‫Welcome back.
‫So I hope that you tried this last exercise in this section yourself, and this is the quick solution
‫to it.
‫So first of all, you take this first cross product term and you go through the exact same steps that
‫we went through when we learned about the cross product, the same procedure, I mean, as J and K,
‫and then you have your determines of two by two matrices that you get from here.
‫For example, if you take your eye unit vector, then you would focus on these four terms which would
‫be here.
‫And then in the case you would focus on these and these and they will be here.
‫And then in the case you would have this part here, you know how to take the two by two matrix determinant
‫first two times M W minus our times and the.
‫OK, so it's like here Q times M W minus R times.
‫In the end you do the same thing for the other cases as well.
‫And don't forget about this minus sign here that you have to incorporate here.
‫And once that is done, then you will extract your cue R and then P, and then you put this thing in
‫this form and you do the exact same thing with this part, and you will end up having this thing here.
‫And then this will be your c.B matrix, where you will have a three by three zero matrix here and also
‫here.
‫And then this y three by three matrix, it will be here and the orange three by three Matrix, it will
‫be here.
‫And so that was your General Newton oilor formulation here.
‫And by the way, to be completely correct, then the c.B matrix, we should actually write it like this,
‫c.B, as a function of the vector in the body frame, because this matrix here, it contains elements
‫from this vector, right.
‫In this vector, you have six elements.
‫You had U, v, w, then P, Q, R, transpose.
‫And so these elements, they appear in this matrix.
‫And therefore you should put Seeb as a function of this vector and then this one.
‫Well, that's your force and moment vector and to be precise, your net force and net moment vector.
‫So this is extra knowledge and I didn't plan to put it in this course, but then I saw this for many
‫times in the literature as well.
‫So I just wanted to show how you get here so that you would see that it's actually the same thing compared
‫to the previous form that we had derived in this concludes this section.
‫What we have done so far, we have derived generic kinematic and dynamics equations that can be applied
‫to a rigid six degree of freedom body.
‫We have been using a drone as an example.
‫However, the derived equations are generic, meaning that they can be applied to other rigid six degrees
‫of freedom systems as well.
‫Now, in the next section, we will take what we have derived here so far and we will apply it to our
‫specific USV quadcopter and derive a specific model for that using the results from this section as
‫a base.
‫So it has been a big section, but it was necessary to lay down the fundamentals for you.
‫In the next section, we will apply our fundamental equations to our specific drone.
‫So thank you very much and see you in the next section.