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‫Welcome back.

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‫So I hope that you tried this last exercise in this section yourself, and this is the quick solution

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‫to it.

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‫So first of all, you take this first cross product term and you go through the exact same steps that

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‫we went through when we learned about the cross product, the same procedure, I mean, as J and K,

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‫and then you have your determines of two by two matrices that you get from here.

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‫For example, if you take your eye unit vector, then you would focus on these four terms which would

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‫be here.

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‫And then in the case you would focus on these and these and they will be here.

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‫And then in the case you would have this part here, you know how to take the two by two matrix determinant

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‫first two times M W minus our times and the.

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‫OK, so it's like here Q times M W minus R times.

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‫In the end you do the same thing for the other cases as well.

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‫And don't forget about this minus sign here that you have to incorporate here.

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‫And once that is done, then you will extract your cue R and then P, and then you put this thing in

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‫this form and you do the exact same thing with this part, and you will end up having this thing here.

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‫And then this will be your c.B matrix, where you will have a three by three zero matrix here and also

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‫here.

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‫And then this y three by three matrix, it will be here and the orange three by three Matrix, it will

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‫be here.

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‫And so that was your General Newton oilor formulation here.

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‫And by the way, to be completely correct, then the c.B matrix, we should actually write it like this,

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‫c.B, as a function of the vector in the body frame, because this matrix here, it contains elements

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‫from this vector, right.

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‫In this vector, you have six elements.

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‫You had U, v, w, then P, Q, R, transpose.

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‫And so these elements, they appear in this matrix.

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‫And therefore you should put Seeb as a function of this vector and then this one.

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‫Well, that's your force and moment vector and to be precise, your net force and net moment vector.

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‫So this is extra knowledge and I didn't plan to put it in this course, but then I saw this for many

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‫times in the literature as well.

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‫So I just wanted to show how you get here so that you would see that it's actually the same thing compared

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‫to the previous form that we had derived in this concludes this section.

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‫What we have done so far, we have derived generic kinematic and dynamics equations that can be applied

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‫to a rigid six degree of freedom body.

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‫We have been using a drone as an example.

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‫However, the derived equations are generic, meaning that they can be applied to other rigid six degrees

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‫of freedom systems as well.

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‫Now, in the next section, we will take what we have derived here so far and we will apply it to our

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‫specific USV quadcopter and derive a specific model for that using the results from this section as

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‫a base.

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‫So it has been a big section, but it was necessary to lay down the fundamentals for you.

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‫In the next section, we will apply our fundamental equations to our specific drone.

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‫So thank you very much and see you in the next section.