‫So what we have done here, we have derived an equation of motion for the translational motion, which
‫is a dynamic equation because it assigns the force to the motion of the body.
‫And here the force vector is the net force, so is the sum of the forces.
‫It's not one specific force.
‫It's the sum of the forces.
‫And then this gunman double that is the second time derivative of your position vector, which essentially
‫is your acceleration vector.
‫Now, that's all very good.
‫However, that is all assuming that we are working in the inertial frame, the forces and the accelerations
‫are measured in the inertial frame.
‫And in fact, the second law of Newton, which is F equals M times acceleration.
‫This law is valid in the inertial frame.
‫However, when we work with a drone, then the control forces and control talks are treated in the body
‫frame.
‫If you remember, then our control force was thrust generated by all of the propellers and we called
‫it You Won, and then you two use three and you four.
‫They were roll pitch in your control talks respectively.
‫If you look at how they are applied to the drone, then you see that they are applied to the drone in
‫the body frame, not in the inertia frame, but in the body frame.
‫The reason for that is that it's easier that way, which makes sense because the propellers of the drone
‫are fixed to the body frame and not to the inertia frame.
‫So all that means that we have to work in the body frame and again, not in the inertia frame.
‫So we have to derive the equations of motion for the translational motion when we are in the body frame
‫and see what we get instead of starting from a position vector.
‫Let's start from a velocity vector.
‫Vehbi vector equals U, V and W transposed in meters per second.
‫And remember, you belong to the body frame x axis like this v belong to the body frame y axis and then
‫w you belong to the body frame Z axis.
‫In fact, there is one remark that I want to make with regards to position vectors, when we work in
‫the body frame position vectors in the body frame don't really make sense to me.
‫If you think about it, let's look at the 2D case.
‫You have your inertia frame here.
‫And then you have the body frame that moves from.
‫P one to P two in this Molex direction, so what is the position vector in the body frame?
‫Well, it should be zero meters, right, because even though you moved, but the body frame attached
‫to you moved with you.
‫So relative to the body frame, your position hasn't changed.
‫However, if you forget about position vectors in the body frame and you define velocity vectors in
‫the body frame.
‫If you say that something moves in the body frame, axis direction and this kind of velocity.
‫Then it makes more sense.
‫So if you measure X dot and Y dot in the body frame, then that's more logical to me.
‫And that's why since now we want to derive the equations of motion, but when we are in the body frame,
‫then we are not going to start from a position vector.
‫We are going to start from a velocity vector.
‫Now, this velocity vector here can also be written like this.
‫The in the body frame equals UI plus Vijay plus W.K..
‫It's just another way to write the same thing, and so again, the I, J and K, they are the unit vectors,
‫but now they are not unit vectors in the inertia frame anymore.
‫They are not these unit vectors, their unit vectors in the body frame.
‫This would be I, this would be J and this would be K here in the body frame X, Y and Z direction respectively.
‫And so to derive the equation of motion in the body frame, you need to have an equation for the force
‫vector in the body frame equals the mass of the object times, the acceleration in the body frame.
‫Which is Vedat B vector, so you take this vector and you have to take the time derivative of this vector
‫in order to get the acceleration in the body frame.
‫So let's find the time derivative for this vector.
‫So the time derivative of our velocity vector in the body frame is the following Yuda I plus V the J
‫plus W dot k.
‫However, the product rule from calculus also tells you that you also have to have another term that
‫takes the derivative of I, J and K while keeping your U, V and W the same.
‫So this is what you will get you idot plus v jadot plus w k dot.
‫However, this is where the difference comes in.
‫The second law of Knewton, which is force equals mass times acceleration or the more general case force
‫equals the time derivative of the linear momentum.
‫This law says that it is true in the inertial frame.
‫It means that this law is true when the X, Y and Z axis are fixed.
‫It means that I dot equals Jadot equals K dot equals zero units per second, meaning that the unit vectors
‫don't change their direction nor magnitude in time.
‫However, when you're in the body frame, then you're not fixed.
‫As the drone rotates, the body frame rotates with it.
‫With respect to the inertia frame, it means that in the body frame, the time derivative of the unit
‫vectors ie that Jadot and K that they do not equal zero.
‫And that means that we have to find our IDOT Jadot and K Dot and incorporate them in our equations of
‫motion.
‫And that's how our translational equations of motion will look like if we work in the body frame, that
‫can rotate with respect to the inertia frame, the idot Jadot and key that they do not disappear this
‫time.
‫So this would be the net force vector or the sum of the forces.
‫This is still your mass.
‫The objects mass doesn't change when you change your reference frame.
‫And this entire thing is your the dart in the body frame.
‫And so our next order of business is to find IDOT, Jadot and Cadart.
‫So let's find them.