- [Narrator] Hello, everyone.
We're sneaking in a bit of eek math into your day
to demonstrate how engineers use algebra and calculus
to advance new ways for robots to work together.
A key objective
of the Ocean Exploration Cooperative Institute.
I sat down with ocean engineer, Val Schmidt
to learn how he approaches a real world problem.
How do we optimize a search algorithm
for a vessel like DriX to use its sensors,
to find an underwater robot friend should it get a bit lost?
Join me in Nautilus' lounge as we get a bit detailed.
- [Val] All right so the scenario is this.
Mesobot is lost, you don't know where he is,
and you gotta go out and find him.
And so what's the most optimal way
to go search for something that you're missing?
And if you roughly know where he is,
one of the most efficient ways to search
is what's called an expanding box pattern
where you spiral out a fixed distance away
from your previous path,
covering more and more area
centered on the place where you think he is.
So then the question is, how far away should these be?
And what speed should you travel along this path?
And why is that not an easy thing to answer?
(talking sped up)
- [Narrator] All right.
So hang in here, 'cause we're about to go deep
with equations and numbers.
Now we don't expect you to learn calculus from this
but I hope you'll stick with us to learn a few principles.
One, underwater detection range can be described
as a function of speed
and two, we can find the best balance of range and speed
to optimize our search pattern using calculus.
Got it.
Okay.
So here's Val again.
- [Val] Okay, so Mesobot is misplaced.
We gotta go find the little guy.
We don't know where he is.
We know that the most efficient way to go find him
is this expanding box pattern.
What we don't know is how far
each of the circles of our expanding box pattern
should be from the next one.
And it's not an easy thing to answer
because it turns out our ability to listen
to the acoustic signals that Mesobot is sending us
is dependent on how fast we're going.
It's like sticking your head out the car window
of a car doing 50 miles an hour.
You can't hear a thing, but the wind going by.
And it's the same thing underwater.
If you go slow you can hear really far distances.
If you go fast, you can't hear a thing.
(propellor whirring)
So we need a mathematical expression
that roughly describes that variation
of the distance that we can hear.
Maybe R in an equation
versus the speed that we're going,
which I've used the variable V here.
So we've got the distance R
is 500 meters plus four times 1500 divided by V squared.
Where and the heck did that come from?
Well that came from some experiments that we did earlier on
and a bit of shooting from the hip, frankly from myself.
But we came up with this expression
so that when you're going four meters per second
it comes out to about 2000 meters.
And when you're going six meters per second,
which is the maximum speed that DriX can go.
It's a little bit more than 500 meters per second.
So now we've got an expression
for how far you can hear
based on the speed that you're going.
Now we need an expression
for how much area we can cover
in our search per unit time.
And that expression is
the area per unit time or per second.
It's equal to two R because we can listen
to the right side and we can listen to the left side.
So that's two R times the speed at which we're going V.
And so for every second,
we listen to some area that's equal to that expression.
Okay so we can take that expression
and we can stick our expression for R in that equation.
So then we get the area per unit second
is equal to two times the quantity of 500 plus,
and then we're multiplying four times 1500,
which is 6,000, plus 6,000 over V squared.
And then we've got all of that multiplied by V at the end.
(talking sped up)
(record scratching)
- [Narrator] Hey there still with us?
- Like this is a different number than I got before.
- [Narrator] The internet will tell you if you're wrong.
(laughs)
- [Val] Yeah, I know that's what worries me.
- [Narrator] I love this moment
'cause it shows an important element of math.
Always check your work.
- [Val] It's definitely a different answer
than I got last time.
So now let me think about what I've done and
if I made some mistake.
Oh, I did make a mistake.
That two has gotta be multiplied by that 6,000
on just that one.
So this is 12,000.
- [Narrator] As Val corrected it
you'll see those corrections in our grand calculus finale.
- [Val] That's why you always
check your work, Jonathan.
- [Narrator] (laughs)
- [Val] Always.
Cause this could save somebody's life.
So calculus gives us this great tool
for figuring out how to optimize one expression
that's a function of another.
So in this case, how to optimize the area covered
per unit time as a function of the speed that we're going,
and that's called a derivative.
And a derivative is a funny word,
but all it means is how fast a function changes
with time or with space or whatever.
So if it's going up really fast,
it may be a big positive number.
If it's going down really fast,
it's a really big negative number.
But if it's not, if it's at the top of the hill,
the slope is zero up there,
but that's super helpful cause we can take a derivative
of our expression and then set that to zero
and that'll tell us
the optimal speed to go
to cover the most amount of distance per unit traveled.
So I take my expression for area covered per unit second,
and the first thing I'm gonna do
is simplify this expression a little bit.
I'm gonna multiply through by the two
and then multiply by the V
and I'm gonna get area covered per unit time
is a thousand V plus--
- [Narrator] 12,000.
- [Val] Divided by V.
And then I'm gonna take the derivative of that
with respect to the speed or V,
and that's gonna be a 1,000 plus minus one times--
- [Narrator] 12,000.
- [Val] Divided by the V squared.
Okay, so then I take that expression,
I set it equal to zero.
I solve for V squared
and do some mental manipulation of all the numbers there.
And I get V as equal to square root of 12.
And I can whip out my calculator
and figure out that that's 3.46 meters per second
or about seven knots or so.
Okay, so now I have the speed at which to go
that optimize the amount of area coverage.
I need to figure out what the range is
so that I know how to do that math
and I come up with a number that's about a thousand.
So that tells me that
each of these spirals going around
need to be about 2000 meters apart.
And now I know the optimal speed to go,
and that will ensure that I cover the most amount of area
per unit time so I can go find my little buddy
who's missing.
Okay, so this is cool.
It's a real world application where we're breaking out our
calculus from high school probably or maybe undergraduate,
and we're solving a real world problem
where we're figuring out
what the optimal way is to go do something
that is really important.
And this could be a robot that's lost
but it could be a person overboard, or a downed plane,
or whatever these tools you keep in your toolbox.
You pull 'em out when you need 'em
to solve problems like this.
Sometimes under duress while you're at sea,
or sometimes in the lab before you go.
But having those tools in your toolkit are really important.
Okay, we've done our math.
We've got our range it's 2000 meters.
Roland has planned up a expanding search bot
search pattern with that line spacing set.
We can execute that search pattern
and DriX will dutifully go and find Mesobot.
How do you like them apples?
- [Narrator] It's pretty spicy.
- [Val] Yeah.
All right you want to hear one of my favorite math stories?