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Oh, everyone, my name is Basil.

In. Today I'll be talking to you about design and

control of industrial robots.

Um, these robots included robots such as warehouse robots or

delivery Jones, but anyhow, the first step is to actually

generate a stream off trajectories that needs to be fed

into a controller.

Therefore, this is what I'll be talking about first.

Then I'll talk to you about the control off six

degrees of Freedom Arms, three degree of freedom arms.

These are open change, and then I'll talk to you

about design of based systems.

These are like the quad rotors or the cars that

will attach.

Use open chains.

Answer on.

Then I'll talk to you about coupling the dynamics of

these open chains and those based systems together so that

they work synchronously.

And then I'll show you some simulations and results that

I've got.

So let's go through this.

So I walk you through quickly the trajectory generation because

I want to have some more emphasis on control.

But essentially the first step is to have a space

plan points in space to go through, and these are

found using different methods such as random sampling methods or

great based methods and then finding the path between these

paths using a star search potential fields different, really mapping

techniques on.

Then what we do is we These two points we

join these two thes consecutive points in spaces and we

for equation number three, which is a trajectory from the

start, point to the endpoint.

And we actually need to.

Paramount tries the valuable s so we need to make.

This is a function of time.

Now I just like to mention that equation Number four

from four and five are the same as that in

three, but it's but it's decoupled.

So the trajectory is split rotation and translation.

What is the significance of that?

If we have ah, decoupled system, then we translate and

rotate around the body frame simultaneously, so we translate in

a straight line.

On the other hand, in the couple system where we

had the X trajectory, we actually rotate around the twist

access and rotate around the body frame simultaneously.

So now we need to time tries the function that

we have in space, and usually the literature addresses two

main ways, which is the minimum snap trajectory.

The minimum jerk trajectory on will actually be discussing the

harder one, which is the minimum snap trajectory case.

Regardless, this is all the drive from calculus of variations

where we should find a function that satisfies, like lunging.

The way we paramo tries is by, um, forming 1/7

order polynomial in time and then solving for the constants

by ensuring the initial and final condition and solving for

the continuity of the derivatives across the whole.

Graf if we do that, we get the system off

this system of matrices Alfa equals B and were able

to solve for Alfa.

I've actually quoted a trajectory generated myself.

It takes in the States that we need to Parramatta

rise on dhe the point in space that you need

to go through.

And it generates that trajectory for you.

Ah, very smooth trajectory that can be achieved without jerky

motion off the motors and yeah, so we account for

the mass effects of the system.

The dynamics of the system.

Yeah. Okay, so that was it for trajectory.

So I'll dig into talking about open chain cinematics now

in open change aromatics.

The first thing that we should note is that it's

very important to know how to find the universe schematics

and universe dynamics.

Universe schematics are found using the equations on the bottom

right corner, where we have the body twist multiplied by

the pseudo universe of the Jacoby in in order to

get all corresponding joined velocities and invest dynamics.

I'll be talking to you about it now.

But regardless I mean controlling these chains.

There are two main methods that are addressed in the

literature Ah, low level control with all other controller, which

is controlled using force inputs, and the other one is

a control with it's more like a high level controller.

The velocity input case on also be just briefly talking

about some other types of controls addressed.

So let's quickly go through the inverse dynamics, while the

inverse dynamics for any arbitrary number of links can be

can be derived using the first equation, which is derived

from from the The significance of this equation is that

it immediately gives you the Tao, the torque, the joy,

the joints, the talks at the joints.

On the other hand, the second equation is the older

YouTube equation on that gives you, um, the wrench be

at that joint.

So this matrix the rich BF b that you see

right there is an is A is M is a

six by M matrix, where m is the number of

links you see.

So if you want to find the torque at that

joint, what we need to do is take the column

corresponding to that to that joint transpose it multiplied by

this crew access associated with that joint.

And then we get the torque.

So there's a case when the in vertebral body Jacoby

in off at the at the end of factor.

Okay, Me.

Okay, well, there's there's a special case when the Jacoby

in in the body frame of the end of factor

is in vertebral.

In that case, we can actually lump everything and the

mass matrix.

And, um, it's when we call it the valuable Lambda.

It's known as the apparent and effective Mass.

The result of this is it's six by one victor.

So in the end, effective wrench f.

And if we multiply that, multiply that by the Jacoby

universe, we get all corresponding, um, joint torques, which we

can use for control.

So let's talk about the first type of control, which

is the control using force inputs.

So essentially, there are two main methods that are addressed

in the literature, the feedback and feedback and feet forward

and feed our controller.

We essentially use the P I.

D controller where the integral term is used to, um,

zero out the steady state.

Better. But the problem is that what I'd like to

draw your attention to is that the integral gain has

an upper bound that upper bound can really stabilize the

system. I think this is called the integral wind up.

So thistle is what an integral wind up is on

DSO. Usually these types of systems tend to use a

P D controller instead with a very high proportional game.

If not, then the integral control.

The integral gain is very low.

I'd like to also draw your attention to the fact

that this type of controls the third order system.

If you notice at the very last equation, this is

opposed to the velocity control which I'll discuss later, which

is going to be a second order system.

Okay, how about feedback and feed forward control er well

in feedback and feed forward control.

What we do is we model the system.

You see, we have em hat and a chat is

a model for the mass metrics.

And a chat is a model for the exploration and

the gravity terms on what we do here is we

apply a p I D controller onto the acceleration term.

I really fucked that it.

Okay, so this is really the feedback control.

Now, is this method off modeling the system better than

the feedback on Lee?

I mean, it depends if we have a very good

model of the system, it will give better results.

You see here, when tracking a desired trajectory data, we

are able to track it at much higher accuracy and

also at lower control effort.

So definitely, yes, it does give better results.

Um, but if the system model is not that good,

then it might actually do harm to the system.

So let me talk to you about the second type

of control, which is ah, high level controller, velocity control

inputs. In this case, we trust local controllers at the

joints to achieve the desired velocity that we need, You

see? So this is why it's called the high level

controller because we're not really controlling the talks were controlling

the velocity.

We trust the local controllers to achieve that.

Well, the basic discussion is done in joint space.

So you see the state of valuable here?

Yeah, that's a, um, and by one matrix.

So where m is the number of things?

Number of joints.

I'm sorry.

So, you know, it's a matrix containing all the joints.

Now, you see, I would like to draw your attention

to something that this, uh, the dynamics of the off

the velocity control is second order.

It's not third order.

And that's why we don't need a derivative.

We don't need a derivative gain on.

So when we use velocity control, we tend to use

a p I controller.

The best performance is achieved by by using a bye

bye critical damping.

Mhm. The best performance is achieved in critical in the

critically damped case.

When K I equals K P squared over four.

And as you can see, it completely zeros out the

study. The studies, it all My God, One second.

Okay, so it seems that mhm the best case of

this controller, the best set of gains for this controller

is that she is the one that achieves critical damping.

And that's achieved one k I equals to keep his

squared over four.

You can see it's really you know, uh, the steady

state later approaches.

Zero. Uh, this is only the case.

When when theater is changing constantly, you see theater there.

The desire data is linear, so one should, when that

is changing very slowly.

So how about some?

How about tasks based control?

We'll all that would happen is that the velocities will

turn into twists and the positions will turn into trajectories.

And in order to find the joint velocities in fatal

dot all we need to do is take is take

the inverse of the body Jacoby and the end effective

frame on Multiply that by the end, effective twist.

Okay. And then we apply the same feet forward p

i controller to the whole thing, you see.

I mean, it's the same controller again.

All we need to do is find the end effect

er twist and then use that to find the joint

philosophies, the corresponding joint velocities using inverse cinematics.

Okay, So how about other methods?

Well, you see the feedback and feet forward Controller is

very competition.

We accept expensive, very computational intensive.

And so there is some other.

There is the other method P, D and graffitti competition

and notice.

There is no I just like I said, because off

the integral upper bound, we tend to remove the I

on. So we use a P D controller and plus

the gravity compensation term where we model only the gravity

term. Other types of controls include hybrid motion force control,

where we use both velocity and force inputs nominal control.

Robust control when we account for modeling, enters its creative

learning control when the tax are being repetitive.

So, for example, we're picking up putting back things in

and, uh, adaptive control when the conditions are not ideal.

So it kind of works like but by retuning the

system every, you know, depending on the conditions, it's it's

something like that.

Yeah. Okay, so now let's talk about the design off

the base systems, which is extremely important

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