‫Welcome back.
‫So now that we have covered the three conventions and derived the rotation matrix for them now, I think
‫it's a good idea to finalize our discussion about these conventions by going back to this exercise here.
‫And I think after this video, your intuition about these conventions will be much better.
‫So in the last videos, we derived rotation matrices for X, Y, Z, Z, Y, X and X, Y, X.
‫In each case, the final rotation matrix is different because you multiply different matrices together,
‫or if you use the same matrices, then you simply multiply them together in a different order.
‫For example, here it's our Z times are Y times R, X, and then here are X, times are Y, times are
‫Z.
‫So obviously these matrices are different.
‫OK, so some videos ago we chose a convention are X, Y, Z, and then we said that all the angles,
‫Fifita and Passi, they were all power to radians.
‫And because of this convention, our body frame ended up in this configuration.
‫And then we said that our drone had this angular velocity, its p q r components were all one radians
‫per second in each body frame dimension.
‫And then we found out that if we take this matrix, our Z times are Y times R X, which is essentially
‫this matrix here because you see the same convention, X, Y, Z.
‫And if we substitute all the angles with PI over to Radians, then this big matrix here will boil down
‫to this matrix here.
‫And so you took the components of your purple vector in the body frame that where one, one, one.
‫And once you multiplied this three by three matrix by it, you found the coordinates of the drones angular
‫velocity vector in the inertia frame.
‫So in the inertial frame, which is this white one, which is fixed in this frame, this purple vector
‫has a value of one in the X direction, a value of one in the Y direction and a value of minus one in
‫the Z direction.
‫And by inspection, you could verify that it's true because if all your angles are pi over two radians,
‫then it's very easy to visually inspect all this.
‫But what would happen if instead of this convention, I used this convention here, Zwick's.
‫What if I had taken this matrix here, this rotation matrix, and used this one to multiply by this
‫green one one one vector?
‫Well, your intuition should tell you that you would get a wrong answer, but we can verify it.
‫Let's take this rotation matrix here and substitute all the angles with PI over to so our Phi Theta
‫and PSI, they're all pi over to radians.
‫Or if we choose to name those angles, gamma, beta and alpha still pi over to radians.
‫The good thing about power to Radians is that whenever you see a cosine then you know that that term
‫is zero.
‫So for example here you know it will be zero.
‫Then here you have a cosine here and you have a cosine here.
‫So this element will also be zero and here you will have two sides and then here you have cosines.
‫So this part will be one and then this part will be zero.
‫So you will have zero zero and one.
‫Then here you have a cosine.
‫So zero then here you have to cosine zero.
‫But then here you have sine so you will have one with a minus sign and then here you will have zero
‫and then here you will have zero as well because you have cosigns here.
‫So you will have a zero minus one and zero.
‫And finally you have a sign here and here and here you have cosines.
‫So these two elements will be Xeros, but here you will have one.
‫So you will have one zero and zero.
‫So now if you take this matrix and multiply it by this peak, you are matrix, then you will get one
‫minus one and one.
‫So you should have positive X, negative Y and then positive Z.
‫Obviously, it's not true because you have positive X and positive Y and negative Z.
‫And it makes sense because this body frame, it's in this orientation because we followed the X, Y,
‫Z convention, we first rotated power to radians about the inertial X axis, then pi over to radians
‫about the inertial y axis and then power to Radians about the inertial Z axis.
‫And that's why your body frame ended up in this configuration.
‫And so when you have your drone's angular velocity, then you measure this omega B in the body frame,
‫which is in this configuration.
‫Now what you need to understand is that your drone's angular velocity vector does not depend on which
‫convention you choose.
‫In this sense that if this vector points in this direction, then it will point in this direction,
‫regardless of which convention you choose.
‫So, for example, if this is my x axis and this is my Y axis and let's say that I'm walking in this
‫direction, then that means that I'm walking in the positive X direction and then the Y dimension is
‫not affected at all.
‫But now, if I all of a sudden decide to change my convention, then I'm still walking in this direction.
‫If this is my home, then I want to go home.
‫And that means that I have to walk in this direction.
‫So if I change my convention, I will still walk towards my home.
‫I'm not going to change the direction of my motion just because all of a sudden the convention changed.
‫No, I will still keep walking towards my home no matter what is just if I use this convention, then
‫I'm walking in the positive X direction and then if I change my convention, then all that means is
‫that now I'm walking in the negative X direction and the same thing with the angular velocity of the
‫drone.
‫If that's the vector of the angle of velocity of the drone, if that's how fast the drone rotates and
‫if that's the direction of the drones rotation, then that's how this vector will be, regardless of
‫which convention I choose.
‫So if I all of a sudden want to change my convention and let's say that I want to use this convention,
‫Tsewang X will, then first of all, I need to understand how this body frame will be oriented with
‫respect to the initial frame.
‫If I use that convention and I rotate power to radiance about all my inertial axis.
‫So this here, the lower one, this is my Zwi X convention, so I first rotate power to Radians about
‫the inertial Z axis, then I rotate power to radiance about the inertial Y axis and then power to radians
‫about the inertial x axis.
‫So if I use the R, Z, Y, X convention, then my body frame will be in this orientation.
‫If I rotate about all the inertial axis by pi over to radians, if the angles are different, then of
‫course this body frame will be oriented differently.
‫But in case of power to radians using the R, Z, Y, X convention, my body frame will be oriented
‫in this way, not in this way that we had for R, X, Y, Z, but in this way our Zywiec.
‫So this is my inertial frame and now our body frame is in this configuration because we used this convention
‫and all our angles were pyra to radians.
‫You see, it's like here, here you have X, B then Y be and then A Zebb XP, Y, B and Zebb.
‫But like I said, the drone itself physically will not start rotating differently just because you went
‫from R, X, Y, Z to R, Z, Y, X.
‫The angular velocity vector of this drone is still like this, just like it's here in this picture.
‫Its components are still in the same direction.
‫And this is your Omega B, but because of the fact that you changed your convention, then you have
‫to make some modifications to your components.
‫So, OK, this is your body frame x axis.
‫So that means that this component is still your P, then this is your body frame Y axis.
‫So this is your Q and this is your body from Z axis.
‫So this is your arm.
‫So if you compare these two situations, then you can see that you have P here and then you also call
‫this P, then you have Q here and you also call this Q and then you have your R here.
‫And in this case your R is also here.
‫So you did not have to rename your components.
‫However, there is something different in this case, X, Y, Z, which was our original case, this
‫purple vector, it's coordinates in the body frame where one one one.
‫But since we changed our convention, now the coordinates of our angular velocity vector in the body
‫frame is not one one one anymore.
‫Now you see that your P is minus one because it's in the opposite direction to the positive body frame
‫x axis.
‫Then the same thing with Q it's also minus one and then your R is positive one.
‫Since you changed your convention, the drone still continues to rotate as it had rotated when you used
‫this convention.
‫So the physical emotional of the drone hasn't changed.
‫But since you changed your convention, you now describe this physical motion in your body frame differently
‫in this convention.
‫This motion is this minus one, minus one and one.
‫And now if we take this matrix that corresponds to this convention, why X?
‫And then we multiply it by the coordinates of the angular velocity vector in the body frame that corresponds
‫to this convention, then you will get one here, one here and minus one here.
‫And these are your inertial frame coordinates and you see now you get the right stuff.
‫Now this purple vector indeed has a positive X component, positive Y component and the negative Z component
‫in the inertia frame.
‫And we got the same result when we use this X, Y, Z convention.
‫It's just in the case of X, Y, Z, the body frame coordinates of this purple vector where one, one,
‫one, and then the corresponding rotation matrix for this convention provided that all the angles are
‫pi over to radians was this.
‫And that's how you found out the inertia from coordinates for this purpose vector.
‫But now since we changed our convention, then our matrix changed and our body frame coordinates changed.
‫But if we stick to the convention that we choose, we will still get the right inertia frame coordinates.
‫So you see, when you move from one convention to another, your inertia friend coordinates do not change.
‫What do change are your rotation matrix and your body frame coordinates.
‫And now to really solidify this understanding, let's go through the same thing.
‫But for this convention are X, Y, X.
‫Let's do that in the next video.
‫Thank you very much.