We're going to shift gears. I have another 20 minutes. Good. We talked about reaction wheels. You've seen some basics of reaction wheels before, 50, 10 a little bit, so we'll revisit them in a more general way. What happened? There we go. Increase this. Can you guys see this now reasonably? I started it up. What we want to talk about is just reaction wheel configurations. This could be an interesting project type because even that simple question can be quite challenging. What do you think is the most popular or one of the more popular ways to align your reaction wheels? If I give you a body, and here is b1, b2, b3, and I give you three wheels, Anthony, how would you align those?You put one here one there and one over here, each lining up with that. Why would you do that Anthony? Because it simplifies your math. It definitely simplifies the math and we will see that when we get to reaction wheel controls next time. We're going to put this all together. There's going to be a projection matrix that becomes just an identity and like, I like it, not zero. But man, actually from controllability, you don't want it zero because zero times any control is still going to be zero. But it's an identity which is like, that's so much easier. That's one. Now, this is cool and you will see this stuff. What if I need a torque that is somewhere here in space and you do it's projections? With this, you can quickly figure out this torque has to be produced by this wheel, this torque has to be produced by this wheel, and this torque has to be produced by this wheel. It gives you a really clean decoupling. What is the challenge with this kinds of reaction wheel configuration though? If one wheel fails, let's say we're going to go to red and go, this is bad. Now we're left with two wheels. That's a challenge because now we can produce some torque, but only the torque component that is in the b2, b3 plane, not the one that is in the b1 because that wheel is just done. As project, that actually makes an interesting control too because if the wheel is broke, how's it broke? How do wheels break? Bearing failure, that might just go, because now it's just metal on metal and it's grinding. You will have awful noise if there was noise and space. Thank you, Star Wars, but let's pretend there's noise. It's more fun, isn't it? Imagine Star Wars TIE fighters coming by without noise, it just wouldn't be the same thing. Metal bearings break metal on metal. What happens to the wheel speed? How much? A lot to zero. What happens to the spacecraft as the wheel speed rapidly decreases and it just comes to a grinding halt relative to the spacecraft? [inaudible]. Whatever we'll speed it had, it's now into the spacecraft. What does the spacecraft do? Off it goes, and then the scientists get upset, and the control operators, mission operators declare emergencies and bad things happen. Go see the Kepler mission. They lost lots of wheels. Those were not good days. That's a problem. What if one fails? Is there anything you can do with three wheels and prevent loss of controllability if one would fail? Not really. I mean with three you're at the bare minimum. Now you just have no reserve. If one of them has an issue, there is no backup, so we can look at others. What are other ways wheels could fail? What is driving the wheel? The motor. Do motors fail? If the motor fails, what happens to the wheel then? [inaudible]. Not enough. If the motor fails, it may have a partial failure where you're asking for one newton-meter, but for some reason this thing is just having issues and they cannot produce it. You might get less torque. That could be an interesting project too that you can look at. For example, power. We talked about how much power it takes, and we're just assuming we get whatever torque we asked for. But if you actually did a circuit, electrical circuit that says, well, I have a battery, it can produce only so much power at a time. If all of a sudden you're drawing 200 watts of power and I can only give you a 100 watts of power, you are not going to get the torque that you want. That would be an issue. That's something you can actually model, power constraints. But besides saturating the wheel speed, if you're asking for more torque than the thing can provide, you're not going to follow the power curve. But if the motor completely breaks, you essentially have a free spinning wheel. I'm no longer producing a torque. Something was able to produce a torque electromagnetically or something, and all of a sudden, all those circuits fail, all the currents going out of the electromagnets. Now it's just spinning freely and the brakes haven't been applied. Let's say that failed. You might have a free spanner which can then absorb momentum and dump momentum back into the body and do whatever their dynamics require. That can be a disturbance all of a sudden. Broken wheel can act like a disturbance. If the wheel is broken, you may not be able to measure and feedback compensate for the wheel momentum. How much does that impact your stability and your control? That's something you can numerically investigate too and play with. If we have three, this is a popular configuration that we do. We don't like three we want at least four. If we have one wheel here, here and here, what is a possible way that you could add a fourth wheel? On the diagonal. Is that what you said? [inaudible] Complete there. He's talking about, if you make this box as a vector from here to here, and if you put a wheel on that diagonal. So it's diagonal evenly with all the three axis. That's the fourth. That's a popular configuration. What's the benefit of that configuration do you think? You could back up any of the three. True. You could backup any of the three, but I could also change this wheel and move it up 10 degrees and I can still backup any of the wheels. Why is this one more popular than just having it slightly off the b1, or b2, or b3. What do you think? [inaudible] Yeah, basically this gives you the balance. I don't know if one, two, or three is going to go back, but I'll do the best I can. Then you have something that's not a nice orthogonal projection, but it's askew projection, and you have to do the right math to get the motor torques that we need and put this altogether. Okay, good. That's one. But now let's look at other configurations and see how that goes. I'm going to use mathematic to do that. Here right now, I'm using three wheels and then later on we'll go to four. What I'm showing you here across three wheels, the best coverage you could have is let's say each wheel produces one Newton meter. The question is, what's my torque envelopes? What torques can I produce? Because if I have three wheels lined up with my primary axis, about which axes can you produce the largest torque? That was here. If I do have these three wheels working, what is my best controllability where I can get the most bang? Is it when I just talk about b3? [inaudible] Again, in that diagonal what happens if you maximize one, two, three wheels? You then get the maximum about the b1 axis, the maximum about the b2, the maximum about b3. In which one does one Newton-meter, just to have easy math. You get square root of three in that direction. You can see with this configuration, you've given yourself the maximum controllability about this diagonal axis. What if you don't want that? A lot of spacecraft have modes where it's like look, we're flying, I'm [inaudible] up and down, but mostly I'm scanning left and right as I'm flying over an area. You need maybe more torque-ability along one axis. If you do it with three, you probably want to line up that diagonal with that axis that needs the most strength. That's what you can do. But you can see your torque envelope is not going to be homogeneous. It's not going to be the same across all possible directions that a torque could act on it. There's actually papers that study torque envelopes for reaction. That could be a cool project to investigate that and how you can maximize it. I'm going to show you my stuff with Mathematica. Here, the best I could have is all three wheels contributing at the same time. Then I'm looking at the [inaudible] long so like a planetary map. You got to azimuth and elevation, so I can look for all possible torque orientations. What's the thing that you would have? Let's see. I'm computing that here, either as a contour plot so I won't do a density plot. The three wheels for all possible orientations. You can see there's regions where I have less controllability somewhere I have more. But this is the one that's fun. Then let me do it live. Here, for example, I've lined up my three wheels to be essentially along three orthogonal directions. I've drawn it such that the diagonal across all three happens to be the blue AP axis. Contained. That's what we're now looking at. What's that angle? This forms a little pyramid, those three. You could always stand them upside down then it makes a pyramid. The question is, is this angle perfect? Or could I go steeper? I can't make my fingers go orthogonal shallow, it's hard. But how shallow, how steep in the middle. What are the benefits of that? There's papers just on this topic, or they often become mission-specific. Here on this sphere, I'm showing you how much you can torque along different axis and one is a darker orange. Those are the regions where if you're torquing around those this is where the wheels aligned up. That's where you're going to get the pure orange, that one. Everywhere else were a little bit brighter and they contribute in different ways. But right behind the wheels, we would have just one because they're here essentially orthogonal. The most torque happens along that diagonal axis, this is somewhere up here. Square root of 3 times 1, which puts you somewhere up there in that color. That's what you have. Again, if you have a mission that has, well, not all axis are created equal. I need more torque around one than another, even with the three configurations. You can start to play with these envelopes and see what goes on. But we can also change these angles. I can make it shallower. All of a sudden with a very shallow but slightly elevated, my three wheels spin axis still span three-dimensional space. I can do with any general control, but I will have much more control ability about though, one and two axis, less on the three. Maybe that's the one that doesn't need as much controllability and that would be a reason may be to go with a shallower configuration of your wheels. Versus if you really have to have a lot of torque around one, you might go with a much steeper configuration. Now you can see the best you could do is three. If everybody lines up, but then you've really mess with the other two axis because now you don't have 3D Control. You need something that spans three-dimensional space and something like this might give you that big torque. What people also do sometimes instead of three wheels there may have, I saw one spacecraft that had six wheels. It was expensive to get a spacecraft that has wheels with two-newton meters because you have to custom build a wheel that was that strong. It was far cheaper to buy two wheels that have a one-newton meter of torque and just stack them next to each other over each other so they control the same axis. That's how we got an extra torque. Sometimes you do tricks like that and then one of them is stronger. Or you can play with the configuration, or you can do multiple by getting bigger and smaller wheels. Maybe you don't need three big wheels. Maybe one bigger one and the other to support and you play with the angles. You can see from a design space, this gets very interesting and very mission-specific. What are the maneuvers that you're going to do and how are you going to put this all together? There we go. Now, the four-wheel one, let me just instantiate all of this. Now I have four wheels, the same pyramid configuration. If you compare this one, there's light and there's red. There's pretty high variations in color, which means you have dips and valleys in your torque envelope. Some of them you get only low, up down to one, other ones, go up to square root of 3. That's a pretty high variation. With 4 just in a standard pyramid configuration at 45 degrees. I'm getting something much more even. Here we see different angles, this is 45 and this one is 36. 45 still variations but not the harsh color changes we're seeing earlier. If I tweak the angles to 36, that one tends to be a popular one with four wheels because it gives you this almost pastel-like. It's not completely even that's not possible with discrete number of wheels, but it's pretty done even. That means I get the same control ability across all possible axis. Whereas here is another configuration which gives you more control ability about one axis, but it's not quite with the dips and valleys that three wheel configuration has. What does that look like? Here's the four wheel and we're in the 45. That's the first plot we saw. If we decrease this to about 36, you can see here. You see it has becomes almost even. That's where you can get much more of an even, not quite as spherical torque. If you plot a torque per axis, this will give you a surface that you can look at. That's cool. There's a lot of things you can play with. This tends to be more of a standard CMG configuration, this 36 degrees four-wheel pyramids. That's what you would have. Here is interesting because I think here I'm looking at it. What if one of them fails? Now instead of four wheels, I only have three. You can see I can still span the space, but all of a sudden you have a dip appeared, all of a sudden it was like, well this one's nice and even but man, losing one. There's a big hole in my torquing capability that might cause saturation, performance issues, and how that goes. There's other studies. This is some optimizations I was running where I'm trying to figure out what angle minimizes these variations across the torque envelope, and if I have one optimal angle, if I have four working ones, with three, I've lost one. It's not going to be as pretty, but is there a better configuration that if I have a loss, I have more of an even distribution? That's what I'm playing with here so we can see what would happen. If I go too extreme, you have lots of variations. Again, if I go too extreme here, lots of variations. But before we had more the 36 and the slightly more optimal was more like 39. You can see a slight variation. It's not a huge change, but it is an improvement in bringing up the lowest torques, springing down the max torques just to get them a little bit closer. That's an interesting topic. Actually, you'll find papers just on this. If you do a mission and study it, how do you line them up? If you have four wheels, that's cool, what happens if one of them fails? How do you configure them in a way that it's easiest to recover with that kind of a system? Anyway, but this is reaction wheel configurations, 50, 10, we just lined them up, assumed this GS matrix or something, and here we're doing the same right now. But I wanted to bring up at least some discussion on this, to put some ideas in your head that just the classic b1, b2, b3 alignment may not be the best that you could do. Even there, you might rotate how you're doing it to whatever axis is the most active, give it the most torque capability, so you don't saturate so quickly. The other ones to keep it stable doesn't need that big torque, but it needs some torque just to keep it there. But you're doing these quick left and right sweeps, give you most torquing capability about this axis. Then with three and four wheels failures, that's the thing what you worry about. Other things people consider r is actually power usage, running four wheels at once. Sometimes people worry about wear and tear. I'm putting more lifetime on my wheels. Then you'd worry about, I'm I making one of them fail quicker versus less letting the fourth be locked? If one fails, you turn on the fourth and turn the other one off, and now we have another one that hasn't been used much. That's one option that sometimes people fly. Another thing to look at is with four torques, the power required to drive your attitude control actually becomes less. The reason needed is, if I have four wheels to torque, I don't have to accelerate the wheels quite as much if you have one in perfect direction. If you have three of them, you really have these torque vectors are all adding up. If I had one wheel in the direction of where I need a torque, that would be the cleanest way to do it. But now I have things fighting each other, especially when they're skewed and adding up to what I need. With four wheels, you tend to be more likely to be closer to a pure spin configuration where you can get a nice clean thing. All the torque required if they're all helping each other, instead of needing one Newton meter, I can get everybody with 0.2 Newton meter and not accelerating as much, and acceleration then relates to power requirements. The more this wheel speeds go up, your power will go up. That's going to be a challenge. So with four wheels we'll see. In the book there's actually sections and papers I've published on power optimal attitude control that talks about these things. If you're thinking of topics, power optimality could be something you could play with in this class and dealing with wheels because you will have all the kinematics to compute the actual power needed and monitor that. I'm just trying to throw out lots of different ideas how this all goes. Any questions on reaction wheel configurations? Whenever, you said adding more wheels to spin at slower speeds of that visual, I would think that they would also benefit because when you are trying to torquing one axis and you have another wheel that's skewed to some other axis, it access the CNG when you find it torqued. Then you introduced some other terms. What you're talking about is momentum bias essentially. If you have three wheels orthogonal and any of the wheels is non-zero, you will have n momentum and non-zero momentum. The bigger that momentum vector of your reaction wheel assembly or the reaction will cluster that you have, the bigger that wheel, as soon as you want to rotate, there's a huge gyroscopic torque that you have to compensate for with these little motors. That's why reaction wheel is going quickly, like 5,000 RPM, that's a physical limit. Typically we never get close to that. Because if I go above 1,500, 2,000 RPM and I do a rapid maneuver, it's going to sap. Those gyroscopic torques are bigger than my motor torques and I lose control and it will go all over the place. Those are things to consider when we do all this stuff like that. It gets surprisingly complicated. All the things that you have to balance and figure out, where are you going to fly and what speeds can you do.
