In this video, we're going to cover the prediction step for the unscented, come and fill it up now.
We will break down the prediction, step into the two cases based on the presence model noise.
So the first case is going to be for additive noise.
And in this case, the process model will have a form that looks like this.
So you can see that we have added noise at the end of the term here is not included inside the precise
model function.
We will also have a look at the second case, which is the general or non additive process of noise.
And in this case, the process model function itself has the noisy input as part of one of the parameters.
So we'll have a look at how the prediction step changes with these two process model functions.
So for the first one, the additive noise, which we're going to look at first, this is just a straightforward
implementation.
Using the incident transform.
However, for the second case, for the general process, model noise is a bit more complicated.
We're going to have to augment the state with the process model noise and we'll cover that later on
in this video.
So the first step in the prediction step for the additive process, more noise is to generate signal
points for the onset and transform for the prediction step.
So basically, we set the first single point for the zero point just to be the main of the current state.
And then for all the other points, for I equals one to two n where and is the size of the state vector.
We're going to evaluate this relationship here where this term here is going to be based on what I know
we're looking at.
So for the first I equals one to N we're going to be looking at this term here.
So this is a square root of the covariance matrix multiplied by the weighting function here.
And this is the eighth column of that matrix and then the Delta term for the rest of the terms from
the end, plus one all the way up to 210 is going to be this term here.
This is just going to be the negative of the square root of the Matrix.
And we're going again for each of the columns.
So this is the straightforward implementation that we've covered inside the enten and transform for
the general and senator transform.
So what this is saying is that if you look at the first single point for the zero point here, this
signal point is just going to be a copy of the Maine or the state vector.
Now, the next sticking point for point number one, and this is just going to be the Maine again,
plus the weighting of the square root metrics of the first column and then for all the other terms for
up to the tenth term is going to be the same thing.
So we're going to have all the different columns of the Matrix square root matrix all the way up to
end.
And that's going to be the end single point.
Now for the next and single points, we're going to do the same thing is that we're going to be subtracting
the columns of the square root matrix rather than adding them.
So I can say the only difference between these terms here and these terms here is just that we're subtracting
the terms.
The next step in the prediction process is then to use the process model of the system to predict all
the single points forward in time.
So the process is pretty much to take all the single points where I is equal zero all the way up to
two and put them into the state transition function this function here and then calculate the transform
to similar points.
So we're looking at all the single points we've calculated before.
What we do is for the first thing, the point we take it, we put it into the state transition matrix.
We evaluate that and then we get the transform single point for the short term.
And then we do pretty much do the same thing for all the other sigma points.
So we put them into the state transition function one at a time, and then we end up with all the different
transform sigma points.
Now the next step is then to recover the main and the covariance.
So to calculate the main, we basically calculate the weighted mean again using this equation here.
So these are the transformed sigma points here.
This is the weighting and the weightings again from the general unscented, transform, adjust these
weightings here.
Sorry, the first weighting is going be for our main.
It's just going to be this kapper over and plus cappa.
And then the waiting for all the other sigma points is going to be this time down here.
And we do a similar process for when we're calculating the covariance.
So we use the same weighting matrixes and then we just evaluate the covariance using this equation here.
But in this term now to get the final covariance for the prediction, we have to increase the uncertainty
based on the process.
Brunoise and then to do that, this is where the key metrics come in.
So this time here is the uncertainty from the transformation.
And then this time he calculates the process, model uncertainty.
So this adds the extra uncertainty for the process model.
So this is very similar to how the linear commas literally expanded my field of work.
We calculated first transform the uncertainty from the process model and then we add in the additional
uncertainty from the transformation.
So those steps were for the process model when we had the additive noise.
Now we're going to look at the general case when we have processed more noise inside the process model
function.
So basically, we want to augment the state and covariates with the noise model and what that actually
means is that we wanted to find a Augmentin state vector.
So this is Aleksei.
And the first and terms are going to be to be a copy of the state vector.
And then the last terms are going to be the noise vector itself.
So we're augmenting the state vector with the noise components.
We also want to do the same thing for the covariance matrix.
So we get this augmented covariance matrix with the diagonal terms being the original covariance for
the state estimates.
And then we had the covariance for the noise.
So this is the key metrics and covariance of our noise vector here.
So then we want to generate the signal points using the augmented state of covariance, so basically
for every sigma point we're going to get the first end states being the state vector and then the last
states going to be the noise into the system.
So what that actually means is that we use a process model of the system to predict all the signal points
forward in time, breaking each signal up into the state and noise inputs.
So when we evaluate the prediction step, we break the argument and state vector of the signal points
up into the state components and the noise components.
And we put the state components into the state input and we put the noise inputs over here into the
noise inputs.
And we evaluate this state transition function to get the transform signal points here.
Once we've got the transform signal points, it follows a very similar process, we want to first calculate
the main of the new state, which is again just calculating the weighted average of the transform single
points.
How about when we calculate the covariance?
We now no longer have to add in the Q matrix because we've calculated the covariance with the noise,
uncertainty inside the precise model.
So we basically just want to recover the covariance straight from the sigma points themselves.
So hopefully this should be a fairly simple explanation of how to go about implementing the prediction
step for the answer to come and filter.
We can also compare this to the extent of Hilda that we've covered in the past.
So for the process model prediction, i.e. the prediction step for the state, the extent to come and
fill it up pretty much just took in the previous main vector for the state estimate, put it into the
price.
This model took the input in and then we predicted it forward to get the main predicted main on the
output.
Now when we do the same thing for the extended coming for the state prediction, it looks a bit different.
We take in the original state estimates and the variance we generate our similar points according to
whatever transformation we're going to use.
Then we take our signal points and put them into the nonlinear process model to predict the Sigma points
forward.
Once we've had these transform signal points, then we can then estimate the main using this equation
here and all of that will come our new prediction.
So the the Appu I mean, we can also look at the same step for the covariance matrix, for the prediction
step.
So the extended comma filter basically took in the covariance matrix for the step.
It used the linear approximation or the Jacobins or the models that this equation here.
And then it gave us the new covariance prediction.
Now, the answer to my filter, again, is very similar as when we calculate the main.
So we have the same first couple of steps of calculating the signal points, transforming them through
the process model.
But then in the difference, we use this equation here to estimate the covariance matrix.
So we calculate that the weighted.
So we calculate the weighted covariance of the different signal points to get our new covariance matrix
for the predicted step.
So a quick summary of the prediction step equations for when we have additive noise, we first want
to calculate the signal points using these equations here.
So the Cirrus Sigma Point is just coming in the main.
And then for the sigma point, one to end is just going to be the main plus each of the columns of the
square root of the covariance matrix and then the remaining single points.
I just give me the main minus the square root of the covariance matrix for the different columns.
Next step is we want to do the state transformation, so we take our similar points, put them into
our process model and transform them into the transform single points from the transform signal points.
We can then estimate the state vector by calculating the weighted mean using these weighting functions
here.
And we can do the same thing for the covariance, making sure we add on the presence of noise at the
end once we've calculated the covariance.
The prediction that equations for when we have non-addictive noise, so in the general, Casey is going
to be slightly different.
The first thing we want to do is to augment the signal points.
So basically we take the main vector and then we want to augment it with the noise vector down below.
And we want to do the same thing with the covariance matrix before my augmented covariance matrix,
with the diagonal matrix being the original covariance for the state estimates and then the covariance
matrix for the noise.
We then calculate our signal points using the augmented state vector and augmented covariance matrix,
using the exact same process to calculate the sigma points, once we've got the same points, we can
use them in the straight transition.
So we take the augmented signal points.
We break them down into the state and into the noise, putting them into the appropriate places inside
the process model to transform the state vector into the new state or to predict the state once we've
calculated these transform single points.
Then again, we recover the main using the weighted estimate and we cover the covariance again using
the weighted estimate.
However, in this general case, we don't need to add on the extra noise term because it's already been
calculated inside the transform signal points inside this calculation here.