update step, we want to use these equations to calculate

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how much we may want to change your state by.

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So we calculate the common again using this covariance matrix, which is across covariance between the

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state and the measurement.

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And we have our update step in the correct form based on how much innovation area we have.

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And then when we want to update the common field of covariance, we use this equation down here.

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So we want to calculate how much we want to shrink the covariance by by how much information the center

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is giving us at that time.

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Step.