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‫Welcome back.

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‫And so we have found our theater angle, and that was the expression for it.

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‫And next, we have to find our fire angle.

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‫If you remember in the previous video, we had two equivalent equations.

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‫We had tangent by equals this expression, and we also had tangent y equals.

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‫This expression here, these two expressions, they are the same thing, they compute the same thing.

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‫They both give you a tangent, FYI, so you can use either of them to compute your attention, Fi.

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‫And of course, you would find your fi angle like this, phi equals and then the inverse tangent.

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‫And then inside these parentheses, you can either put this expression or this expression.

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‫They will both give you a fi angle.

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‫And of course, you have this Seeta angle here and also here and then also here and here.

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‫However, you already have an expression for your seat diagonal.

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‫So you can just put this entire expression here and then you can just put it inside these Seeta variables

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‫here.

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‫I'm not going to do it because that's a lot of writing.

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‫But you get the idea.

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‫You have already computed your theta angle.

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‫You already have a number for that.

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‫So you can just put it into these variables right away in the code.

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‫You will also first compute your theta angle and then you will directly put that number, whatever it

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‫is, you put it here and then here and also here and then here.

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‫However, remember that D equals sine Sysop R and then C equals cosine sysop r and they are in your

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‫denominator.

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‫So tell me what happens when your sysop are equals.

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‫Either plus pi over two or minus pi over to what's going to happen.

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‫Well then your cosine sysop r will be zero, right.

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‫And you will be dividing by zero.

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‫And now tell me what's going to happen when your sysop are equals zero or pi radians or two pi irradiance

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‫or three pi radians or four pi irradiance etc..

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‫Well then your sine psi sabbar will be zero and you will be dividing by zero here.

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‫And so when the denominator equals zero, then your tangent phi becomes indeterminate.

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‫So we need to avoid that scenario.

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‫And so how would we deal with these singularities?

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‫Well, one way to look at it is like this.

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‫So I'm going to have to access here.

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‫This one will be cosine Sysop R and this one will be sine PSI subpar are I'm going to draw a circle

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‫here and this point here will be one.

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‫Also this point here will be one, this point here will be minus one and this point here will be minus

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‫one.

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‫So this is what we call a unit circle.

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‫And so this angle here, this angle would be your sysop are.

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‫So you can see that when you're sysop R equals zero radians, then your cosine sysop R equals one and

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‫you're sine sysop R equals zero because you are in this point.

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‫And then if you're sysop, our angle is pi over to some 90 degrees, then your sign isobar equals one

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‫and then your cosine sysop R equals zero.

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‫And so we have two equations here and what we can do is this.

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‫We can define certain regions.

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‫So, for example, this point here, that's when your sysop up are equals pi over four radians like

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‫this.

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‫Then here, it's three times pi, over four radians, here you have five times Pi over four radians,

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‫and then here you will have seven times Pi over four radians.

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‫So what we can do, we can say that if our size up our angle is in this region here or in this region

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‫here.

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‫Then choose this equation, because you will have a cosine sysop are in the denominator and then you

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‫will avoid dividing by zero because cosine would be zero if you were in this region here.

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‫And then if our sysop are is in this region instead or in this region.

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‫Then we're going to choose this equation and we know that while we are in this region, we won't be

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‫dividing by zero because in this region, sign cannot become zero.

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‫Sign will become zero if I'm in the purple region.

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‫And that's how we're going to avoid these singularities, the code is written in such a way that it

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‫takes this angle and then it determines whether it's in this green region or in this purple region.

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‫And then if you're in the green region, you choose this equation and then if you're in the purple region,

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‫you choose this equation.

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‫And then you take the inverse tangent of it in order to get your fi angle.

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‫And by the way, it doesn't matter whether your size up or is in between.

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‫Zero and two irradiance, or if it's.

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‫Between two pi and four PI irradiance, or if it's between minus two Pi and Zero Radiance, because

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‫remember, we have spiral trajectories so we can have multiple rotations and then we can also have rotations

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‫in another direction.

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‫In that case, your sysop are would be negative.

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‫So if you look at this circle, it doesn't matter how many rotations you have had or in which direction

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‫you rotate.

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‫What matters is that if you are in this purple region, then you need to choose this equation.

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‫Then you are guaranteed not to divide by zero.

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‫And then if you're in this green region, then you have to choose this equation.

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‫Then you will divide it by sine Sysop R and then you know that you will not have a situation where you

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‫will divide by zero.

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‫And so, in conclusion, if you are in the purple area, if you're Sysop are is in this purple region,

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‫then you will choose this formula here to compute your fire angle.

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‫So your inverse tangent and then here you have everything else.

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‫And then if you're Sysop are is in this region here, the Green region will then you will choose this

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‫formula, this one here.

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‫And so you have your sea triangle, you have your fire angle.

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‫And now finally we're going to find you one.

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‫We have an expression, Z, double dot plus G, and that equals cosine five times cosine theta times

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‫you one divided by and now you already know your fi angle and your angle.

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‫And now it's very easy to find your new one.

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‫If you rearrange these equations, then you one will equal mass times, the Z double dot plus G divided

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‫by cosine five times cosine theta.

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‫Now here, of course, you might look at it and you might think that OK, here I can also potentially

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‫divide by zero.

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‫However, in order for that to happen, either PHI or Theta angle must be plus or minus pi over to radians.

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‫However, we will never have a phi angle or a to angle plus or minus pi over to radians.

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‫It's not going to happen.

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‫Your PSI angle could be plus or minus power to radians because it was your angle, it was rotation about

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‫the body frame axis, but this is rotation about the body frame x axis and this is rotation about the

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‫body frame y axis.

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‫And now we're fine, thi tango's will be very small for our drone.

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‫If, for example, our fire angle was pie over to radiance.

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‫Then that means that our drone would be perpendicular to the ground.

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‫So this is our force of gravity here, and then the propellers would give you thrust in this direction.

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‫And obviously this kind of thing is not going to happen in our case.

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‫So we don't have to worry about it.

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‫We only had to worry about the angle because indeed, your drone can rotate like this without any limits.

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‫And one other thing that I want to mention is that once you have your fly and see the angles, they

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‫will be multiplied by an array of five once.

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‫Because remember, the NPC needs an array of these fire and seetha values.

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‫So, for example, this one here is your Feis up are that the NPC receives and then the NPC needs to

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‫track it like this.

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‫So the NPC controller will get these angles in this format, in this Fekter format, the role vector

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‫format.

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‫And of course, the planner will also give its size up our values to the NPC in this format, in the

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‫format of a role vector, so you will have five elements here.

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‫So that will come from the planner.

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‫And there you go, you have found your fire fighter and you won control input.

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‫And now I want to point out one thing.

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‫As you know, the mathematical model of our drone for position control was anything but linear.

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‫It was full of cosine and sine terms.

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‫So if you look at this model and then you can see that it's not linear at all, so the system is not

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‫linear.

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‫However, we could still use the poor placement method that was only suitable for LTI systems because

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‫remember, only in case of LTI systems, your poles were fixed in one place so we could use this technique

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‫in our controller, despite of the fact that we have strong nonlinearities in our model because we had

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‫these intermediate variables, error and error, dot and error double dot that did form an intermediate

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‫LTI system or LTI differential equation.

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‫So this one here.

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‫In our controller as an intermediate thing, we were able to have something that was linear time invariant,

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‫and it was this differential equation and because of that here, we could use our poor placement method

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‫in order to find the right e double dot that would always drive our error and our error dot to zero.

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‫As time goes to infinity and then you would connect the found e double dot with your Phi Theta and you

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‫won variables and that's where the name comes from.

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‫Feedback Lionization.

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‫You use linear techniques on a nonlinear system by utilizing an intermediate form that's in the LTI

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‫form, this one here.

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‫And then you go from the LTI intermediate form to the original nonlinear form, just like we did it

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‫in this section.

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‫All right then.

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‫And that is the end of this section and in fact it is almost the end of the course.

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‫I have covered all the theory that had to be covered here in the next and final chapter.

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‫I will give you a general overview of the code.

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‫Thank you very much.