All language subtitles for 05_agility-and-maneuverability.en

Next, we'd like to start thinking
about how to create agile robots.

Robots that can start
from a position of rest,

accelerate pretty quickly
to a maximum speed,

stop when it sees obstacles, and
then accelerate again to a maximum speed.

You see this in the video
clip that you're seeing.

This is a manually piloted robot and
you can see this is an expert pilot that's

able to drive the vehicle really quickly
through fairly complex environments,

and we'd like to create autonomous
robots that can do exactly this.

Let's see what it means to stop

from a configuration where the robot
is going at maximum forward speed.

First, the robot is pitching forward
when it's going at maximum speed.

When you decide to bring it to a position
of rest, you must pitch it backward,

reversing the direction of thrust so
that you get deceleration.

This means that the robot must be
pitching back at an aggressive angle.

This will cause the robot to slow down.

But as a result, the thrust factor which
now points in a direction other than

the vertical direction will also cause
the robot to loose height because

the component of thrust in the vertical
direction is now less than the weight.

As we maximize agility,

we really want to be thinking about
minimizing the stopping distance.

The other thing worth exploring is
the robot's ability to turn quickly,

now I want you to think of this robot
flying forward at maximum speed and

then turning as quickly as possible.

What we'd like to do here is to minimize
this turning radius which you see

denoted as ro.

In both these examples, stopping from
maximum speed and turning at maximum

speed, is actually sufficient to consider
a fairly simple model of a quadrotor.

What you see here is a diagram of
a vehicle in the vertical plane.

The propellers apply a thrust and
the sum of these two thrusts,

actually four thrusts for
a quadrotor, is the vector you want.

That vector you want
now has two components,

one in the horizontal direction and
one in the vertical direction.

The difference of the thrust contributes
to the moments and that's u2.

If I write down the equations of
motion in the plane, essentially,

I get three equations of motion
that describe how the components

of the thrust, u1, and
how the turning moment,

u2, accelerate the robot in the yz plane,
and

also turn the robot in the direction
of the pitch angled feet.

Again, you have two accelerations,
linear, denoted by a,

with components in the y and the z
direction and angular, denoted by alpha,

which obviously has only one component and
this is the rotation in the plane.

The two key ideas are that you
wanna accelerate quickly and

you wanna roll and pitch quickly.

To accelerate quickly, you wanna maximize
acceleration, I denote that by a sub-max.

And in order to roll and pitch quickly,
you wanna maximize alpha sub-max.

The first quantity is
the linear acceleration.

The second quantity is
the angular acceleration.

To maximize the first quantity,
you want to maximize the ratio of u1 to W.

In other words, take the maximum thrust,

divide that by the weight,
and maximize that ratio.

If you think about the second quantity,
that you can maximize by taking u2,

which is the turning moment,
maximize that divided by

the moment of inertia along the x-axis.

We've developed a very simple
simulation that illustrates this.

You will see the robot starting from
a maximum forward speed and effectively

slamming on the brakes but again,
the brakes are slammed on by creating

a reverse pitch which generates a reverse
thrust, and that slows the vehicle down.

One of the things you can do is then
calculate the stopping distance for

different acceleration rates.

Here we show two curves.

One at 5 meters per second squared, again,

roughly half the acceleration
due to gravity.

The second is 10 meters
per second squared, again,

roughly equal to
the acceleration due to gravity.

In both cases,
we've essentially used a dynamic

simulation to create a graph
of the stopping distance with

respect to the maximum velocity
the robot starts off with.

As the robot travels with a larger
velocity, the stopping distance increases,

and clearly, the higher the ability
of the robot to accelerate or

decelerate, the smaller
the stopping distance.

These are two curves we have
generated that gives you a flavor

of what it means to maximize the agility.

You want to be able to stop quickly if the
vehicle sees an unexpected obstacle, and

this is quite critical to maneuverability.

To explore this,
we've designed a very simple math

lab exercise where you're
given a simulator, and

you're gonna use this simulator
to explore this design

space of operating speeds,
the maximum acceleration and

the inertia and the mass of the vehicle.

The larger the mass, or
larger the inertia, clearly,

the larger the stopping distance.

Likewise, your ability to accelerate
fast is also gonna decrease the stopping

distance.

The larger the velocity you're going with,

the greater the stopping distance and
ultimately,

you wanna decrease the stopping
distance for the same operating speed.