OK.
In this video, we will learn how we will calculate forward kinematics for dramatic manipulator.
Indeed, we can calculate forward kinematics using geometrical approach or by in a manual way, without
any systematic way or without some specific method.
But it would be very hard.
What we can do, for example, we could do assign frames, the robotic manipulator, for example.
Let's take this one.
OK.
We could assign for each joint coordinate system, OK, for example, for this one, for this one.
OK, for this excuse me.
And for this one, for each of the joint.
OK, we could do assign Gordon Frames.
And then we could find, therefore four homogeneous transformation metrics for each one.
For example, this is joint.
This is joint one.
This is drawn to.
This is three or let's do it.
Not joint, but framed.
This is frame zero.
This is one.
This is two.
This is three.
Then we could find, for example, h one zero, excuse me, eight zero one.
OK, well, let me just h one zero.
Then we could find that between this one h two one.
OK.
So from homogeneous transformation from two to one, then we could find for this one h three two OK.
And for the end effector, we also assign the fourth coordinate system and we could run it for between
h between the frame three and four eight four three.
OK.
We could conclude this in a manual way.
OK, and then what we could?
What could we do?
We would mark multiply each of these.
Excuse me, in each of these homogeneous transformations, mattresses with each other.
And this is three two times h for three, and we could find the homogeneous transformation matrix of
h four zero.
OK.
This would give the relation between the base frame and end effector frame.
However, it would be very difficult to do that without in a systematic way, and it would also be very
difficult when we would do it for a more difficult, more complex robotic manipulators.
So what we will do, we will we will use almost the same logic.
But instead of these assigning frames and calculating homogeneous transformation methods, this will
be based on some method.
OK.
It's going to not be in some stochastic method.
OK.
So here is we will use the H Convention in order to calculate forward kinematics.
OK.
We will use.
The convention and we have see, though, why we use the commission because it is systematic approach
and it makes us easier to calculate the four kinematics, even for the difficult robot manipulators.
OK, then what are the steps to calculate the age forward kinematics using the convention?
The first what we have to do, we have to assign a frame to each of the joints.
OK.
As we have said before, we have to assign a frame fixed frame to each of the joints.
Surely this assigning of frame will not be in an arbitrary way, but the age conventions specified method
to assign frames to the joint of the robotic manipulators.
Then what we will do, we will do calculate the age parameters or the age parameter table, which is
four parameters four parameters OK for parameters.
But then we will use by we will calculate this homogeneous transformation method.
Is this using these four parameters?
Then we will surely multiply these homogeneous transformation mattresses with each other in order to
obtain relation between base frame and in the picture frame, or between base frame to some other and
point on the robotic manipulator or between any point any frame of the robotic manipulator.
So, OK, let's continue with how we will assign the rule of assigning frames to the robotic manipulator.
OK.
And this is the rule how we will do this.
First, we will assign zit access.
OK, here is it by axis because we will have more than one joint OK for each eye joint, we will assign
Z I axis for real joins.
Excuse me.
This is not a trivial joint.
We will assign the z axis in the direction of rotation, OK?
Because as you know, the joints provide the rotation about some axis and we will choose this direction
of rotation as a z axis.
Then for prismatic joint, we will choose that axis as the translational motion direction.
OK.
So anyway, for z axis, it has to be in the joint direction of motion again.
So let's see in in this example how we will assign z axis.
So for the first joint, this joint, as you can see, is roll a joint and provide rotation about this
axis.
OK.
It provides rotation about this axis.
So as you let me just say, not about this, OK, this axis, it provides rotation about this active.
So as we have said, so this joint rotates in this way.
OK.
So what we will choose then z axis.
Surely it will be these taxes because they reveal that joined the direction of motion is about this
axis.
So this will be our rotation, our z axis.
So this is four frames zero, this will be zero zero.
OK, direction directional that zero, then what we will do, we will assign the the z axis for the
second joint.
As you can see, the second joint is also rotational joint and it will rotate all about this axis.
OK.
About this axis and what we can do.
We can assign it in this way.
OK, because this is the direction of motion.
OK, the joint will rotate in this way.
OK.
This is theta zero.
This is T to one.
OK.
This will provide this rotation.
And as you can see, the third joint is prismatic joint.
OK.
So it moves in this direction upward and downward.
OK?
It it it provide translational motion like upward and downward.
And so the direction of motion is this one?
Yeah, it translates over this axis, it translates or this axis.
So the z axis will be in this direction and we will choose the z axis for this one as this way.
OK.
This is the direction of the make it a bit shorter.
OK.
This is the direction of translation, so the direction of motion.
So we will do this as z
z to.
OK.
Yeah.
The zeros, it's one and z.
OK.
This the fourth, the joint is also rotational joint, so we will assign and surely we have to find
a full, first of all, what is the direction of rotation?
And in this case, the rotation joint provides rotation in this way.
OK.
This is the rotation it provides.
OK, and I hope you can find the direction of rotation.
Just put your finger in, use your right hand and put your thumb in this way and the other cool other
fingers.
OK.
I cannot draw in a good way, but curl your fingers in this way, and this finger will provide the axis
of rotation and the curl.
The fingers will provide you the the rotation.
OK, so in this case, if you curl your fingers, it will be something like that.
This will be your fingers because it is rotated in this way.
Then if it is rotated, then your sum will be in this direction.
So the direction of rotation is in the same direction, which is like that.
OK, so this was the tool and this is the direction of rotation, and we can choose it in this way.
OK?
You can show that shows it in upward direction or downward direction.
It doesn't differ.
The only thing matters.
It should be if it should be in this direction in this axis.
OK.
This is it.
Three.
And then we will assign one frame for also the end effector.
OK?
So let's take what will happen for the end effector end effector frame.
We will choose the end effector as the same and the vector axis the same as the Z X is the same as the
last drawing.
OK, so it will be downward also.
This is our Z Fold.
OK.
So after we have assigned our zip axis, now we can.
Let's first write this well.
So this will be the translation distance will be d, uh, two.
This will be T to three because this is rotation and the rotation will be in this direction.
OK.
It will rotate.
Excuse me, it will rotate in this way.
Not in this direction, but in this direction.
Because as we have chosen our it in this direction, our sample will be downward.
OK.
Some will be downward and surely there are others fingers will be colder in this direction.
OK, in this direction, they will curve in this direction.
So the rotation is this way, OK?
Because our fingers, other fingers are curled in this way, but our sump is in this axis.
OK, so.
So this will be also the same way as the last joint so that the rotation angle will be offered only
in this direction.
No, let's assign the x axis how we will do for the x axis.
The x axis for the Earth joint will be such that it will be perpendicular to the previous z axis and
that intersect the previous z axis.
OK, but for the first joined us, there is no previous joint before.
There is no previous joint for the first joint.
We will assign x axis as a suitable for us or as we design.
So let's try to assign.
Does it Axis Folsom?
Excuse me, x x is also so for the first one, we have said that we can assign it in in any direction.
So let me note one in here.
During assigning the axis this, we can assign axis in the valid direction for the axis is upward or
downward or, uh, right or left, OK, or inside the page.
OK.
This is this shows the inside the the page.
This is the outside of the page.
OK.
You can assign the valid assign direction is are these one OK?
So we can assign it how we can assign.
Does it axis excuse an x axis?
As we have said, we can assign arbitrarily, but we will choose for convenience in this way for the
x axis.
OK, in this direction, we will choose x axis.
So this is our x zero, then what we will choose for the joint one, we will choose x axis in this way.
Another in think I want to node is choosing the center of the frame, OK, choosing the center of frame
if we can choose the central frame in this way, for example, while choosing the
center of the frame for the joint, we will use this method again for this blue sync represents the
plane.
OK.
So if the z axis previews that axis and the next axis are parallel to each other and they are on the
same frame, then we will choose.
We can choose the central frame for the Z Eye plus one joint.
OK?
Excuse me, I +1 joined over the Z +1 audacity +1 OK axis.
So any point over this axis you can choose as the center of frame.
If the Hollywood, when they are on the same frame, the previews and the next joints are on the same
frame and the same plane and there is it accesses are cut each other or intersect.
So if you make it longer, let's make it longer.
If you make it longer, for example, this is just sync.
And there are also additional note here seemed about 3D here.
Everything is 3D is not in 2D.
OK, so think about the planner.
Everything.
Excuse me, not planner, but in the space, in 3D, not in the 2D.
This is very important.
You have to have 3D visualization.
OK, so you can also make it in this way.
As you can see, these two accesses intersect here, so the center of the frame will be here.
So but what happens if the accesses are skewed?
OK, is it accesses?
Are skill?
What does it mean SKU?
It means that they are not on the same plane, so they don't cut each other OK.
They are not on the same plane as you can see.
Does it eye?
Plus one is a boot?
This is it, I.
OK.
This is another plane and this is another plane.
Think about 3D again.
OK, then what we will do.
We will just again, we will make them longer, OK?
Make them longer this way, and we will also make it in this way.
OK, longer.
Okay, this passes here, but it doesn't cut because this is the length of zip as it is, it is passes
under the plane of the eye plus one.
OK.
They don't cut each other at this, at this part.
OK, because this is not 2D, but this is in 3D and we have different planes.
So what we will do, we will choose the common normal to each of these planes.
OK, what is the common normal is the minimum distance between these two planes and the common normal.
And we will choose the point when this common normal intersects the zip by +1 axis.
And this happens at one place only because there is unique common plane for these two accesses the and
the common plane, excuse me, common normal to these accesses.
And the common normal is the distance between these two axes is that is the minimal distance and perpendicular
to both of them, so normal to both of these accesses and provides the minimum distance between them.
This is the common normal and this is unique for each of these four for these two accesses.
And the point where these common normal cuts that i +1 axis.
This is, as you can see, this plane will be the center of the frame for the eye plus once joint.
OK.
So this is this can be seen know difficult in theory.
But when we put in practice, I mean, in the robot manipulator, it would be easier.
For example, in this way, as you can see, these and these are on the same plane, OK?
At least in this, excuse me, in this picture, they are on the same plane.
As you can see, overall, this robot is in the same plane because it doesn't start to.
Then it will rotate.
For example, when the first George will rotate, then they will not be in the same plane.
But for now they are on the same plane.
OK, robot is straight.
So as you can see, if they are on the same plane, then zero zero and zero one will be what parallel
to each other.
So what?
We can take the center of the center for the second frame.
Any point here, OK?
Or go to the axis, OK?
We can take it over here, here, here and here.
OK.
For example, what I mean this one, for example, you can assign X here.
OK, that is so and there y here, OK?
Or you can assign x axis to here.
OK.
Excuse me.
For example, you can assign x axis to here and y axis like that.
OK?
This is and this is the opposite axis, but it is not convenient for us.
OK, why we have to do it in this way, right in the center of frame has to be has to have an offset
to the center of the joint.
OK.
Why we make the life difficult for ourselves.
Let's make the as we can choose in the axis.
This was our z axis, so let's see if we can choose the center of frame anywhere on the z axis.
Let's choose it here.
OK, so our lives will be easier.
And what we have said for X, we have said that the X one axis has to be in the way, for example,
four in it, it can be in this weight.
Yeah, if we consider the permissible directions, it can be to the page.
It can be out of the page and it can be in this direction again.
So but let's check which are which of these directions are
suitable for the D.H method.
We have said that, as you can see here for the X Axis A has to be perpendicular and intersect.
This is it I minus one axis.
If we choose x by axis in this way, for example into the page, will it intersect the X?
Is it a z zero axis?
So is it a minus one axis?
Surely not because of line, because it think about in 3D.
If you make it longer, as you can see, it will not cut it.
OK?
You can see that it will cut it here, but it it will happen in 2D.
OK.
But if we are in 3D, it will not.
Got it, but it will pass over the pass.
OK.
There will be distance between this access and success in 3-D.
OK, so they will not intersect with each each other, so we cannot choose.
We cannot choose this direction as our X can.
So as our X one direction, the vehicle can be triggered by the same logic.
We cannot choose out of the page because if we make it in this way, OK.
And in this way, it will not intersect the z zero axis.
OK, because we are in 3D again and you have to have 3D imagination for this one.
OK, let admit this one goes to that.
OK.
Make it in this way again.
OK.
In this way, yeah.
So we have two directions.
We can either choose it in the left or in the right.
And then in order to make our life easier during the calculation of the age parameter table, we need
to.
It would be better if we choose the axis in the same direction.
OK.
Making minimal changes.
OK?
We would like we want every time.
The optimal thing is that for us, it will make the calculations easier.
If these accesses don't change.
OK, so that zero is in the direction of Z.
One x zero is in the direction of X one y zero is in the direction of one.
And for this one also, they are as much similar as possible to each other, then it would be easier
for us to make their calculations so we will choose X.
In order to minimize the changes, we will choose our X One Direction as in the same direction of X0.
Because if it does excellent in this direction, it will be perpendicular and will cut the zero zero
axis, which is, um, which we require for the H conversion.
OK.
So for this axis, for this prismatic joint, for joint to what we will do or joint three with what
we will do, we will choose the first of all central frame how it will be the central frame.
Again, they are in the same plane.
So as you can see, is it two and Z are parallel to each other so we can choose the center of the frame,
OK, as any point over the Z to access.
And in order to make our life easier again, we will choose it here.
OK?
At this point, as the joint center as the central frame.
OK, we will choose the x axis.
As we have said x two axis has to be perpendicular and it intersected z one axis.
So we have as in the previous case, we have two possible direction.
Either this way, OK, or either this way, in order to minimize the changes, we will not choose this
direction, OK, because divide to choose it opposite to the X1 axis.
OK.
If we can choose the same as x one axis, choose it as the X1 axis the same direction as X1 because
it will make our the edge parameter calculation easier.
OK, so we will choose our X two direction in this way.
OK, perfect.
Because in this way it will.
If you make it longer, it will come and cut it that one axis, OK, which we require.
Let's think about this three axis, OK?
As you can see, in this case, also Z two and Z three are parallel to each other because they are on
the same plane so we can choose the central frame as at the center of the joint.
OK.
So let's try to make the here also to choose X reduction again.
We can choose in this case, we can choose X Three Direction as part of the page.
OK.
As right as left.
OK?
And excuse me, this is inside to the page.
This is inside to the page.
But and then also to the page.
We can choose all of these.
Why?
Because in each of these cases, if we make it longer and this one is longer as they are on the same
plane, they will cut each other.
OK?
Let's, for example, in order to make it clear.
Let me first, for example, choose inside the page, OK, inside the page.
Let's choose it as the axis as im inside to the page.
OK.
For example, in this way, as we have said there, if you make it longer this and make it this longer.
And they take also into account that these are on the same plane.
OK.
These are on the same plane.
If you make it longer the z axis, it will cut this x axis at this point.
OK, so we can choose our X in this direction also.
OK.
By the same logic, we can choose our x axis.
Let me make it by the same logic we can choose.
The x axis as part of the page, OK?
Yes, because in this case, also it will cut there's a two axis.
OK, so then what we can do, we can choose also the x axis, um, to the left or to the right.
OK, because all of these kids, it's the z two axis and the two axis will intersect each other.
But as we have said before, what we want to make the changes minimum.
OK.
So then the changes will be minimum because as X2 is to the right, we will choose it to the right.
OK, in order to make changes minimum with respect to the previous joint.
OK.
Um, so we will choose this direction because it is a volatile action.
This is our X3 axis.
Then what we will choose, we will choose and we we what we have said, we will.
We have said that we will almost in every case we will just copy the last frame for the end effector
because if we choose it in this way again, we will make the changes minimum with respect to the last
frame.
OK.
And also, this will cut the Z three axis, which is required for the H m convention.
Now what happens with the Y axis?
We can assign y axis or we cannot assign if what we get?
We cannot assign.
It's possible that not assign the y axis because during the calculation of h parameters, we will not
require the y axis.
However, if you want to know how we will assign the y axis, you probably assume this.
You probably know we will choose just as we know that axis x axis of evil, use the right hand convention
for choosing y axis.
OK, what is the right hand role?
You will make your sub impact.
Started showing you all this.
I think this is the middle finger in the Z direction and this finger, OK, this finger will be in the
right direction automatically.
OK?
It will show you right direction.
So based on the right hand rule.
So in this case, if you put a middle finger in z axis, OK, middle finger in the direction of z axis
and your sum in extraction, OK?
So the extraction and the your y will be in this direction.
So like that it will be the y axis will be in this way.
This is why.
Zero?
Yeah, this will be the same for y one.
And what will happen in this case?
The two middle finger in the Z two direction somewhere in the X two direction.
Yes, extradition and Y will be out of the page to us toward us.
This will be y two.
Yes.
And let me do the same and it will be the same for y three.
And it will be the same for Y for.
And here is how we assign the coordinate frames for the.
Yeah.
As in the D.H convention, OK, we will do some examples also again.
And so it will be easy for us.
And now what we have to do in the next lesson, we will calculate the H parameters.