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‫Welcome back.

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‫So hopefully you have found the time derivatives for these equations and the second time derivatives

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‫as well, I simply followed the calculus rules, the trigonometric rules, the chain rule, and these

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‫are the answers here.

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‫So in Orange, you have the first time derivative of these equations and then here in purple, you have

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‫the second time derivatives of them.

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‫And so what the planner does, it essentially creates a table where you have your time, your X are,

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‫Y are and NCAR, X or Y that are, and Z that are and their second derivatives.

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‫And so you have this table here and then here you have time equals zero seconds, then zero point four

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‫seconds, because remember the outer loop lasted for zero point four seconds and then during that time

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‫you had four inner loops, then zero point eight seconds here and so on.

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‫And for each time value you had a value for the reference values, their first time derivatives and

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‫second time derivatives.

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‫So for each time value, you would have all these nine values here like this.

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‫And so to get these values here, for example, if you want to know the values for time equals zero

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‫point eight seconds, then you just take this zero point eight seconds and then you put it here, here,

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‫then here, here as well.

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‫Also here.

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‫Well, here you will have a constant so you don't have a time variable here, but then you put it here

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‫and then here.

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‫And then the second derivative of the reference Z value is zero.

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‫And that's how you compute these values here.

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‫And then once this table is ready, then the planner gives it to the feedback lionization controller.

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‫And now due to the fact that the two controllers run at different frequencies, then that means that

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‫the new one control force that was produced by the position controller will have the same value throughout

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‫all the internal loop iterations.

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‫However, you two, you three and you four will have different values in every inner loop iteration

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‫because they are found by the attitude controller in each inner loop iteration in each in a loop cycle.

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‫So, for example.

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‫You have your you want you to use three and you four and then you one was, of course, in Newtons and

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‫then you two, you three and you four, they were Newton meters.

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‫So this would be your time sample, now K, then K plus one K plus two, K plus three and K plus four.

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‫And let's say that you one is one point three Newtons.

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‫So one outer loop will produce these values here, and you can see that they are all the same because

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‫they were produced by one outer loop.

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‫At the same time, you have had four inner loops.

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‫And so in each inner loop, the controller is going to produce some kind of value for you to Q3 and

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‫Q4.

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‫And let's say that during the first inner loop, the NPC controller will produce these values for you

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‫to use three and you you've had one inner loop.

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‫And so the NPC controller will give you the first batch of your values, you two, three and four.

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‫Then another in a loop happens.

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‫And again, you have new Q2, Q3 and Q4 values in Newton meters, another in a loop and another you

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‫two use three and you four values.

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‫And then the final fourth inner loop will produce these values for the control moments.

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‫These are all random numbers, of course.

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‫I'm just taking them from the top of my head.

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‫And so in one outer loop, you have four identical U.

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‫One values, but you two use three and you four, they're different every time an inner loop happens.

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‫And now you might ask yourself why I didn't put these numbers from Kate plus one to Kate plus four.

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‫Well, remember, from the previous course, if you remember then we used this kind of notation when

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‫we dealt with the NPC controller where a control input at K would influence the states and therefore

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‫it would produce the states at K plus one.

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‫So let's say that here you have states, whatever they are, and so you have states one.

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‫States to states three and states four, and so these inputs here that you have at K will produce these

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‫states here at K plus one.

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‫Then the inputs at K plus one will produce the states at K plus two inputs at K plus two will produce

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‫the States and K plus three and then inputs at K plus three will produce the states at K plus four and

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‫then a new outer loop happens and then the old K plus four will become the new K.

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‫And then from here, it goes to Kate plus one, Kate plus two and et cetera.

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‫Because remember how it was with this graph, for example, when we had our fly, right?

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‫This was our first fly, our.

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‫Then a new alternative happens and then we start from here, and that would be our second fire, our.

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‫And then that, for example, would be our third are so for this point, it would be our Kate plus four,

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‫but then for this point it would be our OK.

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‫And then it would be Kate plus four here, and then again, you would start from K here and then end

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‫up here at K plus four.

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‫So that's the logic behind our notation here.

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‫And finally, let's close up these loops like this from the plant, you get all these 12 states, six

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‫in the body frame and six in the frame.

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‫This inner loop happens every two seconds and the outer loop at every four times two seconds, and then

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‫the state's P, Q, R, Phi Theta and PSI will enter into the NPC controller.

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‫Because after all, for the attitude control, you need the angle of the losses of the drone, and then

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‫you will also need the orientation angles in the inertia frame, and then you will also need the omega

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‫value from the plant.

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‫That will go into the controller.

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‫And then the states that enter into feedback lionization are this X, Y and Z, then Phi Theta and Passi

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‫and then you V and W, after all, it's a position controller.

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‫So you might think that you need information about X, Y, Z, and also about you, the W, but also

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‫you have a rotation matrix there.

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‫So you need these five three time BPCI values as well.

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‫And that's it for this section.

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‫In the next section we will focus on the NPC controller and we treat the feedback lionization controller

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‫as a black box.

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‫We won't care how it works.

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‫We will simply assume that it magically gives you fire.

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‫And Sittar that NPC sees as reference values.

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‫And then in the section after that, we will see what's inside the feedback patronization box.

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‫And then in the final section, we will look at the code just like we did in the previous course in

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‫this series.

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‫Thank you very much and see you in the next section.