Now, moving away from just getting the dynamics. That's the natural motion that would happen if you give it some initial tumble rate. The wheels has this gimbal rate and has this spin rate and it just see what happens. You can also throw in torques and see what happens. But now we want just good things to happen, things that we actually care about. That's where feedback control comes in. Last time I showed you how we can take that very same Lyapunov function that we used for this classic three-axis feedback control where the control variable was u, an external torque and now we're solving for u as switch is a set of reaction, we'll motor torques and we get there. When we derive that, if you do the control with thrusters or you do the control with reaction wheels, this MRP feedback that we've seen before, you've done it in your first homework. Is the closed loop response going to be different just because we use reaction mills now? Why not Anthony? [inaudible] Yeah. Well, this particular, your closed-loop dynamics, how do my errors behave is exactly the same. We always get to that step. The trick is always going to be we have a Lyapunov function. It's in terms of the states and the rates, you differentiate it, then the V dot has rates in both the first and the second term, we can factor out the rates, make it negative semidefinite or negative definite in terms of rates but it's overall negative semidefinite. That's the V dot that we prescribe, set those equal, factor out the rates. This way we can pull out a vector equation that we have here or three by one matrix equation and that's the closed loop dynamics. For all these controls that we keep looking at at least at this stage, this has been the exact same internal response that you would expect. We just achieve the required torque differently and yes, the wheels are spinning to have momentums as we rotate, you have to compensate and that compensation is happening through these additional terms. That was basically a new one. Everything else looked very much the same like the classic ones. We feed forward, compensate for whatever the wheel gyroscopics are doing and then do this. What does that mean? We feed forward compensate for the wheel momenta. What if the wheel speeds are big? How fast can a reaction wheel go? Give me a rough number. How many RPM? Five thousand RPM, 6,000 RPM, somewhere there depending on the manufacturer. If you do the math, if you take any one of those wheels and do 5,000 RPM and then say and, that much momentum times that and you cross it with your angular velocity of the spacecraft. Let's say the spacecraft, what do you think is a decent maneuver speed? Rough numbers. Yeah, easy number. That's not bad. Nobody would cry too much foul. You don't want to do too much but one degree is already. We're moving along. You do one degree per second times 5,000 RPM and whatever momentum they're giving you and actually the manufacturers typically told you the momentum, the maximum maintenance you can store, and you take that value, that cross-product gives you a torque. You take that value and compare it to the maximum torque the wheels can produce. What you will typically find is you have well exceeded what the wheels can actually produce. That's why in theory you can go up to that. But in practice, you don't want to be flying the things with 3, 4,000 RPM steady-state. You may temporarily do it because you're spinning up on one axis and then you have to slow down again, so you put momentum into it and you're pulling it out and you coordinate it and it works. But generally if it's just humbling at 1, 2 degrees per second and you're already going 4,000 RPM on the wheels, you're going to be saturating it like crazy and having all kinds of challenges there. This is the classic feedback control that you might have. This is a torque loop that we did and then we solve for gs times us which means we have to figure out how do we actually unravel this projection. We know what the projection of motor torques onto the spin axis should be so now we do the inverse problem. Given that quantity, given this required torque that has to act on the body, what motor torques do I apply? Depending on the wheel configurations, you can do, we typically use a minimum norm inverse but that's not required. You might use different inverses. If you only have three wheels and the span three-dimensional space there will be a unique answer. There's no wiggle room. Before more you will have wiggle room and all space that you can then use different inverses as well. That's what we derived. This was a torque loop.
