OK, let's start with one of the simplest centralized control methods.
This is simple, but yet are private yet better than previous decentralized control schemes where we
didn't consider dynamic model of the robot manipulator.
You can ask here why we don't utilize simple PD controller.
As we have said before, the centralized control method doesn't take into account tools that are generated
due to different effects in the robot dynamics so that it doesn't guarantee the achievement of the position,
control, objective and gravity talks on coincided.
So we cannot assure that trajectory will be tracked and checked, and desired position will be reached
by just applying simply PD or PD controller because it doesn't take into account gravity time.
Indeed, with simple PD controller, we assume that the final configuration gravity toric will be zero,
which is not correct.
However, with PD and gravity compensation, we will remove this constraint and we'll try to achieve
asymptotic of stability even when gravity vector is not zero.
So as we have said before, our goal is to track the given trajectory with asymptotic, asymptotically
stable aerodynamics.
So we want to tell, though, which is position error and QTL, the DOT, which is velocity error,
the conversion to zero and becomes zero at the end before jumping into the developing of control algorithm.
Let's analyze one important motion.
Let's analyze the steady state of the dynamic system.
As we have said, we want to till the and to tell the DOT to be zero.
So we can achieve desired velocity and configuration we want.
Here is a dynamic model of the robot manipulator.
We can write this equation in steady state as in this form as you can see to become desired and could
becomes kudo desired.
And also gravity term has been compensated fully.
This is our purpose, isn't it?
Because the name of the controller is PD and gravity compensation name and gravity is compensated for.
So in order to achieve this kind of steady state C on, some conditions have to be met from above equation.
You can see that in order the sum to be zero, all of the components have to be zero separately.
Y as m metrics or inertia matrix is always positive, doesn't it surely kill?
The double thought has to be zero.
Otherwise, this term will not be zero.
Moreover, in order other terms to be zero Q, the Dot has to be zero because when it becomes zero,
C Matrix becomes zero automatically and the sum and the search term also becomes zero automatically
from these two constraints.
We did use that in order to achieve both steady state condition Q de la Q, the name the desired configuration
has to be constant.
Don't forget that if the is constant, then Q Tilde Dot will become just minus Q Dot because cutely
dot will become zero.
OK, now we can deep dive or become active or dive deep.
Yeah, let's first determine state vector for aerodynamics.
We want aerodynamics to BS Asymptotically stable, then the second worst to zero because if C converts
to zero, then Q tilde and the Q becomes zero.
So if we achieve desired steady state condition.
In order to develop a control algorithm that will make ever dynamics asymptotically stable, we will
utilize in the optimal analysis method.
So we will first choose short the validly optimal function and it's given in this way, Cape Matrix
is positive, definite and symmetric metrics.
If you look carefully, you will see that the first term of the neptunium function is in the form of
dynamic energy, while the second is potential energy.
As with potential energy as kinetic and potential energies are always positive definite, then the optimal
function will also be positive, definite except in the origin, namely when quiet and security that
becomes zero.
This proves that we have little the Apollo function.
Also note that we have chosen the optimal function such that it includes Q and Q tilde variables.
So when the steady state has been reached, namely when vetoed becomes zero, they will become zero.
Also perfect.
Now let's analyze the derivative of the optimal function in order to see whether it is negative.
Definite so error dynamics is asymptotically stable or not.
I will start with the derivative of the kinetic energy because I want to show the calculation at this
point.
As you can see first, we have applied simple derivative of summation role as an is symmetric and positive.
Definite it's transpose will be equal to itself, so we can write equation in this way also without
changing the result.
Now let's analyze the dimension of green products as you can see the product of Q, the transpose inertia
matrix and Q table both will be grown in the Q.
Excuse me, as you can see the product of Q transpose Enoshima please and could double.
Both will be one by one named Scarlet.
While you saw two green terms are equal to each other and we can write them in this way.
Also so derivative only a form of function can be written like that easily.
If you find into to double talk from Standard Dynamics equation and plug in this equation, you will
obtain that equation where entered minus two C will be equal to zero.
C matrix is constructed by Christoffer Symbols.
So finally, we can write V Dot in this form.
As you can see, first is negative definite.
But what about the second term?
We don't know.
We know that we want.
We thought to be negative, definite.
So what we can do is to eliminate the term with unknown.
So let's see the Vidot again.
If we choose to name the input control in this way, then vetoed will surely be negative.
Definite except Kyoto becomes zero except when you don't become zero.
Saw a birth control method is nothing but linear PD control with compensation of nonlinear gravity terms.
So the name of the control is P D and the gravity compensation.
Here, I want to note one thing if we would choose input control TAL as on the P control and the gravity
compensation, it would also make the negative definite.
But why we have added additional derivative term.
This is because robots are made with as minimum friction as possible.
Mechanical engineers tries to decrease friction in robot components as much as possible.
However, on the other side, friction helps to improve systems time, response and convergence the
equilibrium.
So we add additional derivative term to make convergence to the equilibrium faster, namely improve
time response.
You can see this by comparison.
We by comparing video terms when Paul is chosen as PD and gravity compensation and when chosen as P
Control V the correct compensation.
Another thing to note is, as we have said before, we were both Q2 and Q3 of the DOT to be zero when
the Dot becomes zero.
So to get the desired steady state at the end.
But if you analyze vetoed in above equation, you will see that it contains only CU adult term, namely
only cumulative becomes zero when we don't become zero.
But what about the CU til the term then won't be reached.
The desired configuration by PD and part of the compensation control lets on the.
Still steady state of the system to learn that here I have written dynamics equation again with Tom
replaced by P.D. and the gravity compensation control we have formed before, instead of state kudos
becomes zero and also double dot because robot stops moving and gravity terms will take each other out
and we will end up with that equality.
You can see from here that Churchill will become zero at steady state, so we will not only obtain cute
not equals to zero, but also be able reach desired configuration we want.
That's perfect.
However, keep in mind that this global asymptotic stability belong only or the ideal case where we
can predict the value of gravity took victor exactly.
And so compensation becomes exact.
In practice, we cannot calculate good over the term exactly due to uncertainties in the dynamic parameters
of robot model.
So compensation will be partial, not complete, and we will not be able to get asymptotic stability
but simple stability.
As long as KP and KD mattresses are positive definite, we will see this concept more clearly during
robust control techniques.
And here is the clear diagram of the PD and gravity compensation control technique.
As you can see, the red part is PD control to stabilize aerodynamics and gravity compensation with
green power to provide, as we have said, gravity compensation in steady state.