OK, from this point, we are going to start with advanced controls.
We will start first with the robust controller tower and let me give you an insight why we need advanced
controller before dark jumping to the robust controller.
Here's a dynamic equation of the robot manipulator.
We can write this equation like that.
Also, we are and includes the terms like friction, gravity and Coriolis.
We'll write it in this way in order to simplify the calculations we will do.
As you remember, in previous controllers, we have always use this dynamic model of the robot manipulator
and use these terms during development of the controllers.
And we got the results we want.
Indeed, with control of robot manipulator, our main goal is to achieve asymptotic stability in terms
of trajectory tracking errors.
And we achieved this result with previous control techniques like inverse dynamics control.
However, while this equation of robot dynamics is correct, in ideal case, it is not valid during
practical cases or in implementation.
So what is the problem with that equation?
Firstly, by writing Dynamic Equation as a book, we assume that we know all the dynamic parameters
which makes dynamic terms.
Is that correct?
Surely not.
We don't know all the dynamic parameters of robot manipulator.
Even the ones we know contain some uncertainties.
Additionally, robot dynamic parameters change as the robot handles some objects, so we cannot say
that we know robot dynamic terms.
Exactly.
Additionally, in previous control techniques, we didn't consider the simplifications we made in order
to achieve the above dynamic model.
We simplified many effects in the robot manipulator like friction.
Also, there are other external disturbances that we cannot model.
All of these affect the robot manipulator dynamics and in previous control methods, we didn't take
them into account.
So all the assumptions we previously made are not practical and doesn't work or work poorly for the
real life.
If we take into account the uncertainties in the robot dynamic model, then we can ride the dynamic
model.
In this way, we are terms with hat indicates that they are just estimates, so that means that it is
just an estimate of the term M and so on.
So in reality, we have all estimations of dynamic terms that are constituted by dynamic parameters,
namely, as we don't know exactly dynamic parameters.
We don't know exactly also dynamic terms of the dynamic model of robot arm.
So we also have disturbances due to simplifications in the dynamic model, as we have said before,
and we can also have external disturbances that are not because of the are modeled or simplified dynamics,
but because of the environmental effects.
So here you may ask OK, OK, enough.
I understood the problems we have, but how they cause problems.
So let's analyze the problems we have with previous control methods.
If we take into consideration the nonlinear tis not ideologies, as we have said before, the purpose
of the robot manipulator is to achieve asymptotic convergence in terms of trajectory tracking errors.
So we won't cutelaba tilde and QTL the team, the position and velocity errors to go to zero.
OK, let's consider inverse dynamics control technique we have seen before.
Indeed, it is effective control method, so we will try to make it robust during the series of robust
control.
But now we want to analyze what will happen with this control technique if we consider them online realities.
As you remember, in inverse dynamics, we have taken input control as this.
If you don't remember, please go and repeat inverse dynamics control.
Perfect as you can see, dynamic terms that make input control are considered as ideal.
With this control input, we have achieved exactly initialization and decoupling of nonlinear dynamics,
which led us lets us to use linear control, such as speed control on the linear system.
So we were able to get asymptotic convergence in aerodynamics.
But now let's take estimations into account, which we have mentioned before, which we have mentioned
before, and changed the dynamic terms with the ones we had.
So this is not ideal case control input because of estimated dynamic terms.
Now it's interesting for us what will be the aerodynamics with this control input and to achieve it.
We have to plug Eq. 1.0 into the standard robot dynamics equation, as we have seen before.
OK, let's do this now.
We will do some manipulations with this equation to get aerodynamics.
First, we will add m y and then subtract it again from.
The right hand side of the equation.
So after some very simple manipulations, we can get this equality from here.
We can find cure double talk very easily.
Let's call this term as Tita and write equation in this way, which is much simpler from this term.
You can easily notice the beginning of the problem before you double.
That was, it was equal to Y.
However, in this case, it is not.
We have it a term which constitutes nonlinear terms because we know only estimations of the dynamic
terms.
So we couldn't achieve exact initialization and decoupling.
This means that our new system is still nonlinear and coupled.
OK, let's ignore that and continue by choosing white as linear control and acceleration feedback as
we have done before, in ideal case, inverse dynamics.
Let's plug why into the above equation and obtain this aerodynamics.
As you remember, KP and Key D gains are positive definite mattresses.
So as you can see, aerodynamics is stable, but wait a minute.
While it is stable, it doesn't convert to zero, but ETA, which is not equal to zero.
If you replace estimation with exact terms in ETA term, you will see that ETA will becomes zero.
Unfortunately, we know only estimations of dynamic terms, so we cannot convert to zero.
This means that a simple linear controller like PD is not enough for us any longer because the system
equation is nonlinear and coupled.
So what we have to do, what is the solution?
Here comes the robust control into the help.
What we will do is simple.
I mean, simple to lead to at least, but a bit harder to implement.
We will keep control with feedback the initialization because effective control technique with respect
to others we have seen before, but instead we will add it robustness by designing ultra look for the
control technique so you can see general structure of the robot general structure of the controller
in this picture.
We will have inverse dynamics control to do not exact, but some degree of animation of nonlinear terms,
and we will use PD control for stabilizing purposes.
We have seen that while PD control provides stabilization, it doesn't provide asymptotic stabilization,
namely convergence to zero.
So we add robust control part.
Surely there will be input to the robustness part also, but I didn't draw them here for simplification
purposes.
This robust control part will ensure that it will reject the disturbances, either due to the external
reasons or estimation errors in nonlinear dynamics.