‫Welcome back.
‫In this video, I want to clarify the conventions of the rotation matrices, it can be a pretty confusing
‫topic.
‫So in this video, I will clarify that in total, you have 24 conventions for rotation matrices 12 when
‫you rotate about fixed axis and 12 when you rotate about moving body frame axis.
‫So here you saw the 12 different conventions.
‫These were for the fixed axis and then you have 12 more for the moving frame axis.
‫However, some things need to be clarified, and particularly it is important to understand how the
‫final product of the three rotation matrices are X or Y and our Z is computed for each convention.
‫We will derive these products of matrices for three conventions so that you would understand the procedure
‫and the conventions will be R, X, Y, Z, R, Z, Y, X and then R, X, Y, X..
‫So here we are rotating about the inertia from Axis.
‫But as you know, you will get the same product of matrices if, for example, for this convention you
‫first rotate about the moving body frame Z axis, then about the moving body from Y axis and then about
‫the moving body frame x axis.
‫And then for this convention, it would be rotating about the body frame x axis, then body frame y
‫axis, and then about the body frame Z axis, and then in this case, rotating about the body frame,
‫x axis, body frame, y axis and then body frame x axis.
‫So even though we have twenty four different conventions in total, we have 12 different products of
‫these matrices.
‫So if we only deal with rotation matrices, then it doesn't really matter if we use this convention
‫or this convention, because in this case, this would be the product of these three matrices for both
‫of them.
‫In this case, it would be this one.
‫And in this case, it would be this one for both of them.
‫But very soon we will start deriving a transfer matrix.
‫And as we derive it, then you will see that this transfer matrix will be different depending on whether
‫we choose this convention or this convention.
‫So in reality, it matters which convention you choose.
‫It matters whether you choose a fixed angle approach or other angle approach.
‫Either you rotate about the fixed axis or moving frame axis.
‫And by the way, so far I have only claimed that this product of these rotation matrices is the same
‫for this and for this.
‫I haven't shown it mathematically.
‫When we derive our transfer matrix as a bonus, you will also mathematically see that this product of
‫these three matrices is also valid for this convention here.
‫This is the name of the lecture in which I'm going to mathematically show you that this product is also
‫valid for this convention.
‫But don't go there yet.
‫This lecture will only make sense if you don't skip any videos in between.
‫You first need to start deriving the transfer matrix and then there will be an added value to doing
‫it because it will become apparent that for this convention, this is the product of these three matrices.
‫Now, it's not enough to know how we multiply these matrices.
‫We also need to know which angles we put into which matrices, because that will change depending on
‫the convention you choose.
‫In this course, we use these three angles, Phi Theta and BPCI.
‫However, in this particular video, I will say that my PHI equals Gamma, Theta equals better and BPCI
‫equals Alpha.
‫I do that because the book that.
‫I used to verify the correctness of my rotation matrix product derivations, that book used the angles,
‫Gambetta and Alpha, and since it's very easy to get confused and make a mistake in this topic, I decided
‫to use the nomenclature of the book that way.
‫I'm certain that what I'm going to show you here is actually correct.
‫However, it's just a name we can call our angles Fifita and BPCI or Gamma, Beta and Alpha or something
‫else.
‫An angle is an angle and how we named those angles.
‫It will not change the logic of these rotation matrices.
‫And by the way, the book that I'm following for this rotation matrix topic is called Introduction to
‫Robotics, Mechanics and Control.
‫Third Edition by John J.
‫Craig, it's a great book, by the way.
‫I highly recommend it.
‫If you're interested in robotics in this book in Appendix B, you will find the final rotation matrix
‫products for all the twenty four conventions and then you will see that even though there are twenty
‫four products, there are 12 different products because a product, for example, for this convention
‫and for this convention will be the same.
‫But after we have derived these three products, then you will be able to derive the rest of them as
‫well because the procedure is the same.
‫And so in the next video we will derive our first product, which is this one here.
‫See you in the next video.