‫Welcome back.
‫Let's now substitute our fire and see tangles with zero radiance and see what we get.
‫So if you have zero radiance here and zero radiance here, you know that you're sign the Zero and Tenjin
‫zero, they will give you a zero element.
‫So this entire thing will be zero.
‫Cosine zero will give you one, but then Tenjin Zero will give you zero.
‫So you will have one here, but you will have zero here.
‫So one time zero, it will give you zero.
‫And here, of course, this was zero and then this was zero.
‫So you would have zero times zero, which would be zero.
‫And then here five zero and cosine zero is one.
‫Here you would have zero and minus sine zero will be zero.
‫If I make this five zero and then this theta zero, then this numerator will be zero and then this denominator
‫will be one.
‫So zero divided by one will give you zero and then five equals zero and theta equals zero.
‫So the numerator will be one because cosine zero is one and the same thing for the denominator.
‫So you will have one over one, which means that this entire thing, this entire element will be one.
‫And so that means that with this small angle assumption, your transfer matrix will become an identity
‫matrix.
‫And now something very interesting happens.
‫This relation that you have here that relates the body frame, angular velocity's with the inertia frame,
‫angular velocity is this a relationship will become like this.
‫So you multiply an identity matrix by the body frame, angular velocities.
‫And so if you write this equation out, then you will find that your Phi Dot, which is here, equals
‫your P.
‫Your seat that will equal your Q.
‫And your side, that will equal or safai, that equals this one theater that equals this one and that
‫equals this one.
‫And then it is also logical that your fi double dot equals p dot theater, double dot equals your cute
‫dot and then side double dot equals your R dot.
‫And so then you can take these three states based equations here for the attitude.
‫And you can replace all your PS Qs, R.
‫S P dots, Q dots and R dots with these values here, and if you do that, then this is what you will
‫get.
‫You see, instead of P Q hours, you will have five dots, Seeta dots and side dots, and then instead
‫of P does Q dots and or dots you will have five double dots, theta double doesn't side double dots.
‫So just to be clear, these equations here are used only in the NPC controller.
‫In the plant, you still use these original equations alongside with the transfer matrix where you did
‫not have that PHI and Theta equals zero radians assumption, where you did not have that small angle
‫approximation.
‫So this entire form, these three equations here and then this transfer matrix relationship that would
‫be in the plant still.
‫But because in general, the financial tangles will be small throughout the entire maneuver, then because
‫of this small angle approximation in the controller, in the NPC controller, we will use these equations.
‫And the performance of the controller because of this assumption will not change significantly because,
‫see, tangles throughout the maneuver will be small as they should be, because if they are not, then
‫the U.S. simply won't be able to fly.
‫Because the propellers of the oveI are not that powerful, that they would be able to produce enough
‫vertical components thrust in order to counteract the force of gravity when you're fine and FITA angles
‫are large.
‫And also, if we think about it logically, then the feedback lionization controller shouldn't really
‫give big theta reference angles to the NPC because if you're fired, see tangles are large, then you
‫will have a very huge component in the horizontal direction.
‫And then if you have some kind of trajectory that you want your you have to follow.
‫Then with a large horizontal velocity, it will be difficult.
‫You know, it might drift from the trajectory.
‫Whenever there is a sharper turn, for example.