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‫Welcome back.

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‫So now let's finalise our discussion about this rotation matrix conventions by doing what we did in

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‫the last video.

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‫But for this convention, again, we assume that the physical rotation of the drone is this purple vector.

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‫This is a rotation vector.

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‫So if my rotation vector is like this, then according to the right hand rule, it means that your object

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‫is rotating like this as shown in green, but then the rotation vector is perpendicular to the plane

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‫of the rotation.

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‫And so if my drone rotates with this purple vector, let's say at five radians per second, then it

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‫means that this drone rotates like shown in green at five radians per second.

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‫And of course, you can decompose this rotation vector like this.

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‫And so, for example, completely unrelated to this convention.

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‫If this is my body frame axis, this is my y body frame axis and then this is my body frame axis, then

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‫this would be my P, this would be my cue and this would be my arm.

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‫So I have decomposed my rotation matrix into three components because then it's convenient for me to

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‫keep track of them in the vector form.

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‫So that's how we measure rotations using these rotation vectors.

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‫OK, let's go back to our R x, y x convention.

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‫The physical rotation of the drone is still this as shown with this purple rotation vector.

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‫That doesn't change because we changed our convention.

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‫But if we rotate our body frame with respect to the inertial frame using this convention and then these

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‫are my angles, then the first thing that I need to find out is how my body frame is oriented with respect

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‫to the initial frame.

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‫So this is the beginning when our body frame completely overlaps with the initial frame and now you

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‫rotate piver to radians about the inertial x axis.

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‫Right?

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‫You first rotate about the inertial x axis.

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‫So your body frame ends up in this configuration, then you rotate power to radians about the inertial

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‫y axis like this.

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‫So you will end up in this configuration.

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‫And finally you rotate about the inertia from X axis again by PI over to radians and it will be like

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‫this.

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‫So this is your inertial frame and this will be your body frame.

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‫So if you compare this orientation with this orientation, then you can see that it's different right

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‫here.

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‫You had X be here, y be here and zib here and now you have X be here, ybe here and Zib here.

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‫Well let's see what it means.

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‫The rotation vector still has three components in the same direction, so the angular velocity vector

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‫is still the same.

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‫Is this one here.

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‫You see the rotation of the drone hasn't changed, not in terms of its magnitude nor in terms of its

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‫direction.

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‫The Vector or Hormiga B is exactly the same.

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‫However, now we have to rename our components in the case of X, Y, Z convention.

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‫This component was P because it was in the direction of body frame X, but now this component is in

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‫the direction of body frame Z.

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‫So now we have to rename this component and we have to say that this component is R then in the case

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‫of the X, Y, Z convention, this was our Q component because it was in the body frame Y direction.

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‫But now this component is the same component physically.

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‫This component it's the same.

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‫But now in this new convention, this component points in the body frame X direction.

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‫So now we have to call it P.

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‫And in the original case this component was R, but now the same component in the new convention needs

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‫to have a different name.

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‫So it's.

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‫Q So because of the fact that we changed our convention, we have to rename our.

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‫And so what would be the body frame coordinates of this vector in this convention, X, Y, X?

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‫Well, it will be very simply one one, one, just like in this case, actually.

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‫However, the difference is that now the components have different names.

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‫So in this case, in the X, Y, Z case, this one that you have here, the first one, which is P,

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‫since it's the same component, it's the last one.

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‫If this R component had been, let's say, three times longer, not one, but three, then in this X,

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‫Y, Z convention, this three would be the first number.

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‫But in the X, Y, X convention, it would have been the last number.

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‫So in the X, Y, Z convention, it would have been three one one provided that the other components

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‫stayed the same.

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‫And then in this convention it would have been like that.

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‫And so now this is the rotation matrix for this X, Y, X convention.

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‫And last substitute all the angles with poverty, radiance.

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‫You know that whenever there is a cosine, it means zero.

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‫So that means that in this first column you will have zero one and zero like this.

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‫Then the second column you will have one here and then zero and zero.

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‫So one zero and zero.

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‫And then finally you will have zero here.

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‫You will have zero here.

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‫And then you see that you will have minus one here.

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‫So zero zero and minus one.

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‫And so to compute the coordinates of this properly vector in the inertia frame, we take this matrix

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‫and then we multiplied by the coordinates in the body frame.

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‫And as you expect it, you will get one, one and minus one.

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‫But now let's really assume that this R component is three times longer.

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‫So this component is still one, this component is one, but then the R component is three.

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‫Then if we use this convention are X, Y, X, then your angular velocity vector would be one, one

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‫and three.

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‫So you see P, Q, R would be one one and three.

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‫And I really need to project this purple vector down here.

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‫So this purple vector, it comes from the origin.

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‫And so by inspection you can see that in the inertial frame.

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‫The coordinates of this vector are one one minus three.

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‫If we take this rotation matrix that came from R, X, Y, X, and then we multiplied by this vector

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‫one, one and three, then we'll get one one and minus three as expected.

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‫But now if we choose another convention, if we choose R, X, Y, Z, which is in this case, then

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‫the purple vector in this convention would be three, one and one.

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‫Right.

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‫Because you're three now this longer component, it's in the body frame X direction and then the other

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‫components are still one.

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‫And now if we still want to get the inertial frame coordinates for this purple vector, then we need

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‫to get the rotation matrix for this convention.

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‫And this rotation matrix was this one here.

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‫Right.

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‫And so I have it here.

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‫And then I multiplied by this vector.

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‫Remember, this matrix is for the X, Y, Z convention, and also this vector is for the X, Y, Z convention.

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‫And then what I will get is one, one and minus three.

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‫As you can see, the same thing, the math works out.

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‫So remember, you have a fixed inertial frame.

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‫The physical motion of your object doesn't change, but when you change your rotation matrix convention,

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‫then two things change your rotation matrix and your body frame coordinates.

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‫But in the end, all the.

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‫Conventions, if you do it right, we'll give you the right inertial frame coordinates now, you know

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‫that if you have information in your body frame.

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‫In other words, if you have body frame coordinates and you want to find those coordinates in the inertial

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‫frame, then you use the rotation matrix.

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‫But if you want to go the other way around, let's say you have information in your inertial frame,

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‫but you don't know your body frame coordinates, then you need to take the inverse of your rotation

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‫matrix and then you will find your coordinates in the body frame.

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‫So how about we test that theory?

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‫If we take our convention, which is R, X, Y, Z, and then we rotated power to radians about all

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‫the inertial axis, then this was our rotation matrix and then these were the different coordinates

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‫when we used this convention and these coordinates here, they were my inertial frame coordinates.

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‫So let's assume that I don't know my body frame coordinates.

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‫I only have my inertial frame coordinates.

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‫In that case, this would be my inertia frame coordinates.

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‫Then this here, this would be my inverse of the rotation matrix and here I should have my body frame

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‫coordinates.

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‫So what would be the inverse of this rotation matrix?

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‫Well, since rotation matrices are also normal matrices, then the inverse of these matrices are the

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‫transpose of these matrices.

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‫In that case, the first row here is zero zero one.

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‫It would be the first column here.

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‫The second row here would be the second column here, and then the third row here would be the third

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‫column here.

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‫And so if I performed this matrix vector multiplication, then I will have three, one and one, just

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‫as we expected, three one one.

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‫If we had used another convention, for example, are X, Y, X, then remember, two things changed

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‫the rotation matrix and then the body frame coordinates.

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‫So let's take this one here, which follows the convention are X, Y, X again, we assume that we don't

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‫know the body frame coordinates.

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‫We only know the initial frame coordinates.

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‫And this is the rotation matrix that came from this convention, x, y, x, since rotation matrices

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‫are also normal, therefore it is very easy to take their inverse.

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‫You simply take their transpose.

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‫And the transpose of this matrix is this one.

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‫This is the R x, y x rotation matrix inverse, which is actually the same matrix like the original

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‫rotation matrix.

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‫So if you take the transpose of this matrix, then it will be the same matrix.

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‫And I guess that has to do with the fact that we have this interesting convention here, X, Y, X.

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‫We rotated two times about the inertial X axis.

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‫But anyway, these are our coordinates in the inertia frame.

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‫So we put them here and as a result we get our body frame coordinates one, one and three.

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‫And that's what we have for our R, X, Y, X convention.

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‫So the math works out.

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‫And of course, as mentioned before, in this course, we will use this rotation matrix that could be

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‫applied to two conventions, the fixed angle approach and then the oilor moving frame approach.

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‫In this course we will be using the convention are sub Z, Y, X.

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‫These are small letters, so meaning that we are rotating about the body frame axis.

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‫So we are using the oilor angle approach, meaning that we will first rotate about the body frame Z

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‫axis, then we will rotate about the body frame Y axis and then about the body frame x axis.

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‫And that's it about these rotation matrix conventions.

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‫I hope that this extra explanation about these conventions has been helpful.

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‫Thank you very much.