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‫Welcome back.

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‫So now you know, the logic behind the control law, e double dot equals K one times E, which error

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‫plus key two times the E dot, which is the time derivative of the error.

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‫The same control law is applied for controlling and driving the error to zero in the X, Y and Z dimension,

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‫essentially by choosing the right pulls.

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‫So in our case, Lunda one equals minus one and Lunda two equals minus two, and in our case for all

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‫X, Y, Z dimensions.

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‫By choosing those polls, we computed our key values.

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‫So our K one ended up being minus two and key two was minus three.

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‫And so that means that our control law became like this error double dot equals now K one was minus

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‫two, so minus two times the error plus and then Katou was minus three, so minus three times the error

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‫dot.

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‫So in the end you can write it down like this.

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‫And so through this control law, you found an E double that that drives the error and the error dot

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‫to zero as time goes to infinity.

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‫So whatever your E and E dot are, the E double that that you find with your K one and K two values.

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‫This error double derivative with respect to time will drive your error and the first derivative of

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‫your error with respect to time to zero in this error, double that, it can do it because it influences

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‫both.

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‫Remember, it's a chain, the E double that influences the IDOT and then the E that influences the error

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‫itself.

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‫If your error is zero and then error that is zero and then error double dot is zero, then if you're

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‫error double that goes up, then also your error that goes up and then because of that your error will

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‫go up as well.

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‫So they are linked with each other like a chain because error that is a change of error with respect

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‫to time and error.

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‫Double that is a change of error dot with respect to time.

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‫And that's what you want, right?

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‫You want your error and error does not go to zero and you want it for your X, Y and Z dimensions,

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‫because then that actually means that your drone is tracking the trajectory.

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‫If the errors for all dimensions go to zero, then that means that you're doing very well in tracking

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‫what you need to track.

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‫And by the way, you need both error and error dot to go to zero, because if your error is zero but

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‫your error dot is not zero, then you are not going to stop at zero error.

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‫You will overshoot.

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‫You see, this is your reference line and then you're trying to track it.

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‫And right now your error is zero, but your error dot is not zero.

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‫So you will overshoot like this.

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‫You are not going to be able to steadily approach zero error if your so-called error velocity does not

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‫jump out as well.

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‫However, if both of your polls are negative, then your error and error that will both damp out.

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‫So is the problem solved now?

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‫Well, not quite.

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‫So far, the feedback lionization comptroller has produced error, double that in the X dimension,

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‫in the Y dimension and in the Z dimension.

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‫However, in order to actually physically achieve that, you need to have certain control, input values.

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‫You won you two, you three and you four, it's very cool to compute the necessary E double dots that

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‫would make your dream track the trajectory.

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‫However, now we need to find the necessary control inputs that will actually make it happen.

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‫And when you find your actual control inputs, then ultimately you also need to find the corresponding

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‫omega one, Omega to Omega three and Omega four, because ultimately these are the things that will

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‫physically move your drone and ultimately the Omega's, they will give you the necessary double dots.

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‫And so we already know how to compute our Omega's from our control inputs.

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‫And our feedback immunization controller computes you won.

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‫But you, too, you three and you four are computed by the model predictive controller, but for that,

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‫the NPC needs to get a reference your wangel.

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‫This is what it will get from the planner and then it also needs to get fi are and to are.

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‫And that will come from the feedback lionization controller.

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‫So in our controller, the E double dots, they are the intermediate variables, the end products of

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‫our feedback lionization controller.

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‫Are you one fi, r and Theta are.

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‫And that means that we have to find a relationship between our erodable dots and you one fire and see

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‫to r and so how to do that, how to find this relationship.

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‫Well we know that E double dot X equals X double dot the reference values.

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‫So the second derivative of our reference X values minus the true X double dot, which is the acceleration

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‫in the X dimension in the inertial frame, then e double that in the wider mentioned equals Y double

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‫dot reference values minus the true Y double dot and then E double dot in the Z dimension equals Z double

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‫that reference values minus the true Z double dot values.

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‫Remember these reference value, double derivatives, they come from the planner.

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‫We computed them in the planner and so we can rearrange these equations like this X double dot equals

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‫X double that R minus the error double dot in the X dimension then Y double dot equals Y double dot

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‫subha minus E double dot in the Y dimension and then Z double dot equals z double dot sub R minus E

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‫double dot in the Z dimension.

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‫So these terms here, they come from our control and these terms here, they come from our planner.

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‫So you automatically know what's your X double dot, y double date and Z double dot.

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‫But if you remember at the beginning of the section you derived some equations that related these X,

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‫Y and Z double dots with the variables that you need.

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‫You want fire and theta are so x double dot equals cosine five times sine theta times cosine psi and

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‫then plus sine five times sine psi and then all that you multiplied by you, one divided by the mass

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‫of the drone.

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‫So this equation was for the X dimension.

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‫This equation was your X double dot, your acceleration in the X dimension in the inertial frame.

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‫So this equation was your Y double dot, your acceleration in the Y dimension in the initial frame,

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‫and this was your acceleration in the Z dimension in the initial frame since you know, the double derivative

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‫of the reference values and the arrow double dots that were computed using the controller, this one

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‫here, thanks to that, you know, your X double that, Y double that and Z double that.

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‫In addition, you know what your CI is as well.

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‫That's your reference value from the planner.

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‫So here as well, you get this number from the planner.

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‫And so what you have is three equations and three unknowns and your unknowns are you one fire are and

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‫seats are.

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‫So from this equation, you need to find your you one, then you also need to find your files like this

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‫and you need to find your Cetus that are here.

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‫And using algebra, you can find them once you have found them, then your new one will go to the plant

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‫and then your Fi and Seeta, they will go to your NPC controller.

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‫And for the NPC controller, they will be your reference values, fire and Sittar, and that will be

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‫your exercise.

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‫Try to manipulate these equations in order to find your Fifita and you one, try to find them in terms

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‫of other variables.

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‫I'll give you a hint to get started.

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‫Put the equations in this form X double dot divided by Z, double dot plus G.

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‫So take this G and put it on the other side of the equation sine and then take this X, double that

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‫equation, this entire equation, and then divided by this equation.

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‫So if you're Z double dot plus G equals this cosine five times cosine theta times you one over the mass

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‫then you would take this entire equation here and then you would divide it by this thing here and then

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‫do the same thing with y double dot divided by Z, double dot plus G.

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‫So put your equations in this form here and then go from there.

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‫We're going to discuss the solutions in the next video, however, I first want to finish our schematics.

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‫OK, so this is our schematics and now let's finish it so you know how you get your error double dots

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‫in the X, Y and Z dimension.

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‫And now when we're going to do we're going to take our reference value second time derivatives out of

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‫our planner.

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‫So this here will be your Z double dot sub are this here will be your Y double dot subha and this here

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‫will be your X double dot subha.

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‫And remember we had to subtract our error double dots from our reference value double dots.

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‫So we're going to have plus here and plus here and plus here and then we're going to have minus here,

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‫minus here and minus here.

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‫Because essentially what you need, you need X double dot sub R minus E double dot sub X.

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‫Right.

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‫That's what you need.

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‫And the same thing for the Y and Z dimension.

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‫And so here in this box, you have these three equations that we had X, double dot Y and Z double dot.

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‫So these equations here.

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‫These are the equations that you have in this box here.

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‫And so if you have your X double dot sub R minus E, double dot, sub X, then you will have your X

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‫double dot here that will go into this box.

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‫And the same thing happens here.

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‫You get your Y double dot, if you take your Y double dose of R and then you subtract E double Dunstable

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‫Y from it.

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‫And then the same thing happens here as well.

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‫You get your Z double dot in the same way, Z double dot, sub R minus E, double dot subs Z.

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‫And then as an exercise you will find your fi, r and fita r that will go into your NPC controller and

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‫then you will also find the you want control inputs that will go directly into your plant.

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‫So that's your exercise, these are your equations, and just try doing it yourself.

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‫Try doing some algebra and then we'll see the solutions in the next video.

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‫Thank you very much.