What becomes of G_s if you do that? If your G_s axes line up with B1, 2, and 3, what does G_s become? [inaudible]. The G_s is equal to g_s1, g_s2, g_s3, and it's got three wheels. A popular choice is still just like, well, where do I put the wheels? Well, this first one, put them on the first axis. This becomes equal to b_1 hat, b_2 hat, b_3 hat. Now in all the math and we actually solve it, we put all this stuff, all these g_ses, g_ts, and other things all into the body frame. Paul, what is b_1 in the body frame? Give me the numbers numerically b_1 and b frame components. [inaudible]. Yeah, so that would be this. This one is going to be 0, 1, 0, and this is going to be 0, 0, 1, so it gives you the identity matrix. Boggles my mind how I'm off today? If this becomes that, then it's just going to be Omega cross a [inaudible]. it's a three set of wheels. It's Omega till the HS, that's all you have to do. There's no g_ses and if you want to plug in equations and solve for a motor torque, your motor torques have to be Point 1, Point 2, Point 3. You no longer have a job to project the requested torques onto the set of stuff. Whereas here, if you solve for the motor torques, you really solve for gs, us. Then you have to take an inverse. Let me see, I thought was going on, we'll do that next time. I'm just going to go here. With reaction, we also have the G_s matrix times us, and in the end we will come up with a feedback control. But this is the term we going to solve for. I'm just going to call it the required torque, maybe plus or minus you can use either sign it doesn't matter. How are we going to find that? That's the required torque that our control needs 0.1 Newton-meter by the one axis, 0.2 About the second minus 0.1 about the third. If I have three wheels, and this is an identity, so if G_s is equal to the identity matrix, then U_s is equal to L_r and you're done. If G_s is a three-by-three, then what is U_s? How do you solve this for U_s? You take the G_s We take G_s, don't give out on me here. inverse, Good grief. Inverse L_r. Now, does this inverse exist? I have three wheels, you're shrugging, what does it depend on? The rank. The rank. Abby, is it possible to have three wheels and not the ranked 3. Yes if the two and one is the matrix. Some joker made this point. You want the three bills in him. That's easy. I'll just bolt them all down. I gave you three wheels and you laugh, you never know how cheap helper are you going to hire to put the spacecraft together. That would be bad because then, but as we joke about this, but the takeaway is a really important point. You control the rank of G_s. If you make it rank deficient, you have nobody to blame but yourself and you laugh, but we're going to go to c and G_s and you will no longer control the rank of that matrix. It depends on dynamics and that's what the gimbal lock and singularities are. But at least with wheels and joy, you get to control the rank of that one and therefore, you can put it there. Do we ever put two wheels on the same axis? Andrew. [inaudible]. Exactly. I've seen spacecraft do that. They literally had six wheels 2, 1, one, 2, 1 another, 2, 1 a third, and they weren't actually orthogonal. They skewed them as well, because they really needed a lot of torque about one primary. If one failed, this was their backup, or maybe just financially was the reason, because you can buy off-the-shelf wheels for a tenth of the price. This gives you 10-newton meters of torque. Great. I need 11. That's not going to cost you 10 percent more, it's going to cost you 100 percent more or 500 percent more to get a custom-designed wheel. The more you can go cots commercial off the shelf, you might just go, you know what, it's cheaper. With the mass, it might be slightly heavier, but we can take that hit. This does happen, but you still want to make sure you control the wheel orientations, and that this is full rank, and everything is good. Smiley face. Now, what if G_s is a 3 by N, and N is greater than 3? Same question, are we guaranteed that it's full rank? The answer is it depends. You might have put N wheels along one axis. It's like, yeah, well, that was a waste of money. But you get to control now. But what is different with taking an inverse of a 3 by N versus a 3 by 3? [inaudible]. Yeah. This inverse, what did I call that? The pseudo-inverse, is that typically this symbol? Maybe plus or minus depending on which pseudo-inverse you substitute. Yeah, times L_r. The pseudo-inverse will give you two possible inverse types. What are the options? Do you remember? One is minimum energy solution? Minimum norm. Minimum norm. What's the other? Forget. Least-squares. If you review quickly, the least squares one is you have y is equal to ax plus b. Here's all your data points, and you're fitting a straight line. I only have two knobs to tune. I get to pick an a, and I gets to pick a b, but I don't just have two points. If I have two points, it's like the identity matrix. Then with two points and two knobs, two points always describe a line, unless the two points are the same point, in which case, again, you're out of luck. But two points is fine. But if I have more than two points, it will find the least squares fit, typically. That's not the only answer, you could fit otherwise across them. But that's typically what you can get. Whereas this problem is more, how do I describe that? I'm trying to think how to write this in a way that makes sense. I'm going to have a naught plus a_1 x plus a_2 x squared. I have one and two data points. I have three knobs to tune, but I've given you two data points. Are there multiple answers? Is there a single answer? Is there no answer? Drow, what do you think? No, they've got the multiple. Multiple. Because you can now write this kind of a curve, you could have something that comes in and accelerates, maybe you have a curve that does this. There's an infinity of curves that you can pick, and that would fit that property, all of a sudden. The minimum norm inverse will give you the one set of parameters where a naught a_1 and a_2 are minimized. So the smallest set of parameters that satisfy this condition. Makes sense? Wait, now, I have to go here. Here, what type of inverse are we doing? Say, we have four wheels. Are we doing a least squares fit or are we doing a minimum norm? Probably. Probably.