From this lesson, we will start to analyze stiffness control.
It will be divided in several parts in order to grasp the concept much more better.
The name is stiffness or compliance control because, as you know, compliance is nothing but inverse
of stiffness.
So if you control stiffness, then you control compliance and vice versa.
Anyway, let's jump to the core concept.
Let's consider this simple.
On the left, the robot arm is given while on the right is the object or more generally environment
that the robot is interacting with.
Here is the input force that's applied by the robot.
Next is the position of the environment.
This can be obtained from the model of the environment.
X is a steady state position of the end effector.
It is nothing but the desired position of the end effector.
Don't forget that the robot and the environment have some stiffness.
Based on these given variables, we can get the interaction forces between robot arm and the environment.
This is simple springform forced formula because as robot and the environment has some compliance,
we can accept them as springs.
Additionally, we can ride the system dynamics easily like that.
Now, let's assume that our force input is nothing but simple PD controller.
Then we can write Equation 1.1.
After doing some algebraic manipulation, we can obtain Eq. one point two.
We are interested with what will be the steady state position of the end effector and what will be the
steady state interaction forces.
A steady state acceleration and velocity will be zero clearly, so we can easily get equation one points
three from Equation one point two and based on Eq. one, we can get steady state interaction force between
the robot and the environment, which is given by Eq. one point four.
If you observe a bit carefully, you will see that equation 1.4 is simple spring force equation with
equal and brigance of key e q.
Let's see the equations of 1.4 and equal and spinning this again from equals.
Bring this equation.
You can see that the robot's spring and in warm spring is in serious because, as we have said before,
we accept the robot and the environment as two springs.
The robot's stiffness can be controlled by adjusting key, and this will surely affect the interaction
force.
Also, here's the important thing if you analyze equation one point carefully, you will see that when
k the name environment stiffness is much higher, the robot's stiffness will get stiff environment but
compliant robot.
So interaction force is generated mainly due to the robot because it is the elastic and deformable one.
However, when the robot's stiffness is much higher than the environment stiffness, then increment
is compliant and interaction force is generated mainly due to the environment because it is the compliant
and deformable work.
In practice, general condition is correct because robot is more compliant with some kind of damages
while the environment is stiff.
So based on a book considerations, we can see that we increase the stiffness of robot on directions
where position has to be controlled.
Precisely because when we choose keep a high robot will be stiff and trajectory will be tracked precisely.
KP is proportional control term.
So by choosing higher vehicle for trajectory errors, we decrease CP in the directions where there is
interaction forces.
And so these contact forces have to be controlled, so we decrease keeping in them the robot's stiffness
and so increase robot compliance.
This will increase effectiveness of interaction force control while decreased trajectory control preciseness.
So we had to choose KP at first peak until we reached the point that interactions happen.
At this point, we decrease KP to get compliant robot.
In general, our robotic system is stable because of friction forces.
However, if it is not the how or if it is not, we can provide stability by adjusting KP and Kiwi terms,
which are positive.
So our system will be stable and we will have steady state.
As we have seen before, the steady state interaction force is given by this formula.
And if environment is much more stiff than the robot, then this equation 1.4 can be approximated like
that.
As you can see, in order to control interaction forces, the first fix the robot stiffness keep based
on environment.
Stiffness, which is known approximately how do you have to know environment position precisely.
Be careful.
Choose Cape is such that it is much less than key.
Then we adjust to and based on it, we can control interaction forces.
So as you can see, first control is obtained by means of positional control.
Namely, this is indirect force control.