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‫Welcome back.

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‫So how would you write this G in the body frame as a six by one vector?

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‫Well, first, you know, the force of gravity equals minus the mass of the drone times G.

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‫Right.

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‫And it's minus because it points down.

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‫Now, you have to be careful here because this force of gravity is in the inertia frame.

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‫So even if your drone is tilted like this is the case here where your body frame Z axis does not align

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‫with the inertia frame Z axis, then this force of gravity still points down in the negative inertial

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‫frame Z axis.

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‫But since we work in the body frame, we have to represent the force of gravity in the body frame as

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‫well.

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‫So this here, this is our inertial frame, and then in red, you have the body frame and then you see

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‫the force of gravity goes in the negative inertial frame Z direction.

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‫Therefore, if your initial reaction was to write something like this, then I'm afraid it's not quite

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‫correct because this is in the body frame, but this would be in the inertia frame.

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‫So we have to convert that into the body frame, but we already have the tools to do that, right.

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‫We built it in the previous section.

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‫It's called rotation matrices.

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‫So the force of gravity in the inertia frame equals.

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‫And then we had our rotation matrix that had the convention body frame, Z, Y, X, so the oilor angles

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‫and then you would multiply that by the force vector.

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‫That would be the gravity force vector, but in the body frame and that would be your equation, right?

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‫By the way, let me put a lower case G here, because that's how we know that here.

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‫But the thing is that we already have our force of gravity in the inertia frame.

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‫So what we need is our force of gravity in the body frame.

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‫So we really need the opposite equation.

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‫So we inverse it.

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‫We say that our F g in the body frame equals.

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‫The inverse of our rotation matrix times the force of gravity in the inertia frame so you can rewrite

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‫our Lambda G in the body frame in this way, lambda G in the body frame equals.

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‫Force of gravity vector in the body frame, and then this zero three by one vector for the moments.

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‫And now we replace the body from gravity force with this expression here that comes from here.

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‫And so the final answer would be like this, this would be your force of gravity in the inertia frame,

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‫but then you multiply your inverse rotation matrix by this inertia frame force of gravity.

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‫And here you will have three elements then and then here you would have three zeros as well.

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‫So if you perform this vector matrix operation, then you will have a three by one force of gravity

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‫vector.

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‫But then in the body frame, because this would give you the force of gravity in the body frame.

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‫This expression here, and this is where I wanted you to stop.

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‫So I wanted you to get something like this.

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‫However, if you make the real calculations, then you will end up with this kind of expression.

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‫You will have many times sine theta here, minus M.G. Times cosine theta times sine phi here as the

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‫second element and the third element is minus M times G times cosine theta times cosine phi.

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‫So if you multiply together the three year rotation matrices that we had, according to our convention,

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‫then this is what you would get.

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‫So this answer is when you.

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‫Multiply these three matrices together like this in this order, because, first of all, you rotate

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‫about the moving from the axis, then moving from y axis and then moving from x axis.

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‫And so as a result, your rotation matrix is a product of these three rotation matrices here.

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‫And if you multiply them together like this.

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‫Well, then you get this as a result, and so if you take the inverse of it, if you want to find this

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‫where you have the product of your three matrices and then you take the inverse of it, then you will

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‫get the inverse of the rotation matrix, which is simply the transpose of this.

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‫Remember the rotation matrices you could get there in versus if you just take their transposes because

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‫they are also normal matrices.

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‫So this here, this would be the inverse of your rotation matrix.

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‫All right.

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‫And this would be the force of gravity in the inertial frame.

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‫All right.

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‫So F g in the inertial frame.

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‫And so if you multiply this matrix by this vector, then since these are zeroes here, then the only

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‫thing that's going to happen is that in each row, only this column matters here.

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‫All right.

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‫Because if you take this element, then you have to multiply it by zero.

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‫And then if you take this element, you have to multiply it by zero.

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‫And the same thing here and here and also here and here you would be multiplying them by zero so they

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‫don't matter.

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‫However, here you would multiply minus sine theta times, minus M.G. and you would get plus M.G. times

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‫sine theta that here you would multiply sine five times cosine theta times minus M.G. and that would

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‫be minus M.G. times sine five times cosine theta.

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‫And lastly, this element here, you would multiply it by minus M.G. So it would be minus Meji times

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‫cosine, five times cosine theta.

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‫And so that would be your force of gravity in the body frame.

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‫This would be the body frame X direction.

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‫This would be the body frame Y direction, and this would be the body frame Z direction.

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‫And here I've simply rearranged the multiplication order.

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‫So here now it's cosine theta times sine phi and then cosine theta times cosine phi and that's it.

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‫And then you will have three zeros for the moments here and that's how your vector would look like.

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‫As you can see, it's a six by one vector.

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‫Now in many cases I'm going to use some abbreviations.

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‫So in order to write less, I'm going to sometimes write s sub theta, which would be sine Seeta.

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‫As flying would be signed, fly Essi would be signed.

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‫So these will be the abbreviations for the cosine, cosine PHI cosine theta cosine BPCI.

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‫And these will be for the tangent, so tension, fi, tangency, c10 tangent.

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‫And there you go.

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‫You have your first force and moment vector lambda g in the body frame.

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‫And now let's look at other forces in moments now.

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‫Thank you very much.

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‫And see you in the next video.