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‫Welcome back.

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‫So now that we have an expression for our thrust force, for one murder, we can compute the control

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‫inputs you one, you two and three.

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‫And remember, the thrust force for model one was the thrust factor.

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‫See T times the angular velocity of the model squared.

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‫So first, let's look at you one.

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‫So this is our drone here and here you have all these four mortars and of course, this is your body

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‫frame by frame X-axis and body frame y axis.

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‫And let's put here a body frame, a Z axis.

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‫The thrust force produced by more than one will be this one.

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‫So time one equals C. Sometimes Omega squared, and now since we're dealing with multiple murders here,

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‫then let's specify the angular velocity here.

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‫So this angular velocity, it belongs to more one because each motor can rotate differently at a different

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‫angular velocity.

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‫Then this one could be the thrust force for murder two, which is seize up the times omega two squared.

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‫So that's the angular velocity form or two.

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‫And note that we assume that for all the murders, the trust factor will be the same because essentially

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‫the blades are the same for each model is just each murder rotates at a different angular velocity than

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‫this one will be thrust force four more three, which is again the same thrust factor times Omega three

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‫squared and finally time four equals C 70 times Omega four and squared.

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‫So to get you one, which is the total thrust force generated by all the models of the drone, you can

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‫find it like this.

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‫You want equals thrust from model one, from model to from three and from more four.

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‫You can also rewrite it in this form here.

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‫And now you can just factor out the trust factor.

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‫And then in the parentheses.

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‫You will simply have the squared values of all your angular velocities for all your Maurer's.

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‫And you one, of course, has the units of Newton, because it's a control force, and even though omega

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‫two and Omega four have negative rotation, but since you square them.

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‫In this way, then they will still be positive omega two squared and positive Omega four squared.

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‫So that's your total control thrust force generated by the four models that you have.

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‫You simply sum up each individual thrust force from each model, and then you will have your total thrust

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‫total, you won.

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‫Now, if you remember then you was your control moment about the body frame X-axis.

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‫So the drone would angular really accelerate about the body frame, X-axis like this, so the rotation

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‫would happen about the body frame x axis and about the body frame x axis only, not about the X and

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‫Y axis now, only about the X axis.

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‫And to create this situation, what we had to do, we had to have equal thrust force by model three

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‫and by more one.

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‫Both of these thrust forces, they are equal and they have to be equal because if they were not equal,

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‫then the drone would start rotating about the body frame y axis.

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‫But we don't want that.

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‫But we do want the drone to rotate about the body from x axis, and so in order to do that, we're going

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‫to apply less thrust here and we're going to apply more thrust here.

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‫So your T sub M2 is bigger than your T sub M4 and also from the center of each mortar to the origin

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‫of the drone.

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‫You have a distance and let's call this distance El.

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‫So the moment from more to is about the positive x axis, if you follow your right hand rule, so it

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‫would be t sup m two times L because the distance from this point, up until the origin of the drone,

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‫that's also L so it will really equal C T times Omega two squared and that's the thrust part.

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‫And then times the length of this rod from the center of the drone to the center of the model.

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‫Now there is also a moment generated by motor for but this time, according to the right hand rule,

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‫your thumb would point in the negative X direction and therefore the moment generated by M4 will be.

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‫Minus T, up M, four times L equals minus C, D times Omega, four squared times L.

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‫And so the first equation was the moment generated by M2 and this was the moment generated by M4, and

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‫so youtoo equals M sub M to minus M sub M for.

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‫And when you do that in your moment from M-2 is bigger than your moment from and for because you're

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‫thrust force from M-2 is bigger than from and for, but your links are the same.

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‫If that's the case, then there will be a positive net moment about the body from X-axis, and that's

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‫when the universe will start rotating about this axis.

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‫And so you can rewrite this equation in this form and then what you can do, you can factor out see

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‫safty times now and then in the parentheses you will have omega two squared, minus Omega four squared,

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‫and that will be your Yutu or your pitch control moment.

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‫And similarly, if we want to produce the control roll moment or U3, then what we have to do, we have

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‫to keep the thrust forces equal in M4 and M2.

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‫And that means that the moments that these two task forces generate, they will cancel each other out

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‫and then the drone will not rotate about the body frame x axis.

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‫However, we do want to have rotation about the body frame Y-axis.

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‫And to achieve that, we're going to have a small thrust force in M1 and we're going to have a big thrust

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‫force for M three this time your T sub M3 is bigger than your T sub M one.

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‫However, both lengths they will be the same.

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‫So a positive moment about the body frame y axis will be generated by more three M's up M three.

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‫And you know, that is positive because if you take this thrust force and you multiplied by this distance

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‫here, then according to the right hand rule, you will start rotating about the body from y axis,

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‫meaning that your thumb will point in that direction.

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‫And then the moment from Model one will be in the opposite direction, it will rotate about the negative

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‫body frame y axis and now you simply rewrite them like this.

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‫This is moment three and minus, this is moment one.

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‫And so if you factor them out, then this is what you will get, this will be your U3, your role control

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‫moment, and that will make the drone rotate like this as shown here.

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‫And note that U2 was a moment, so its units are Knewton meters and the same thing applies to U3.

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‫It's also a moment.

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‫But you one was a force and therefore its unit is Newton, so now we have related you won you two and

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‫you three with the Omega's.

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‫However, to find the control moment called your, which is EUFOR.

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‫For that, we have to dig a bit deeper.

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‫And we're going to do that in the next video.

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‫Thank you very much.