‫And now I want to draw your attention to this green bean, if you notice that this green beam in this
‫trajectory six is tilting down more and more.
‫So you see right now your green beam is like this, but as time goes by, it tilts more and more.
‫So you see before it was here and now it's already here and it keeps tilting down more and more.
‫So why is it like that?
‫Why does this green beam tilt down more and more?
‫So it seems like you need more and more centripetal acceleration.
‫And of course, this thing is not up to scale.
‫Obviously, the drone itself is not tilted like that.
‫So these subplots here they are just to show you the direction in which direction, the way we still
‫think it's not measured in magnitude.
‫So you see, your horizontal scale is like this, but then your vertical scale is like this.
‫So in reality, if you were to make these two scales up to scale, then it wouldn't be tilted that much.
‫But at least it gives you an indication of the direction of the tilting.
‫And you can see that it tilts more and more.
‫So let's explore that.
‫Well, what's the formula for centripetal acceleration?
‫The formula is like this.
‫The centripetal acceleration equals velocity squared over radius.
‫And in our case it would be like this.
‫Our use squared.
‫That's your tangential velocity and then divide it by the radius.
‫So as your USV travels in a spiral in which your radius is increasing and so that's your x y plane here,
‫you're looking at the spiral from the top.
‫Then approximately your tangential velocity is you and then your radius would be this one here, and
‫why do I say approximately?
‫Well, that's because your drone actually goes up as well.
‫Right.
‫So if you really want to be precise.
‫Then this is your body frame X direction, then this is your body from the Z direction and this is your
‫inertial frame Z direction.
‫And so in reality, your velocity would be something like this because you have to go forward and then
‫you have to go up as well.
‫So you need to have some kind of component in the inertial frame Z axis and then also some kind of horizontal
‫component.
‫So like this, the horizontal component would make you fly forward and then this vertical component
‫would make you fly up, but then we can also decompose this vector like this.
‫We can decompose it in the body frame X direction and in the body frame Z direction is just a matter
‫of how I decompose my vector.
‫And so that would be my view velocity, and then that would be my W velocity, but just to make the
‫matter simpler, I'm going to neglect it because I'm going to assume that the velocity is a lot greater.
‫And so I'm going to assume that this green vector here, I'm going to assume that this is my U velocity.
‫And that's why I said that it's approximately two meters per second.
‫But it's a simplification at this point.
‫The precision doesn't matter that much because what I'm trying to show you is the concept.
‫And so if this is your centripetal acceleration here.
‫Then as our are our radius goes up in time, then our centripetal acceleration actually should go down
‫as well because your arm is in the denominator and the bigger R will give you a smaller centripetal
‫acceleration.
‫However, this you velocity also becomes bigger and you is squared and it's in the numerator.
‫So that part increases the centripetal acceleration.
‫So if I print my centripetal acceleration like this, so you squared divided by square root, X squared
‫plus Y squared, then this one in the denominator, that would be my radius if I printed here.
‫You see, this is my U.
‫Squared, divided by the square root of X squared plus Y squared.
‫Then you see that my centripetal acceleration goes up in time because you square it contributes more
‫to it than this square root denominator.
‫You see, this is your centripetal acceleration here that I have printed out and it goes up in time.
‫And you can only increase your centripetal acceleration if you tilt your positive Y side down more and
‫more.
‫And well, actually, I can make this equation a little bit more precise so I can say that the centripetal
‫acceleration equals square root, you squared plus W squared.
‫So now I am taking the body from the Z velocity into account.
‫And I can divide that by the radius, which is this one here, this is the initial frame X dimension,
‫and this is the initial frame Y dimension and the same thing here.
‫This is your initial frame X and inertial frame Y.
‫And so I have written this equation here.
‫And let's now print it.
‫You can see that it did not change much.
‫You still end up with a centripetal acceleration of one point something, but the point that I'm trying
‫to make is that your centripetal acceleration is growing.
‫And that's why if this is your drone here and this is your body from X-axis and this is your body frame,
‫why access?
‫Then this part here needs to tilt down more and more in order to give you this higher centripetal acceleration
‫as your radius becomes bigger.
‫So a bigger radius in time results in a bigger centripetal acceleration, which means that this side
‫of the oven needs to tilt down more and more in time.
‫So now you know the reason.