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‫Welcome back.

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‫So now that you know how the NPC controller works, we have to cover one final thing in our control

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‫architecture.

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‫We have to fully understand how the feedback lionization controller computes you, one that goes directly

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‫into the plant and how it computes fire and Seeta are that go into the NPC controller as reference values.

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‫And in this section you will fully understand how it all happens.

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‫First and foremost, we have to realize that this is a position controller.

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‫Therefore, we don't need the equations that relate control moments you to you three and four with the

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‫angular accelerations.

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‫The last three equations here, we don't need them.

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‫In fact, we don't even need the first three equations.

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‫Yes, they relate Phi Theta and you won with translational accelerations.

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‫However, these equations are in the body frame.

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‫It is much easier to perform feedback penalization when your equations for position control are in the

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‫inertial frame.

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‫And so that will be our first order of business to obtain the equations of motion for position control

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‫in the inertial frame.

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‫In fact, I'm going to leave it for you as an exercise.

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‫Your task now is to obtain the equations of motion in the inertia frame, but only for the position

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‫control.

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‫In other words, you need to have this X double that equals something Y double dot equals something

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‫and then the Z double dot equals something.

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‫You have to find what they equal to in the equation form.

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‫So when you obtained you that equals something.

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‫V dot equals something and then W dot equals something.

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‫Then those where your state base equations in the body frame that related the body frame translational

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‫accelerations with the control input you one and also the angles, Fi and Seeta.

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‫And if you remember from the second section, then your starting point for deriving these equations

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‫was the Newton's second law of motion.

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‫So the net force vector equals the time derivative of the linear momentum and the linear momentum was

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‫M times the velocity vector, so mass times the velocity vector.

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‫Now since your mass was constant, then this term here became zero and your force vector, which was

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‫a net force vector composed of individual forces, it was mass times acceleration.

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‫That was your starting point.

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‫And this is a screenshot from the previous videos in the second section, since you were in the body

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‫frame, you had to take the rotation of the body frame with respect to the initial frame into account.

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‫It meant that the unit vector time derivatives ie that Jadot and K that were not zero.

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‫They equal this.

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‫You see, you have the cross product here.

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‫And so in the end you're Newton oilor equations for describing the relationship between net forces and

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‫translational accelerations in the body frame which are here.

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‫They looked like this.

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‫So that was your equation of motion.

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‫And from there you went into the state based form.

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‫The cross product here comes from the fact that we were working in the body frame and not in the fixed

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‫frame.

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‫And then you went from the equations of motion form to the equation form like this.

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‫That was your procedure here and this is your space form here.

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‫And remember, you had to rewrite your net force vector in terms of individual forces.

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‫And we had this gravity force moment vector, then the gyroscopic force moment vector and then the input

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‫force moment vector here.

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‫They also included the three dimensions for the moments, but now we only need the dimensions for the

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‫forces.

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‫So the first three elements in these vectors here.

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‫So that's the stuff from Section two and three.

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‫And now in order to find X double that Y double dot and Z double dot and what they equal to you start

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‫from the same starting point, the Newton's second law of Motion.

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‫You followed the same derivation procedure.

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‫However, now you have to remember that you will be working in the inertia frame, not in the body frame.

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‫Your position vector is Gunma.

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‫In the inertia frame, which is X, Y and Z meters, then that's your rotation matrix here in case you

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‫need it after all.

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‫Now you're working in the inertia frame, so maybe you don't need it, but maybe you do.

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‫I will let you decide that yourself.

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‫I'm going to give you the final answer now.

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‫However, your job is to get from the Newton's second law to that answer.

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‫So that's your final answer.

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‫This is what you need to get in the end.

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‫That is your end result.

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‫So you have to get from here.

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‫To hear using this derivation procedure that we had in Section two, but now remember, you are in the

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‫initial frame, not in the body frame.

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‫So make your decisions accordingly.

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‫Take your position vector here in the inertia frame and apply the second Newton's law to it.

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‫Think of the different forces that the drone experiences.

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‫We covered them in Section three and then remember that now again, you are in the initial frame.

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‫Don't forget that.

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‫Not in the body frame.

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‫So try out this exercise and I will show it to you in the next video.

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‫Thank you very much.