Until now, you should have a solid knowledge about robust control, especially variable tribal structure
controller and chattering them after we have learned all of these concepts.
It's time to apply them on industrial robot manipulators before continuing.
Let's remind us some concepts that we have seen during introduction to robust control.
Firstly, we have said that we couldn't achieve exact compensation in inverse dynamics as we have uncertainties
in the parameters of the dynamic model of robot arm.
So we couldn't achieve exactly an organization and decoupling which have seen by this formula, which
we have seen by this form.
In this formula, as you can see, which represents uncertainties in the dynamic model doesn't let exactly
unionisation and decoupling because data itself is non-linear time learning and coupled.
We have also say that as linear and decoupling aren't exact, it's not enough to apply this simple linear
input control to obtain asymptotically stable aerodynamics, namely by choosing this as input control.
We will get this aerodynamics, which is stable but not converges to zero, which we wanted, which
we wanted to convert to zero.
But it converges to ETA.
This means that we will not have zero error in trajectory tracking at the end.
If we remember that due to the double dot is nothing but the difference between desire desired double
and triple double talk, then we can manipulate the equation 1.0 and obtain this equation, which will
help us to determine our dynamic states.
So we can write this equation in matrix form like this, and from here, you can see that the state
vector of aerodynamics is seen.
If we called these two mattresses as H.A., we can write equations simpler.
As one point two, as we have said, C is nothing but state vector of aerodynamics.
So Equation 1.3 is 1.2 is the error of dynamics like 1.1 but written in matrix form.
Our aim is to design the control y to make aerodynamics in Equation Point Asymptotically stable, namely
see if it'll go to zero.
Be careful aerodynamics is nonlinear and time varying because of ETA.
So we cannot apply simple linear control.
We will instead develop the robust controller to make the inner dynamics asymptotically stable.
So our purpose is to suppress ETA uncertainty term to make the dynamics asymptotically stable by using
high energy input control, as we have seen before.
In order to be able to do that, we have said that we have to obey some conditions like condition A.
Where we have to note that ETA is bounded because we have said before that robust control can be applied
successfully.
If we know the bond on disturbances, know the equation is that and now the equation, the question
is that is eta bonded.
And the answer is fortunately yes.
ETA uncertain to term is bound because it includes dynamic terms like inertia, Coriolis gravity friction
terms, and we know that they are bonded.
We have seen this while discussing robot manipulator dynamics and also acceleration velocity and generalized
coordinates like rotation angles.
Interval of chance and length in prismatic joints are all bonded because they cannot be in finite.
So, all in all, uncertain the term ETA is bounded and this means that we can apply a robust controller
to solve the problem.
Here are bonds on the terms that constitute uncertain the term ETA Bond film.
While these equations seems intimidating, they just mean and that desired acceleration is bounded by
cu m until the term is bounded by.
And as it contains dynamic model terms and Eq. one point four is bounded as it includes inertia matrix,
which is not bonded.
Now we have design, control action y to be able to get asymptotically stable aerodynamics.
Let's choose it like that.
As you can see, it consists of previous y with the new term W.
This W term is added to compensate for ETA uncertainties.
You will see that we will design W such that to get asymptotically stable system and as we have seen
before, q the table is just acceleration feedforward term to the system.
Them PD control part is to stabilize aerodynamics as Cady and keep mattresses are positive, definite
and can be designed based on dynamic system requirements, namely initial frequency and damping coefficient.
We will do some manipulations or aerodynamics we have seen before, so let's plug Eq. 1.6 into 1.2 to
get this equation.
We can manipulate this and obtain this equation where KP and Cady are combined into one game matrix.
If we can gain Matrix as a tilde, we can write equation in simple form as in one point seven.
As you can see, H Tilde is negative definite matrix with negative eigenvalues because it has triangular
structure, and so CP and key determines define its eigenvalues as h tilde has negative eigenvalues.
This means that if atom would be zero, then we would get asymptotically stable aerodynamics by setting
a term to zero.
But we have uncertainties in the parameters, so not p zero.
So we have to choose W such that it will take into account also atatu and will make dynamic system asymptotically
stable.
We can determine such rule for W by the optimal analysis method.
Let's choose of function like this.
As you can see, it is positive definite except origin where C is zero, and this means that the optimal
function we have chosen is valid.
We also included Term C because we won't when we thought becomes zero.
C also becomes zero.
And surely, as always, Q metrics will be chosen, can be chosen by us and it is positive, definite
and symmetric matrix.
Now let's analyze the derivative of the optimal function in order system to be asymptotically stable,
it has to be negative, definite, as you know.
Let's plug Eq. 1.7 into this equation and manipulate it and write in this way and know if the group
returns together and green terms together by taking into account the transpose of Q is equal to itself.
Equation can be written like that.
As we have noted before, each tilde is negative definite matrix and Q is positive definite.
So we can write this equation for positive definite p matrix.
This is valid because h tilde is negative definite and the Q is positive definite matrix.
The sum will always be negative definite.
So for a positive definite matrix p, we will have a unique solution of Q so we can plug P into the
equation and write it in this almost final form.
We will adjust to a simplification by calling D Transpose C term, uh, as it perfect.
No, analyze the terms that constitutes.
We don't.
We know that first term is negative definite because it consists of positive, definite matrix of p
and quadratic relation of C with minus in-front.
But what about the second term?
We don't have any idea about it signed.
It can be zero.
It can be positive or negative as it is unknown and changing.
But we want to make the negative definite.
We have several options for that.
Let's analyze them.
The first is that C will be in the null space of D Transpose Q, which will make z zero.
So second term, the total will be zero.
But is that realizable?
No, because C is changing, while d transpose Q is a constant matrix and we cannot always ensure that
C is in the space of D transpose Q matrix.
The second option is making W equals to ETA, which we'll make again the second term of the total zero.
But this is not possible also because in order to do that, we have to move ETA on certain to term.
Otherwise, how we can say it w equals the ETA.
However, we don't know ETA term, so this option is also unrealistic.
And the last and valid option is that to make the second term of it not negative, definite also, then
we will become some of two negative definite terms and itself will become negative definite.
We can make sure that the second term is negative definite by choosing W in this way.
This is indeed very similar to the equation we have seen before in the huddle structure control, namely
k times sine of ethics, which was sliding surface.
Here, sliding surface is Z and the bi driving it by right.
It excuse me, by dividing it with its more the obtained unit are rich in Vector, which is like signs,
signs seeking them function OK, not Red Cross and 3km function, but signal function because seeing
them function was needed for us to get the knowledge of in which the action, we have to apply input
control action.
And this this is provided by zero divided by law of Z because it gives us unit direction.
Vector and the row is the same as gain of K, which has to be higher than zero.
What I want to say is that this equation shouldn't be strange to you because we have analyzed it deeply.
We have said before that in order to achieve asymptotically convergence with this, input control key
has to be higher than the magnitude of the disturbance.
Remember condition eight?
So in this case, we can apply.
The same Sor rule has to be higher than the upper bound of ETA uncertain to term.
Let's see that here we have written previous equation with the new definition of W by using Triangle
Inequality Row, we can write equation like this.
We can take, normalize it as common, multiply all side and write equation in this way.
From here, you can immediately see that if we choose the wrong greater than the normal ETA Z transpose
atom minus W will be always negative definite.
So it's not.
So we thought now let's do final manipulation and describe a row with the bounce we have seen earlier
because we can determine these bounds from which we can determine bound for ETA.
And if we choose to greater than this bound, we will ensure that we don't is negative definite.
So if we apply Eq. one point six in the equation of ETA, we have seen before.
And by using triangle inequality rule, we can write this as you can see clearly v all of these terms
are bounded so we can remember the bounds we have defined in Eq. one, one point 1.4 and one point five
and plug them here and obtain this equation.
From that equation, we can easily obtain minimum value for rule.
So if beeches row in this way, we ensure that we don't, we'll be negative definite.
So aerodynamics will be asymptotically stable, even with uncertainties in the parameters or ETA.
So all of the control input to the manipulator consists of inverse dynamics control, which consists
of uncertain dynamic parameters or provide partial compensations or partial generalization and coupling
the coupling.
Excuse me, feedforward and PD control together to stabilize aerodynamics.
Be careful not to make aerodynamics asymptotically stable, but simply stable.
And finally, robustness term, which ensures asymptotically the asymptotic stability.
Finally, we can apply boundary layers method simply to avoid or more precisely reduce the effect of
chattering.
As you know here, Epsilon is nothing but bits of boundary layers.
And here is the overall picture of the controller, which will make it easier for you to understand
the whole control.
You can see the robustness time is added to the overall controller.
Before closing, let me not something, as we have said, is it is our sliding surface which contains
cue tricks that can be chosen by us.
So a sliding surface dynamics and the aerodynamics can be adjusted by us as we have seen previous lessons
by adjusting Q metrics.
Additionally, aerodynamics also will be affected by Cheryl CP and CD terms, which are for stabilizing
aerodynamics.