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Let's talk about the problem that I have mentioned on the last lesson, namely Kimball look, Gimpel

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look is one of the most important problem of three angle representations of rotations, namely aler

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and roll pitch yaw representations.

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As we have mentioned before, three angle representation have advantages of being compact.

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It's easy to understand and visualize them, and also they are minimal representation of the rotation

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because we have three parameters three quantities.

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While they have such superior sides, they also contain some problems like gimbal look, which is not

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a small problem.

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So when does gimbal look happen?

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The problem arises when the axis of inner and ultra gimbals are aligned.

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Let's go to the MATLAB and try to visualize the problem.

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Now, let's try to visualize the Gimbel problem.

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Again, we will use the Pethokoukis Library in order to visualize gimbal log problem.

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Let's first write triple.

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Triple angle comment.

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OK.

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As you can see, this is our plane.

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And these are street gambles, and they represent different or in the rotation of a plane.

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So what do I mean?

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The red one represents the rotation about zip access.

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The green one represents the rotation about Y-axis, and the blue one represents the rotation about

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x axis.

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So and they are related with each other hierarchically.

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What does it mean?

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They are related with hierarchically.

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So the red one or the rotation of z z axis rotation has the highest priority or others priority.

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I mean, hierarchical.

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So the Y is in the middle and the X has the lowest priority.

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So if we try to rotate the z axis, as you can see, the plane is rotated both in that axis and the

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rotation of the rotation of the axis affects the also the rotation of the blue one and the green one.

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As you can see.

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So let's try to now rotate about what axis you will see that it doesn't affect the red one because it

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has the highest priority in hierarchy, but it affects the blue one because the blue one or the rotation

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about x x axis rotation has the lower hierarchical priority than the green one.

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And if we rotate about the x axis, you will see that it doesn't affect, uh, any of these accesses

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because it has the lowest hierarchy of priority, it affects only to the plane.

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So let's reset that.

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OK.

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What we have said, we have said that Gimbel look arises when the axis of inner and outer Kimball's

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are aligned.

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As you can see, the red one is our outer gimbel and the blue one is our inner gimbal.

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So when these are the axis of them are aligned, then we have gimbal problem.

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So let's try to do that in order to make them align first.

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Let's try to rotate about x axis.

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OK, let's make them align and then try to to make OK.

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As you can see now, let's try to make it in this way look a bit like.

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OK.

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As you can see now, the rotation axis of inner movement and alter red one are aligned well, so we

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are in we have now given the look problem.

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So, OK, let's see.

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What does it mean?

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Give them a look problem if they try to rotate our plane about z axis?

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There is no problem, as you can see.

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OK.

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However, if we try to rotate our, uh, plane unlocks axis, as you can see, it changes the orientation

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in the same way as z axis does.

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So what does it mean?

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It means that we lost one degree of freedom, which is not a good thing because we cannot achieve some

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orientations that we want to achieve.

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So in order to achieve these rotations or orientations, we have to do some additional movements in

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order to leave the Gimbel problem.

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For example, we have to change first of all this one in order to leave the Gimbel look problem.

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And then, as you can see, we can achieve different orientations.

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So this is the problem of Gimbel.

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Look, Gimbel OK, problem exists in all three quantity representations of our rotations.

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So in order to have a solution to this Gimbel problem, we can use quaternary or rotation axis representations

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of orientations.

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So now let's talk about more mathematical problems of Kimball look and also its effects on robots.

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So we have seen Gimbel look in action and understand how it happens.

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Now let's try to see what are the difficulties that arise because of the problem of Gimbel look.

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There are several problems that are happening because of Gimbel look one and abuse of them is we lose

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a degree of freedom, as we have said before, which don't let us achieve some orientations without

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doing additional rotations in order to leave.

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But look, mathematically, Gimbel look means that we can have infinitely many a solution for the two

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angles because we can establish on the linear relation between these two angles because we have determined

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one angle by ourselves in order to align inner and outer gimbel axis and the rest of two angles don't

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have exact solutions.

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The best we can do is to determine the sum of these angles.

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This generates a big problem in robots, which is called singularity.

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Don't worry, we will talk about robots singularities on our future lessons during this singularity.

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There are many, infinitely many solutions for the robot to achieve a desired orientation.

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However, this is possible mathematically, not mechanically, because there are some solutions.

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While the sum of the angles meet the requirement, the individual values of the angles of them are impossible

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for the robot to achieve.

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Additionally, during singularities, some drawings of robot can be made to rotate large degrees in

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infinitesimally small amount of time, which could damage both robot's mechanical parts and also motors.

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So because of these problems, several solutions both in terms of software and hardware, have been

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developed.

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One of these solutions, as we have seen on the last lesson, is to use Saturnian or rotation mattresses

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in order to represent orientation as they don't have the problem of Kimbo look.

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See you on the next lesson.