In previous lessons, we have seen stiffness control for one day.
But in practical case, we use more degree of freedom.
Robotic manipulators.
So let's develop the stiffness control for any degree of freedom robotic manipulators.
Let's first see the formula for city-states interaction force and equal stiffness formula for any degree
of freedom robotics manipulators.
We don't have only forces, but also talks, so we have both linear stiffness due to the translational
forces and rotational or torsional stiffness due to the talks.
So we will utilize range in our formulations instead of just force, as you can see here.
If it is nothing, but it is nothing but the linear spring can see here is nothing but linear spin constant
and Kate and is rotational spring constant as the second diagonal of The Matrix is 0s.
We don't have coupling between translation and rotation.
PD and Omega D.T. are just infinitesimal translation and rotation.
Accordingly, Eq. one point one is in workspace coordinates because interactions happen in operational
space, so it's logical to represent formula in workspace coordinates.
This creates no problem in terms of translation, but rotational part creates a problem because it's
represented either in R, P, Y or all angles.
And as you know, we cannot find angular velocity by just finding simply their derivatives.
With respect to time.
So Omega D is not valid for RPI or rectangles.
In order to get valid form, we need to multiply the transformation matrix.
Here is the corrected formula.
As you can see it, transformation metrics consist of identity matrix, which corresponds to translational
motion and T transformation matrix, which is configuration dependent.
Now we have one additional problem translational and rotational coordinates are about x y z axis in
terms of W, but all orange.
But on the other side, translational coordinates are about x y z.
In terms of x one.
Rotational angles are r p y or earlier, as we have said before, so we need to make them compatible
in order to do that.
We use a game transformation matrix, but we transpose in that respect matrix dimensions by multiplying
final formula is given in Equation 1.3.
As you can see, the equation has the same structure as in one DB.
We have seen before also, stiffness Matrix K is positive somewhere definite because the environment
may not be able to generate forces on all directions.
Additionally, by using green space of Matrix K, you can have information about the directions in the
workspace along which forces may be applied.
Because along some directions, we have only position control while along some directions on the force
control.
Now we want to find steady state position and range values during the robot's interaction with the environment.
First, let's ride dynamics of robot dynamic equations robot manipulator, which we have seen many times
here.
G Transpose W is interaction forces and u is nothing but input force by the robot manipulator.
This input force is controlled by PD control and gravity compensation.
You may be.
This formula seems intimidating, but let's see how it is formed.
So as we have said before, force control is in workspace coordinates because interactions happen in
operational space.
However, dynamic model of the robot is in joint space, as you can see from Eq. one point four.
So we need to convert input control.
You also took the joint space coordinates first.
Let's see PD control and gravity compensation formula in workspace coordinates, which is easier and
more familiar to us.
No, we need to convert this force vector into joint space by using Jacobean transpose because as you
remember, Jacobean transports convert workspace wrench into joint space.
However, we have one small issue here in this formulation.
X2 term is workspace velocity, but we don't have sensors to measure workspace velocities.
We have sensors in joints to get joint velocities.
So we need to convert this term in the German space velocity using Jacobean matrix.
So this is how we get Eq. one point.
Five, let's analyze what will happen instead is state of the system in steady state acceleration and
velocities will be zero and gravity turns will cancel each other in perfect assumptions.
And we can get formulation for steady state error by just manipulating Equation 1.6 here in Red Circle.
You can see compliance metrics, which is diagonal based on the structure of the transformation matrix.
We can see that compliance matrix is linear and configuration independently in terms of forces, while
nonlinear and configuration dependent in terms of talks.
By combining equations 1.3 and 1.7, we will get below Eq..
From this equation, we can obtain steady state and end effector position and range.
You can get Equation 1.9 by just manipulating Eq. one point eight As this of the vehicle equation one
point nine, we can get Equation 1.10 through Equation one point three from equations one point nine
and one point ten.
We can see that steady state values depend on environment position, and these aren't in the vector
position equation.
One point is interesting in terms of control of the behavior of the robot manipulator.
Interaction forces and talks depend on K and CP.
As we have said before, we want to get rid of B to get rid of behavior of the robot in position control.
Excuse me, as we have said before, we want to get rid of the behavior of the robot in position control
directions.
We can get this behavior by increasing the value of CP and to get compliant behavior of the robot.
Enforce control actions by decreasing value of KP.
Now let's see this example.
As you can see, this is two degrees of freedom robot manipulator moving in x y plane desired position
coordinates are specified by X and Y.
Then there is no stiffness in the right direction.
You can see that by analyzing a mattress, which is system stiffness.
You can see that, um by.
You can see that you can see this by analyzing key metrics, which is system stiffness or stiffness
is given by CP metrics.
As we have said before, we choose CP x low because environment stiffness is high in X direction, so
we need compliant robot manipulator.
We choose high volume for CP y stiffness in y direction, which makes robot manipulator stiff because
environment is compliant.
These are the steady state and the effector position and interaction forces.
As you can see, a robot manipulator reaches desired position in y direction because in wide reaction,
there is no stiffness and we don't need force control to get this position.
Here are the statistics position on, first of all these again, let's analyze two cases.
If CP X is much higher than K X, this means that environment is more compliant with respect to the
robot.
So our environment is compressed up to XD because there is almost no resistance.
The robot the compliance is passive because environments acts like a spring and spring first is generated
because of environment stiffness.
Otherwise, escape is much less than K X. This means that the robot is more compliant with respect to
the environment and environment.
Almost not compressed.
Robot acts like a spring and spring force is generated due to the robot.
So this is active compliance.