OK.
Let's continue with the hybrid position for control.
This is the last part for this topic.
Let's listen.
We have seen these two forms where we have tried to get control, both on posts from general six dimensional
force and velocity vectors.
Please note that by seeing controllable, I don't mean the in control.
Sorry.
Instead, I want to mean that we can control without being constrained by the environment geometry.
OK.
As we have said before, we can do this selection with the help of selection mattresses, namely a V
and a F.
Now we have to decide what we want to do or what's our control objective.
We want to impose design task evolution to the parameters of motion, namely as an parameter of force,
namely lambda.
So we want to steer them to desired values through control algorithms we have seen before.
We can achieve this in two steps in.
The development of the method is somehow similar to the impedance control.
Namely, we will first try to linear rise and decouple the system in the task frame by feedback.
Legalization normalization of the system equation will help us to apply linear controls, while decoupling
will help us to apply appropriate control in appropriate directions independently and in the second
step, we will apply linear control algorithms to achieve desired task evolution.
OK.
Let's first note the dynamic equation of the robot manipulator and forward differential kinematics you
term in equation one point zero is our input control that we will design.
Let's write Equation 1.0 in 1.3 form by the help of selection matrix of F.
We can do the same for Equation 1.1 and convert it to Equation one point two with the help of selection
matrix of a Z.
Now let's try to find a derivative of Equation 1.2.
This will help us to get equality for could double load, which is given in Equation 1.4.
Now what we need to do is to plug Equation 1.4 into robot dynamics equation, which will give us this
new equation in matrix form.
This is nothing new, and you can confirm that if you plug Equation 1.4 into one point three, you will
get Equation 1.5 by just expanding Equation 1.5.
Now we want to choose such control input that can linear rise and decouple our linear and coupled system
equation.
Here's the design choice for you.
This is of the nothing new because we have done this several times for inverse dynamics and impedance
control.
A s and a lambda is our new control inputs in DeNiro's and decoupled system equation.
If you plug Eq. 1.6 into Equation 1.5, you will obtain linear nearest and decouple system equation,
as you can see in one point seven.
Here is a small nod.
As you can see, the relative degree of a s is to excuse me.
The relative degree of S is two.
Namely, we have to differentiate output of s two times to get input of S in output equation.
However, the relative degree of lambda is zero, so we don't have to differentiate it to get input
a number in the input equation.
Excuse me, in the output equation, this is good news because lambda comes from force torque sensor,
which is noisy and so we don't want to differentiate noisy signal after we have linear rise and decoupled
system Eq., we can now choose control inputs to achieve desired task evolution.
NIM asymptotically stable aerodynamics.
Let's first design a s for a simple PDA control is enough because we just want to control position.
The aerodynamics will be as in 1.9, which is asymptotically stable as KP and Kadia are both diagonal
and positive.
Definite four force control a lambda has been chosen as simple integral control.
If we choose it as in 1.10, then force aerodynamics will be as in equation one point eleven, which
is also asymptotically stable with positive, definite quality of care.
Here is again a small knot equation.
Lump 10 would be also correct if it would take Integral Tum out, but we added it to increase the robustness
against constant disturbances.
We also don't want to add derivative time because then we would have to differentiate lambda, which
is noise a signal from for strong sensor from above equations.
You can see that we have to find as s taught and lambda.
The extraction of these parameters are very easy.
It's obvious that in order to find, as we can use forward kinematics, we are given joint angles.
Then it's easy to find this.
We can get SE and lambda through selection mattresses here.
Be careful that this is not normal inverse of Matrix, but so the inverse of Matrix.
This is because selection mattresses can be non square.
Here's a block diagram of hybrid position forms control.
This is the control inputs we have designed for position and torque control and heavy use, so the inverse
of a F and a V selection matters is to get control.
I was, as we have said before, I want to show you this block diagram also, which is a simplified
form of previous block diagram, but contains some key points.
As we have said before, robot outputs, namely Estcourt and Lambda, are and with respect to the base
frame of f zero, but we need to convert them to the tusk frame.
But to apply by upload selection method to apply selection mattresses and get artificial and natural
constraints separation.
So the use our rotation matrix that will help us to get in the task constraint frame of EFSI, as we
have said previously of the switching task frame the upslope so.
The emergence of selection mattresses to select controllable parameters, and here is again, the union
force and position or velocity controllers.
It's not some small thing also and then finish.
The lesson won't be considered until here that we don't have natural constraints that don't let the
excuse me, that we have natural constraints that don't let robot either move or apply force torque.
So initial constraints, constraint, motion and force two exactly zero.
Because we have said in the beginning that robot and object are very rigid by object.
I mean, environment and the environment is frictionless.
But this is not correct in reality, and this caused some inconsistencies with our model.
Namely, we have friction force in pure motion directions.
And in the case of on friction, the force is increasing as normal force increases.
The second inconsistency is due to the fact that the robot is not rigid but compliant.
So when it touches the environment, it deforms and caused motion in no motion directions.
How are these disturbances are small?
Then they can be neglected.
Additionally, if we know the geometry of the environment, precisely, these inconsistencies will automatically
be removed due to the selection mattresses.
So what's the general had done with the hybrid position?
Force control technique is some kind of real time environment geometry estimation algorithm to get rid
of these inconsistencies.