All language subtitles for final_620b8e7968e8ea00e093428f_578627.mp3

Speaker1:
Hello, my name is Carlos Osorio.

I'm an application engineer here at the
mathworks, and this is part one of a series

of recorded sessions that I am doing for the
power differential equation becomes a robot

seminar. In this section, I'm going to focus
on rigid body dynamics.

So we're going to talk about mechanisms and
how to build the differential equations for

these mechanisms. We're going to look a
little bit at analytics.

Then also, I'm going to show you how to
interface with cat tools and import the

information directly from cat tools to build
full dynamic models of your three dimensional

mechanisms. So hopefully, you watch the
overview section.

But just to recall the problem, we're trying
to build a multi degree of freedom mechanical

manipulator. In this case, it's a seven
degrees of freedom robotics manipulator.

So the argument I was making is this idea of
to solve a complex problem.

We have to decompose it in simpler, more
manageable problems.

So we're going to concentrate on the
mechanics right now, even though there's

electrics and there's controls and all kinds
of other things.

So let's just concentrate on the mechanics
for now.

So even when we look at the mechanics, we
want to look at the simplest possible

situation to try to understand and gain some
insight into the problem.

So let me look at a single pendulum.

There's a multi-link multi-link mechanism.

So what are the primary variables that are
involved in this are there's an angle of

rotation, there's length of the making.

There's a mass that is pushing it down or
bringing it down.

I can think of now, once I have this
established, I might want to grab a piece of

paper and start writing some equations.

But the problem is like manual approaches
are always prone to error.

And I mean, I've written it here as an
example, but even writing this in a clean

manner, it took me a little bit.

And so we have a symbolic analysis tool that
can a very powerful, symbolic analysis tool

that can assist you with this kind of task.

So let me go into MATLAB with MATLAB.

In order to open the symbolic analysis tool,
you can go on the start menu toolboxes and

look for that symbolic math toolbox.

So if you have the symbolic mark toolbox in
your license, you will have this interface,

which is called Muppet Newport is the
notebook interface that allows me to build my

symbolic analysis, or you can call it
directly from from the MATLAB environment

here. So this is a blank notebook.

It's a full editing notebook so I can do
text.

For example, I can bring in Sara title
dynamic mix of simple pendulum.

Pendulum, I can insert images, for example,
if I wish, I have an image library somewhere

here, so let me look for a pendulum picture.

Ok, there you go. This is going to help me
keep track of what I'm building.

If I'm going to use a first principle
equations or I'm trying to follow a paper or

a book in this case for like the dynamics of
a pendulum.

The first law that I'm going to use is
Newton's laws.

Or let me say, Typekit Newton's law.

Newton's law says sum of talks is equal to
the inertia times, the angular acceleration.

So I'm going to define an expression.

This is going to be a symbolic expression,
not notice that I'm using the call on equal

notation. And I'm going to say this is some
of torques is equal to the inertia, which

would be the inertia around the z axis.

The rotational axis multiplied by the
angular acceleration.

So the angle is theta.

The angular acceleration will be the second
derivative of theta.

So I can evaluate them directly.

Of course, nothing very interesting is
happening is happening here because I

haven't. I haven't. I haven't defined any of
the terms.

And this is a very simple problem because
what I want to do is just give you a sense

for how this tool operates.

So let's let's just define some of the
terms.

So let me find the the moment of inertia of
a thin road, for example, around the center

of mass. You can look it up in any book you
find that that is supposed to be more square

over 12, so the mass multiplied by the the
length squared and more squared over

12. So that's the moment of inertia around
the center of mass.

But at the moment of inertia I want is around
Z, so I have to use the parallax theorem.

So I ICC will be actually that ICM and
translated by the parallax theorem.

So mass times the distance which would be in
this case are over two squared times mass

times square. So not as even in simple
things like this, it's actually doing all the

factoring and computing the whole operation
for me.

And so that starts the more complicated my
operations start becoming, the more helpful

this becomes. Let's find the talk.

The talk will be produced by whatever the
the gravity force in this case, which is mkg.

But the force that is producing the torque
is the one perpendicular to the rotation.

So mkg sine of the sign of theta.

So it will be minus m g sine of theater of T
minus because the force is opposing

the the direction of theta multiplied by the
moment arm, which is half the length of the

road, so multiply by R over two.

So I am creating expressions right now.

They are all this is very different than
programming in MATLAB or programming in C.

I'm Eq..

One is a symbolic expression that contains
these values.

So note is that now the equation looks a
little bit more like what we want it to look.

Although I still have Theda there, that is
kind of bothering me so I can replace that.

Let me define Theta as the Greek symbol
corresponding to see it as a symbol of theta.

For example, once you once you get a little
a little practice on this with a notation, it

becomes pretty, pretty straightforward at
first.

Maybe if you are MATLAB programmer, it might
be a little bit different.

So if I look at Equation one again now, now
it looks more like the kind of equation that

I want to look at. Let me get clear double.

Let me get this second derivative of cleared
from this equation because I want theta

double equal to something so I can.

I can do things like this.

For example, let me define a second a second
expression that would be the first

expression, the first symbolic expression.

So equation one, if I want a clear m r
squared over three, I'm going to divide this

by m multiply by r squared, divided by
three.

So I am dividing both sides of the
expression by M.R.

Square over three.

So now it's clearing out three double dots.

Theta double dot is equal that a function of
sine of fidelity.

This is a differential.

Second order nonlinear differential
equation.

I can try to solve it analytically.

And by the way, there is some menus here
that allow me access to some functions

directly without me having to memorize
everything.

So there solve all these.

This is the form of the equation for of the
expression, for solving differential

equations. So by the way, this is all like
MATLAB.

So let me open, bring it up a little bit.

So if I put my cursor on top of one of the
functions.

Notice how it will give me the form of the
expression.

So this tells me, oh, in curly brackets, I
have to put the equation and the initial

conditions to solve the differential
equation.

Or you can also use the f one command.

To go directly to the documentation, same as
you do in Malta, where there's examples,

explanation, everything.

So the equation I want to solve again, curly
brackets.

So just remember that the equation I want to
solve is equation two.

And the initial conditions, because it's a
second order, I have to give two initial

conditions. I have to give an initial
condition for Theda.

In order for our pendulum to flip around, we
need to give it a little bit of an angle to

start moving. Otherwise everyone is just
going to hang there.

So I'm going to give it a small angle pi
over 12, and then I need the initial

condition for the velocity to because it's a
second order thing.

I'm going to say the velocity is equal to
zero.

And here is what variable I am solving for,
which is speed of T in this case.

So as in any symbolic analysis tool to solve
differential equations, this is going to look

for patterns. And this form this this form
of this equation, this particular form of

this equation c the double dot equals sign
of theta.

It doesn't have a closed form solution.

So what this is telling me is that there's
no close frame solution.

It's just giving me back the equation.

And so if I want to solve this in this
particular case, let me add a little comment

here. I'm going to use a small numerical
trick.

So when you have very small angles, you can
you can use what is called the small angle

approximation, which means which says that
for a small angle, that's the value of the

sign of theta is is the same as the value of
theta in radiance.

So let me create a third expression here.

I'm going to create Equation three, Equation
three, and that's going to be what I'm going

to do is I'm going to use the command subs,
which substitutes substitutes in equation to

substitute in equation two.

What is it that I want to substitute?

I want a substitute sign of theta of T sine
a fleet of T.

I want to substitute that for just three of
the.

So what I'm saying in this with this command
is wherever in this expression equation to

wherever you find sine of feed of T, just
replace it by by thede of TW.

So notice how it always has done it has done
that, that substitution.

And now I have a second order linear
differential equation, which hopefully I can

solve analytically.

Actually, I will be able to solve this
analytically, so let me use the same command.

Only in this case, I'm going to solve
equation three same initial conditions.

Let me run it, and it's giving me a general
solution of the differential equation, any

symbolic analysis to do this.

So it's giving me all the possible
conditions, but we have some physical insight

into this. We know that gravity is not equal
to zero.

So what I'm going to do is I'm going to add
a little a little additional command here.

I'm going to say assuming solve it, assuming
that G is greater than zero because I know

that and also we can help it a little bit
more.

Even we can say we know that our the length
of the road is greater than zero.

So now it gives me a better form.

So as a slightly prettier solution, I can
replace the values of G.

For example, G I know is nine point eighty
one.

I know. Ah, let me define our SEM value.

Let me say VAR is equal to one.

Now, if I look at Saul again, I will have an
evaluation of the numerics here.

Notice that it's in curly brackets because
it's giving me a vector of solutions, really.

So if I want, I'm going to define Redefine
Saul as just the first element of that

vector. And so now I'm extracting it out.

That's the element I want.

And once I have this solved, I can actually
just do all kinds of things I can do

plotting, for example, I'm going to plot
Saul.

And it's a function of T, so I need to tell
it.

I want T to vary from zero to five seconds.

For example, let me press enter, and that is
going to evaluate the the expression for T

equals zero, for T equals zero to five this
analytic expression.

Once I have this, not only I have a clean
solution that I can change and modify and

correct or whatever, but I can also.

Well, first of all, save it as a notebook
that I can share with other people that have

MATLAB and the symbolic tool.

Or I can export it.

I can export it where it's going to put it
in my images, director.

I'm going to export it as like my pendulum,
for example, my pendulum.

And you can export things as PDFs, as HTML,
as just text.

I'm going to just select the default, which
is PDFs and stay safe.

So what I has done is it has saved this
notebook into a PDF format.

Actually, let me go back one here.

It's in my images directory now.

So here I should have some PDF.

There go my pendulum PDF.

So let me open it outside of MATLAB and I
have a fully documented version of the

analysis that I just did with all the
plotting, all the commands, all the

expressions. And it is a.

The file, so I can email it to whoever I
want, and they don't need to have MATLAB to

take a look at it. The one thing that I
wanted to show you to let me go back to the

directory where I was before.

So what I have here is a MATLAB script that
shows some of the interconnection between

mobile and MATLAB and the big a big part of
the power of this tool is the fact that it

runs on MATLAB so you can once you have your
script, you can call it from MATLAB.

You can do you can evaluate all the
functionality.

This is exactly the same script that that
same archive that it just created before.

Only I have a few more comments, but the
important thing is here at the end, we are

defining a variable. So it's a symbolic
variable and I want you to notice, let me go

back to my script.

There's commands like get VAR MATLAB
functions like get VAR and set VAR that allow

you to communicate back and forth, either
from the Muppet notebook into MATLAB or vice

versa. I'm going to use that notebook that I
created.

I want to bring up the variable Salz, and
it's going to put it into a variable my soul.

So when I run that, if I look at MATLAB now,
I have my sole variable, which is a symbolic

expression. I can look at it here.

It's the symbolic expression of what I just
calculated there.

This is still a symbolic expression, so if I
want to actually use it, there's commands

like MATLAB function that allow you to
convert that symbolic expression into a

numeric expression that can be evaluated by
MATLAB.

You can create function handles, or you can
let me run this second one here.

So this is going to create actually a
function file, for example.

So it just created here my func file Durham,
which is a fully operational numerical MATLAB

evaluation function.

So you can call it from your MATLAB script
so you can put it in a block in signaling,

for example, there's output.

There's automated automatic ways of doing
this.

And so if you look at the script again here,
there's a commands like MATLAB function block

that allows you to directly from the
symbolic expression, create a simulant block

that you can use for a dynamic simulation,
for example.

Let me go back to my slides right now, so I
have shown you how you can use a symbolic

interface to analytically solve a problem
and create equations.

This symbolic analysis tool has multiple
animation commands.

It's very powerful to do visualization and
graphics, so I would recommend you guys take

a look at it. So once I have the
differential equation, I can solve it.

If it has an analytic solution, I can solve
it in the new power interface or I can solve

it numerically.

So Nimue, but also has numerical solvers that
allow you to to solve a nonlinear

differential equations, for example, or you
can obviously bring it back to MATLAB.

Matlab has all kinds of 0d solvers, ordinary
differential equations, solvers that can be

called once you define the function that you
want to solve the set of differential

equations that you want to solve.

You can call the odd solver, and that would
solve that will give you the solution of that

differential equation in this case.

This example, for example, I'm here.

You noticed that the equation that I'm
solving and I'm calling an odyssey solver to

do it, or I can bring this up to simulate,
for example, let me go, let me go back to my

my script here, and I'm going to open a
single example.

This is a similar implementation.

This is a similar implementation of that
differential equation that I calculated

analytically simulating such all kinds of
mathematical capabilities to build any kind

of linear, nonlinear, continuous,
discontinuous differential equation and

simulating already has the notion of time
and all that all the numerical solvers

embedded in it. So I don't have to worry
about calling the different the the numerical

integrators. I just build my equation
simulating press play, and in this case, it

will be running it with an O.D.

or the rancor of four or five algorithm, and
it's giving me the solution of that

differential equation.

The advantage of the graphical nature of
simulation is that you can componentes things

and start very quickly changing things.

If I want to add additional nonlinear it is
there's all kinds of blocks that allow you to

do saturations and backlashes and all lookup
tables.

If you start, if you start programming
systems that are much bigger in MATLAB

becomes very complicated to manage one.

The other part of the power of this is the
fact that signaling us, as I showed you with

MOBA runs on top of MATLAB so it can make
use of the programming capability of MATLAB.

So let me close this and show you one more
example where I have a MATLAB script.

This is a MATLAB script where you have
MATLAB commands that allow you to talk

between MATLAB and signaling.

So there's full communication here.

I'm opening a system.

It's just the same equation that I show you.

Only I added like a dumping and dumping
element.

This is a full, full dynamic simulation
environment, so all the integration is

happening automatically.

Now I am seeing seeing the behavior of the
system in this case has damping.

So the oscillation is.

Coming down. But the important part here is
that because I can run this from a script and

outside of all this stuff, that is a lot of
graphics, what I want to wear you, I want you

to pay attention to this line in the in the
in the script, the same line, same is a

MATLAB function that allows you to call a
dynamic simulation, so simulates a dynamic

system. So you tell tell MATLAB what model
you want to run and what are the parameters

that you want to use for a particular
simulation.

And the cool thing is that you can put that
inside a for loop, for example, and very

quickly run a parameter sweep or multiple
parameters.

You can see how you can have like nested for
loops where you switch many, many parameters

and then automatically, I can play here and
this is going to use the simulation model in

the background is going to run it many, many
times, and it's going to automatically

capture all the result.

And I'm putting them all in a printed matter
plot, for example.

So a big value of the environment is the
fact that all these tools can talk to each

other, as I just showed you.

Now let's go back to our original problem.

We have a full full six degrees of freedom
mechanism, and I just started with a single

pendulum. So let me extend the approach to a
double pendulum.

And very quickly when you start doing this,
it becomes it starts becoming complicated pre

pretty easily. And because now you have
relative motion and the inertia are varying.

So the equations for solving this are more
are much more complicated than that than just

using Newton's law.

Actually, I did the exercise just to figure
out how complicated will be.

No, and I'm not trying to exaggerate here.

I'm just using Euler LeoGrande equations,
calculating the the energies, kinetic and

potential energy to to express the different
differential equations and notice that it

took me seven or eight pieces of paper to
write those equations for just a double

pendulum. Of course, you can do this still
with me, but let me show you actually that

one will be worth showing you.

Let me go back to my script here.

So this, again, is a mute notebook.

But in this case, I have done all the
analysis.

Let me evaluate it.

I have done all the analysis for a double
pendulum in this case.

While this is solving it, what I want to
mention is, again, I am trying to get insight

into the problem that I'm trying to solve,
and that's why I'm getting and I'm trying to

get an idea of the effects of the lengths or
the effects of the the initial conditions,

maybe and see what my ranges of motion are.

But notice here I have.

I am. When this becomes a big advantage, now
I'm telling you from doing it on paper, which

is super prone to error.

I cannot tell you how many hours I spent
writing it in paper.

So I have some say my clean solution.

But getting I'm making a mistake on a sign
or on a simple derivative is very easy.

So here I define positions as length times,
cosine of angles, for example, and to

calculate velocities and accelerations, I
simply differentiate positions, differentiate

velocities. I am using Euler, Lagrange,
Lagrange equations of motion to solve this

problem. No, because it's the generic
equation that solves that allows me to solve

multi degrees of freedom mechanisms.

I have to calculate the Lagrange point,
which would require me to calculate the

kinetic energy and look.

So even for a double pendulum, this
expression start getting really, really ugly.

Potential energy I calculate the Lagrangian,
which is the difference between those two.

Then I have to start taking partial
derivatives with respect to each one of the

states. So this is this is really cumbersome
when you're doing it without the help of some

kind of symbolic analysis tool.

And eventually, I will get to the to the
basic equations of motion to a connected

second order, nonlinear differential
equations that you are seeing down here once

I replace all the values.

So now I can use this for my similar model,
for example, or I can solve them here.

I'm showing here the use of the numerical
solvers in Newport.

I am. This is again, this form equation
doesn't have a closed form solution, so I

need to solve it numerically.

And so this is doing the numerical solution,
and I'm showing I want to show a little bit

of the animation capability so you can
create objects, all kinds of access.

What this animation is showing you is it's
animating the solution of that pair of

differential equations, for example.

Let me just stop that.

Of course, as I said, once I get the
equations, you can actually build them in

simulation. So once I have those equations,
I can use Smerling to build those

expressions. The more complicated the
problem, the more complicated the models are

going to become. So there are many, many
terms in this case, but I can component so I

can start adding elements and notice that
I'm doing the integration from three double

dot to fit a dot to Theda in this very
simple form, just by adding integrator blocks

and signaling. And the solution that you're
going to see here is exactly the same as the

solution that you get.

If you do the analytics, it's the analytics
or if you're doing a numerical source using a

numerical solver. But the problem that we
are noticing is that the more complicated the

problem becomes, the more complicated these
equations become.

So it gets to a point where those equations
are not going to give me very much insight.

So there are there's this saying about using
the appropriate tool for the appropriate

problem. So if we want to think of this
problem, this mechanics, so if we want to

think of this problem in a slightly
different way now, for example, think of

mechanisms as just bodies, rigid bodies that
have inertia and masses, and they have

degrees of freedom that are giving back
given by certain joints.

We have a specific tool called simple
mechanics that is a part of signaling that

allows you to model this in a very straight
manner.

So let me go to the tool to show you how a
little bit of how the mechanics works.

So first, I'm in MATLAB.

I can open simulant using the simulant icon
or by typing Citilink at the MATLAB command.

This is the basic signaling library browser.

Now what you're seeing here is the space
signaling product that has all the

mathematical blocks to do integration
systems of differential equations, transfer

functions. You can do all kinds of nonlinear
charities.

You have all kinds of mathematical
operations that you can implement and

underneath. Here you see a list of add on
products or signaling.

So under the same scope heading here, you
will find some mechanics.

So same mechanics is our three dimensional
mechanical modeler, and it's all based on

rigid body dynamics.

So you see libraries for bodies and joints,
and the way you model things here is let me

open a blank canvas here.

Let me dragging a body and a body is
defined.

A body is defined by its mass and its
inertia tensor.

And what you see on this table here are the
coordinated locations of hookup points.

So where those see us one and see us to see
a stance for coordinated systems?

And so I am going to define key one as zero
zero zero with respect to an adjoining

reference. What that means is that this
coordinate system is going to fall on top of

whatever it is is connected to.

And what I am going to do is I am going to
define the CG.

I'm going to have very simple. Simply create
a bar.

For example, I'm going to say the CG is one
meter on the X direction with respect to that

c one. And that as to the second hook up
point is going to be two meters to the right

of that kes one point, for example.

So I am just I'm doing a relative notation,
for example, in this case, I'm creating a two

meter long bar.

The CG is in the middle and then one extreme
is the zero or the other extreme is up to

four. Because for the sake of argument here,
what we're doing is we're let's let's imagine

that X is a horizontal axis here and Y is a
vertical axis here.

So then that means that C would be the z
axis will be coming out of the screen,

essentially. So I have a body I can bring
in.

I'm going to bring our ground a reference
because that's what this is.

This is going to be all these measurements
are going to be referenced to some kind of

ground. This is just a coordinate location
of that ground.

I need an environment port because every
time you, you create a machine in mechanics,

you have to bring one of this solver point.

What one of the solver blocks.

Let me flip it around, and this environment
is going to control how the solution of the

equations of motion for the mechanism are
happening.

So here you define your constraints and
tolerances for that solver.

And also, you define the direction of
gravity.

By default, gravity is negative in the y
direction.

So it's vertical here.

So we're happy with that, but we can input
it from the outside.

If I connect this to, nothing is going to
happen because as I'm connecting my bar to

ground, so the way you give motion to things
is by adding degrees of freedom and the way

degrees of freedom are added is by using
this libraries of joints.

So notice that the library of joints has all
kinds of combinations here.

Bushings one degree of freedom, revolution
or prismatic joint up to six degrees of

freedom. In this case, what I want is a
revolution.

A revolution is how you will think of a has
opened a door, for example, a hinge on a

door, and also on one side, they have the
frame of the door and the other side I have

my moving pendulum.

And the way joints are defined is by an axis
of action.

So notice that the axis of action for this
joint is zero zero one is around.

It's a rotational degree of freedom around
the z axis.

So around the axis coming out of the screen,
which is what I want.

So I have a bar that is going to rotate
around that point.

So once I have set all that I can, and
because this is a three dimensional

mechanical modelling tool, it has a
visualization.

So I'm going to just set it.

So it it starts when I run the simulation
here.

Some mechanics show animation and display
machines when we update the diagrams.

That's all I need to do to turn on the
animation.

And when I press play here, what you're
going to see is the animation is going to

start. The animation is going to start and I
have a bar that is freely rotating.

Let me run it again here.

So this is a single pendulum rotating back
and forth.

This is the zero zero where the ground
location is my CG and that system is over

here. But notice that this doesn't look very
realistic because I'm just rotating.

I this will keep on rotating no matter how
long I'm simulating it, because the joint

that I have defined is an idealized joint.

There's no no friction, no no damping, no
losses.

So this will keep on rotating forever.

So the way I can affect joints, for example,
is by opening additional ports so I can open

however many ports. I want notice that a
couple of extra years appear on this, and

when I go in the library, there's a section
for sensors and actuators so you can activate

on bodies and measure things from bodies.

Or you can activate on joints and measure
things from joints.

Let me bring a joint actuator and I joined
sensor so I can activate on this with any

kind of signaling calculation and I can
measure things, for example.

So this actuators allow me to activate with
either talks or forces, or I can do motion

profiles, for example, if I want to send a
motion for a specific motion profile.

I can set it like this now if I want to
apply a damping force, for example.

To make this more realistic, a damping force
is going to be proportional to the velocity

in which at which this is rotating.

So in my measurement side, I can select
notice that it recognizes it's a revolution

or it gives me all the possible measurements
that I might want from a revolution.

I can measure angle, angular velocity talks,
for example.

I'm going to measure an angular velocity in
this.

Case that is going to come out as a symbolic
signal.

And I can actually feed it back if I want to
as a force is going to be fed back as a force

proportional to the velocity.

In this case, I want it proportional to the
velocity.

So let me let me go to my simple math
operators and bring in again here.

So it's going to be proportional to the
velocity and it's going to be opposing the

motion.