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‫And now I want to draw your attention to this green bean, if you notice that this green beam in this

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‫trajectory six is tilting down more and more.

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‫So you see right now your green beam is like this, but as time goes by, it tilts more and more.

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‫So you see before it was here and now it's already here and it keeps tilting down more and more.

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‫So why is it like that?

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‫Why does this green beam tilt down more and more?

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‫So it seems like you need more and more centripetal acceleration.

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‫And of course, this thing is not up to scale.

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‫Obviously, the drone itself is not tilted like that.

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‫So these subplots here they are just to show you the direction in which direction, the way we still

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‫think it's not measured in magnitude.

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‫So you see, your horizontal scale is like this, but then your vertical scale is like this.

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‫So in reality, if you were to make these two scales up to scale, then it wouldn't be tilted that much.

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‫But at least it gives you an indication of the direction of the tilting.

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‫And you can see that it tilts more and more.

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‫So let's explore that.

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‫Well, what's the formula for centripetal acceleration?

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‫The formula is like this.

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‫The centripetal acceleration equals velocity squared over radius.

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‫And in our case it would be like this.

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‫Our use squared.

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‫That's your tangential velocity and then divide it by the radius.

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‫So as your USV travels in a spiral in which your radius is increasing and so that's your x y plane here,

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‫you're looking at the spiral from the top.

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‫Then approximately your tangential velocity is you and then your radius would be this one here, and

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‫why do I say approximately?

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‫Well, that's because your drone actually goes up as well.

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‫Right.

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‫So if you really want to be precise.

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‫Then this is your body frame X direction, then this is your body from the Z direction and this is your

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‫inertial frame Z direction.

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‫And so in reality, your velocity would be something like this because you have to go forward and then

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‫you have to go up as well.

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‫So you need to have some kind of component in the inertial frame Z axis and then also some kind of horizontal

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‫component.

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‫So like this, the horizontal component would make you fly forward and then this vertical component

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‫would make you fly up, but then we can also decompose this vector like this.

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‫We can decompose it in the body frame X direction and in the body frame Z direction is just a matter

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‫of how I decompose my vector.

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‫And so that would be my view velocity, and then that would be my W velocity, but just to make the

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‫matter simpler, I'm going to neglect it because I'm going to assume that the velocity is a lot greater.

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‫And so I'm going to assume that this green vector here, I'm going to assume that this is my U velocity.

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‫And that's why I said that it's approximately two meters per second.

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‫But it's a simplification at this point.

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‫The precision doesn't matter that much because what I'm trying to show you is the concept.

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‫And so if this is your centripetal acceleration here.

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‫Then as our are our radius goes up in time, then our centripetal acceleration actually should go down

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‫as well because your arm is in the denominator and the bigger R will give you a smaller centripetal

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‫acceleration.

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‫However, this you velocity also becomes bigger and you is squared and it's in the numerator.

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‫So that part increases the centripetal acceleration.

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‫So if I print my centripetal acceleration like this, so you squared divided by square root, X squared

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‫plus Y squared, then this one in the denominator, that would be my radius if I printed here.

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‫You see, this is my U.

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‫Squared, divided by the square root of X squared plus Y squared.

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‫Then you see that my centripetal acceleration goes up in time because you square it contributes more

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‫to it than this square root denominator.

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‫You see, this is your centripetal acceleration here that I have printed out and it goes up in time.

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‫And you can only increase your centripetal acceleration if you tilt your positive Y side down more and

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‫more.

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‫And well, actually, I can make this equation a little bit more precise so I can say that the centripetal

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‫acceleration equals square root, you squared plus W squared.

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‫So now I am taking the body from the Z velocity into account.

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‫And I can divide that by the radius, which is this one here, this is the initial frame X dimension,

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‫and this is the initial frame Y dimension and the same thing here.

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‫This is your initial frame X and inertial frame Y.

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‫And so I have written this equation here.

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‫And let's now print it.

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‫You can see that it did not change much.

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‫You still end up with a centripetal acceleration of one point something, but the point that I'm trying

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‫to make is that your centripetal acceleration is growing.

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‫And that's why if this is your drone here and this is your body from X-axis and this is your body frame,

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‫why access?

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‫Then this part here needs to tilt down more and more in order to give you this higher centripetal acceleration

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‫as your radius becomes bigger.

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‫So a bigger radius in time results in a bigger centripetal acceleration, which means that this side

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‫of the oven needs to tilt down more and more in time.

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‫So now you know the reason.