what we need to come out for with G and h. But again,

from the previous video,

we had these signals, when the inputs,

these output being in this case the angle and we will have a model that

corresponds to

the following equations.

So, what are G and h?

We can use first principles to derive this model.

The idea is that,

since the model is discreet,

first principle will give us a continuous time model,

which will arrive in a future video.

And the discretization of

that model will give us the model that I'm going to write, right here.

The intuition is very simple,

if you think about the angle,

what's going to happen is that the model is going to give us the current angle,

plus some variation of the angle which will be captured by V2,

and then multiplied by sum constant.

I'll tell you later more about the constant.

But, basically what this is saying,

is that the variation is zero of the angle,

and then when the speed is zero,

then the angle keeps the same at every k. Similarly,

we can do a model for X1,

and for X2 dynamics.

And for those models, we will have also some other constant,

call it d, this could be different.

And the change will be according to the sinusoidal function of the angle,

for X1 and the cosine for X3.

So, what this is basically saying is that if I have a particular rate of change,

the sinusoidal will project the variation over the right plane,

and the cosinusoidal will project the variation over

the right plane according to the angle of the vehicle.

And necessarily, we can tune these with the velocity

or the force thrust that will correspond to moving on that particular line.

So, if now we look at this dynamical model,

which again we will look at we will come up with the continuous time version,

from here you can read out the function G. So

the function G is essentially this expression that you see right here,

d and c are constants that depend on

what is called sampling time.

As we said before given initial state,

in this case initial position and initial angle,

given this constant that corresponds to the model itself,

and given that inputs V1 and V2 as a function of k,

we can now generate the change of the state according to these difference equation.

And generate the output according to this equation.

And, in this case,

generate how the angle will change over time.

As you probably realize,

this model is not very sophisticated.

It's a good discrete time model of a vehicle.

But, the benefit of now wrapping these around cyber component,

which will in particular implement a control algorithm,

will be that we will assign V2 in particular,

let's say go from a desire point into space no matter where we start.

So, they in particular a control problem will

be to drive the vehicle to this particular point,

with this particular orientation from any region of the space.

And that will require the second piece,

the cyber component that we have right here,

the interfaces and a way to design

the entire system correspond to a cyber physical system.