Know, I know this is a luxurious topic, became too long, maybe, but I'm willing to give you every
important detail you will need on this topic, really.
I have discarded some long mathematical proofs and so on that you will so that you will not need.
But, you know, we cannot discard a whole myth anyway.
After coming up with a beautiful excuse, let's continue with the last video on the grunge dynamics.
And this is linear permits possession of a little grunge dynamics and parameter identification.
These words can be intimidating for now, but don't worry, we will see them clearly.
Here's the famous dynamic model of the robot manipulator that we have come up with during the duration
of this formula.
We have seen that this formula contains some variables, namely angular position, velocity and acceleration
terms, or cue to note and double load.
As you know, already, formula is non-linear function of these variables, and this formula contains
dynamic parameters.
OK, what are these dynamic parameters?
Let's assume this one thing.
Namely, if I think of the robot manipulator in reveal joint for this thing, we can determine at least
these parameters must MRI of the link, which is a similar parameter distance up to the center of mass
of the link, which is indicated by RC, AI and the Vector Victoria.
And it's a victory, a quantitative dimension of three by one with respect each axis.
And finally, inertia tensor for the link with respect to the Frame FC, which is connected to the central
mass of the link.
While it has dimension of three by three, we know that it is symmetric and constant because it is defined
with respect to the central of mass frame, and the only six elements are different.
So all in all, we have ten parameters for each link.
Why I have said that at least 10 parameters because we can take into account friction parameters also.
So for any degree of freedom, a robot manipulator, there will be 10 m dynamic parameters in total.
OK, but we don't know their values, so they have to be determined accurately.
Y you will know this better in future lessons.
But for now, let me say that if we have a good model in our hands, then we can use it in model based
control of the robotic manipulators where we will utilize model of the robot in order to develop some
control algorithms that control the robot.
And effectiveness of these control algorithms depend on the accuracy of this model.
Indeed, it is not correct to say that we don't have any knowledge about these parameters, because
if we designed and built the cars, consider the robot manipulator, then we know these parameters from
the cat model of the robot manipulator.
However, when we realize this cad model, the parameters change because we have some imperfections.
Not only this during usage of the robotic manipulator, we can add some sensors or gripper to the robot,
which will also affect inertia mass parameters.
Also, a robot manipulator is not designed to show off, but it will manipulate some objects which,
as Gripper holds firm to these objects, they become part of it and surely changes some of the parameters.
Moreover, as the time passes, lubrication of the robot also changes, and this affects also the parameters
in the friction parameters.
So we have to identify these parameters in order to have use a full model of the robot manipulator.
Indeed, in adaptive control, we will use these concepts because adaptive control tries to adapt because
of the changes in dynamic parameters.
OK, now the question is that how we will determine these parameters, let's investigate that.
No, that is the moment we have to study the linearity of dynamic model with respect to the dynamic
parameters we have seen before.
As we have said before, dynamic model is nonlinear function of Q, Cunard and tunable.
However, it can be written as a linear function of dynamic parameters.
So here is the formulation.
As you can see in the red part, model is written as a function of dynamic parameters and the relationship
is linear.
Here, pie is nothing but a vector of dynamic parameters, and its dimension is 10 m by one as the robot
is and degree of freedom, its vector of dynamic parameters, dynamic parameters vector, which has
dimension of ten for each link.
So what's why PI, then it's called regression metrics and is an upper triangle and is an upward three
angular metrics each YJ component of its vector or dimension?
Excuse me.
Each white IG component of it is vector of Dimension one by 10.
As you can see, regression matrix if nothing but the function of Q could not and could double that
as regression matrix has an upper triangle structure.
It means that eith dynamic equation depends on me or excuse me, for pronunciation.
I dynamic equation depends on the on the dynamic parameters of links to N.
You can observe this by multiplying regression Model X y time with X y Pi with PI.
However, there is an issue here in the previous equation we have considered all the ten and parameters,
and by doing that, we mean that all of the dynamic parameters for the people to participate in the
dynamic equation of the remote manipulator.
However, this is not correct.
Some of these parameters will appear in the dynamic model of the equation, and B will not appear in
the dynamic model of the equation.
And because of that, corresponding columns or Y Pi regression meetings will be zero.
Not only that, but also these.
Some of these parameters will be a combination of other dynamic parameters.
So some columns of the regression Model X will be linearly dependent.
These will make regression matrix rank deficient.
Both of aforementioned issues caused problems during Parameter ID because we will have to find inverse
of the regression matrix, as you will see in a minute.
So we have to get rid of these issues.
How will we do that?
We will isolate p of the dynamic parameters, which are independent and appear in the dynamic model
of the robot manipulator.
The number of these parameters are way less than the number of total dynamic parameters because because
of this, this isolation, we can separate columns of regression metrics into two parts, namely P columns
that are independent, which correspond to P isolated dynamic parameters, which we can indicate with
PI independent and and 10 and minus P columns that correspond to the dependent parameters which are
indicated with PI dependent.
We can write Y dependent as the combination of the columns of the Y Independent.
So then we can write our new dynamic model equation in this way.
As you can see, both regression models and dynamic parameters are separated as dependent and independent.
Then we can write dependent part of the regression Model X as a combination of the columns of independent
part of the regression matrix.
And after multiplication of regression metrics with dynamic parameters, Victor, we get this new formulation,
as you can see here, regression metrics has dimension of not to end by 10 end, but end by P because
we have all the independent columns and the Alpha has dimension of P by Ron.
And it is called dynamic or efficient.
Be careful, not dynamic parameters, but dynamic coefficients, which are combination of dynamic parameters.
Indeed, as our dynamic equation consists of these new regression matrix and dynamic core efficiency.
Then why we would try to determine each of the dynamic parameters, which is way more difficult.
It's not for us to identify dynamic coefficients only not dynamic parameters, because dynamic coefficients
constitute dynamic equation of our robot manipulator.
So let's do that then.
Oh, don't forget, robot dynamics is also a linear function of these dynamic or efficiency.
And in the case of deiner, as in the case of dynamic parameters, after we have manipulated dynamic
model of the manipulator and wrote it as a linear function of dynamic coefficients, we can convert
our problem statement from identification of dynamic parameters to the identification of dynamic coefficients,
and we will do that using least squares ID methods, which is very simple.
So let's see new regression matrix and dynamic parameters.
Excuse me, dynamic the quotations again, the want to identify by name dynamic coefficients.
If you multiply the regression matrix with dynamic or efficient vector, we will get any linear equations
with P unknowns, which is a problem because number of unknowns is higher than the number of equations
we have to make it such that the number of equations are higher than the number of unknowns, so we
can determine the value of unknowns that meets each of equations in the best way.
So with minimum error, that is the purpose of least squares identification.
So what we have to do?
Let's see the process of identification step by step.
First, we have to choose sufficiently exciting motion trajectory for our robot manipulator.
What does it mean?
Sufficiently exciting in a good way.
It means that we have to choose our motion of trajectory in such a way that it will excite all the dynamic
parameters by exciting.
I mean, it will include.
So we have to choose such a trajectory that will make robot to move over the whole workspace.
We can control the robot by using more model based control methods as we don't have good model of the
robot manipulator.
We try to achieve it so we can use speed control, for example, in order to follow trajectory as good
as possible.
So with minimum error, at each instance, we will sample angular position, cue and combine it with
input talk of you or toll.
In our case, which will make together tuple of angular position and input torque v no input torque
because v commanded and at the end we have n c number of angular position and input tuple.
We are in C is sampling times.
We can determine angular velocity and acceleration from Q by using numerical integration.
Excuse me, and by using numerical derivation differentiation.
Yeah, I'm sorry for the terminology.
After this sampling process.
We have not one Q Q Dot and Q double dot, but in C times Q Cunard and Q double dot samples so we can
write about Eq..
We have seen for one sample of Q, Q and Q Double Toad for NC samples, and we will have regression
metrics of dimension of NC times and by P.
So we will have Mott and linear equations, but in C times and linear equations, which is surely higher
than the number of P dynamic or efficient.
So we can apply least squares method here.
Let's call new regression metrics as y ma.
It will be full column rank, namely ranking BP if they had used sufficiently exciting trajectory,
and let's call this stack of trucks as tall bar as why Matrix is full column rank, we can find it's
so the inverse, which will give us the best values for dynamic or efficiency we use.
So the inverse, because as you can see why bottom matrix is not square, but told metrics namely number
of rows are higher than the number of columns.
So this is how we identify dynamic coefficients for the construction of dynamic model of robot manipulator.
I want to stress that again, we don't identify dynamic parameters, but dynamic coefficients, which
are a combination of dynamic parameters.