OK, last time we have said that we need to find a new control algorithm that will make the process
of making the quantum algorithm adaptive easier.
Now let's say this algorithm here is the control algorithm.
If you're asked why this, then why not the expansion of the terms curado and you are double dot is
indicated also here.
Sigma is nothing but the difference between reference velocity, QR code and actual velocity.
Not.
We will talk a bit curado.
Don't worry here.
Cady's sigma term is like PID control action if you expand it.
Namely, derivative gain is KDDI and the proportional game is Lambda Times CD.
Lambda and Katie are positive def. mattresses, as you can see different from the previous algorithms.
Here we have used reference velocity of curado instead of Cunard, because if we put Q Kudi Dot instead
of curado, this algorithm will achieve a sum total convergence to zero in terms of velocity tracking,
but the position alone will not be zero.
But we want both of them to be zero.
So we change reference velocity to Q ordered, which also includes the position error term.
So a robot moves until desired position until desired position is achieved.
Perfect.
Now let's try to get aerodynamics.
We can achieve this by substituting Eq. one into the standard robot dynamics equation.
Then we will obtain Equation 1.1, which represents aerodynamics.
We want to prove that the aerodynamics is asymptotically stable, so our position around, namely QTL
and velocity aerodynamic kubert tilde converge to zero.
Surely we will use the option of method to prove that, as we have done before.
Here is the opponent function.
As you can see, it is a valid Liverpool of version because it is positive definite in all values of
Sigma and Q tilde except zero, and it becomes zero when sigma until then becomes zero.
It contains only sigma and curtail the terms because we want these to be zero at the end.
If Sigma and the Q Tilde becomes zero, then tell us that the arrow tilde automatically becomes zero.
Because of the construction of sigma, namely sigma is equal to the QTL dot dot lambda
equals the Q till the dot.
Minus number to tilt up if Cutelaba and Sigma becomes zero til the thought has to be zero automatically,
no, let's find a derivative of the Liverpool function.
I will not go in the deeper calculations because we have seen the same before.
We can extract M Sigma the term from Equation 1.1 and plug it here.
After some manipulation, we can reach this step.
No, I want you to remember this equality we have seen before.
This is true because we have chosen.
This is true because we have chosen crystal symbols to construct semantics by considering this property.
We can write equation in this way.
Here we have three terms.
Two of these terms are negative death, and we know that because they are quadratic terms with negative
signs.
What about the third one?
We don't know, because it consists of multiplication of two different terms that can have different
signs.
So we need to eliminate this term if we choose a in this way, we can write equation like that, which
is surely negative.
That's not perfect.
We get that the derivative of the Apollo function is negative definite and it becomes zero only one
position and velocity.
The error becomes zero.
Because in order we thought to be zero, Q tilde and sigma have to be zero.
We get asymptotic stability with this algorithm.
Now what we have to do, we have to first take into account uncertainties.
The exact terms will be replaced by estimates.
Then we will estimate the parameters and adapt the controller to these estimations.
We will do it by choosing correct updates.
Rule four Alpha had to name the update of the estimations or the dynamic parameters.
Here we have written the same control algorithm with the estimations of the dynamic terms so we can
use parametric unionisation property again.
Where I had is nothing but estimation of the dynamical efficiency.
Here, I want to note that regression matrix does not depend on Q double the term, which is good because
acceleration term is always noisy.
No, as you guess, we have to obtain error dynamics by substituting Equation 1.0 into the standard
robot manipulator dynamics.
Here we have done this.
If you consider these two equations, you can substitute their values in the equation and obtain this
equality.
This manipulation of equation is very easy.
You can do it by yourself.
Tilde terms here shows that there are deviations of terms from their true values.
So until the terms show deviations had terms, show estimations and the terms without hat or tilde shows
the correct values.
You can again use parameter linear regression property and write equation like that.
As you can see, this term does not equal to zero, which is normal because as we have used estimates
of the terms aerodynamics, we do not convert to zero, but we want it to convert to zero asymptotically.
We can make this term zero, either making alpha tilde zero, which means we have to find correct estimates
of the dynamic parameters, which is difficult and is not our goal.
The second is much easier and trivial where we want to make over tilde to be in the null space of regression
matrix, right?
So their product will be zero.
We can do that by finding correct update room for dynamic parameters that will ensure that they are
inside the null space of Y matrix or regression matrix.
If they want to achieve a sum total convergence, we can do that using the optimal method so people
choose newly appointed function and make its derivative negative definite to get us in total convergence
in error dynamics.
Let's do this.
Here is the newly appointed function.
As you can see, it's what part is the old Liverpool function that we have seen earlier.
The only new term is added is green term.
This is valid Liverpool function because it has a positive definite for all the values of Sigma Q tilde
and alpha tilde and zero when these three variables are.
Europe, as you can see it, the included over her term as the third variable indicating we want to
be it to be, we want it to be zero, but you will see that we will not be able to do that, but we
will use it for making vetoed negative.
Definitely.
Let's analyze derivative of the Liverpool function.
I didn't try hard to obtain derivative of these because these are simple calculations that you can do
by yourself.
Additionally, we have seen how we can do this before.
As you can see, its first part is like the derivative of the old, the Apollo function, and we know
that it's negative definite.
But the second part is new indicated with green color, and we don't know it's on.
So we either need the excuse me, so we need to eliminate this term.
Before doing that, let me point that out.
Over tilde is deviation of dynamic parameter estimation from their true values.
As you can see, the second part of the derivative of the opponent function contains derivative of the
A4 till the term.
So this is the derivation derivative of alpha tilde, as alpha had, will be constant because it is
true value.
It is its true values.
It is true of values and it will be constant, so its derivative will be zero.
So derivative of error in dynamic parameters is the same as a derivative of the estimates of the dynamic
parameters.
So we need to choose such a betrayal of the dynamic parameters, namely, if had thought, that will
make green terms zero.
So we will get negative.
Definitely opponent function that negative definite derivative of the Apollo function.
OK, we want this term to be zero.
We can find.
If I had thought from this equation easily like that and this is the dynamic parameter update rule that
they wanted to get.
So by choosing Alpha had thought in this way, we will make the derivative of the Liverpool function
negative definite.
Here's the interesting thing as you can see, the note doesn't contain alpha hot term.
So when we do not become zero, Kittila and QTL, the Dot will be zero.
Namely, we will get desired trajectory tracking, but we cannot see that four tilde will be zero,
so we cannot see that estimation of dynamic parameters will convert to their correct values.
Also, estimates of dynamic parameters will be bounded, namely not equal to zero.
So with the aforementioned controlled low and dynamic parameters of tetlow, we guaranteed that aerodynamics
view converts to zero.
As we have seen before, this can be happen in two cases.
Either Alpha Hut will be in the space of Y, which we guaranteed that or alpha becomes zero.
While we have said that, we cannot guarantee happening off that this can happen if trajectory to be
tracked is persistently exciting, which we have seen this topic when we talked about the linear parameterization
of global dynamics equation.
So as you can see by adaptive control, we get smoother control action and opposite to the robust keys
because in this case, the dynamic parameters are updated at each sampling time.
Instead of like switching from high control, high power control, which was which caused noise and
also was harmful for the structure of the robot manipulator.
Additionally, other than opposite to the robust control case for adaptive control, um, just accept
disturbances in the robot manipulator.
And for example, if it grabs some object or the friction parameters change or other parameter dynamic
parameters change and or other changes in the robot structure it all.
Adaptive control accept all of these disturbances as change in dynamic parameters and tries to adapt
to these dynamic parameters instead of producing high control action in robust control keys.
So.
Every disturbance is accepted as a dynamic parameter change in adaptive control and adaptive control
tries to adapt or update their estimates of these dynamic parameters, such that it will get a sympathetic
trajectory control so a thought asymptotically convergence in terms of trajectory tracking error.