Stability, now we've discussed stability in this class some. Maurice, if we talked about rigid body or we had tubule spinners, we talked about stability. Could you quickly summarize for me what we're talking about there. >> Well, we're talking about the [INAUDIBLE] changes between rotating it up and [INAUDIBLE]. >> Why does total energy change if you rotate, [SOUND] >> [INAUDIBLE] [SOUND] >> If we're torque free, what happens with total energy, of a single rigid body? Stays the same, actually that's one of our constraints. Same thing with momentum. We just put momentum in body coordinates and then it gave you the whole parts. Where momentum was written as just a sphere, that constraint, and then energy is rewritten as a ellipsoid. So we had those arguments which were actually good. And we talked about stability there. With the did we ever do any linearizations? Andrew. >> Can you repeat that? >> Did we do any linearizations when we looked at the plots and argued stability and so forth? >> No. >> No actually which is kind of nice. It's a very graphical way, but it wasn't, this is going to be a much more mathematical way now to approach that. But that was kind of showing yes, if you're doing this, that's how this constrains. This must be your one-dimensional omega curve that you will have, right, which was kind of a cool thing. We did do a mathematical approach when we went to the dual spinner. Let me derive the equations of motion for a dual spinner, right, and we ended up linearizing them about certain states. So Robert, what does it mean to be stable, and those already are problems. Just in plain words, no math. >> Spinning around the first or third principal axis or the second with no perturbations? >> So the second one, okay. You brought that one up. Where is my, I don't have anything good here to throw around. Here we go, one of these. It has nice distinct inertias, right? Least inertia, max inertia, intermediate inertia. So if somebody says, look, if you throw this about the intermediate inertia, it is stable. Is that the correct statement? Abdul no Brian, what do you, okay. >> No >> Brian why not? >> Well, it's at an equilibrium point but there is no driving force pulling it back towards that equilibrium. >> Right, so if you're talking stability, that's the key thing I hope you're getting here is the stability it's about departures over reference. What Robert was talking about is the reference itself, if we do the pure spin that would just continue to do that. That just makes it an equilibrium, it doesn't guarantee its stable. Stable means you take something, that's your reference. And now I bump it slightly, infinitesimally, and does that stay close. And what we've shown and also mathematically, we linearize it all and show hey, that's where the stiffness of that spring mass system becomes negative. And the system is definitely unstable. And the practical examples showed very quickly. It just takes a little delta and it goes unstable. So there's this distinction between equilibria and well, your reference point as we'll discuss here, and the stability. The stability is the motion about the reference. So if you have this path you want this robot to follow and you program in, be here at 0, be here at 1, be there at 2. Then your reference keeps moving, and the stability is define that if you're off a little bit do you stay close? So that's it, now for linear systems once it's stable typically then what we have actually the roots. All on the left hand side of that imaginary plane, you may have imaginary roots but negative real parts. That means all the expected components will decay. So if it's stable, it is exponentially stable. You can always put this exponential curves above it. That's it, it will converge. It is asymptotic, you know. It's always those wonderful things. If it's marginal stable, that means you have a system that basically is like a pure swing mass without damping. Then the roots go in the imaginary axis. The real parts is zero and it will oscillate forever, and that's one of the challenges. So if you deal with the linear system that is that is the form then that is your response, your sines and cosines. However if you're starting with a non-linear system and your linearized form is an [INAUDIBLE] oscillator, it's not quite rigorous to say well it's stable. Because the higher order terms could actually make that oscillation slowly grow, in which case it's unstable. It could decay, in which case it's asymptotically stable. Or it may do nothing and it just keeps wobbling along forever. So that's where things get a little bit more distinct. So if you're coming from classical linear control ideas just it's either stable or unstable. There's a whole new world you've been [INAUDIBLE] exposed to.