‫Welcome back.
‫So you have two general equations now you have the first one, which is for translational motion in
‫3-D, and then you have the second one, which is for rotational motion in 3D.
‫And so if we want to put them together in one vector, matrix form, then first of all, we have to
‫rewrite the first equation like this.
‫We're going to open the parenthesis and then here we're going to add an identity matrix, which is this
‫one.
‫This is this I three times three, the identity matrix, and you can do that because in the Matrix world,
‫the identity matrix is like one.
‫So it's like multiplying something by one.
‫And then in this term, I'm going to put the mass here on this side of the cross product and then all
‫that equals the net force vector.
‫The second row will be the same.
‫And remember, I superscript b here is the mass moment of inertia matrix about the body frame axis.
‫We are measuring these quantities about the body frame axis and we assume that our drone is mass symmetric
‫about all the body frame axis, which means that all your products of inertia are zeros.
‫So that's your mass moment of inertia matrix.
‫And now we can take this chunk here and rewrite it like this.
‫So that's how you can rewrite it.
‫And now you can also see why you needed this identity matrix here in order to make all the dimensions
‫match.
‫So essentially, this is a six by six matrix here and these are the three by three zero matrices here.
‫And then you multiply it by this vector here.
‫Then you take this chunk here.
‫And you rewrite it like this, and this is your net force and moment vector.
‫So this thing here is a six by one vector.
‫And if you truly want to ride it out, then it would look like this, as you can see, it's quite massive,
‫but it's the same thing.
‫So the sum of the forces in moments equals and then you have this part that assumes that your body frame
‫is not rotating with respect to the initial frame.
‫And then this is your correction factor.
‫So whenever you have some kind of object and that object is subject to all kinds of forces and moments,
‫then you sum them all up and then you get the net force and moment vector.
‫And that equals all this.
‫And this formulation is called Knewton oilor.
‫Formulation and with this formulation, it is possible to describe the motion of six degree of freedom.
‫Rigid body.