Let's listen.
We have learned about the Jacobean matrix.
However, this is just one of the types of the Jakobsson matrix, namely geometric Jacobean matrix,
as you know, it relates that piece, namely linear and angular velocity of the end, the with the joint
velocities.
However, what if the orientation of the in the vector is represented by role pico or eyler nodes?
Then will our Jacobean will be the same?
Let's check that out.
Let's assume that we have given all pitch your angles of Peter are TTP and Y.
Then we can ride the cumulative rotation matrix in this way.
Name the multiplication of three rotation mattresses.
On the next step, we try to find derivative of the rotation matrix.
Be careful here because overall pitch your angles are not constant, but time dependent.
So if you want to calculate your, you have to.
If you so if you want to calculate the rotation matrix monopoly in manually, you have to consider that
as we know, the derivative of the rotation matrix equals to the multiplication of skill, symmetric
matrix of angular velocities and rotation matrix.
From from here, we can find skewed symmetric matrix by inverting our rotation matrix if it expands
symmetric matrix and compare it with the result from multiplication of derivative of rotation matrix
with itself.
Then we will be able to relate the rate of change of role.
Pitch your angles with angular velocities.
I don't know you, but I don't want to calculate this derivative manually.
So let's jump to the mop up and handle this problem.
So let's continue to calculate the derivative of our rotation matrix and obtain as filmmaker.
Let's first define some symbolic variables which are time dependent.
So that's why we write R t p t.
These are all pitch your ankles that we will utilize.
OK.
Of the defining that let's find the cumulative rotation matrix or equals the rotation X or OC times
for rotation.
Y.
P.
Excuse me.
P times for rotation.
Z Y.
OK, let's visualize it.
As you can see, it's a bit complex, so that's why I put that semicolon.
Anyway, let's continue.
Then let's calculate our result.
OK.
By finding different, by differentiating our rotation matrix, however we have, we have to differentiate
it with respect to T.
OK.
So that's why we right here T because our variables are time dependent.
OK.
Now, as you can see, it's also very complex, but we can make it a bit easier to read by writing,
by first simplifying the relation.
So result.
OK.
You can first simplify it.
Let's show you.
Yeah, it's a bit.
Uh, it's a simplified, but you know it as it is complex.
It not affects too much.
So let's write it in.
Also, we can do pretty and simplify.
Combine these comments in order to, you know, have better visualization on our problem.
Because as you can see in instead of Y Dot or D or D Y T it right, this why this is because these are
symbolic variables so we can visualize it more in a print way by using printed comment.
And OK, let's calculate OK, it is better as you can see it right?
D or d t p t OK.
And uh, as you can see, each column is separated by
OK, this side, I forgot its name.
OK.
So after we get after we obtained the derivative of our rotation, not.
Let's now calculate our S Omega, so this is our excuseme metric of angular velocities, OK?
We will do what we will multiply the derivative of our rotation matrix with inverse of our rotation
matrix.
OK.
And now we obtain our exclusive metric matrix.
OK, let's switch it.
OK?
This is also complex, as you can see.
Anyway, let's also do the same in order to visualize it.
Let's use the common pretty with simplified common and s w.
OK, now, as you can see, we got very simple.
Not OK, very simple, but much more simpler than this matrix, OK?
As you can see, a whole of these mattress is nothing but this one.
This is our matrix.
You can ask What is this?
This is five.
OK, sharp five.
Sharp two.
Oh, OK.
These are nothing, but their meaning is they are like a variable.
OK?
This sharp five is nothing but this one.
OK.
Just take it from here and put it inside here and you will get your matrix equation.
As you can see, this is a scale symmetric matrix because in diagonal we have zeros.
OK.
And here we have minus z plus set.
OK, y minus y and minus x x OK.
Now we can go to again our presentation and try to get the results, so get the relation between omega
angular velocities and rate of change of our p y angles.
OK.
Let's go with the metrics that we have obtained by map of the separating derivatives of RPI angles from
the metrics and the write the equation in matrix form and compare it with the previous scores symmetric
metrics, we eventually get the relation between the derivatives of road Piaggio angles and angular
velocities.
So as you can see, derivatives of R P Y angles are not the same as angular velocities.
Let's call that matrix in between B, which is a function of r p y angles.
So it's not constant and changes as the configuration of the robot manipulator changes.
Now we can write the new relation in this way, which relates joint velocities with linear velocities
and rate of change of r p y angles of end effector the matrix in between its analytical Jacobean.
And it is related with geometric Jacobean.
By this relation, analytical Jacobin is much more initiative than geometric Jacobean for engineers
because it relates the rate of change of RPE y angles, which is much easier to understand than angular
velocity vectors.
Also, I'm sure that you have seen the inverse of The Matrix and you think that is it always invisible?
This is the topic of our next lesson, which is Jacobean singularities see you on the next lesson.