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So in the previous exercise, you had to go through and implement the prediction step of the common

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Philidor.

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It's always good practice that if you have a simulation of the model that you're trying to estimate,

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then run the simulation without any noise from a known initial condition and then run the common field

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of prediction, step from the same initial condition and see that the two are consistent.

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Now, this is an important validation step.

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It shows that the common field of model that you're using is appropriate for the system dynamics that

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you're trying to estimate.

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And we want to run this without the update step from the common filter.

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This is because if we run it with the update step, sometimes the correction step can hide any errors

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in the model.

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So it's always good to make sure that the underlying prediction is as best as possible for the model

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that we want to try to estimate so we can have a look at this when we look at the code.

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So now let's switch to the code.

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So to start off this, let's first start running the simulation with a zero initial state and zero covariance

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and noise.

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So now let's set our initial state.

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So let's set our state just to be four zero zero zero zero zero.

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So this is zero for the exposition, for the Y position, for the V, X and VI.

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So the effect has being defined in this form here.

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We're also going to use zero for the position, standard deviation and for the velocity standard deviation.

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We're also going to use zero for the noise standard deviation.

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So our random variable for the acceleration is going to be zero.

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So now let's build this and then run the simulation and we should be able to see what the simulation

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does.

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So we can see as we start running the simulation, you can see that this is where the Red Cross is,

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the estimated state of the system, and it starts and stays at zero because we have zero input into

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the system.

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You can see up here is our vehicle state.

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So we're traveling at five meters per second at a 45 degree angle.

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And you can see the field of state say zero.

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And as we run, we are increasing X and Y and velocity error.

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So this is to be expected.

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So this shows that.

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So this shows us that.

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We've profiled one.

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The contest drives 45 degrees at five meters a second.

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So now let's actually put in the true side of the system so we know the origin that the so we know that

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the car starts at the origin.

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So zero zero.

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And we know we're traveling at five meters a second at 45 degrees.

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So let's change our velocity here to be five meters a second and at a 45 degree angle.

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So now let's make this again.

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So let's build it and then let's run it.

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OK, so now we can see that the common field estimate.

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So the red line here with the Red Cross being the estimated position for the current frame, you can

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see that it's tracking the truth.

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So it's moving along at the correct speed and velocity.

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And you can see down here, we have very small we have zero any Armus error.

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So this is because we put in the true state for the initial condition.

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We are not adding any noise into the system.

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And the system itself is just going on.

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It's just undergoing a constant velocity at five miles per second at heading 45 degrees.

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So this just proves that the prediction model that we're using is working correctly.

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If we had any errors in the model, we can we will start to see errors creeping up here.

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So the variance in the filter must always represent the approximate level of the uncertainty inside

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the field or estimates, and this needs to be done through a correct model and correct tuning.

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What this means is that the true state of the system is within the uncertainty bounds around the estimated

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state.

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If this is not the case, then the field is called inconsistent.

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And if the filter is inconsistent, this leads to bad and incorrect estimates and it degrades the whole

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estimation process.

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In some cases, it pretty much means that the estimated state from the filter is worthless.

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It's useless if you're trying to use it for anything, it will just cause problems.

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So we've checked that the state transition model is working correctly, it can correctly predict from

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a known initial condition the true state response.

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We also want to check that the prediction model correctly transforms the uncertainty.

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The uncertainty in the system should not be increasing.

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If our key metrics process model noise is equal to zero.

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Rather, it should be transformed, appropriate with the system model.

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So on the slide here, we can see the common field, our prediction set for the uncertainty.

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So this is the current uncertainty.

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This is the predicted uncertainty.

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These are the state transition matrixes.

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And this is the process model noise.

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So this step here takes the process model.

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So this step here takes the uncertainty at the time came on minus one, transforms it through the process

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model to get the new uncertainty at the current time.

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So this transforms the uncertainty from the time set.

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It does not actually inflate the uncertainty, does not increase the uncertainty in the system.

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It just transforms what's appropriate with the system dynamics.

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Whereas this key metrics here, this actually increases or inflates the uncertainty inside the system.

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So if cuz equal to zero, we're just transforming the uncertainty based on the process model.

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If CU is non-zero then we're actually increasing the uncertainty in the system.

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So what we can do is that we can check the position uncertainty is operating as expected, and to do

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this, we set the initial position uncertainty to be a non-zero value.

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We set the initial velocity, uncertainty to be zero, and then we can run the filter and see that the

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position uncertainty stays the same.

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So let's do that with the code that we have developed.

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So now let's set up the initial covariance to be a non-zero value for the position, but for the velocity,

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uncertainty will keep it at zero.

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So let's set our position on Sunday to be five metres.

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So now we can build this and we can run it.

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And in this situation, we should see that the position uncertainty stays constant.

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So you can see as we run the simulation, we have a the three sigma uncertainty for position.

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This red circle here stays at a constant value.

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So in terms of the position, uncertainty, prediction for the covariance update, it is working correctly.

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This is what we expect to see.

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So the invariants prediction for the position and certainly seems to be operating as expected, so now

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let's check that the velocity uncertainty is operating as expected.

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And to do this, we're going to set the initial position on the any to zero, but we're going to set

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the velocity uncertainty to a non-zero value.

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And when we do this and run the field, we should see that the velocity uncertainty stays at the same

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value.

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But the position uncertainty in this case should grow linearly with time.

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And this is just due to the dynamics of the system.

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So if we integrate diversity, we should get a change in position.

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We can also change the velocity uncertainty to be a non-zero value.

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So let's change the velocity, uncertainty to be one and would change our position and certainly back

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to being zero.

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So let's make this and run it.

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So now we should see the effect of the Awasthi uncertainty inside the simulation, inside the covariance

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prediction.

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So we can see as we run the simulation, the uncertainty for position starts off very small and then

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it grows at a constant rate in accordance to the velocity uncertainty.

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So this is what is to be expected.

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So we've looked at the uncertainty due to the process model.

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Now we want to let you have a look at the process model noise.

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So this is the key metrics, part of the uncertainty equation.

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So we want to check that the process model noise, the acceleration uncertainty is operating as expected

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to do this, we're going to set the initial position and the uncertainties to zero.

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And we want to set a non-zero acceleration stand deviation.

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So we want the key metrics, the process model noise matrix to be non-zero.

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So when you run the filter, we should say that the total level of uncertainty in the system should

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be growing with time.

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Now, this means that the velocity uncertainty should be increasing linearly and the position uncertainty

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should be increasing quite radically as expected due to the system dynamics, because we know that position

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is a second integral of acceleration.

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So let's carry out this experiment, Velcade.

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And then lastly, we can change our initial velocity and position uncertainty back to being zero and

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we'll start off with a non-zero acceleration noise value.

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So we're going to assume that the random variable modeling the acceleration as a non-zero value.

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So let's build this and see what what happens.

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So in this case, we should see that the error or the uncertainty grows exponentially.

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So let's run this so we can say that the initial position, uncertainty starts off at zero and we can

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say we get this increasing uncertainty in position as time goes on and we can actually see that our

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position uncertainty.

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So the three sigma balance here is actually growing quite radically with time.

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This is because we get a first order integration from acceleration, diversity and then from velocity

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to position.

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So it is a quadratic exponential response for the covariance.

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So the process model noise increases the uncertainty in the system to capture the stochastic nature

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of the system dynamics.

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Now, in this simple case where we have an object that starts at a known origin and moves at a constant

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velocity and there's no randomness, randomness inside the system then is not really all that important.

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This is why we the.

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So this is why we can set a key matrix to be zero, because there's no uncertainty in the system.

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The only uncertainty at the moment is around our initial guess of the position and velocity.

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So in general, when we're running a common field, the key metrics or the process of noise can compensate

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for a common Philidor process model that does not correctly or accurately model the true system dynamics.

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And this works by increasing the uncertainty inside the system, which helps to cause a filter to stay

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consistent so that it helps to make the uncertainty around the estimates to actually include where the

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true state actually is.

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The presence of noise ties closely and with the measurement of the noise and the performance of the

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filter, so we only cover this in a lot more detail later on.

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So hopefully now you have a better understanding of why we went about the different exercises in the

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way that we did, a key takeaway point from all this is that whenever the designing or implementing

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a common filter, if we have a simulation of what the true system is going to do, we should run that

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and run the company through the prediction step to make sure the two are consistent or as consistent

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as possible.

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If we do this, then there's a good chance that the model we have selected for the common filter should

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be appropriate for the system in that mix that we're trying to estimate.