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‫Welcome back.

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‫So these are your complete states based equations here and now we are going to use wrongly Akutan method

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‫to integrate the states and by integrating the states.

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‫I mean, I want to know the evolution of states as a function of time.

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‫For example, let's take the state party, which is the angular velocity about the body frame, X-axis.

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‫You want to know what happens with P. radians per second as you move along the timeline?

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‫With steps of peace, which in our case is zero point one seconds.

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‫In order to start with the integration process, you first need to define the initial states for your

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‫system.

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‫That's the very beginning of your evolution at time, equals zero seconds, and then from this beginning,

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‫you will have the evolution of your state.

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‫So at each sample time, you want to know how many radians per second you have for your state.

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‫But of course, the initial state doesn't have to happen, starting from time equals zero seconds.

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‫You can define your initial states, for example, at time equals zero point two seconds right over

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‫here, so with a delay and then your evolution will start from here.

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‫But what matters is that at the very beginning, you have concrete values for all your states.

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‫So at the very beginning, when you start your simulation or your maneuver, you have to have some kind

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‫of value for all these states here, sticks in the body frame and then six in the inertial frame.

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‫And so your initial states would be your ex up, Kay, and then you use either Runga, Kouta or Oilor

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‫method to get your ex up K plus one, then in the next loop, this ex up K plus one will become your

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‫ex up K and then you compute your new ex of K plus one.

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‫And you do that until you finish your simulation or your maneuver.

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‫And, of course, integrating accurately so that your predicted state evolution.

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‫Would match the true state evolution as closely as possible.

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‫Or I would say as closely as necessary, because you always can go more precise with your models, however,

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‫at some point you have to ask yourself, is making the model more complicated?

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‫Worth it or not, if your model already gives you satisfactory results with small errors that you can

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‫tolerate, then there is no point of making your models more complicated in order to gain a little bit

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‫of more precision, because that would just make your problem harder and it will probably increase your

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‫computational cost.

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‫Meaning that your computer will have to make more computations, which means more time, which in many

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‫cases is a big factor.

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‫But still, you want your predicted evolution from your oilor or wrong Akutan method to match the true

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‫state evolution very closely.

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‫Because you're going to use your plant to tune your controllers, right?

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‫Remember, controllers, they need to be tuned, for example, in pide, you had your Cape Cod and Kaixi

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‫constants, and then in NPC you had your weights and so you had to tune your controllers.

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‫And so you would tune your controllers based on your plan model.

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‫And then you take your tuned controller and you apply it to the real system.

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‫And the real system has sensors that measure it states.

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‫And if your plan model predicts the behavior of the real drone, well, meaning if the sensors in the

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‫drone receive values for states.

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‫That are very close to what you get when you try to predict them using your state's base equations and

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‫wrong Akutan integration.

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‫Then your controller will control the real drone.

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‫Well.

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‫But of course, if your plan model is too imprecise, so then what will happen is that you're going

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‫to tune your controller based on that imprecise model.

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‫So when you apply that controller to the real drone and then if what happens in real life differs from

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‫what happens in the plan model by a lot.

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‫Then the controller will be badly tuned for the real drone.

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‫Because it was tuned for the planned model, but the plan was very different from the real drone.

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‫And of course, like we talked in the previous course when working with real drones, you also have

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‫to take into account the signal noise that is produced by sensors and the uncertainty of the mall itself.

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‫And one method to smooth these noise and uncertainties is to put them through a common filter.

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‫So that's additional information for you, things to watch out for when you start working with a real

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‫drone.

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‫Oh, and you will probably need to program the real drone in C++.

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‫Usually hardware like drones, robots, etc, are programmed in C++ because it is faster than Python.

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‫Python is more user friendly and good for simulations.

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‫But either way, the fundamental concepts and modeling that I am teaching you here is the same.

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‫The laws of physics don't change based on what programming language you are using.

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‫And one more remark that I want to make, when you use your drone and you use it sensors to measure

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‫the status of the drone in real time, then if you use an NPC controller, then remember and NPC controller

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‫was a model based controller, meaning that your NPC controller would take the mathematical model of

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‫your drone and it would use it to compute the future status of your drone.

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‫If you remember then NPC had Horizon period and then when Horizon period equals, for example, 10,

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‫that means that starting from now NPC would use the mathematical model of the drone to try to predict

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‫state's 10 sample times ahead.

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‫But it's not going to use sensors for that because sensors just tell you what is happening now.

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‫So NBC will need to have a good mathematical model in order to predict how your system will behave in

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‫the next ten sample times from now.

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‫And so if your mathematical model is not good enough, then that means that your NPC will make poor

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‫predictions.

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‫And then since NPC uses those predictions to give you the best control inputs, then if the models are

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‫not good enough, then that means that the NPC will give you control inputs that are not very good for

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‫the real system.