In the last lecture, we saw that we have really two choices when it comes to

combining behaviors. We can either switch between them or we

can blend them. And what I want to do in this lecture is

really discuss a little bit further this issue of, of maybe not how we should

combine them but if we need to introduce something more than just our dynamic duel

of behaviors in order to deal with the navigation problem.

And as a hinted act at the end of last lecture, was the induced mode,

the sliding mode is somehow going to be important.

But what I want to do today is start by discussing different kinds of worlds or

environments. And I'm going to start simple with worlds

where everything is a point, the obstacles are all points.

And then, I'm going to start building up more and more complex scenarios to see

when do we actually need to introduce another mode or another behavior in order

to be able to act well in the world. So, I'm going to start with points then

I'm going to add a little bit of spatial dimension to these points so they're

going to be circular obstacles. Then, we're going to generalize that to

something called convex obstacles. we're going to make it harder than by

making them nonconvex. And finally, we're going to look at

labyrinths. So, basically the maze problem. If you're

a, if you're a rat, a lab rat trying to get to the cheese,

what kind of behaviors do you need to actually find, find the cheese? So, let's

start like this and what I'm going to do now is basically discuss the different

worlds at the rather high level by drawing cartoons of what's going on.

And then, in the next lectures, we're going to be a little bit more precise

about what's going on. But let's start with point worlds.

As always, blue means the robot, here's the robot.

Green means go and red means obstacle. And this little disk around the obstacle

is simply me saying, this is the distance when it's unsafe to, to travel towards

the obstacle and just switch, somehow, to avoiding obstacle.

Either with a hard switch or a bland and in this case, it really doesn't matter

too much which one we do. So, the robot, well, it's simply going to

start [SOUND] so here is go to goal. Start moving towards the goal and then

here, right? We're now hitting the, the bad region.

Well, we know that the obstacle avoidance is going to push us away like this

somehow so, you know, we do, we're going to do something like this and we got

away. Let's say, there's a little epsilon disk

around it. So, here's avoid obstacle and then, now,

we're going to switch back and start [SOUND] going towards the goal.

So, this seemed like a slam dunk, we did well somewhat at least.

so, maybe we can declare successful point worlds.

in fact, I want to show you another situation where I move the point a little

bit. Now, [SOUND] I'm going towards the goal and here,

go to goal, we want to move in this direction and avoid obstacle, you want to

move in exactly the same direction. So, what could potentially happen in this

set up is when you answer here, the robot just keeps oscillating back and forth and

it's never really finding a way around itself.

There is a really, really, really unlikely scenario that could happen if

you have this bilinear line up where everything is on the same line, where you

just end up switching back and forth without really making, making any

progress. Now, that is extremely unlikely when an

actual robot is out in the world. That ain't going to happen because the

world is always a little bit noisy, the sensors are always a little bit noisy and

even the slightest amount of noise is going to push you out, away from this

situation. So, the punch line, when we look at point

worlds, is these two behaviors seem to be enough unless we're super unlucky.

And I call this some kind of mild Zeno. basically you may end up trasitioning

back and forth with no progress but like I said.

in the real world, we're going to have noise, so points we don't really worry

too much about. Well, let's go up in complexity.

Let's look at circles. Well again, let me draw a cartoon on

what's going on. Now, the red dot is big but I'm going

towards the goal. [SOUND] Here, I'm going to start avoiding

the goal and then let's say here, I'm going to start maybe, maybe I end up

switching back and forth a couple times and then

save. So, circular obstacles here seem pretty

reasonable. it seems like our two behaviors are, are

basically doing the same thing. Now, if the circular obstacle was moved

down, again, so the center of mass of the obstacle was bilinear, on the same line

as the goal and the robot, the again, we would get stuck in this highly unlikely,

never-gonna-happen-in-real-life scenario. But basically, circular obstacles are

point obstacles, just larger. But philosophically, and complexity-wise,

we can deal with them. Now, the next level up in complexity is

convexity versus non-convexity. So, on the left here, you see what's

called a convex obstacle. It's basically a, it looks like a circle

or a disk that's being squished a little bit.

what makes it convex is that if I take any two points inside this obstacle and I

draw a line in between them, that line lays entirely inside the obstacles,

right? If I take the right obstacle, which is in

fact, not convex then, you know what, if I take two points, we'll find this line

lies entirely with in the obstacle. But if I take a point here and a point

there, then this line actually falls outside of the obstacle.

So, an obstacle or a set is convex if every line in between two points in the

set lies entirely inside that set itself. So, we're going to start with convex

obstacles as the first step up from circles or disks and then we're going to

look at non obstacles. So, I have a nonconvex obstacle, right?

It's not a circle anymore and what I have here is, well, now my red marker doesn't

show up, but any line I can draw here, of course, is entirely within this this

disk. So, let's see what's happening here.

the robot starts moving. [SOUND] A straight line.

And then here gets, let's say that they, they now want to starts avoiding the

obstacles so maybe avoid obstacle is going to push it there, it may be pushed

back. What's going to happen is there is a

point when avoid obstacle is going to point straight away from go to goal and

now and we've ended up in this situation that was super unlucky and unlikely when

we had a point. But in the convex with noncircular

obstacle situation, that situation is no longer super unlucky because we ended up

in it. Let me again point out what's happening

is that we're getting here, let's say, here is actually the point

where avoid obstacle wants to go in this direction,

go to goal wants to go in this direction and by switching, we're ending up there,

which is the unlucky point, but in this case, it's actually not unlucky.

So, these two behaviors themselves, no matter, no matter if we blend them or

switch between them, we actually get stuck.

So, we need someway of you know what, getting around the obstacle and somehow

we need to follow the boundary of this obstacle and not just switch back

absent-mindedly between going into goal and avoiding obstacles.

So, here is a situation where we clearly need to introduce another behavior in

order to be able to navigate this obstacle.

Well, if you have a nonconvex obstacle, this is known as a robotic flytrap.

what it is, it's, it's a nonconvex obstacle.

It's a cul-de-sac. And you know what, this robot is going to

get down here. First of all, it's going to get stuck

here right, right, because here, we have this go to goal and avoid obstacle

pointing in different directions but even if we were able to continue here, well,

now all of a sudden, we're going away from the obstacle and, no, sorry, from

the goal. Clearly, we need something more than just

go to goal and avoid obstacles to do this.

we need some clever way not only of following the boundary of this obstacle

but following it in such a way that we actually progress towards the goal

globally even though we're locally in the short run and getting further away.

So, nonconvex obstacles, it's even worse than convex obstacles.

Two, two behaviors certainly aren't going to work.

Well, what if you have a labyrinth? So this is, you know, you have a bunch of

different things. You may have something extra here.

You may have something here. You have a bunch of things.

And the robot needs to figure out that it's supposed to do this, right? Which.,

I mean, [SOUND] now I'm going to sound, try to

sound like I'm from New Jersey, forget about it. That was probably not so close

with my Swedish accent. But anyway, it ain't going to happen with

just two behaviors. Clearly, no way of dealing with that with

only avoid obstacle and go to goal so what we need is really at least one more

behavior that allows us to deal with worlds that aren't just circular

obstacles or point obstacles. That are, in fact, convex or nonconvex or

even, you know, labyrinths. We want to be able to solve this problem,

we want our robotic rat to get the, the cheese.

And that is going to be the topic of the next lecture.

Convex and Non-Convex Worlds

Control of Mobile Robots

Meet the Instructors