Let's talk about the problem that I have mentioned on the last lesson, namely Kimball look, Gimpel
look is one of the most important problem of three angle representations of rotations, namely aler
and roll pitch yaw representations.
As we have mentioned before, three angle representation have advantages of being compact.
It's easy to understand and visualize them, and also they are minimal representation of the rotation
because we have three parameters three quantities.
While they have such superior sides, they also contain some problems like gimbal look, which is not
a small problem.
So when does gimbal look happen?
The problem arises when the axis of inner and ultra gimbals are aligned.
Let's go to the MATLAB and try to visualize the problem.
Now, let's try to visualize the Gimbel problem.
Again, we will use the Pethokoukis Library in order to visualize gimbal log problem.
Let's first write triple.
Triple angle comment.
OK.
As you can see, this is our plane.
And these are street gambles, and they represent different or in the rotation of a plane.
So what do I mean?
The red one represents the rotation about zip access.
The green one represents the rotation about Y-axis, and the blue one represents the rotation about
x axis.
So and they are related with each other hierarchically.
What does it mean?
They are related with hierarchically.
So the red one or the rotation of z z axis rotation has the highest priority or others priority.
I mean, hierarchical.
So the Y is in the middle and the X has the lowest priority.
So if we try to rotate the z axis, as you can see, the plane is rotated both in that axis and the
rotation of the rotation of the axis affects the also the rotation of the blue one and the green one.
As you can see.
So let's try to now rotate about what axis you will see that it doesn't affect the red one because it
has the highest priority in hierarchy, but it affects the blue one because the blue one or the rotation
about x x axis rotation has the lower hierarchical priority than the green one.
And if we rotate about the x axis, you will see that it doesn't affect, uh, any of these accesses
because it has the lowest hierarchy of priority, it affects only to the plane.
So let's reset that.
OK.
What we have said, we have said that Gimbel look arises when the axis of inner and outer Kimball's
are aligned.
As you can see, the red one is our outer gimbel and the blue one is our inner gimbal.
So when these are the axis of them are aligned, then we have gimbal problem.
So let's try to do that in order to make them align first.
Let's try to rotate about x axis.
OK, let's make them align and then try to to make OK.
As you can see now, let's try to make it in this way look a bit like.
OK.
As you can see now, the rotation axis of inner movement and alter red one are aligned well, so we
are in we have now given the look problem.
So, OK, let's see.
What does it mean?
Give them a look problem if they try to rotate our plane about z axis?
There is no problem, as you can see.
OK.
However, if we try to rotate our, uh, plane unlocks axis, as you can see, it changes the orientation
in the same way as z axis does.
So what does it mean?
It means that we lost one degree of freedom, which is not a good thing because we cannot achieve some
orientations that we want to achieve.
So in order to achieve these rotations or orientations, we have to do some additional movements in
order to leave the Gimbel problem.
For example, we have to change first of all this one in order to leave the Gimbel look problem.
And then, as you can see, we can achieve different orientations.
So this is the problem of Gimbel.
Look, Gimbel OK, problem exists in all three quantity representations of our rotations.
So in order to have a solution to this Gimbel problem, we can use quaternary or rotation axis representations
of orientations.
So now let's talk about more mathematical problems of Kimball look and also its effects on robots.
So we have seen Gimbel look in action and understand how it happens.
Now let's try to see what are the difficulties that arise because of the problem of Gimbel look.
There are several problems that are happening because of Gimbel look one and abuse of them is we lose
a degree of freedom, as we have said before, which don't let us achieve some orientations without
doing additional rotations in order to leave.
But look, mathematically, Gimbel look means that we can have infinitely many a solution for the two
angles because we can establish on the linear relation between these two angles because we have determined
one angle by ourselves in order to align inner and outer gimbel axis and the rest of two angles don't
have exact solutions.
The best we can do is to determine the sum of these angles.
This generates a big problem in robots, which is called singularity.
Don't worry, we will talk about robots singularities on our future lessons during this singularity.
There are many, infinitely many solutions for the robot to achieve a desired orientation.
However, this is possible mathematically, not mechanically, because there are some solutions.
While the sum of the angles meet the requirement, the individual values of the angles of them are impossible
for the robot to achieve.
Additionally, during singularities, some drawings of robot can be made to rotate large degrees in
infinitesimally small amount of time, which could damage both robot's mechanical parts and also motors.
So because of these problems, several solutions both in terms of software and hardware, have been
developed.
One of these solutions, as we have seen on the last lesson, is to use Saturnian or rotation mattresses
in order to represent orientation as they don't have the problem of Kimbo look.
See you on the next lesson.