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As we have said before, while the rotation mattresses are easy to implement and manipulate, they contain

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nine elements that only three of them are independent.

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This is quite initiative as a rigid body possesses at most three rotational degrees of freedom in 3D.

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This means that we can describe the orientation of a body in 3D using only three independent quantities.

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So in this lesson, we will try to observe such kind of representations of rotations that uses only

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three independent quantities.

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Let's start with a little angled representation of rotation.

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This is one of the most common method of representing a rotation.

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Let's assume that we have a fixed frame of zero and the rotated frame of one.

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We can describe the orientation of frame or one with respect to frame, or zero using three successive

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rotations about current frame.

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First, we rotate all zero above the z axis by the angle C.

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Then we rotate the newly obtained frame by angle theta around the current y axis.

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Find the larvae, rotate the frame B around the current that axis by angle of C and get our desired

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frame of four one.

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As you can see, we can specify in the relative orientation by only three subsequent rotations.

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The angles of Feed, Peter and C are called Aler angles.

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Now we will try to describe these subsequent rotations using rotation mattresses, which will help us

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to deduce Aler angles.

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Before doing that, I would like to mention that the subsequent rotations are about the current frame,

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so the rotation mattresses will be just multiplied.

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Let's multiply subsequent rotation mattresses in order to get the resultant rotation.

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This is the resultant rotation matrix that we have got from multiplication.

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This matrix is called that.

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Was it a lower angle transformation.

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The key problem is how we can get your angles from this rotation matrix.

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Let's copy the metrics here again in order to find a solution to our problem.

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We will analyze two cases for the first case.

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We will suppose that not both of R1 three and R2 three are zero.

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By this assumption, we can deduce that since that is not zero.

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And so both of our three one or three two are also not zero, as since data is not zero, we can deduce

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that R3 three is not plus or minus one because it can happen only when it goes to zero or PI.

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So we can read that customers to equals two or three three.

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And since data equals the plus minus square root of one minus R3 three squared after we obtain cosine

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data and since Theta, we can easily get it up by using a to function.

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If you don't know what is attached to function, then don't worry for now, except accept it as an inverse

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of tangent or act tangent.

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In order to get theta on the next lesson, I will briefly touched on this function.

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So let's continue.

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As you can see, we have two solutions for the Theta.

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If we take the first solution for Theta, then sign Theta is greater than zero and we can get fit and

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see using below formula.

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It's not for you to find the given values in the formula inside the Matrix in order to understand how

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we get these formulas for five feet and see if it will take on the second from you.

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But it is important for you to grasp the concept anyway.

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However, if we take the second solution for the T to, then sine teeter will be less than zero and

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we will find below solutions for fee and see angles.

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So we have two solutions to our problem because of the sine difference of Peter in terms of the second

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case, if both are ones three and R2 three are zero, then sine teeter must be zero, which means that

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it can be zero or point.

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This also means that R3 three is plus minus one because our three three is cosine of theta.

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Also, our three one and our three two are also zero because they contain contains sine of TITA.

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Then our rotation matters becomes like that if we accept our three three as plus one.

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Then the TITA is zero and sine theta is also zero.

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So we get below solution four feet and see, as you can see, we can only have a solution for the sum

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of these angles.

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This means that we can have infinitely many solutions for feet and see how or by convention we can take

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a zero.

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If we accept our three three as minus one, then Peter is PI and Sun is also zero.

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Then we get solution for the subtraction of B and C.

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This also means that we can have infinitely many solutions for feet and C.

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Let's go to another way of representation of rotation.

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This representation is called roll pitch your angles.

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And if you have done even a bit of robotics or mechanics, I am sure that you have heard these terms.

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This method is similar to the earlier angles representation, except the rotation on rotations are not

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about the current frame, but fixed frame.

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So we will use pre multiplication of rotation mattresses to describe the final rotation after doing

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some matrix algebra.

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We can get the final result.

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You can get the roll pitch and your angles from this mattress by following the same methodology as we

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have done for or angles.

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Before finishing this lesson, I would like to note that in general, we don't do all these calculations.

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These are just the theoretical concept which will help you to grasp the other concepts more easily and

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deeply instead of just writing a code without knowing anything.

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These procedures have been applied by libraries and given as readily when we called our robots.

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Don't worry, we will apply these concepts during our practical lessons and you will see what I mean.

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See you on the next lesson.