OK, after we have seen one of the simpler, centralized control techniques, namely PD plus gravity
compensation control, let's not not learn inverse dynamics control.
This is very important control technique indeed, which constitutes just of many other control algorithms.
So let's try to understand this as much as possible.
If everything OK, let's continue.
Let's ponder about why the control of the robot manipulator is complicated.
What makes what makes it difficult?
Indeed, we have seen robot manipulator dynamic model before.
We have seen that it is coupled and nonlinear.
Coupled means when we try to control one joint, it affects others also.
So this makes control complex because while we are trying to control one joint, you have to take into
account the other, the joints also.
Additionally, the nonlinear the robot dynamic model requires nonlinear control algorithm to be implemented,
which is surely harder than the simple linear control algorithm.
So it would be better and easier for us to get rid of the nonlinearity in the robot dynamics, decouple
it and apply linear control or separately for each joint.
The big question is that can we do that?
Ideally, yes, and we will see how we will accomplish this mission.
In this lesson, we will see at the end of the lesson why I have said ideally, OK, here is a dynamic
model of the robot manipulator that we have seen before.
We can write it in more compact form in this way, as we have seen it before.
Robot dynamics is non-linear in terms of joint positions velocities.
However, if you look at the equation 1.0, you will see that the dynamics equation indeed linear in
terms of input control.
Paul, just accept M and M as some variables and you will see that you are getting linear equality.
So what does this mean?
This means that we can indeed linear ize our dynamic model.
By specifying input control of toll after linear transition and decoupling, we can simply apply linear
control.
Let's see how we will do that.
Let's choose Tal like that where y is new control input to the nearest and decoupled system?
If we plug Eq. 1.1 into 1.0, we will get this equation, which represents new system dynamics.
This new dynamics has been obtained by exact compensation because we have assumed that we know the values
of inertia, matrix M and other dynamic terms.
And exactly so we construct the tunnel as in Equation 1.1.
Keep this in mind.
This is key concept, and the new system dynamics is decoupled and linear, which is what we wanted.
As you can see, each y i component influences with a double integral relationship only to the Q I joined
variable and there is no end dependency, so we have accomplished first task.
Now we have to design new control input y as linear controller.
Let's do that.
Here is the definition of Y.
As you can see, it's simple p d linear control with reference input of R, we will see that what is
R exactly?
Now, if you plug Eq. one point three back to the Equation 1.2, you will obtain this second order dynamic
system, which is asymptotically stable because CP and K the mattresses are positive.
Definite you can design, copy and KDE Matters is indeed based on the specification of your required
system parameters like natural frequency and damping ratio.
Now we can obtain error dynamics and analyze it.
In order to do that, let's assume that we are on a given trajectory with desired position of cure the
velocity of the Q.
But the acceleration of double boat.
The then you can write a reference are like that.
And if you plug our back into the equation 1.4, you will obtain aerodynamics.
As you can see, it is asymptotically stable because of positive, definite Cadee and KP mattresses,
which is what we wanted.
This means that QTL and Cure Tilt the board will convert to zero irrelevant of the initial project through
error, so we will get the desired orientation and, excuse me, desired configuration and velocity
tracking.
Desired velocity does not have to be constant as in the case of PD and gravity compensation.
Additionally, I want to note that aerodynamics convert.
It is complete.
Convergence is controlled by CP and Keady mattresses, as you can see.
So by tuning them, you can get different system response in terms of aerodynamics.
Perfect.
We have obtained what we wanted.
Here's a complete picture of the control algorithm, namely in inner loop.
We are linear rising a decoupling system, dynamics nonlinear and coupled system dynamics by nonlinear
state feedback as we are using inverse dynamics during comp. This control algorithm is called inverse
dynamics control after system dynamics as being the nearest and decoupled we can apply all through look
are to loop linear control, which is a combination of PD control, action and desired acceleration.
Fit for the purpose of ultra loop is to stabilize the aerodynamics.
OK, indeed, we get very good result.
Ideally, we have some problems with inverse dynamics in practice.
My friends, first of all, we have to compute inverse dynamics in real time, which requires many computing
resources.
Secondly, as I have said before, we have assumed exactly innovation in inverse dynamics control.
Indeed, it was the fundamental concept of inverse dynamics in which the whole control method is based
on.
However, in reality, we cannot get exactly near position and decoupling because of uncertainties in
the dynamic parameters of robot dynamics on land and the fact that a payload that can change dynamic
parameters and the existence of a model dynamics as we can not model every dynamics of robot manipulator
exactly like fiction, we cannot do exact compensation, which caused non-linear rotors and couplings
in the final dynamics, namely that this control method is not robust against disturbances and uncertainties.
But don't worry, in the future lessons, we will see how we will make this control method robust against
disturbances and uncertainties by using robust control techniques.