‫Welcome back.
‫So this was our state's basic equation in a very compact form.
‫Now, of course, we have to dig deeper than that and express this equation in a less compact way in
‫order to be able to work with that.
‫However, that's just a matter of representation.
‫You just expand vectors and matrices and see what's inside.
‫So, for example, instead of writing this vector like this, we can write it like this where you have
‫each time derivative of the state equals and then something else.
‫But before we do that, we have to do something else.
‫You see this lambda B vector here?
‫It's a vector that contains the net force and moments.
‫So you can write it out like this.
‫You have your F net and then M-Net in the body frame and then transpose.
‫But these are the net forces and moments that in itself does not tell you much.
‫The net forces and moments are composed of individual forces.
‫In moments like this, you have your F one plus F two plus F three plus.
‫How many forces you have there equals your FNET and then you have your M one plus M two plus M three
‫plus.
‫How many moments you have there equals your net moment.
‫That's why we also call them the sum of the forces and the sum of the moments.
‫So one thing that we have to do before we express our Newton oilor formulation in the states space form,
‫we have to identify the individual forces and the individual moments that our drone is exposed to.
‫So this is your little drone here and you can be exposed to, for example, two F one, and maybe then
‫later to F to.
‫And then you can have moment one and then perhaps moment two, so all these forces in moments need to
‫be added together to get your net force and net moment.
‫But we have to know the individual forces in moments as well.
‫So what is our drone exposed to?
‫Well, first of all, as you can imagine, the drone is exposed to force of gravity, right?
‫So, again, that's your drone here, and since the drone is mass symmetric, then the equivalent force
‫of gravity vector acts through the center of mass of the drone.
‫So this is your force of gravity here.
‫And the center of mass of the drone is itself aligned with the origin of the body frame, so the origin
‫of the body frame is also here where you have the center of mass.
‫See that and dot.
‫Therefore, you can say that the first lamda and let's call it Lunda one vector B, we can say that
‫this is for gravity and lets for clarity also call it like this, Lunda G in the body frame.
‫So G stands for gravity so we can rewrite this lambda G in the body frame in this way here, here this
‫f g in the body frame is a three by one force vector that has three elements.
‫So it has effects F, Y, and then F, Z.
‫And remember, this is in the body frame and now it's important to note that there is no moment of gravity
‫because the equivalent gravity force vector goes through the center of mass of the wave and the origin
‫of the body frame is where the center of mass is.
‫If the body frame origin was somewhere else, like, for example here.
‫And let's just imagine that this is our body from x axis and then that would be our body frame y axis
‫and then body frame Z axis.
‫Then you would have a distance here between the origin of the body frame and the center of mass of the
‫drone.
‫And then with respect to this purple body frame, your force of gravity would create a moment.
‫But we don't want that.
‫So we do it in a smart way.
‫And we put the body frame in the best place where its origin matches the center of mass.
‫So we put the origin here in the center of the drone, that means that no movement is created in our
‫body frame and what you have is a zero three by one vector.
‫As your gravity moment elements, so this zero three by one vector looks like this zero zero and zero,
‫that's how it looks like.
‫By the way, what do I mean when I say equivalent gravity force vector?
‫Well, gravity is actually a distributed force.
‫Imagine that you have a square plate here, then force of gravity acts on every single point on this
‫object.
‫Let's say that the mass of the plate is 50 kilograms and let's say that the area of the plate is five
‫square metres.
‫You know that the force of gravity equals F G equals mass times G.
‫Well, that's approximate G to be 10 meters per second squared, even though in reality it's nine point
‫eight one meters per second squared.
‫But just to make it easier, let's just say that it's 10 meters per second squared.
‫Then that means that f some G equals mass times G.
‫So 50 times 10 equals five hundred Newtons.
‫In reality, though, this is not a point force, it's a distributed force.
‫So it would be F g divided by the area of the plate, and that would be five hundred divided by five,
‫and then that would equal 100 Newton per square meter.
‫That's how it is in reality.
‫However, since we assume that the plate is homogeneous, which means uniform in terms of mass distribution.
‫So you have constant density, constant kilograms per meter cubed, meaning that this part of the plate
‫is not heavier than this part of the plate, but the mass distribution is equal in all the locations
‫of the plate.
‫And this plate has all sides with equal length, then, for example, this side here has length L and
‫also this side here has length L and the same thing here L and then this one here as well.
‫L Then for modeling purposes, you take the total F G, which is five hundred Newtons, and you say
‫that if this force vector acts through the center of mass, which in our case is the middle of the plate,
‫let's say right here, then if this plate is a rigid body, then it has no difference whether you depict
‫this case in terms of distributed force of gravity or equivalent force of gravity f g that goes through
‫the center of mass of this plate.
‫It makes no difference in terms of what kind of behavior your mathematical model will predict.
‫No difference whatsoever.
‫It's just easier to model things with one force going through one specific point, like in this case
‫here s.m, but it's an equivalent situation.
‫Hence I'm using the word equivalent force vector going through the center of mass in the case of this
‫drone and also in the case of this plate, careful, though, these two representations are true for
‫rigid bodies.
‫Only if you remove this rigid body assumption or constraint however you want to call it, you are allowing
‫this plate to bend and it will bend differently when it is subjected to a distributers force compared
‫to a point force.
‫But that's more for structural engineers to worry about.
‫So I just wanted to clarify this point.
‫If your plate is rigid, then this representation and this representation, they're equivalent.
‫They will behave in the same way.
‫However, if they can bend, then they will bend differently.
‫But in our course, we don't worry about bending.
‫So that's your lamda gravity here.
‫So the question is, once the drones force of gravity, well, you know, that force of gravity equals
‫M times G, right?
‫So here is your exercise.
‫How would lamda G in the body frame look like as a six by one vector?
‫Again, avoid any lengthy calculations?
‫It is a one minute exercise and that's it.
‫Try it and I will show you the solution in the next video.
‫So essentially, if you know that F G equals M times G, right.
‫And you know that the force of gravity points down in the negative Z direction, then how would you
‫rewrite this vector as a six by one vector?
‫How would it look like?
‫See you in the next video.