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In this video, we're going to look at depriving the Tuti tracking system process model.

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So looking at how we can derive the equations of motion and put them into the matrix form that we want.

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So here we have the equations of motion, we have the acceleration, we have velocity and we have position.

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So we know that acceleration here is a second order differential equation.

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So what we can do is break it up into two first order equations like we've shown in the past, one four

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velocity, one four position.

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So solving this equation, assuming a constant acceleration, we can get the standard relationships

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that we've seen before.

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So these are basically saying the velocity of an object is equal to the initial velocity plus the acceleration

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and the amount of time that's applied.

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The same thing can be said about position.

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The position of an object is equal to the initial position, plus the time at the initial velocity,

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plus a half, eight squared.

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So half of the acceleration value eight times the time squared of the time it it's accelerating.

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So if we look at the acceleration, this is basically saying if we have a constant acceleration, we

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get a profile that looks like this.

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So we have one value that's held over a time period.

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Now, over this time period, we have the initial velocity v note and it'll be accelerating at a constant

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rate.

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The acceleration up to this kind of the V here at time T and if we have a look at the position, we

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can see the same thing.

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We can start with initial position, but this time we have a quadratic relationship to the final position

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at T, and this is because we have a constant acceleration.

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We have a first order increase in velocity, which means we get a second order increase in our position.

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So these are our first order equations.

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Now, these aren't differential equations anymore because we've solved the differential equation, but

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we can put this into a matrix form.

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So to do that, we can basically break it down into our position and velocity, so these become our

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states, we get a state matrix that looks like this.

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So this is basically saying the position in the final time is equal to pay, not plus tee times velocity,

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initial Qawasmi, plus an acceleration value multiplied by these coefficients here.

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And the same thing can be said about velocity.

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Velocity is going to be Alvino plus our acceleration value eight times the time it's applied.

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So this matrix equation form here, it's just the matrix form of these equations.

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Now we can basically convert this into a discrete time by saying are we not here?

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Is going to be OK minus one Alvie here is going to be OK and we're going to have a timestep of Doherty.

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So we set is a utility and basically we just make the substitutions here, here.

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So it's exactly the same equations except now we have to timestep k k minus one and timestep delta t.

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So these become our discrete time equations that they're going to use for the common Philidor.

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So we have to durata the equations for a one dimensional example that has a versity and a position in

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one dimension.

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So now we're going to extend this into a two dimensional example.

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And to do this is fairly straightforward.

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All we have to do is just put the systems together for the X and Y, so the X and Y are just going to

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be independent of each other, but they're just going to be replications of the one dimensional example

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that we had before.

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So we have an API.

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So the position in the X position in the Y, we have a V, X and a V, why?

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So the velocity in the X velocity in a Y and as you can see, if you look at the sub matrix for P,

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X and the X, it's just going to be the same one as we had before.

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An assembly for the P p, y and the VI is just going to be another replication of it before.

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So this matrix has no cross coupling between the X and Y axes, but that's fine to do in this example.

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And we can see for the input.

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Again, it is the same thing here.

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So now we have a X and Y for the acceleration, the extraction and acceleration in Y direction.

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They have one problem with this example is that it requires an input vector input vector of the accelerations.

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So this becomes a problem because we can't have acceleration sensors on board the object that we want

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to track for this example.

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So basically, we can't have an input because it doesn't fit the problem.

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So what we do instead, instead of having a known deterministic input, we assume that the input is

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just going to be due to noise.

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So basically we model the same matrix instead of a matrix.

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We have an L matrix, which is exactly the same.

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And instead of X and Y being a deterministic input, now they're random variables with a certain probability

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density function and we assume it has a normal distribution of zero main and a sigma of this X and a

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signal of a Y.

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So basically, these two premies sort of become attuning parameters, they say they don't tell the system

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how much we think the acceleration is going to vary in the extraction and how much we think it's going

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to vary in the Y direction.

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So we can't measure the acceleration directly.

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So we have to assume that it is a random variable and let the common filter work it out implicitly.

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So our precious motto becomes the process model shown on this slide here.

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We have dropped the input and we assume that the only input into the system is going to be from the

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noise value.

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So we have a noise value of Iwai, which are going to be part of a normal distribution with standard

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deviation shown here.

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We have our noise sensitivity metrics, which is just going to be now the copy of the input matrix.

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But now we're going to use it as a noise matrix.

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So we're going to have the process model here is going to be based on the current state, plus some

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random input, which is going to be from the noise.

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So this formed this great process model that we're going to use in the common field of the state, Vector

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X is going to be a four dimensional vector.

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It's going to have states X, P, Y, the X, V, Y.

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So the position and the X, Y and Vosta in the X, Y, the state transition matrix is going to be the

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simple concept matrix here, which is going to be a function of the timestep.

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And then we're going to have this noise input matrix here, which also is going to be a function of

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the time.

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Skip step.

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And it's going to be the input is going to be based on these two noise distributions here, which each

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have a standard deviation of some value for the noise density.