‫OK, but now let's go back to our Akutan method.
‫You have your first order, Odie's, here, or ordinary differential equations, and let's just assume
‫that we know our current status.
‫This is our current state vector in the body frame.
‫And these are our current state vectors in the initial frame, and now we're going to use the wrong
‫Akutan method to find these state vectors, but at K plus one.
‫And to do that, we follow the steps.
‫We first compute the first slope's for our state vectors, this, this and this, so we need to get
‫the time derivative of this state vector, which will be the dot at K, then we need to get government
‫dot at K.
‫And then we need to get see Dot at Kay.
‫That's what we need to get and of course, this is in the body frame and it's very easy to get them,
‫it's very easy because you have all your state values here at Kay.
‫You take all the state values and you plug them into these equations, you plug them into the body frame
‫equations and on the right side, you're going to plug them into the rotation and transfer matrices.
‫And also into these vectors as well, you, VW and P, Q, R.
‫So you have everything you need to compute all the 12 derivatives here.
‫And just like that, you have all these time derivatives.
‫Now, that's the same thing like it was for the oilor approach, but now that's where the difference
‫comes in.
‫Now what I will do, I will take the Oilor method and I will compute a new state, but not at time T
‫plus T.
‫S instead, I'm going to compute new states at time T plus T.
‫S over to.
‫So here I will make a graph for one state as an example, but the same logic applies to other states
‫as well.
‫So I have the slope here that I computed at time equals T. So the slope is for this point here.
‫And then I project it until T plus takes over to to hear.
‫So this is the first slope and piece of cake.
‫So it's this red line here, the red straight line.
‫And so when I projected till this point, then it's the same thing like using the oil or method to compute
‫a new state, but until this time here.
‫So I'm going to cold this point a little bit differently.
‫I'm going to say that this point is.
‫P, sub K plus one.
‫But then here in the index, I'm going to put T plus T.
‫S over to so that you would know that this point is computed at this time period here at T plus T is
‫over to and I'm going to put one here, which means that I computed this point using the first slope.
‫So mathematically, it will look like this.
‫So this point pops up, Kate, plus one at T plus two or two using the first slope.
‫It's this point here, you get it if you take your present, state your piece up cake and then plus
‫and then here I'm going to put the first derivative p dot subquery, which is this slope here and then
‫times T.
‫S over to it makes sense, right?
‫You have your slope here that you computed at time equals t where your present states are that slope
‫at this point and then you projected it till T plus T.
‫S over to and you reached here and mathematically you computed with this formula your present state
‫plus the derivative of your state.
‫With respect to time times, it's over two, because this interval here is your Ts over to and that's
‫where you put it here.
‫And then you do the same thing for all the other states, and here they are, you see this is your first
‫slope.
‫And this is your new state, but at T plus two or two, and it was computed using the first slope and
‫you will see in a moment why I use this notation.
‫And then what you do, you take all these new states that you have at T plus PTS over to using the first
‫slope, you take all these new states.
‫And you're going to put them here, so whenever you see a state variable, here we are CU or Phi Theta
‫PSI, you replace them all with these new states here.
‫So all these new states, they will go into these equations here.
‫And when you do that, you get a new set of slope's.
‫You will have another 12 slope's here.
‫Another 12 time derivatives and these will be your second slope's.
‫And so we need to somehow name our second slope's.
‫And I'm going to name them like this.
‫P dot, sub K plus one.
‫At T plus T.
‫S over two, so that's how I'm going to name them and I've put T plastics over to here to indicate that
‫this derivative was computed at time plus T.
‫S over to.
‫In other words.
‫We used the state values at T plus is over to that, we computed using the first slope, we use these
‫state values to compute our second slope.
‫And by the way, these notations, they are my own notations.
‫I'm not saying that they're some kind of official notation somewhere in books.
‫It's just I'm trying to give you the meaning of Kouta.
‫And right now I'm just using this kind of notation to differentiate between slope's and state values.
‫So that you would know which slopes I'm talking about and so that you would know whether your new state
‫values were computed using the slope one or slope two or slope three.
‫And so in the next video, we're going to see what we are going to do with our second slope, we're
‫going to need that second slope to compute new state values and then our third slope.
‫But more on that in the next video.