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The punch line with last lecture was that we can, in fact, make the unicycle model

or a differential drive mobile robot, act like a simple x dot equal to u.

What this means is that if we have some desired direction we want to go in,

direction and magnitude, we can come up with v's and omega's, speeds and angular

velocities that make the differential drive robot execute this controller.

And we did it in two ways. First, we saw that we can actually use

some kind of tracking controller to follow this.

But then, in last lecture, we came up with a rather clever transformation that

allowed us to, to actually track this u vector more or less perfectly.

And when I say more or less, what I mean is that we really ignored orientation.

So, we only cared about where the robot actually was,

not particularly which direction it was pointing.

And we couldn't track or used the original point, we had to accept this

small offset error. But it was rather elegant, because no

longer did we have to design trackers. the question one should ask then, of

course, is does this apply to other types of robots? Meaning, this idea of starting

with a unicycle and then somehow making it behave well,

how, how relevant is that to other kinds of robots? So, we have dealt with the

differential drive robots. We know that it's super important in

relevance to that but you know what, what if you have flying things? What if you

have snake bots? What if you have, this is called a snake board, what if you have

fixed-wing aircrafts or blimps or underwater robots? Does this even apply?

Well, what I want to do today is basically see

that it does supply but, it does supply if you squint your eyes a little bit and

focus on what these types of robots have in common as opposed to what they don't

have in common. So, what I want to do today is simply go

through a smorgasbord of different robot classes and see how they actually are

like unicycles or maybe not. We'll see.

Okay. Here's the unicycle,

right? We have, as always, position, x y, and it's pointing in some direction,

phi, this is the standard robot model. So, the states are position and

orientation. The inputs to this model, this

abstraction, are angular and transalational velocitiies.

Meaning, how quickly do you turn and how fast do you go.

Well, let's start expanding this model to encompass slightly more elaborate

systems. So, let's start with what's called a

car-like robot. A car-like robot is not a unicycle

because all of a sudden, we have more states.

We have our old friends which is x and y, which is the position.

We have phi which is the orientation, but then we also have a new angle known as

the steering angle. So, here is the picture we should have in

mind. Here's a car, here are the rear wheels.

Well, the front wheels are going to be turned because we're turning the steering

wheel. So, if I call this point here, x and y,

as always, then this, of course, is, as before, the heading.

And now, we also have this angle here, which is psi, known as the steering

angle, how much are we actually steering it. And

all I wanted to show here is that we have almost the same equation for x and y, the

only difference is that we are now going in this direction, which is phi plus psi.

and then, the orientation has its own dynamics that's a function of the

velocity and the steering angle. And then, what we can do is we can

control how quickly we're steering. So, psi dot is equal to u.

So, this is the next level up in complexity where we no longer have a

unicycle but a car-like robot, four wheels we can turn the front wheels.

Well we've already seen the Segway robot. So, the Segway robot is really a unicycle

or a differential drive robot at the base.

And then, you know, this, this mess here. I drew it like this because it actually

didn't fit in the box if I didn't make it small and sideways.

The point is that, we had a rather complicated equation describing the

pendulum part or the part that's sitting on top of the unicycle.

So, we had a bunch of states, position, orientation, tilt, angle, and here, we

actually had to deal with the tilt velocities and other kinds of velocities

because we actually need to get into dynamics.

Many accelerations of what's happening for this thing not to fall over and the

inputs were the wheel torques to this model.

The point now is that this still has a unicycle underneath and then this mass

sitting on top of it. So, that would, would be the Segway robot

that we actually already seen. Well, what about flying things? Well, a

standard model for a fixed-wing aircraft is ta-da, a unicycle, right? Which means,

it's moving. And then, it can change altitude.

So now, no one in their right mind would actually say that this is what a,

an aircraft is actually doing but from a high level obstruction point of view,

it's not a bad idea. An aircraft can't go sideways but it can

kind of move like a unicyle in, in the plane.

And then, you can change altitude. now, you would actually have to couple

this to the real dynamics of the aircraft if you wanted to do it for real.

the one thing to note though is that there is something slightly fishy about

this and that's the fact that the fixed wing aircraft cannot hover.

When you can't have v=0, but there's really no constraint or no way of

capturing this in, this model, we see later how to do it, but the point here is

that there are ways of thinking about flying things as unicycles plus an

altitude component. Now, let's look at another model.

This is the standard model of an underwater glider,

and, well, let's go back a slide. Aircraft, underwater glider, nothing

changed, right? So, an underwater glider, people

typically model as a unicycle. And then, you can go up and down.

So, you can go down to a lower depth or you could climb up towards the surface.

So, again, unicycle plus an altitude or a vertical component that describes how

high up or high down the, the robot actually is.

Well, all of these robots have some notion of

where it is at least in the plane and where it's pointing.

And if I call position plus heading for POSE, the punchline, at least based on

the models I showed you, cars, segways, flying things, things on the water, and

as we know, differential drive robots, they have this key thing to it known as

POSE. And POSE is x, y, and phi.

You know, that's just, that's just what's going on, where is it and which way is it

pointing. It's pointing in the phi direction

instead of the x and y. Well, you know what, if you ignore the

dynamics, the actual dynamics and say at the certain high level abstraction, we

already know how we should think about this,

the unicycle. So, almost everything involves a POSE.

And almost everything with POSE is almost a unicycle and I will make this more

precise. But the point here is that whenever you

start dealing with postion and heading, the unicycle model is applicable, maybe

not perfectly, but fairly well. So, this model is still very, very rich.

So, the punchline then becomes that we can almost use everything that we've

already done and then possibly add another layer to to what's going on.

So, what you have, is you have, let's say, your, your u plan, right, that comes

into your unit cycle, and out of that comes our old friends, v and omega, and

then, whatever actual robot you have is going to move like this.

[SOUND] And the point is that, in here, we may be forced to somehow make the

actual robot look like a unicycle a little bit.

But it's just another layer to our already elegant and well-layered

architecture that we've discussed. And, in fact, next time, what I want to

do is exactly this for the car robot, to see how do you make a car act like a

unicycle and this is something I had to do when I was dealing with the autonomous

self-driving cars. We were reasoning about them at the level

of position and orientation and speeds and angular velocities but then we have

to map it back onto the, onto the, the car robot.

Before we do that though, I need to talk a little bit about constraints.

What I mean, mean by that is if you're flying, for instance,

you can't hover. You're going to have a certain minimum

velocity or a lot of times, it's convenient to assume a fixed velocity and

then you can turn. But here's an aircraft.

It can't immediately or very quickly just spin either.

So, there are caps on how quickly you can actually turn.

Well, the most famous of all constrained unicycle type models is known as Dubins

vehicle. And all it is is a unicycle with the added caveat that we can only fly

with v=1 or drive with v=1, meaning we're going to go with a constant velocity and

we have a max and a minimum angular velocity.

If you want, you can say, you know, v0 there and omega v max or omega max plus

omega max instead of -1 and 1, but the point is that we're constraining the

velocities. This means, that Dubins vehicle can

basically execute straight lines, it can go along paths like this so, so it can go

along angles, I mean, along circular arcs.

Or it can go anything in between, right? But basically, you have, you

cannot go arbitrarily tight with a Dubins vehicle.

So, this is one way in which people incorporate constraints.

And this makes the planning part significantly harder.

We're not going to cover it, but I encourage you to look up Dubins vehicles,

for instance, to see how people think about curvature constrained versions of

unicycles, for instance. Well, another version or another standard

constraint that people deal with is what's known as the Reeds-Shepp model,

where omega again, is between -1 and 1, or negative omega max and omega max, and

v can now be, well, the absolute value of v is 1, which means that v is either +1

or -1, meaning it's a Dubins vehicle that can also go backwards.

That's all that this means. You can go forward along these arcs or

you can actually pull back in in reverse. And Reeds-Shepp is also a standard type

of model that people like to deal with when they're dealing with mobile robots.

Now, aircrafts typically can't back so it's not a par, all that relevant of a

model for fixed wind aircraft, for instance.

But there are, for instance, robots, at the end of the day, we have a top

speed, for instance, so we can't go faster than v=vmax, so,

but we can indeed go backwards. So, the Reeds-Shepp model would not be a

bad model if you actually want to take max velocities into account.

So, remember that this may or may not come into play.

The last thing I want to say though is that there are, of course, robots where

POSE is not a reasonable way of describing them.

You know, you have a humanoid, well, people will have maybe a position and

orientation but there is so much more going on in a humanoid.

A snake, you know, which way is the snake pointing, well, you know, if the snake

looks like this, what's, what's the orientation of the snake? It's very

unclear. What's the position of the snake? Who

knows? It's not clear what we mean by that.

same thing, we would like mobile manipulators like this robots that we

have here at Georgia Tech, where, well, the base is a unicycle.

So, if you only care about the base, fine.

But if you want to do more things like carry things around or opening doors and

so forth, we have to care about more than the just the POSE,

more than just position and orientation. So, there are situations where we have to

worry about more things, but in a surprising amount of situations, POSE is

indeed the way to go. And like I said, the next lecture will be

about cars and how to actually make a car behave like a unicycle.