‫Welcome back.
‫So let's see our exercise.
‫The difference between the force of gravity and the control input force and torques is that the control
‫inputs are in the body frame now, not in the inertial frame.
‫And since we work in the body frame, we don't have to worry about rotation nor transfer matrices.
‫Therefore, this vector here looks like this.
‫That's how it looks like thrust acts in the body frame, the Z direction.
‫That is why it's the third element.
‫And then you to use three and four.
‫They are control moments that are the fourth, fifth and sixth element of this vector.
‫So that's it.
‫You have another force and moment vector.
‫Okay.
‫So do we have other significant forces or moments?
‫Well, in a matter of fact, we have to consider one more thing.
‫And that thing is called a gyroscopic effect.
‫And I think here we need to do a bit of explaining.
‫When I tell you gyroscopic effect, I want you to associate it with angular momentum change in direction.
‫So just remember this expression angular momentum, change in direction.
‫And angular momentum is a vector that has a direction.
‫And so when the direction of that angular momentum vector changes, then that's when you have your gyroscopic
‫effect.
‫First, let's use a classical mechanism to explain it.
‫Imagine that I have a rod from the original.
‫Oh, till the desk.
‫This desk rotates at omega radians per second.
‫In the positive y direction.
‫So your right hand thumb points in the positive y direction.
‫The entire structure is connected to 0.0 in such a way that it is free to rotate about the x axis.
‫The disc and the rod have a mass.
‫Their combined center of gravity is a shown in the image.
‫And so through that point you have the force of gravity of the entire structure which points in the
‫negative Z direction.
‫And so you have a moment ahm d from 0.0 until CG, thus you have a moment vector due to gravity about
‫the negative x axis its magnitude is f g times.
‫DX.
‫Now normally you would expect the entire structure to swing about the negative x axis, and that would
‫be the case if the disk didn't rotate.
‫However, the disk rotates about the positive y axis and this disk has a mass moment of inertia.
‫Therefore you have an angular momentum vector h equals I times omega about the positive y axis.
‫And now something interesting happens when you introduce a torque or moment into the system like we
‫did with the moment due to gravity.
‫The angular momentum vector h starts chasing that moment vector due to gravity, the h vector will try
‫to align itself with the moment vector.
‫As a result, the entire structure starts rotating about the positive z axis.
‫Let's say that it rotates by the alpha radians.
‫So the alpha is the differential angle, the instantaneous change of alpha.
‫And so there is a differential change in the H vector in the direction of M minus X, the moment about
‫the negative x axis due to gravity and that change is d h how do you get that?
‫D You take the de alpha vector that points in the positive Z direction because that rotation is about
‫positive Z axis and then you cross it with the h vector.
‫So the h equals the alpha cross.
‫H right hand rule shows you that the cross product gives you a direction in the negative x direction.
‫If you divide both sides by the t, you will have d h over d t equals the alpha over t cross h or simply
‫h dot equals alpha dot cross h.
‫But remember the h vector is also a torque vector.
‫And so when you have rotating elements like for example, rotors in the drone, then you have to account
‫for that h dot and now we know that in our this case h dot that points in the negative x direction it
‫equals alpha dot in the positive z direction, cross the h vector in the positive y direction.
‫What happens next, of course, is that the moment vector will also rotate by the alpha radians about
‫the positive z axis.
‫And so the h vector tries to catch the moment vector, but it never catches it.
‫The entire structure keeps rotating about the positive z axis.
‫I can show it to you in a small experiment.
‫I have a small gyroscope here.
‫You can see the disk.
‫It rotates.
‫If I hang it as shown in the video, then we pretty much have our disk case.
‫When the gyro is not rotating, then it wants to rotate about an axis that points into the wall.
‫That's because of gravity.
‫Of course it will not stay in the horizontal position.
‫However, I will now rotate the gyro disk very fast.
‫You can see from my right hand thumb that the disk rotation and thus the angular momentum vector point
‫towards you.
‫And now when I make the gyro disk rotate and when I let go.
‫Then as expected, the entire thing rotates about the positive z axis, which is your alpha dot.
‫So the h vector is chasing the moment vector that is caused by gravity, but it never catches it.
‫And the change of h vector with respect to time h dot which points in the direction of the moment due
‫to gravity and which itself is a moment too.
‫You can get it like this.
‫H dot equals alpha dot cross.
‫H But let's do another experiment.
‫I will now rotate the gyro disk in the opposite direction.
‫Now the angular momentum vector will point away from you.
‫Watch what happens.
‫The moment vector due to gravity hasn't changed.
‫It still points in the direction of the wall initially, but the h vector that points in the opposite
‫direction away from you.
‫It starts chasing the moment vector.
‫As a result, the entire structure rotates in the opposite direction.
‫Alpha dot points in the negative z direction.
‫You can see in the image that the h vector points in the negative y direction.
‫M minus x and h dot still have the same direction, but the entire structure rotates about the negative
‫z axis.
‫Thus alpha dot also points down in the negative z direction.
‫To get h dot, you apply the same formula.
‫H dot equals alpha dot cross.
‫H The right hand rule will give you the same direction.
‫What I want to emphasize is that physically you first need to introduce a torque into the system like
‫this M minus X, and then the angular momentum vector H will start chasing it.
‫For example, consider this case.
‫You have this disk and the counterweight on the other side.
‫So C g is at 0.0.
‫The gravity does not make the system tilt.
‫There is no moment due to gravity.
‫You have the disk rotation and thus the h vector in the positive y direction.
‫If you now apply a talk about the negative z axis, let's call it M minus Z.
‫Maybe because you apply forces as shown in the image, then the H vector will start chasing the m minus
‫z vector.
‫Then your alpha dot is about the negative x axis, but still h dot equals alpha dot cross h.
‫So always think in terms of h vector chasing the applied moment vector, then it's pretty intuitive.
‫Thank you very much.