0:06

And what we're going to do next is we're going to spend a little bit of time,

Â at least two lectures, talking about torque free motion, because for

Â spacecraft this is a pretty common situation.

Â You're not always thrusting.

Â You're not always reacting, because you're using lots of resources, energy, fuel.

Â So how do we understand torque free motion.

Â In fact there's many missions that exploit torque free motions to spin stabilize or

Â do all kinds of stuff.

Â And then we'll look at some disturbances like gravity gradients,

Â we'll derive that from scratch using very much the same kind of notation.

Â And dual spinners, that's another one.

Â And then we'll jump into the control stuff afterwards.

Â Okay, so I didn't

Â put that one off.

Â Let's talk about torque-free attitude motion.

Â As we've been saying today, H dot is equal to L.

Â If it's torque-free the L just becomes zero.

Â That says the nerve, whatever this vector is, H is just a vector in space.

Â 1:08

And a vector has a magnitude times a direction.

Â All we're saying with H dot equal to L is that as seen by an inertial observer,

Â this vector is fixed.

Â So that means, what you just seen before, H is equal to I omega, we compute

Â all this stuff, we have I in the body frame, we have omega in the body frame.

Â We very often write H is equal to something, something,

Â something in the body frame.

Â The body frame components of H for torque free motion are not zero.

Â 1:38

The inertias are fixed, but the omegas are going to vary, go back and forth.

Â But if you reconstruct in the body frame,

Â this is what the vector is, now you know what the body attitude is.

Â So times BN, right.

Â You map it back into the end frame.

Â Those, the three inertia frame components have to be fixed.

Â And that's essentially what we're doing here.

Â This is in the body frame.

Â This is typically how we write it.

Â Here we've assumed what type of body frame to get this form.

Â 2:05

Principle, exactly.

Â Now it's diagonal.

Â We'll do this a lot for analysis.

Â We can always do this for anybody.

Â It just simplifies our basic stability analysis that we're looking at.

Â So that we do that.

Â Then times some DCM in whatever coordinates you wish.

Â Some people that use something besides MRP.

Â I just want to highlight that.

Â That does happen.

Â So you get this and now you have to take the derivative.

Â This is only the N frame components.

Â Now you have to take the derivative of all this stuff and set it equal to zero.

Â And you get a lot of theta one dots and

Â omega dots and yeah that has to be true but what do I do with this dots.

Â It's not very insightful.

Â 3:28

It's good for numerical checks, it's not good for analysis.

Â So the next step is now for analytical insight.

Â How do we look at this stuff.

Â So really, the key component is we want to do everything in the body frame.

Â It helps us.

Â And we want to look at, we want to make use that momentum is preserved.

Â If we're doing torque free motion is there something else that preserved as well.

Â [COUGH].

Â Any momentum.

Â Yes, Spencer, energy actually.

Â Do you remember the power equation T dot.

Â Ended up being omega dotted with l, and at l0.

Â That was one of the cases, how we can get 0 power on the system.

Â So [INAUDIBLE] has to be preserved as well.

Â That's a result we'll jump into next.

Â So if you do everything in the body frame, H was equal to i omega.

Â Omega, in the body frame is the classic omega one b one,

Â omega two b two, and so forth.

Â 5:28

So that's a nice trick.

Â We go, okay, how do we get the magnitude.

Â Well, we're going to get the magnitude squared by just dotting H with itself.

Â Or in a matrix form we do H transpose H.

Â Same math, gives you the same stuff.

Â So this times itself just gives you I one squared, omega one squared, I two squared,

Â omega two squared.

Â And you start adding them up.

Â And all of this has to give you a constant.

Â Because whatever spin you gave the system whatever gyrations are going on

Â these things have to add up.

Â Omega ones twos and threes can vary with time but

Â they have to add up to give you the same number in the end same momentum.

Â So good this gives me one equation.

Â 7:42

>> Okay, not a circle because this is a three dimensional shape.

Â It's not a sphere.

Â >> [CROSSTALK] >> A, B, and C are different.

Â If A, B, and C were the same it would ge a sphere.

Â So what is the shape then.

Â >> Ellipsoid.

Â >> It's an ellipsoid that you actually, this is a classic equation for

Â the surface of an ellipsoid.

Â So that means, this is kind of like stuff,

Â where we have this 4d hyper sphere, that's hard to visualize.

Â A 3d surface, I can draw, so we'll see visualizations of this.

Â But this equation actually means that your solutions for

Â omega 1, 2, 3, reside on this ellipsoid.

Â 9:00

If a, b, and c are just constants is this an ellipse of a fixed shape,

Â or is this ellipse varying with size?

Â >> Fixed shape.

Â >> Fixed shape, right?

Â And so here we just have a lot of coordinates that.

Â Where people get quickly confused is what is the independent stuff that

Â we're looking at.

Â So the omegas would be independent things, like x, y and z.

Â You see in the classic ellipsoidal, Wiki notes or something.

Â I1, 2 and 3, those are principle inertias that are fixed in time because we have

Â a rigid body.

Â If these weren't rigid, these would not be fixed things.

Â And we'll see that actually with dual spinners, so you have multiple buttons.

Â Same thing here, this also will make a squared y.

Â So that's x squared y squared z squared times some constant

Â equals to something that's constant.

Â This is also an ellipse, so that that George was talking about right,

Â that's the space we are actually looking for.

Â We have two constraints, this is an ellipse, that's an ellipse.

Â This two has to be intercepted and that's the answer, and that's not very intuitive,

Â it's not easy to intercept two ellipsis to.

Â So people came up with this other approach.

Â Instead of treating omega one, two, three,

Â as the independent coordinates we're trying to figure out what's happening.

Â We use scaled versions of omega one, two, three.

Â Basically you multiply in times of principle inertias so the new independent

Â coordinates are simply your body frame angular momentum vector components.

Â So I'm treating H1, 2, 3s.

Â I can find H1, 2, 3 time histories, and then to map them back to omegas,

Â all I do is divide them by the principal inertias and you get omegas again.

Â But if you do this, and you get momentum, constraint looks like this,

Â what type of geometric shape is this?

Â 10:40

This is a sphere, all right,

Â it's basically constants equal to x square plus y square plus z square.

Â So this is definitely a sphere, that's easy to draw, as you will see.

Â And then the energy constraint,

Â in terms of this, I will let you do this on your own.

Â You just have to plug in that h1 is i1 omega1, yeah we arrange it

Â you end up getting the energy constraint of divided by t gives you this.

Â This is kind of a classic normalized forward where the ellipsoidal

Â form is something old square equal to one.

Â And then this these terms if you go look up ellipsoids which you could do easily.

Â These become related to your semi-minor, semi-intermediate, and

Â semi-major axes of that ellipsoid that you're going to have, right?

Â This is just a unit, not a unit sphere.

Â It's a sphere of size H, and then you get these energies that come in there,

Â and this is something we can start to put together.

Â So for torque-free motion,

Â to consider what happens to the rates over time as you're jumbling and tirading for

Â anybody, this is nothing cylindrical or symmetric, it's any inertias.

Â These are the two constraints, we've moved from Omega space to h space, and

Â that gave us a simpler way to geometrically find these things.

Â So that's a sphere, and this is the classic ellipsoidal equations,

Â that you can find your a, bs and cs.

Â Which are you're semi-axes for

Â the three things, so yep, that's what you can find there.

Â So we'll have three, if we have three distinct inertias,

Â you will have three distinct axis, and

Â a key element again is well let's see, can t vary on a spacecraft?

Â 13:11

H dot equal to l still holds,

Â if there's no external torque acting on the system, H has to be preserved.

Â So for all the torque-free motions, we always treat H as a constant.

Â That's religion, I mean, it doesn't change, t, though, the energy.

Â We say this is rigid, but

Â how rigid is any structure, everything has a little bit of flex.

Â There's always something, maybe very stiff, but

Â there is always some amount of flexing.

Â 13:46

Yeah, you're going to start dissipating energy.

Â There's always a resistance to flexing and there's some damping coefficient.

Â It might be really, really small but

Â if internal friction there's stuff rubbing against ach other there.

Â There's walls, there's always friction ways to dissipate energy, so

Â even though it might be almost rigid and you now nothing is truly rigid.

Â So in practice in this math we are assuming right now it's perfectly rigid

Â but that's an artefact of our mathematics.

Â In real life as an engineer we realize nothing is perfectly rigid,

Â in fact what was the first satellite that went up for the U.S.?

Â >> Explorer 1.

Â >> Explorer.

Â How did explorer 1 do?

Â It was kind of in this shape, they go hey,

Â we looked at this equations and if we spin about this axis.

Â I heard spinning stabilizes things, so they spun about this axis right?

Â Had big antenna sticking out, that's how they communicated.

Â Those antenna were all flopping,

Â doing stuff, anybody remember how long before it went unstable?

Â 14:49

After that they started to listen to astrodynamicist they go

Â to help if you do it by the right axis and the right stuff, and all right?

Â So that was one because it is flexing it definitely lost energy and

Â you will see here shortly why with all these constraints.

Â This leads to something that can be very unstable, actually, and

Â what does it move towards?

Â There's definite stable equilibrium, but it's not this spin,

Â where you're spinning about this nice symmetry axis, the skinny one, right?

Â So energy can vary, that means, momentum doesn't,

Â if it's torque-free, our momentum sphere is locked.

Â But for that amount of momentum there is a finite range of energies that you can

Â have, this ellipsoid has to intercept with momentum.

Â 16:18

Good, now, let's look at this.

Â This is my first of many figures.

Â We're going to assume, without loss in generality,

Â that our principal in our body frame, the principal frame,

Â it's lined up such that B1 gives Axis of maximum inertia.

Â That would be like kind of like this axis here on this box.

Â Then you have B2 is axis of intermediate inertia.

Â That's the one in-between.

Â That would be this kind of a skinnier side.

Â And then axis of least inertia is here.

Â As we said Principal frames is to find that permutation.

Â I can always choose a frame where this is true.

Â So, I don't have to do this math six different ways.

Â I can just do it once.

Â And you can flip your frames when if needed.

Â And now we're plotting these things in momentum corner space.

Â Not omega space but we realize that omega 2 omega H,

Â sorry H3 H2 Are just omega 1 times I1 and so forth.

Â The momentum shows up as a sphere, that's nice.

Â The ellipsoid shows up here, and it has to intercept.

Â So, we said earlier we've got three coordinates we're looking for, omega 1,

Â 2, 3, subject to two constraints.

Â The answer must be a curve.

Â And that's where our curve is.

Â Is that the only intersection?

Â 17:32

>> There's one I'm not showing.

Â It's on the backside.

Â Thank you. >> [LAUGH]

Â >> So, even here, so

Â if you have energy specified and

Â momentum specified, there's actually two possible trajectories you could be on.

Â They're related.

Â They're kind of mirror images.

Â But the shape isn't flat, this is kind of potato-shaped.

Â Things, ellipsoid, but wrapped around a sphere,

Â that's what the intersection curve looks like.

Â And that's what you would have.

Â And here you would wobble around in this direction,

Â over there it's in this direction.

Â So, this gives you something, this scale body inertia,

Â you can tell right away, this is where my omegas have to reside.

Â This can also be used as an integration check.

Â 18:22

With H, let's look at the smallest Ellipse you can do.

Â If you shrink your energy then remember these axes are proportional

Â to the principal inertias so this axis b1 that has the largest principal inertia.

Â It's the longest elongation.

Â If you shrink the ellipsis small as you can make it, the longest elongation is

Â going to be the only point that just barely touches the momentum sphere.

Â That's kind of the geometric interpretation.

Â So that means, what kind of spin do you have?

Â That's a single intersection point here or here.

Â So that means, in this case,

Â you have a spin that's doing either you're rotating positive or

Â negative about your B1 axis, but you're doing a pure spin about a principle axis.

Â 20:08

Now, it's torp free It's still, probably whatever is flexing

Â going on in the structure might still be dissipating energy but it will never

Â ever come to rest unless you have some external force acting on the system.

Â You need an external torque to change a total angle momentum, all right.

Â So even with damping in the system that you have fuel slosh,

Â it'll dissipate it but it can only dissipate it down.

Â to this, that's the limiting case.

Â Any lower energy that ellipsoid and the sphere don't intersect.

Â You're violating momentum.

Â Now you've changed momentum of the system and

Â we must have applied an external force.

Â Maybe SRPs could do some of this stuff.

Â That's one thing we do with space debris tracking.

Â SRPs Atmospheric drag.

Â I'm looking at electrostatic drag forces, as well, and torques.

Â They can change momentum over time,

Â which really complicates the analysis of the objects.

Â But, this is one of them.

Â So, good. Do you think this is going to be stable?

Â This is an equilibrium spin.

Â That means, if I put my body in this condition my omega 2, and 3 are 0.

Â If you go back to look at the other differential equations, what you end

Â up getting is omega 1 data zero or omega 2 data zero, or omega 3 data zero.

Â All right, let's actually look at, let me just go back a few.

Â 21:56

So if I take this object,

Â and I do a pure spin about this, I'm never going to physically give it a pure spin.

Â I'm just not that good.

Â All right?

Â I'm always off by a smidgen.

Â So, stability is always well.

Â This particular spin rate would be perfect to give you zero rates in everything.

Â But if you're off by a little bit, are you going to stay close?

Â Or is that going to drive it upside down?

Â Gravity gradient is the easy example, right?

Â Here, like this kind of a pendulum would be stable, if I do this,

Â anyhow, it keeps wobbling, then have friction,

Â it just keeps wobbling here to stay close versus this one.

Â This is in equilibrium.

Â But if I wouldn't hold it tight and I'm off by a little bit, it drives it away.

Â That's an unstable one, all right?

Â Here, if we have neighboring motion, we can't make energy any smaller,

Â we wouldn't intercept.

Â So we just make energy slightly bigger, what happens to the interception curves?

Â Do they stay tight and bounded around that equilibrium?

Â Louis?

Â Yes, everything looks rounded.

Â >> So, if you make it a little bit bigger, you would think, okay,

Â you've got this little bit of a curve there.

Â Okay, you're wobbling some.

Â If you're not doing a fierce spin, you would wobble some.

Â And we see that, right?

Â If you take this object and I take it here and spin it,

Â that spun pretty well, no biggie.

Â Let's see So I've got some videos here.

Â Same inertias, I'm showing an object that kind of scales with these inertias.

Â I'm giving it a sphere spin, which would be in equilibrium.

Â I should never see it wobble by other axis, it's just doing that pure B1 spin.

Â Here I'm giving it some slight offsets with half a degree, and

Â you can see, they look almost identical.

Â They are not perfectly identical but they stay,

Â these little wobbles they stay very small and bounded and life behaves well,right?

Â Which kind of makes sense because we're doing deviations about a minimum

Â energy states.

Â And as you've often learned in physics, nature loves minimum energy.

Â Everything kind of converges to minimum energy state.

Â Right, that's the stable one That we would have.

Â So, that's this spin.

Â 23:51

If we now look at intermediate energy, we increase our energy, right?

Â Everything scales evenly, now of a sudden,

Â the intermediate access which is along B2 intercepts You would have

Â a pure spin about this point, or a pure spin about the other.

Â That means when these object were spinning about this axis in a positive or

Â negative set.

Â Those are the two options if you have this specific energy state.

Â And as before, you said H1 and H3 equal to 0.

Â And then, you can quickly solve that intermediate energy state is momentum

Â squared over 2 times the intermediate inertia.

Â If you have that energy state, these are the two points you could be spinning.

Â But it's not the intersection line isn't just 2 points,

Â like with the minimum energy state.

Â There's also this whole curve that happens,

Â it's a saddle point that wraps around itself.

Â That is a physical motion as well, and that's called the separatrix motion.

Â 25:36

Those spin conditions, you can put yourself on it mathematically perfectly.

Â But if you're on the separatrix motion,

Â you will get there with infinite amount of time.

Â It's an asymptote, right?

Â Otherwise mathematically you can see you have a decision point all of a sudden does

Â the stuff go left or right?

Â And that never happens, because you just asymptotically,

Â it would take an infinite amount of time to actually reach that point.

Â And who is that patient?

Â So if you're off slightly,

Â you will see interception lines that kind of look like this.

Â But anytime I get close to this interception point,

Â that's when I know this actually takes a long, long, long time.

Â It'll hang out there,

Â and then, eventually, it goes back over to the other side.

Â 26:13

This is why,

Â if you try to flip an object about the axis of intermediate inertia, right?

Â It flipped, and now it didn't, there we go, it flipped again, and

Â it flipped again.

Â In one twist and I'm not trying hard, it immediately does 180, 180, 180.

Â It's very unstable in that sense, right?

Â That becomes because of these kinds of behaviors.

Â So, intermediate axis spins are never stable.

Â They're highly unstable, but they may be deceiving.

Â They may look stable for a short period of time.

Â So, let's see if I can get this video to work.

Â 27:07

>> That's so cool [LAUGH].

Â It's perturbed very slightly and goes on.

Â >> It's definitely a spin as they're doing it here,

Â this is a spin about the axis of inner median inertia.

Â But there's a little bit of a wobble, it's not a perfect spin.

Â I guess astronauts aren't that well trained, all right?

Â >> [LAUGH] >> So, this is hard to do, but

Â you can see.

Â For short periods of time, it almost looks stable, right?

Â It gets close and it stays close, then crap.

Â All hell breaks loose, right?

Â And you flip upside down again, and then you hang out for awhile.

Â No, YouTube, thank you, who knows what it's going to show us.

Â >> [LAUGH] >> Okay, but it goes back and

Â forth, yep, you can stop.

Â So that's what's happening here.

Â You can see, it looks stable for short periods of time, but

Â it's getting close to here, hangs out for a while.

Â But then, eventually, it's going to get over here and then hang out for

Â a while and then come back.

Â It never settles at one point, that's why it's not stable.

Â You may look like you're staying close for a finite period of time.

Â But as we go through, later on, the control proof, or the stable mean,

Â and so forth.

Â Once you enter a bounded region, you have to remain in that region forever,

Â and you will not.

Â because if you're off, you may stay there for 10,000 years, because you got really,

Â really, really, really close.

Â But eventually you will, you give it enough time, you will bounce back.

Â And it will go off to the other side.

Â Hang out for a while and then come back.

Â So that's kind of a really cool video, I like that one.

Â It kind of illustrates what's happening, right?

Â This is where you're going one direction,

Â and then it floats around and comes back here.

Â And, except for the air drag in that space station, it's really a torque-free motion.

Â And it's just bouncing back and forth satisfying, this stuff.

Â This is actually what leads to chaos as well.

Â So with a single rigid body,

Â he can proof completely rigorous mathematic that this is a chaotic system.

Â If you're spinning nearly around a separatrix kind of energy state.

Â because a infinitesimal change here has you spinning this way.

Â And another one has you spinning this way, which is different orientations.

Â And so infinite sensitivity to small changes is what redefines chaos.

Â And so separatrix motions is a chaotic type of motion.

Â So you see all kinds of papers published on it.

Â So chaos is fun, here we look at it,

Â I've just done a minor minor [INAUDIBLE], again 0.5 degrees.

Â And even after less than one revolution, hugely different orientations, right?

Â That's what you're seeing when you flip this thing and

Â let it spin and its already tumbled.

Â 29:56

it's reasonably stable, it's just an awkward way to spin it.

Â Little bit of a tumble, but it looks stable.

Â Again, you make your other Other 2, 0.

Â Now from this term, you can derive energy is h squared over 2, minimum inertia.

Â That makes sense,

Â you divide by the smallest inertia to get the biggest energy state, right?

Â So that's the maximum energy you could have with this amount of momentum.

Â To have any more, you have to apply external torques and increase your sphere.

Â But you intercept at 2 points, which is good.

Â Now, here, if you have, we can't have more energy.

Â That's a maximum energy state, if we have slightly less, Nathan,

Â is that motion going to be stable or unstable?

Â >> Unstable.

Â >> Why?

Â >> Because it'll go to the minimum.

Â >> Why?

Â >> because it's going to lose some energy, it's going to dissipate some energy.

Â >> If you don't consider energy loss, and this was really Explorer 1,

Â how bad could it be?

Â Really, it's just a little dappy, nothing, it's very small, right?

Â So without energy loss, if you're like this, that's exactly what Explorer 1 was

Â doing, spinning about the skinny axis of the rocket.

Â They just got it going, you would just be wobbling here.

Â This looks perfectly stable, but as soon as you count the energy loss,

Â you know this ellipse has to get smaller and smaller.

Â And at some point, you're going to go through the separatrix.

Â At that point, who knows what happens, and you might end up upside down,

Â right side up.

Â It's impossible to predict, it's a chaotic thing, right?

Â 31:28

>> [INAUDIBLE] >> Yeah, this ellipse will go smaller and

Â smaller and smaller.

Â And the smallest one would be the motions here.

Â So, in the end, everything in nature with rigid body converges to a flat spin.

Â So for those of you doing space situational awareness,

Â tracking debris objects and so forth,

Â this is where things would converge assuming null external torques, right?

Â So once you include SRP, and as a recent graduate we did this a lot.

Â We're looking at the statics with Joe's work,

Â things can go quite different all of a sudden.

Â because now you need an external influence to pump up momentum and energy level, and

Â it takes you out again.

Â And things get way more complicated, much, much harder to track.

Â But if it is torque free, these are the classic results.

Â Good, and here's an example, there's no energy dissipation here.

Â So this one, yes, it has a slight difference.

Â You can see it a little bit more pronounced,

Â but overall, nothing like the intermediate axis one.

Â That one really went hog wild and went all over the place.

Â So those are the three, you can look at all infinite [INAUDIBLE],

Â that's what I'm looking at here.

Â So minimum energy would be here, maximum energy would be here.

Â Separatrix energies is here, and you can see the arrows how they go.

Â So if you look at your H1, omega 1, 2,

Â 3 coordinates spaces, you can actually quickly tell.

Â Actually, we're wobbling about an axis of maximum inertia.

Â Because I'm wobbling around in a positive sense.

Â If I have my omegas or Hs move around this axis, and

Â you can see that's actually orbiting in a negative.

Â So this is a positive sense, this is a negative sense.

Â I know I am spinning about an axis of least inertia, or near to it.

Â And if it's flipping back and forth your closer to the separatrix,

Â it's doing some crazy stuff.

Â