OK.
Before switching to Adaptive Controller, let's remind ourselves, why did we learn the robust control
they have seen model based control techniques before?
But the problem was that the robot model that we have came up with was not very correct and this was
deteriorating the effectiveness or performance of model based control techniques we have seen before.
Other than that, we didn't consider many other effects during modeling such such as friction.
All these and other external disturbances were creating a problem in the control of the robot.
In order to solve these issues or more technical, it reject these disturbances.
We have to come up with a robot.
We have come up with robust control, which was using high direct control, more like switching action
to reject disturbances.
But this switching like action, especially due to chattering, is not good for the robot manipulator
structure, and it also create noise if you want to avoid these problems by using smoother control technique.
And here it is, they can achieve this using adaptive control.
So let's see it as we have saved for robust control, we have some problems with model based techniques
that we have seen before, namely the large uncertainty on the robot dynamic model.
So robot dynamic parameters, because essential part of the robot dynamics are dynamic parameters that
we either don't know or they are known poorly.
These uncertain dynamic parameters cause uncertain dynamic or efficiency.
Then we have poor knowledge about inertial payload to the body that we are, namely the body we are
hunting using a robot manipulator, as we have seen before, when the gripper grabs some object, it
becomes part of the last link and objects.
Inertial parameters combines with robots, robots, inertial parameters and cause the cause the change
in dynamic coefficient as we don't know what is the payload.
This cause uncertainty in the model.
So again, the well-known question how to solve these issues without writing a robot like a horse in
case of robots control.
So named the boat so named by adapting to the changes instead of resisting them.
So the key steps are firstly, estimate the value of unknown parameters online.
We know how we can do that and then adapt the control parameters to these estimations.
So namely, be aware of the disturbances and adapt the robot's parameters to these disturbances to achieve
its aim, namely trajectory, trajectory and checking.
Like we humans do in difficult situations.
There are two kinds of adaptive control, namely direct adaptive control and indirect adaptive control.
Indirect adaptive control.
We make trajectory tracking errors go to zero asymptotically by adopting, by adapting to the changes
in the dynamic coefficients.
However, it's not needed for us to get correct values for the dynamic estimations of the dynamic coefficients.
So while tracking errors, while tracking errors will converge to zero dynamic coefficients, estimation
error will not convert to zero, but they will be bound.
That the exact opposite is true for in the adaptive controller, where we not only make trajectory tracking
error go to zero asymptotically, but also dynamic coefficients to convert their true values.
This is more difficult to achieve, and the thrust is that we don't need correct values of dynamic coefficients
if we can achieve the same control without estimating them correctly, so we will use a direct adaptive
control.
So let's see the dynamic parameters that cause problem.
We can categorize this dynamic parameters.
Namely, we have the ones that we are know that we know, namely the parameters which namely, excuse
me, the H parameters, which are for parameters for each link.
So namely for M for the whole robot and then link lengths, which are in total and uncertain, but constant
parameters that can be identified off-line, namely dynamic coefficients.
And if you remember, there are p times the.
Each weigh less than 10 and total parameters and slowly varying parameters.
These are fiction elements as fiction parameters as the robot works, its lubrication changes which
effect friction, surely, and the parameters that are changing abruptly and we don't have any knowledge
about them.
These are the objects that we are handling and we don't know their parameters.
So after detecting the problems and the cause of these problems, we have to determine what's our goal.
Our goal with adaptive control is to execute the given trajectory by achieving asymptotic convergence
in tracking errors.
So both position and velocity errors have to go to the zero asymptotically and we have to provide global
stability.
It means we have to achieve our goal.
What whatever the amount of deviation of dynamic parameters from their true values and whatever is the
initial trajectory errors.
So we are not bounded with small amount of deviations and initial tracking errors.
In this case, it would be logical that stability, but we want global stability.
And finally, we do need convergence to the correct parameter values, as we have said before.
The key concept to achieve this adaptation is utilizing linear parameterization of the robot dynamics,
and we have seen this before.
As you remember, we can ride this robot dynamics equation of robot manipulator in this way where Vivo's
regression matrix and alpha was that dynamic or efficient.
However, as we have said before, we have partial knowledge about dynamic parameters, so coefficients.
So we can write dynamic equations in this way.
Instead, the terms of it had to represent estimations of the terms, namely method is estimation of
term M and so on.
So overhead is estimation of dynamic parameters of alpha.
We need to find appropriate update rules for dynamic parameters, estimation of alpha in order to make
the controller we have seen.
In order to make the controller, we have seen Adaptive, which controllers we have seen before, and
we have seen control feet for birth control and feedback minimization in the Universe Dynamics Control.
We can get good model based controllers by combination of these like inverse dynamics, feedforward
controller, PD control action and nonlinear control that's based on feedback to negotiation.
Let's analyze these two control methods and discuss whether we can make them adaptive.
Let's start with inverse dynamics forward feedforward plus PD control.
You can see the two parts call it differently.
The good thing about this controller is that as the feedforward term contains only desired values,
it can be calculated off-line, which increases sure the speed of the control algorithm.
However, as you can see in the control algorithm, we have used the exact term values.
In this case, we can get asymptotic stability in terms of tracking errors, but we do not know exact
values of these terms.
So the right control algorithm, with terms indicated by hand saw not exactly obvious, but their estimations.
Now we have a problem when we use estimation of the terms, we cannot get decoupling and initialization
we have achieved before with inverse dynamics.
Please review inverse dynamics if you can not remember the linear regression process and also the copy
before with its exact values of terms.
We were able to get M.C terms as common terms outside and get rid of and G terms.
But with estimated values, we cannot do the same because terms with heart are not the same with the
ones we thought had as we can look to these cancellations.
We cannot achieve decoupling and initialization, and we have these aerodynamics, which is stable as
kidI and CPP gains are positive definite, but they convert not zero, but ETA, which is nonlinear
function of unknown, uncertain parameters.
Let's analyze the second one.
Indeed, as you can see, this control algorithm contains feedback minimization not feedforward as previous
case with PD.
Again, we have to write it with estimated terms instead of exact loans, and the same thing happens
in error.
It doesn't convert to zero, even if it's stable.
This is normal that aerodynamics don't convert to the zero because of uncertainties, and we knew that
beforehand.
Love, you have to make these two algorithms adaptive by finding appropriate dynamic parameter updates
rule.
If you find this rule, then dynamic model can adapt to these changes, and aerodynamics can converge
to zero asymptotically.
The issue with these algorithms is that the process of making these control algorithms adaptive is not
trivial because of below reasons.
During a transient estimation of inertia, matter is can be non positive or single.
It can happen because we define an updated rule, and we don't put some constraints to this update.
It will try to adapt the changes by whatever way it can.
This will cause problems during inverse dynamics because inverse dynamics, as you can remember, so
inverse dynamics control, I mean, because inverse dynamics control, as you remember, we have to
find inverse of inertia matrix.
If inertial metrics become similar, then this will create a problem during inversion, then instability.
It can be the problem because as inertia matrix becomes non positive, it will cause non positive PD
gains, which will surely cause instability in aerodynamics.
You can see this from the second control algorithms, namely M Times CP and M Times Cadee.
Terms will be new PD games, which can be negative so cause instability.
Finally, a complex analysis is needed to determine bound on PD gains in the first control algorithm.
So because of these reasons, we will try to choose another control method which doesn't contain these
issues and make the process of adaptation easier.
Then we will find correct then, if I correct update for the dynamic coefficient.
But let's stop here and continue on further in the second part.