‫Welcome back.
‫So I hope that you've tried this little thought, exercise yourself and let's look at it now.
‫If you're PHI and Theta are zeros, then this element here will be zero.
‫Cosine zero will be one size zero over cosine zero will be zero because you would have zero divided
‫by one.
‫And now in this term you would have one times zero.
‫So zero minus sine zero is also zero.
‫And here you will have coastline zero over cosine theta.
‫So one over one so that it will be one.
‫And so you will have a one here, here and here and the rest of the elements will be zeroes.
‫So with this assumption, your transfer matrix becomes an identity matrix.
‫And that is very good, because if the transfer matrix is the thing that connects your P, Q, R and
‫find our Theta and apply that, then if you have an identity matrix here, then that means that you're
‫Phi, that equals P Seton, that equals Q and sign, that equals R.
‫And so when we reach the section in which I teach you the NPC controller, you will see that this result
‫here will simplify our NPC math greatly here.
‫I want to point out something here.
‫With this transfer function, I literally assume that my sine Phi and Tangent Theta are zeros.
‫This is a little bit different from a small angle approximation where I would say that my Phi's approximately
‫zero and my seat is approximately zero.
‫In that case, sine Phi and Tangent Theta could be approximated like this Phi and Theta.
‫This kind of approximation is more precise than what I'm doing here.
‫However, if I do it in this way, then I will not be able to express my transfer matrix as an identity
‫matrix.
‫And then these relationships here, they would not happen.
‫However, if a drone operates close to its hovering position, which means that the tilting angles,
‫PHI and Theta are very close to zero all the time, then if my velocity and acceleration requirements
‫are small, then this rough approximation for the controller will still get the job done.
‫I will lose in precision, but I will gain in simplicity.
‫In other words, in the plans where this assumption is not made, if you apply a controller to it,
‫where this assumption is made, then the drone will still be able to follow the trajectory in a very
‫smooth way.
‫It's important to understand is that this simplification that we have made, this is for the controller.
‫I'm not doing this simplification in the plant.
‫That's not the case in the plant.
‫I want my mathematical model to be as realistic as possible.
‫So I keep my transfer matrix inside the plant in its original form that contains all its PHI and theta
‫angles in the plant.
‫I'm not simplifying anything, but in the controller I do.
‫And then in the controller I will be able to get this identity matrix in this relationship and in the
‫controller I will replace my p q r variables with fi that theatergoer and that not in the plans but
‫in the controller.
‫And then in the simulation you will see that the plant, without the simplifications and without the
‫substitutions, can be stabilized with this controller.
‫That does include this simplification and the substitutions provided that the velocity and acceleration
‫requirements for the drone are small and the drone mostly operates close to its hovering position,
‫meaning that there is tilting with very small fire theta angles.
‫And now you see why it is convenient to choose as this convention rotation about the moving from Axis
‫first Z, then Y than X, even though we have to derive one additional matrix, which is the transfer
‫matrix, but now you see the added value of it.
‫So for translational velocities, you VW that we measured in meters per second here, we do use our
‫rotation matrix and then we get this vector in the inertial frame.
‫That's translational velocity.
‫But with angular velocities, what you measure with radians per second, this convention will give us
‫a very convenient transfer matrix that relates my oilor angle time derivatives with my angular velocity
‫in the body frame.
‫And when your velocity and acceleration requirements are small, then it will allow you to use this
‫assumption.
‫If you had chosen another convention about moving from Axis, then you would still have two angles.
‫But maybe you would have PHY and Passi or Seeta and BPCI and BPCI was your motion.
‫Right?
‫And so you could not have said that.
‫OK, let's assume BPCI to be zero.
‫That would have been impossible.
‫I mean, if you have to fly in a spiral then obviously your sci angle will vary a lot.
‫And also if you had chosen your R, X, Y, Z convention, the rotation about the fixed axis, then
‫you would have had three angles here and then this result would have been impossible as well.
‫And starting from the next video, we will actually derive the transfer matrix mathematically.
‫Thank you very much and I'll see you in the next video.