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‫Welcome back.

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‫In this video, I want to clarify the conventions of the rotation matrices, it can be a pretty confusing

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‫topic.

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‫So in this video, I will clarify that in total, you have 24 conventions for rotation matrices 12 when

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‫you rotate about fixed axis and 12 when you rotate about moving body frame axis.

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‫So here you saw the 12 different conventions.

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‫These were for the fixed axis and then you have 12 more for the moving frame axis.

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‫However, some things need to be clarified, and particularly it is important to understand how the

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‫final product of the three rotation matrices are X or Y and our Z is computed for each convention.

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‫We will derive these products of matrices for three conventions so that you would understand the procedure

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‫and the conventions will be R, X, Y, Z, R, Z, Y, X and then R, X, Y, X..

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‫So here we are rotating about the inertia from Axis.

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‫But as you know, you will get the same product of matrices if, for example, for this convention you

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‫first rotate about the moving body frame Z axis, then about the moving body from Y axis and then about

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‫the moving body frame x axis.

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‫And then for this convention, it would be rotating about the body frame x axis, then body frame y

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‫axis, and then about the body frame Z axis, and then in this case, rotating about the body frame,

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‫x axis, body frame, y axis and then body frame x axis.

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‫So even though we have twenty four different conventions in total, we have 12 different products of

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‫these matrices.

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‫So if we only deal with rotation matrices, then it doesn't really matter if we use this convention

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‫or this convention, because in this case, this would be the product of these three matrices for both

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‫of them.

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‫In this case, it would be this one.

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‫And in this case, it would be this one for both of them.

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‫But very soon we will start deriving a transfer matrix.

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‫And as we derive it, then you will see that this transfer matrix will be different depending on whether

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‫we choose this convention or this convention.

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‫So in reality, it matters which convention you choose.

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‫It matters whether you choose a fixed angle approach or other angle approach.

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‫Either you rotate about the fixed axis or moving frame axis.

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‫And by the way, so far I have only claimed that this product of these rotation matrices is the same

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‫for this and for this.

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‫I haven't shown it mathematically.

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‫When we derive our transfer matrix as a bonus, you will also mathematically see that this product of

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‫these three matrices is also valid for this convention here.

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‫This is the name of the lecture in which I'm going to mathematically show you that this product is also

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‫valid for this convention.

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‫But don't go there yet.

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‫This lecture will only make sense if you don't skip any videos in between.

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‫You first need to start deriving the transfer matrix and then there will be an added value to doing

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‫it because it will become apparent that for this convention, this is the product of these three matrices.

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‫Now, it's not enough to know how we multiply these matrices.

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‫We also need to know which angles we put into which matrices, because that will change depending on

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‫the convention you choose.

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‫In this course, we use these three angles, Phi Theta and BPCI.

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‫However, in this particular video, I will say that my PHI equals Gamma, Theta equals better and BPCI

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‫equals Alpha.

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‫I do that because the book that.

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‫I used to verify the correctness of my rotation matrix product derivations, that book used the angles,

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‫Gambetta and Alpha, and since it's very easy to get confused and make a mistake in this topic, I decided

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‫to use the nomenclature of the book that way.

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‫I'm certain that what I'm going to show you here is actually correct.

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‫However, it's just a name we can call our angles Fifita and BPCI or Gamma, Beta and Alpha or something

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‫else.

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‫An angle is an angle and how we named those angles.

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‫It will not change the logic of these rotation matrices.

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‫And by the way, the book that I'm following for this rotation matrix topic is called Introduction to

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‫Robotics, Mechanics and Control.

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‫Third Edition by John J.

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‫Craig, it's a great book, by the way.

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‫I highly recommend it.

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‫If you're interested in robotics in this book in Appendix B, you will find the final rotation matrix

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‫products for all the twenty four conventions and then you will see that even though there are twenty

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‫four products, there are 12 different products because a product, for example, for this convention

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‫and for this convention will be the same.

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‫But after we have derived these three products, then you will be able to derive the rest of them as

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‫well because the procedure is the same.

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‫And so in the next video we will derive our first product, which is this one here.

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‫See you in the next video.