In this video, we're going to look at how to calculate the measurement innovation, and this is going
to be the first step in the process of understanding the onset and coming through the update step.
So just like all common filters, the common filter corrects for the state estimate errors by fitting
back a way to time based on the observed measurement errors.
So the first step in the update process is to calculate the measurement innovation, which is just another
name for the measurement error and the uncertainty inside the amount of error.
So this is the innovation covariance.
We can predict the measurement.
So I said hat and innovation covariance is using the unscented transform of the measurement model function.
Then we can calculate the innovation vector.
So this means to be the difference between the measurement.
So I said hat that we used inside the unseen transform and then the true measurement that we get from
the center said such that the innovation is just going to be equal to the difference.
So the innovation is just the true measurement, minus our predicted measurement using the best information
of the state at this time.
We're going to break down the measurement innovation, step down into the two cases based on the measurement
model noise, and then the first case is going to be again, for additive measurement model noise.
So this is when we have noise on the sensor, which is just additive, so we can separate the nonlinear
model of the sensor into the nonlinear model and then the noise.
And then the second case is again going to be for the general or out of the measurement noise, noise.
So this is when the measurement noise is part of the nonlinear function for the measurement model.
So we can't just separate it out into an additive term.
So just like we did with a prediction set, the first method is just going to be a straightforward implementation
of the unseen to transform.
And then just like before, to handle the second type of noise, we're going to have to augment the
state with the measurement in the model noise.
So first for the case when we just have additive noise, so when the noise is added on at the end,
we can generate the signal points and weights for the Empire State and covariance matrix.
We then can use the measurement model function to predict measurements for all the generated signal
points.
So basically for each signal point that we've generated, we can put it into the nonlinear measurement
model and calculate a measurement predicted for that sigma point.
Once we've done that, we can estimate what the actual predicted measurement is by taking the weighted
mean.
So this is just the same equation for the answer to transform.
We're just calculating the weighted mean and that's going to give us the predicted measurement.
Once we have attained the predictive measurement, then we can calculate the measurement innovation.
So this is just the difference between the true sense and measurement and what we've just calculated
for the predicted measurement.
And then lastly, we can also calculate the measurement covariance.
So we just take we calculate the weighted covariance, which is going to be our matrix.
And then we also have to add on our noise uncertainty.
So this is the sensor model noise.
So we have to add on our metrics at the end to compensate for the amount of noise in the center.
So overall is a pretty straightforward method for calculating the measurement innovation for when we
have additive noise.
So now let's have a look at the case when we have a general noise or non-native noise.
So the noise is part of the nonlinear input to the measurement model function.
And in this case, we have to augment the state a covariance with the measurement noise model.
So first we want to generate an augmented state vector.
So that's just the current state and then we have the noise component.
At the end, we want to do the same thing to generate a augmented covariance matrix.
So we have the original state covariance and then on the diagonal we have the noise covariance inside
the sense of noise.
Once we form the Augmentin state and covariance matrices, we can then generate the Sigma points based
on these augmented state and covariance.
So our Augmented Sigma points are going to have a component for the sigma point for the state and a
single point for the noise.
We then can use the measurement model function to predict the measurements for all the augmented generated
sigma points.
So basically inside our nonlinear sensor model, we take the augmented state vector.
Break it down into the state and into the noise components.
We can put them into the nonlinear measurement model and then we get our measurement for this.
Each sigma point in turn, once we've got the single point, we pretty much do the same steps as we
did before.
We actually use the weighted mean of the signal transform signal points to estimate our predictive measurement.
Once we have a predictive measurement, we then calculate the innovation.
So the difference between the measurement and our prediction and then lastly, we calculate the innovation
covariance matrix just based on the weighted covariance of the prediction sigma points based away from
the main prediction.
And in this case, we don't have to add in the metrics because we've generated these signal points based
on the augmented state vector, which already has the uncertainty inside the measurement taking into
account.
So now for some extra clarity, let's compare the process for calculating the measurement innovation
between the extended Kamasutra and the unscented comment filter.
So like he would have a in the past, the extended homefield calculates the innovation by taking the
current state, putting it into the measurement model, assuming zero noise and then just subtracting
that from the true sense of measurement.
So this is how we calculate the innovation in the extended common field.
So it uses the nonlinear measurement model.
However, we're estimating about the current state, the antenna come a filter is a bit more involved.
So we take in the estimate the state and covariance.
We generate our signal points based on this information here.
We use these sigma points inside our known in the measurement model to get a number of predictive measurement
points.
Once we have all these predictive points we then use, we then estimate the main using the way to Maine
to calculate, I predict a measurement.
Once we had this predictive measurement, we then can calculate the innovation, so just the difference
between the true measurement and a predicted measurement and we get the innovation at the end.
We can also compare the steps for the calculation of the innovation covariance, so the matrix inside
the extended filter, it's fairly straightforward.
We take in the original covariance, we use the Jacobean approximation to the first order approximation
of the nonlinear model, and then we can add in our added anois term at the end.
So our matrix and out of this will come out.
Innovation covariance.
Now for the unscented Kamasutra, just like before, is a bit more complicated.
We take in the current state of the system, the state and covariance generate the single points, put
them into the nonlinear process model, and out of that we get a number of predicted measurements for
the Sigma points.
We calculate the way to Maine and then we can calculate the weighted covariance.
So to summarize all this up, when we have additive noise, the first step is to calculate the signal
points for the unseen and transform.
Once we've done that, we can then put them into the measurement model to transform them into the measurement
space.
Then we can recover the measurement prediction by basically taking the weighted mean.
And we can also then use that way to Maine to calculate the innovation and then we can recover the innovation
covariance from taking the weighted covariance of the transform signal points and add on the additional
time for the noise at the end for the case when we have non additive noise or general noise or a nonlinear
transform, first thing we want to do is to augment the signal points with the noise model.
So we on the to get the augmented state vector, we add the noise components to the end and for the
augmented covariance matrix we add the measurement covariance at the end.
Once we've done that, we then use the unscented and transform on these augmented points to find these
augmented signal points, we then use the augmented signal points inside the nonlinear measurement model
to calculate these transform measurement points for each single point.
We then can use the way to Maine again, just like we did before, to calculate the measurement predicted.
We can use the measurement predicted to calculate the innovation, and then we can also recover the
innovation covariance by using the weighted covariance of the transform measurement points.
However, in this case, we don't have to add on the additive noise termed The Matrix, because we've
already taken into account when we calculated the signal points because we use the augmented covariance
matrix up here.
So this whole process should look very similar to the prediction step because it's pretty much just
the unscented transform just applied to the measurement model rather than the process model.