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‫So what we have done here, we have derived an equation of motion for the translational motion, which

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‫is a dynamic equation because it assigns the force to the motion of the body.

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‫And here the force vector is the net force, so is the sum of the forces.

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‫It's not one specific force.

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‫It's the sum of the forces.

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‫And then this gunman double that is the second time derivative of your position vector, which essentially

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‫is your acceleration vector.

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‫Now, that's all very good.

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‫However, that is all assuming that we are working in the inertial frame, the forces and the accelerations

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‫are measured in the inertial frame.

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‫And in fact, the second law of Newton, which is F equals M times acceleration.

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‫This law is valid in the inertial frame.

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‫However, when we work with a drone, then the control forces and control talks are treated in the body

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‫frame.

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‫If you remember, then our control force was thrust generated by all of the propellers and we called

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‫it You Won, and then you two use three and you four.

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‫They were roll pitch in your control talks respectively.

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‫If you look at how they are applied to the drone, then you see that they are applied to the drone in

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‫the body frame, not in the inertia frame, but in the body frame.

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‫The reason for that is that it's easier that way, which makes sense because the propellers of the drone

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‫are fixed to the body frame and not to the inertia frame.

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‫So all that means that we have to work in the body frame and again, not in the inertia frame.

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‫So we have to derive the equations of motion for the translational motion when we are in the body frame

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‫and see what we get instead of starting from a position vector.

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‫Let's start from a velocity vector.

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‫Vehbi vector equals U, V and W transposed in meters per second.

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‫And remember, you belong to the body frame x axis like this v belong to the body frame y axis and then

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‫w you belong to the body frame Z axis.

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‫In fact, there is one remark that I want to make with regards to position vectors, when we work in

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‫the body frame position vectors in the body frame don't really make sense to me.

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‫If you think about it, let's look at the 2D case.

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‫You have your inertia frame here.

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‫And then you have the body frame that moves from.

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‫P one to P two in this Molex direction, so what is the position vector in the body frame?

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‫Well, it should be zero meters, right, because even though you moved, but the body frame attached

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‫to you moved with you.

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‫So relative to the body frame, your position hasn't changed.

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‫However, if you forget about position vectors in the body frame and you define velocity vectors in

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‫the body frame.

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‫If you say that something moves in the body frame, axis direction and this kind of velocity.

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‫Then it makes more sense.

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‫So if you measure X dot and Y dot in the body frame, then that's more logical to me.

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‫And that's why since now we want to derive the equations of motion, but when we are in the body frame,

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‫then we are not going to start from a position vector.

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‫We are going to start from a velocity vector.

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‫Now, this velocity vector here can also be written like this.

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‫The in the body frame equals UI plus Vijay plus W.K..

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‫It's just another way to write the same thing, and so again, the I, J and K, they are the unit vectors,

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‫but now they are not unit vectors in the inertia frame anymore.

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‫They are not these unit vectors, their unit vectors in the body frame.

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‫This would be I, this would be J and this would be K here in the body frame X, Y and Z direction respectively.

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‫And so to derive the equation of motion in the body frame, you need to have an equation for the force

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‫vector in the body frame equals the mass of the object times, the acceleration in the body frame.

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‫Which is Vedat B vector, so you take this vector and you have to take the time derivative of this vector

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‫in order to get the acceleration in the body frame.

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‫So let's find the time derivative for this vector.

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‫So the time derivative of our velocity vector in the body frame is the following Yuda I plus V the J

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‫plus W dot k.

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‫However, the product rule from calculus also tells you that you also have to have another term that

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‫takes the derivative of I, J and K while keeping your U, V and W the same.

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‫So this is what you will get you idot plus v jadot plus w k dot.

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‫However, this is where the difference comes in.

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‫The second law of Knewton, which is force equals mass times acceleration or the more general case force

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‫equals the time derivative of the linear momentum.

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‫This law says that it is true in the inertial frame.

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‫It means that this law is true when the X, Y and Z axis are fixed.

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‫It means that I dot equals Jadot equals K dot equals zero units per second, meaning that the unit vectors

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‫don't change their direction nor magnitude in time.

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‫However, when you're in the body frame, then you're not fixed.

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‫As the drone rotates, the body frame rotates with it.

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‫With respect to the inertia frame, it means that in the body frame, the time derivative of the unit

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‫vectors ie that Jadot and K that they do not equal zero.

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‫And that means that we have to find our IDOT Jadot and K Dot and incorporate them in our equations of

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‫motion.

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‫And that's how our translational equations of motion will look like if we work in the body frame, that

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‫can rotate with respect to the inertia frame, the idot Jadot and key that they do not disappear this

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‫time.

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‫So this would be the net force vector or the sum of the forces.

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‫This is still your mass.

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‫The objects mass doesn't change when you change your reference frame.

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‫And this entire thing is your the dart in the body frame.

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‫And so our next order of business is to find IDOT, Jadot and Cadart.

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‫So let's find them.