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‫Welcome back.

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‫So you have two general equations now you have the first one, which is for translational motion in

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‫3-D, and then you have the second one, which is for rotational motion in 3D.

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‫And so if we want to put them together in one vector, matrix form, then first of all, we have to

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‫rewrite the first equation like this.

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‫We're going to open the parenthesis and then here we're going to add an identity matrix, which is this

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‫one.

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‫This is this I three times three, the identity matrix, and you can do that because in the Matrix world,

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‫the identity matrix is like one.

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‫So it's like multiplying something by one.

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‫And then in this term, I'm going to put the mass here on this side of the cross product and then all

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‫that equals the net force vector.

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‫The second row will be the same.

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‫And remember, I superscript b here is the mass moment of inertia matrix about the body frame axis.

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‫We are measuring these quantities about the body frame axis and we assume that our drone is mass symmetric

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‫about all the body frame axis, which means that all your products of inertia are zeros.

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‫So that's your mass moment of inertia matrix.

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‫And now we can take this chunk here and rewrite it like this.

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‫So that's how you can rewrite it.

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‫And now you can also see why you needed this identity matrix here in order to make all the dimensions

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‫match.

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‫So essentially, this is a six by six matrix here and these are the three by three zero matrices here.

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‫And then you multiply it by this vector here.

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‫Then you take this chunk here.

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‫And you rewrite it like this, and this is your net force and moment vector.

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‫So this thing here is a six by one vector.

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‫And if you truly want to ride it out, then it would look like this, as you can see, it's quite massive,

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‫but it's the same thing.

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‫So the sum of the forces in moments equals and then you have this part that assumes that your body frame

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‫is not rotating with respect to the initial frame.

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‫And then this is your correction factor.

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‫So whenever you have some kind of object and that object is subject to all kinds of forces and moments,

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‫then you sum them all up and then you get the net force and moment vector.

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‫And that equals all this.

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‫And this formulation is called Knewton oilor.

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‫Formulation and with this formulation, it is possible to describe the motion of six degree of freedom.

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‫Rigid body.