Now, next set. Rodrigues Parameters. We can go reasonably fast through the classical ones. We will spend more time on the modified. The classical are classical. The modified are the younger, more hipper sets that everybody else uses, so we'll stick with the old stuff for a little bit just to see the history. They're important you will find them in different fields but then you understand the classical, the modified will make sense that quickly, why did they get derived, why are we so excited about that and what can we all do with that, right. And this is in fact most of my attitude controls these days that I do, everything is in terms of modifying Rodrigues Parameters. And there's a lot and lot of papers published, many of the attitude control papers these days are published in modified Rodrigues Parameters. For the control, for the description It's a very convenient set, even estimation. We're starting to make in the roads in estimation now with these things. So, Rodrigues he lived about the 1880s, 18870s, somewhere around there. He published a whole series of papers. It's interesting when you go back then actually as a grad student lifetimes ago, I read one of his original papers from back then. I found a copy and this is prior to vector notation, prior to matrix math. It's just tons and tons of scalar equations. It gets quite interesting as a classical historian and looking at these things. And it's just a + b component + c. And you have to remember a, b, c happen to be of this set and d, e, f happen to be of this set Typesetting was very difficult. These days, we're spoiled. And matrix math, vector math, makes our life so much more compact to derive it. So we're going to do this the modern way. But Rodrigues was an amazing mathematician, did a lot of very interesting work. So how do they define these and the easiest way to define them is really in terms of the Euler parameters. You can see here the definition where we can say compared to betas, I'd take the factorial part of the betas. So beta 1, 2, or 3 and I simply ratio it by the scalar part, beta naught that's it. And then you going to get, three parameters. With three parameters, I know right away, there must be singularity. So Walter, where is singularity going to be here, if you just look at this definition? >> [INAUDIBLE] >> Which orientation is that if beta not goes to zero. >> 180? >> 180 exactly, right. So upside down,these parameters actually blow up. The Euler angles we defined, they never blow up,they were all defined between plus minus 180 or plus minus 90. They were just mathematical ambiguities, right. There was sort of an orientation, but all of a sudden, there's an infinite set. And the differential kinematic equation had 0 over 0 terms which you couldn't mathematically resolve. These will behave very differently. These will literally blow up to infinity. Their magnitude grows to infinity as you tumble upside down. So again if you think of spacecraft attitude control. If you have a real strong aversion to tumbling, you easily get motion sick. [LAUGH] If you want to throw in a feedback control that's going to get increasingly aggressive as you start to tumble upside down. This will actually do it, without doing any crazy gain scheduling or anything elsewhere here you're so, the gain is one if you go far you gain is 10 if you go a really far your gain is 1,000, you don't have to do that, a fixed gain will give you a behavior at and as you tumble close to 10th effort going to to infinity assuming you have infinite thrust capability. Slight assumption. But you can see right away with these coordinates, we can actually, if they go to infinity it could be an issue or it could also be an asset. If you doing constrained attitude control. So benefit q i, so I'm using q here, again that's the classical thing for them that's why the betas are nice, otherwise I'd have q's and q's and things get really complicated. So if you want to invert this, you can solve for beta not, there's 1 over square root 1 plus the norm of q squared. Or scaled this way you get beta i's. I had you drive this in the homework. A little bit of Algebra. The key is remember the Caperion Identity, that's one that you have to use to kind of get in this form. But this just takes a few lines. Once you see how to do the math but definitely something I think is fair game in an exam as well. These kind of simple relationships, can you apply these properties? So Caperion we have three of them. Or quaternions. CRPs, classical Rodrigues parameters. Singular of 180, that goes to 0. Same thing here. To inverse mapping we'll have issues. This goes to infinity and this goes to infinity. And in fact we know that the norm is finite. The betas never go to infinity. But you something that would be very difficult to evaluate. You have infinity divided by infinity. So we don't want to apply this to a problem if its tumbling and you have to account for a multi revolution problem this would be a poor choice of coordinates. But compared to the Euler angles, Charles, with Euler angles how far away were we from the singularity at the best case. >> Is it 90? >> 90 degrees. You can pitch up and down 90 degrees. So the best is zero pitch. Or inclinations, you go from zero to 180. So a polar orbit, 90 degree inclination, is the furthest away from those singular behaviours. That's it. Here at least we have a three parameter set that moves a singularity 180 degrees away. So you could see it right away using these if you lean your eyes dome, their domain of linalization where it's valid it's much bigger about this twice the motor oiler angles are doing. So we're going to speed up to looking at how lineariable these coordinates are. The relationship back to if you plug in the definition this EI times sine for two this is cosine over two, so sine over cosine just gives you tangent X this vectorial part to E1, E2 and E3. So, that's a nice vector matrix notation that puts it together. And if you linearize CRPs. If you're just doing little wiggles and tracking errors and stuff. And q1, 2's, and 3's, you take the values times two, that's roughly how many radians error you have, in roll, pitch, yaw, that your kind of more familiar with. So there are ways to interpret it, and again if the magnitude gets large I know by the way this is close to 180. That point so, there's direct ways here that we can map easily between this coordinates. You can always go to and in from the DCM. But these are much faster direct relations, especially for analysis. And again, in this function, if you have 180 over 2 is tangent of 90. And tangent 90, that's where that curve goes off to infinity, right? That was the issues that we have. Good, so much better suited for large rotations compared to all other angles, but is still doesn't quite allow a completely tumbling body without hitting a singularity, right? That's still the remaining issue with this, but definitely an improvement. Now, from the from a DCM, here's some convenient formulas. I show you another one in a second. But if you have to program it up, there's different ways to compute this particular term in terms of betas. And you take C transpose- C divided by that squared. You can get this tilde, or here's another way to get it, from the DCM you can get the q1, 2s, and 3s. They're all kind of piggy backing on, this looks similar to how we got order parameters from a DCM. There was difference in the of diagonal terms to get the vectorial part but the scaling looks different. So these things are all related. But those are some convenient form as you might see if you're using it.