0:06

Stability, now we've discussed stability in this class some.

Â Maurice, if we talked about rigid body or

Â we had tubule spinners, we talked about stability.

Â Could you quickly summarize for me what we're talking about there.

Â >> Well, we're talking about the [INAUDIBLE]

Â changes between rotating it up and [INAUDIBLE].

Â >> Why does total energy change if you rotate, [SOUND]

Â >> [INAUDIBLE]

Â [SOUND]

Â >> If we're torque free,

Â what happens with total energy, of a single rigid body?

Â Stays the same, actually that's one of our constraints.

Â Same thing with momentum.

Â We just put momentum in body coordinates and then it gave you the whole parts.

Â Where momentum was written as just a sphere, that constraint, and

Â then energy is rewritten as a ellipsoid.

Â So we had those arguments which were actually good.

Â And we talked about stability there.

Â With the did we ever do any linearizations?

Â Andrew.

Â >> Can you repeat that?

Â >> Did we do any linearizations when we looked at the plots and

Â argued stability and so forth?

Â >> No.

Â >> No actually which is kind of nice.

Â It's a very graphical way, but

Â it wasn't, this is going to be a much more mathematical way now to approach that.

Â But that was kind of showing yes, if you're doing this,

Â that's how this constrains.

Â This must be your one-dimensional omega curve that you will have, right,

Â which was kind of a cool thing.

Â We did do a mathematical approach when we went to the dual spinner.

Â Let me derive the equations of motion for a dual spinner, right,

Â and we ended up linearizing them about certain states.

Â So Robert, what does it mean to be stable, and those already are problems.

Â Just in plain words, no math.

Â 1:59

>> Spinning around the first or third principal axis or

Â the second with no perturbations?

Â >> So the second one, okay.

Â You brought that one up.

Â Where is my, I don't have anything good here to throw around.

Â Here we go, one of these.

Â It has nice distinct inertias, right?

Â Least inertia, max inertia, intermediate inertia.

Â So if somebody says, look,

Â if you throw this about the intermediate inertia, it is stable.

Â Is that the correct statement?

Â Abdul no Brian, what do you, okay.

Â >> No >> Brian why not?

Â >> Well, it's at an equilibrium point but

Â there is no driving force pulling it back towards that equilibrium.

Â >> Right, so if you're talking stability, that's the key thing I hope you're getting

Â here is the stability it's about departures over reference.

Â What Robert was talking about is the reference itself,

Â if we do the pure spin that would just continue to do that.

Â That just makes it an equilibrium, it doesn't guarantee its stable.

Â Stable means you take something, that's your reference.

Â And now I bump it slightly, infinitesimally, and does that stay close.

Â And what we've shown and also mathematically, we linearize it all and

Â show hey, that's where the stiffness of that spring mass system becomes negative.

Â And the system is definitely unstable.

Â And the practical examples showed very quickly.

Â It just takes a little delta and it goes unstable.

Â So there's this distinction between equilibria and well,

Â your reference point as we'll discuss here, and the stability.

Â The stability is the motion about the reference.

Â So if you have this path you want this robot to follow and you program in,

Â be here at 0, be here at 1, be there at 2.

Â Then your reference keeps moving, and

Â the stability is define that if you're off a little bit do you stay close?

Â 3:51

So that's it, now for

Â linear systems once it's stable typically then what we have actually the roots.

Â All on the left hand side of that imaginary plane,

Â you may have imaginary roots but negative real parts.

Â That means all the expected components will decay.

Â So if it's stable, it is exponentially stable.

Â You can always put this exponential curves above it.

Â That's it, it will converge.

Â It is asymptotic, you know.

Â It's always those wonderful things.

Â If it's marginal stable, that means you have a system that basically

Â is like a pure swing mass without damping.

Â Then the roots go in the imaginary axis.

Â The real parts is zero and it will oscillate forever, and

Â that's one of the challenges.

Â So if you deal with the linear system that is that is the form

Â then that is your response, your sines and cosines.

Â However if you're starting with a non-linear system and your linearized form

Â is an [INAUDIBLE] oscillator, it's not quite rigorous to say well it's stable.

Â Because the higher order terms could actually make that oscillation slowly

Â grow, in which case it's unstable.

Â It could decay, in which case it's asymptotically stable.

Â Or it may do nothing and it just keeps wobbling along forever.

Â So that's where things get a little bit more distinct.

Â So if you're coming from classical linear control ideas just it's either stable or

Â unstable.

Â There's a whole new world you've been [INAUDIBLE] exposed to.

Â