‫However, now we will make an important assumption, you see, this June cannot have a large fire angle
‫and it cannot have a large city angle.
‫And why is that?
‫Well, let's look at the way these side view.
‫Let's establish our coordinate frame here, the inertia from Z axis is up, the inertia frame Y axis
‫is this one.
‫And then the axis that points towards you, that comes out of the screen, that would be the inertia
‫from x axis.
‫And let's say that this here, it would be the body from the axis.
‫This here would be the body frame Y-axis and then the body frame X-axis would go straight out of the
‫screen towards you.
‫That means that this here is more or to you have more for here.
‫And actually, you would be looking at moral one from the side, which would be this one here, and
‫the more three would also be not seen because you're looking at the drone directly from the side view,
‫from the body frame perspective.
‫But it would be behind that drone.
‫So this here would be your model, too, which is this one here, then you're more four would be here,
‫more one would be here, which you cannot see because it's from the side.
‫And then behind the drone, you would have more three, which is from this perspective on the other
‫side of the drone.
‫But we can draw more one and more three separately.
‫So this is more one and then this is more three, and so we assume that we have a big fire angle like
‫this, this would be our fire angle.
‫And you know that this is a fire angle because you can see that your body frame X X-axis has the same
‫direction, like your initial frame x axis.
‫And so if you rotate about the body frame X-axis and then your body frame, X-axis has the same direction,
‫like the initial frame x axis, then that means that this has to be a fine angle here.
‫And it's a pretty big angle.
‫And so the thrust vectors, the tease that I'm going to draw here and here and here and also here.
‫They are produced by the rotor's and they are perpendicular to the rotors.
‫All right, so this is your model and your trust factor is perpendicular to it if you not decompose
‫this t vector.
‫Along the inertial y axis and the inertial z axis.
‫And let's call it T, Y, and then T, Z, then you see that the component in the inertia frame Z direction
‫is quite small.
‫And I will also do the same thing here for this murder and for this murder.
‫And for this mother as well, and so the bigger the fire angle you have here, the bigger this angle
‫is.
‫The smaller the thrust vectors vertical component is, it makes sense, right?
‫The bigger the inclination, the smaller the vertical component becomes, because the thrust vector
‫will become more and more horizontal.
‫And also you have a force of gravity in the negative inertia from the Z axis.
‫Right.
‫So this is the positive axis and down it would be a negative inertia from the axis.
‫So that would be a force of gravity.
‫That acts through the center of mass of the drone.
‫And so if the angle fi is not small, then the vertical components of all the four motors will not be
‫able.
‫To counteract the downward force of gravity.
‫And so your net vertical force will be downwards and you can computed like this, you have your four
‫vertical thrust forces four times.
‫They are our ports, so they are positive they're these ones here you have four of them, four times
‫two minus the force of gravity that the UAE is experiencing.
‫And so if you're four times Tisza is less than your F g.
‫And we're talking about the magnitudes here.
‫Well, then your net vector will be downwards.
‫And then you will also have a component in the positive inertial y direction.
‫You have T.Y. here to here, here and also here, so it would be four times T.Y. like this.
‫And so that means that the drone will start sliding downwards and then the downward sliding direction
‫will be like this.
‫Your F net vector will be in this direction.
‫Four times, T.Y. will be in this direction.
‫And that would be your result in force.
‫OK, your let's say F, r and so your drone will start sliding down in this direction.
‫Of course, if you had some mega powerful propeller's, then they could produce such an awesome thrust
‫that would produce enough vertical component to counteract gravity even when and Seeta angles are not
‫small.
‫However, that's not the case with us.
‫Besides, in that case, the horizontal velocity of the UAE would be pretty large because of this horizontal
‫component of the thrust force.
‫And this is something that you might not want.
‫And so that means that in order to follow the trajectories, obviously there will be some PHY and Seeta
‫angles, however, they will be small, they have to be small.
‫Because only then the UAE can counteract the downward force of gravity and it won't start sliding down
‫like this.
‫And because of that, for control purposes, we can assume that fire and Seeta angles are zero radiance.
‫And this assumption won't affect the NPC controllers performance significantly, we are not going to
‫touch the transfer matrix in the planned model because we want the plan model to model the real system
‫more realistically.
‫However, if we can expect that the fire and see tangles will be small.
‫Then we can simplify the mall in the controller and we can use this simplified version in the controller,
‫and the way we simplify the model for the oveI is by assuming that PHI and Theta Angles both equals
‫zero inside the transfer matrix.
‫So if we now assume that this is our oves system and this is your feedback lionization controller,
‫so it goes here as well and your planner is here.
‫And then the planner also sends a reference sci angle here to the controller and then the feedback lionization
‫controller will send it's you one into the plan.
‫So what we want to know is that if in the planned box, in the transfer matrix, the fly and then the
‫Seeta angles, when they are not zero.
‫Then if we assume the fire and then the Seeta angles in the transfer matrix to be zero radians in the
‫controller.
‫Will this controller with this assumption still be able to control this plan without this assumption?
‫Well, will this simplified controller still control the plan?
‫Well, in which you did not have this assumption in the transfer matrix and since.
‫The U.S. needs need to maintain small fire, see tangles in order to follow a trajectory, then the
‫answer is yes, it can.
‫The controller with this assumption will simplify our lives.
‫But you will sacrifice very little in its performance.
‫And so this is an exercise for you.
‫Now, what will the transfer matrix equal to if your are and theta angles equals zero radiance?
‫So just make all five Seeta angles zero here in all the elements and see what you will get for your
‫transfer matrix.
‫Try it out yourself and then see you in the next video.
‫Thank you very much.